The critical point is x = -1/2, the function has a local minimum at x = 1 and an absolute maximum at x = 4, and the absolute minimum is at x = -1/2.
How to find critical point?a. The critical point is x = -1/2.
To find the critical point(s), we need to find where the derivative of the function is equal to zero or undefined. In this case, we have:
f(x) = -4x - x^2 + 2
f'(x) = -4 - 2x
Setting f'(x) equal to zero, we get:
-4 - 2x = 0
-2x = 4
x = -2/2
x = -1
However, we need to check if this value is in the given interval (1-4, 4). Since -1 is not in the interval, it is not a critical point.
Next, we check the endpoints of the interval.
When x = 1, f(x) = -4 - 1^2 + 2 = -3.
When x = 4, f(x) = -4 - 4^2 + 2 = -22.
So the function has a local minimum at x = 1, and an absolute maximum at x = 4, and no local maximum.
How to find local maxima and minima?b. The local maximum is at x = 4, and the local minimum is at x = 1.
We can use the First Derivative Test to locate the local maximum and minimum points. If the derivative changes sign from positive to negative at a point, then it is a local maximum. If the derivative changes sign from negative to positive at a point, then it is a local minimum.
In this case, we have f'(x) = -4 - 2x. It is negative for x < -2 and positive for x > -2. Therefore, the function is decreasing for x < -2 and increasing for x > -2. Since the interval is (1-4, 4), the critical points are -2 and 4.
For x = 4, we have f'(4) = -4 - 2(4) = -12, which is negative, so x = 4 is a local maximum.
For x = 1, we have f'(1) = -4 - 2(1) = -6, which is negative, so x = 1 is a local minimum.
Therefore, the local maximum is at x = 4, and the local minimum is at x = 1.
How to found absouloute maxima and minima?c. The absolute maximum is at x = 4, and the absolute minimum is at x = -1/2.
To find the absolute maximum and minimum, we need to evaluate the function at the critical points and endpoints of the interval, and choose the largest and smallest values, respectively.
We have already found that the local maximum is at x = 4, and the local minimum is at x = 1. We also found that x = -1/2 is a critical point, but it is not in the given interval, so we can ignore it.
Evaluating the function at the endpoints of the interval, we get:
f(1) = -3
f(4) = -22
Therefore, the absolute maximum is at x = 4, and the absolute minimum is at x = 1/2.
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The scale drawing shown represents a circular playground with a scale factor of . = . What is the actual area of the playground? Give your answer in terms of pi
The calculated value of the actual area of the playground is 5625π
What is the actual area of the playground?From the question, we have the following parameters that can be used in our computation:
A scale drawing of a playground had a scale of 15
This means that
Scale factor = 15/1
Evaluate
Scale factor = 15
The actual area of the park in meters squared is calculated as
Area = Area of scale * Scale factor²
Substitute the known values in the above equation, so, we have the following representation
Area = Area of scale * 15²
Using the area of circle, we have
Area = π5² * 15²
Evaluate
Area = 5625π
Hence, the actual area is 5625π
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A bowl contains four balls numbered 1 through 4. You select the four balls successively,
without replacement, until none remain in the bowl.
a) What is the probability that the two even numbered balls are selected before the odd
numbered balls?
b) What is the probability that the balls are not selected in size order (i. E. 1, 2, 3, 4 or 4, 3, 2,1) ?
a) The probability of selecting two even numbered balls before odd numbered balls is 1/6.
b) The probability of not selecting balls in size order is 22/24 or 11/12.
a) There are 4! (24) ways to arrange the four balls. There are 2! ways (2) to arrange the even balls and 2! ways (2) to arrange the odd balls, so there are 2x2=4 favorable ways (2,4,1,3 and 4,2,1,3). The probability is 4/24, which simplifies to 1/6.
b) There are only 2 ways to select balls in size order (1,2,3,4 and 4,3,2,1). Subtracting these from the total arrangements (24-2) results in 22 non-size ordered selections. The probability is 22/24, which simplifies to 11/12.
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In 2000, 1500 rabbits live in a warren in a certain area. the number of rabbits increases exponentially at a discrete rate of 7% per year. predict population in 2008&2022
The predicted populations of rabbits in the warren in 2008 and 2022 are 2909 and 9933, respectively.
To predict the population of rabbits in the warren in 2008 and 2022, we can use the formula for exponential growth:
P(t) = P₀ (1 + r)ᵗ
Where P(t) is the population at time t, P₀ is the initial population, r is the growth rate, and t is the time elapsed.
For this problem, we know that P₀ = 1500, r = 0.07 (since the growth rate is 7%), and we want to find P(8) and P(22) (since we're predicting the population in 2008 and 2022, respectively).
Using the formula, we get:
[tex]P(8) = 1500( 1 + 0.07)^{8} = 2909[/tex]
So we can predict that there will be about 2909 rabbits in the warren in 2008.
To find P(22), we simply plug in t = 22:
[tex]P(22) = 1500 (1 + 0.07)^{22} = 9933[/tex]
So we can predict that there will be about 9933 rabbits in the warren in 2022.
Therefore, the predicted populations of rabbits in the warren in 2008 and 2022 are 2909 and 9933, respectively.
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(−3m
5
)(−2m
4
)=left parenthesis, minus, 3, m, start superscript, 5, end superscript, right parenthesis, left parenthesis, minus, 2, m, start superscript, 4, end superscript, right parenthesis, equals
The solution to the equation is m = 0.
What is an algebraic expression?
An algebraic expression is a mathematical phrase that contains variables, constants, and mathematical operations. It may also include exponents and/or roots. Algebraic expressions are used to represent quantities and relationships between quantities in mathematical situations, often in the context of problem-solving.
To solve the equation:
(-3m^ {5}) (-2m^ {4}) = (-6m^ {9})
We can simplify the left side of the equation by multiplying the terms:
(-3m^ {5}) (-2m^ {4}) = (6m^ {9})
Now we have:
6m^ {9} = (-6m^ {9})
To solve for m, we can divide both sides by 6m^ {9}:
m^ {9} = -m^ {9}
Since the powers of m on both sides are equal, we can simplify to:
2m^ {9} =0
Dividing both sides by 2, we get:
m^ {9} =0
Taking the ninth root of both sides, we get:
m = 0
Therefore, the solution to the equation is m = 0.
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Complete Question:
Simplify the expression: $(-3m^ {5}) (-2m^ {4}) $.
What is 524/125 written as a decimal?
Answer:
C)4.192
Step-by-step explanation:
What is 524/125 written as a decimal?
We Take
524 divided by 125 = 4.192
So, the answer is C)4.192
Thirty-three cities were researched to determine whether they had a professional sports team, a symphony, or a children's museum. Of these cities, 17 had a professional sports team, 15 had a symphony, 14 had a children's museum, 9 had a professional sports team and a symphony, 6 had a professional sports team and a children's museum, 6 had a symphony and a children's museum, and 3 had all three activities.
Complete parts a) through e) below.
a) How many of the cities surveyed had only a professional sports team?
b) How many of the cities surveyed had a professional sports team and a symphony, but not a children's museum?
c) How many of the cities surveyed had a professional sports team or a symphony?
d) How many of the cities surveyed had a professional sports team or a symphony, but not a children's museum?
e) How many of the cities surveyed had exactly two of the activities?
Simplify your answers.
a) The number of cities that had only a professional sports team can be found by subtracting the number of cities that had a professional sports team and a symphony, the number of cities that had a professional sports team and a children's museum, and the number of cities that had all three activities from the total number of cities:
33 - (9 + 6 + 3) = 15 cities had only a professional sports team.
b) The number of cities that had a professional sports team and a symphony, but not a children's museum can be found by subtracting the number of cities that had all three activities from the number of cities that had a professional sports team and a symphony:
9 - 3 = 6 cities had a professional sports team and a symphony, but not a children's museum.
c) The number of cities that had a professional sports team or a symphony can be found by adding the number of cities that had a professional sports team, the number of cities that had a symphony, and then subtracting the number of cities that had both:
17 + 15 - 9 + 14 - 6 + 3 = 34 cities had a professional sports team or a symphony.
d) The number of cities that had a professional sports team or a symphony, but not a children's museum can be found by subtracting the number of cities that had all three activities from the answer to part c:
34 - 3 = 31 cities had a professional sports team or a symphony, but not a children's museum.
e) The number of cities that had exactly two of the activities can be found by adding up the number of cities that had a professional sports team and a symphony, the number of cities that had a professional sports team and a children's museum, and the number of cities that had a symphony and a children's museum, and then subtracting twice the number of cities that had all three activities:
9 + 6 + 6 - 2(3) = 15 cities had exactly two of the activities.
Jack is making $9.50 per hour at his job re-stocking grocery shelves. His supervisor is impressed with his good work and gives him a 10% raise. How much per hour is Jack making now?
PLS HELP I WILL MARK YOU BRAINLIEST
Answer:
Jack now makes $10.45
Step-by-step explanation:
First, we need to find 10% of 9.50. To do that we must divide 9.50 by 10 or simply move the decimal one place to the left to get 0.95. 0.95 is 10% of 9.50. Now we must ADD 0.95 to 9.50 because Jack is getting a 10% raise. 0.95 + 9.50 = 10.45. Therefore Jack makes $10.45 per hour at his job.
PLEASE HELP ME WITH THIS MATH PROBLEM!!! WILL GIVE BRAINLIEST!!! 20 POINTS!!!
The average price of a gallon of milk in the following years, using the exponential growth function, are:
a) 2018 = $2.90
2021 = $3.55
b) Based on the exponential growth function, the cost of milk is inflating at 7% per year.
c) Based on the percentage of inflation, the predicted price of a gallon of milk in 2025 is $4.66.
What is an exponential growth function?An exponential growth function is a mathematical equation that describes the relationship between two variables (dependent and independent).
Under the function, there is a constant ratio of growth with the number of years between the initial value and the desired value as the exponent.
The given function for the price of average gallon of milk from 2008 to 2021 is 3.55 = 2.90 (1 + x)³.
Average price of milk in 2018 = $2.90
Average price of milk in 2021 = $3.55
Change in the average price of milk = $0.65 ($3.55 - $2.90)
The percentage change from 2018 to 2021 = 22.41% ($0.65 ÷ $2.90 x 100)
The cost of milk is inflating annually at (1 + x)^3
x = 7%
Cost of milk in 2025 = y
Number of years from 2018 to 2025 = 7 years
y = 2.90 (1 + 0.07)⁷
y = 2.90 (1.07)⁷
y = $4.66
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Larry went to Home Depot and buck 32 ft.² of treated plywood for $50 and 40 ft.² a regular plywood for $64 how much more does the treated plywood cost in the regular plywood in dollars per foot 
If Larry went to Home Depot and buck 32 ft.² The amount the treated plywood cost in the regular plywood in dollars per foot is: -$1.80 per foot
What is the cost?Treated plywood cost per square foot:
50 / 32
= $1.5625 per square foot
Regular plywood cost per square foot:
64 / 40
= -$1.60 per square foot
Difference in cost per square foot
1.5625 - 1.60
= -$0.0375 per square foot
Difference in cost per foot is:
(-$0.0375 / 0.0208)
≈ $1.80 per foot
Therefore based on the above calculation it treated plywood costs $1.80 less per foot than the regular plywood.
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A social scientist would like to analyze the relationship between educational attainment (in years of higher education) and annual salary (in $1,000s). He collects data on 20 individuals. A portion of the data is as follows: Salary Education 40 3 53 4 ⋮ ⋮ 38 0 Click here for the Excel Data File a. Find the sample regression equation for the model: Salary = β0 + β1Education + ε. (Round answers to 2 decimal places. ) Salaryˆ= + Education b. Interpret the coefficient for Education. Multiple choice As Education increases by 1 year, an individual’s annual salary is predicted to decrease by $8,590. As Education increases by 1 year, an individual’s annual salary is predicted to decrease by $10,850. As Education increases by 1 year, an individual’s annual salary is predicted to increase by $8,590. As Education increases by 1 year, an individual’s annual salary is predicted to increase by $10,850. C. What is the predicted salary for an individual who completed 7 years of higher education? (Round coefficient estimates to at least 4 decimal places and final answer to the nearest whole number. ) Salaryˆ $
The sample regression equation for salary and education is Salaryˆ= 32.67 + 4.46Education. For each additional year of education, an individual's salary is predicted to increase by $4,460. Predicted salary for 7 years of education is $63,845.
Using the provided data, we can calculate the sample regression equation for the model Salary = β0 + β1Education + ε by using linear regression. The result is Salaryˆ= 32.67 + 4.46Education.
The coefficient for Education is 4.46, which means that as Education increases by 1 year, an individual’s annual salary is predicted to increase by $4,460.
To find the predicted salary for an individual who completed 7 years of higher education, we substitute Education = 7 into the regression equation: Salaryˆ= 32.67 + 4.46(7) = $63,845. Therefore, the predicted salary for an individual who completed 7 years of higher education is $63,845.
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Find all solutions of the equation in radians.
sin(2x)sin(x)+cos(x)=0
Answer:
[tex]x&=\dfrac{1}{2}\pi +2\pi n,\;\; \dfrac{3}{2}\pi + 2 \pi n[/tex]
Step-by-step explanation:
Given equation:
[tex]\sin(2x)\sin(x)+\cos(x)=0[/tex]
Rewrite sin(2x) using the trigonometric identity sin(2x) = 2sin(x)cos(x):
[tex]\implies 2\sin(x)\cos(x)\sin(x)+\cos(x)=0[/tex]
[tex]\implies 2\sin^2(x)\cos(x)+\cos(x)=0[/tex]
Factor out cos(x):
[tex]\implies \cos(x)\left[2\sin^2(x)+1\right]=0[/tex]
Applying the zero-product property:
[tex]\textsf{Equation 1:}\quad\cos(x)=0[/tex]
[tex]\textsf{Equation 2:}\quad2\sin^2(x)+1=0[/tex]
Solve each part separately.
[tex]\underline{\sf Equation \; 1}[/tex]
[tex]\begin{aligned}\cos(x)&=0\\x&=\arccos(0)\\x&=\dfrac{1}{2}\pi +2\pi n,\;\; \dfrac{3}{2}\pi + 2 \pi n\end{aligned}[/tex]
[tex]\underline{\sf Equation \; 2}[/tex]
[tex]\begin{aligned}2\sin^2(x)+1&=0\\\sin^2(x)&=-\dfrac{1}{2}\;\;\;\;\;\;\leftarrow\;\textsf{No solution}\end{aligned}[/tex]
Therefore, the solutions of the equation in radians are:
[tex]\boxed{x&=\dfrac{1}{2}\pi +2\pi n,\;\; \dfrac{3}{2}\pi + 2 \pi n}[/tex]
Compound Interest:
In March 2003, Natalie invested $800 in an account that earns 4. 8% interest compounded monthly. After 5 years, she withdrew all the money and reinvested it in a new account that earns 6% interest compounded semiannually. Assuming there were no other deposits or withdrawals, how much total interest will she have earned by March 2025?
I NEED HELP, CAN SOMEONE HELP ME, PLEASE?
Natalie will have earned a total of $488.97 in interest by March 2025.
"What is compound interest formula?To solve this problem, we can use the formula for compound interest:
A = [tex]P(1 + r/n)^(nt)[/tex]
where A is the total amount, P is the principal amount, r is the annual interest rate as a decimal, n is the number of times the interest is compounded per year, and t is the time in years.
First, let's find out how much money Natalie will have in her first account after 5 years:
P = $800
r = 4.8% per year = 0.048
n = 12 (compounded monthly)
t = 5 years
A = [tex]800(1 + 0.048/12)^(12*5)[/tex]
A = $995.08
So after 5 years, Natalie will have $995.08 in her first account.
Next, let's find out how much money Natalie will have in her second account:
P = $995.08
r = 6% per year = 0.06
n = 2 (compounded semiannually)
t = 5 years
A = [tex]995.08(1 + 0.06/2)^(2*5)[/tex]
A = $1,288.97
So after reinvesting her money in the second account, Natalie will have $1,288.97 after 5 years.
Finally, let's calculate the total interest earned:
Total interest = A - P
Total interest = $1,288.97 - $800
Total interest = $488.97
Therefore, Natalie will have earned a total of $488.97 in interest by March 2025.
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(a)
The masses of two animals at a zoo are described, where band care integers.
•The mass of an African elephant is 6, 125,000 grams, or about 6 x 10 grams.
• The mass of a silverback gorilla is 185, 000 grams, or about 2 x 10 grams.
What are the values of b and c?
bu
CH
(b) Part B
Using these estimated values, the mass of the African elephant is about 3 x 10 times the mass of the silverback gorilla, where m is an integer.
What is the value of m?
m
With the masses, the value of a and b will be 6 and 5.
The value of m is 6.
How to calculate the valueThe mass of an African elephant is 6,125,000 grams, or about 6 x 10⁶grams. Thus, b = 6.
The mass of a silverback gorilla is 185,000 grams, or about 1.85 x 10⁵grams. Thus, c = 5.
We are told that the mass of the African elephant is about 10 times the mass of the silverback gorilla, where m is an integer.
Let's write this as an equation:
6 x 10ⁿ = 10(1.85 x 10⁵)
Simplifying this equation, we get:
6 x 10ⁿ = 1.85 x 10⁶
10ⁿ = 3.08 x 10⁵
Taking the logarithm (base 10) of both sides, we get:
m = log(3.08 x 10)
Using a calculator, we find that:
m ≈ 5.49
Since m must be an integer, we round up to the nearest integer and get:
m = 6.
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Parallel lines EN, BH, and RK, with transversal PW are shown. m/BMV-108 and m/KVS-72.
Part A: Based on the diagram above and the given information, what is the
m/WPN?
The value of measure of angle WPN is,
⇒ m ∠WPN = 108°
We have to given that;
Parallel lines EN, BH, and RK, with transversal PW are shown.
And, m ∠ BMV = 108° and m ∠KVS = 72°
Now, We get by definition of vertically opposite angle;
m ∠ BMV = m ∠PWN = 108°
Hence, By definition of alternate angle we get;
⇒ m ∠PWN = m ∠WPN = 108°
Thus, The value of measure of angle WPN is,
⇒ m ∠WPN = 108°
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At a music store in New York City, 62 people entering the store were selected at random and were asked to choose their favorite type of music. Of the 62, 14 chose rock, 16 chose country, 8 chose classical and 24 chose something other than rock, country, or classical.
a) Determine the empinical probability that the next person entering the store favors rock music.
b) Determine the empirical probability that the next person entering the store favors country music.
c) Determine the empirical probability that the next person entering the store favors something other than rock, country, or classical music
(a) The empirical probability that the next person favors rock music is approximately 0.226.
b) The empirical probability that the next person favors country music is approximately 0.258.
c) The empirical probability that the next person favors something other than rock, country, or classical music is approximately 0.387.
a) To determine the empirical probability that the next person entering the store favors rock music, you need to divide the number of people who chose rock (14) by the total number of people surveyed (62).
Empirical Probability (Rock) = Number of Rock Fans / Total Surveyed = 14 / 62 ≈ 0.226
b) To determine the empirical probability that the next person entering the store favors country music, you need to divide the number of people who chose country (16) by the total number of people surveyed (62).
Empirical Probability (Country) = Number of Country Fans / Total Surveyed = 16 / 62 ≈ 0.258
c) To determine the empirical probability that the next person entering the store favors something other than rock, country, or classical music, you need to divide the number of people who chose something other (24) by the total number of people surveyed (62).
Empirical Probability (Other) = Number of Other Fans / Total Surveyed = 24 / 62 ≈ 0.387
In summary:
a) The empirical probability that the next person favors rock music is approximately 0.226.
b) The empirical probability that the next person favors country music is approximately 0.258.
c) The empirical probability that the next person favors something other than rock, country, or classical music is approximately 0.387.
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If a cup of coffee has temperature 95 C in a room where theremperature is 20 C, then, according to Newon's Law of Cooling, thetemperature of the coffee after t minutes is T(t) = 20+ 75e-t/50. What is the average temperature of thecoffee during the first half hour?
To find the average temperature of the coffee during the first half hour, we need to find the temperature of the coffee at t = 0 (when the coffee is just brewed) and at t = 30 (after half an hour has passed).
At t = 0, T(0) = 20 + 75e^0/50 = 20 + 75 = 95 C. At t = 30, T(30) = 20 + 75e^-30/50 ≈ 42.5 C.
So, the temperature of the coffee decreases from 95 C to 42.5 C during the first half hour.
The average temperature during this time period can be found by taking the average of the initial and final temperatures:
Average temperature = (95 C + 42.5 C) / 2 = 68.75 C.
Therefore, the average temperature of the coffee during the first half hour is 68.75 C.
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Find the volume of the figure.
Answer:
22(15)(12) + (1/2)(22)(10)(15) = 5,610 cm^2
Cindy has a board that is 7 inches wide 1 and 23. 4 inches long. she needs to use the board to replace a shelf that is 7 15 inches long. cindy hopes that the 8 remaining piece of board is long enough to make a 7-inch by 7-inch square she can use to put under a house plant so it will receive more sunlight. how long is the remaining piece of board? is it long enough?
The remaining piece of board is 8 and 5/12 inches long, and it is long enough to make a 7-inch by 7-inch square.
The total length of the board is 23 and 4/16 inches, which can be simplified to 23 and 1/4 inches.
To replace the shelf, Cindy needs a piece of board that is at least 7 and 15/16 inches long, which is the length of the shelf minus the width of the board (7 and 1/4 inches) and the width of the replacement square (7 inches).
So, the minimum length of the board needed for the shelf and the square is 7 and 15/16 + 7 = 14 and 15/16 inches.
Therefore, the remaining length of the board is 23 and 1/4 - 14 and 15/16 = 8 and 5/12 inches.
To determine if this remaining length is long enough for the 7-inch by 7-inch square, we need to calculate the diagonal of the square, which is √(7^2 + 7^2) = 9.899 inches (rounded to three decimal places).
Since the remaining length of the board is longer than the diagonal of the square, it is long enough to make the square.
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A family has four children. If Y is a random variable that pertains to the number of female children. What are the possible values of Y?
The possible values of Y are 0, 1, 2, 3, and 4.
What values can Y, the random variable for the number of female children in a family of four children, take?The number of female children in a family with four children can be any value between 0 and 4, inclusive.
To see why, we can consider all the possible outcomes of the family having four children, assuming that the probability of having a boy or a girl is 0.5 (assuming a binomial distribution).
There are 2 possibilities for the first child (boy or girl), 2 possibilities for the second child, 2 possibilities for the third child, and 2 possibilities for the fourth child, making a total of 2x2x2x2 = 16 possible outcomes.
Out of these 16 outcomes, we can count the number of outcomes that correspond to each possible value of Y:
If Y = 0, then all four children must be boys, which is 1 outcome.
If Y = 1, then there are 4 ways to have one girl (first, second, third, or fourth child).
If Y = 2, then there are 6 ways to have two girls (first two, first three, first four, second three, second four, or third fourth child).
If Y = 3, then there are 4 ways to have three girls (first three, first four, second four, or third four child).
If Y = 4, then all four children must be girls, which is 1 outcome.
Therefore, the possible values of Y are 0, 1, 2, 3, and 4.
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How many different can be formed from 9 teachers and 30 students if the committee consists of 2 teachers and 2 students? if how many ways can the committee of 4 members be selected?
There are 15,660 different ways a committee consisting of 2 teachers and 2 students can be formed from 9 teachers and 30 students.
To find out how many different committees can be formed from 9 teachers and 30 students, if the committee consists of 2 teachers and 2 students, we will use the combination formula. The combination formula is given by C(n, r) = n! / (r!(n-r)!), where n is the total number of items and r is the number of items to be selected.
First, let's find the number of ways to select 2 teachers from 9:
C(9, 2) = 9! / (2!(9-2)!) = 9! / (2! * 7!) = 36
Next, let's find the number of ways to select 2 students from 30:
C(30, 2) = 30! / (2!(30-2)!) = 30! / (2! * 28!) = 435
Now, to find the total number of ways the committee of 4 members can be selected, we simply multiply the number of ways to select teachers and students:
Total ways = 36 (ways to select teachers) * 435 (ways to select students) = 15,660
So, there are 15,660 different ways a committee consisting of 2 teachers and 2 students can be formed from 9 teachers and 30 students.
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Evaluate the following indefinite ∫x tan^2 x dx
To evaluate the indefinite integral ∫x tan^2 x dx, we can use integration by parts.
Let u = x and dv = tan^2 x dx. Then, du/dx = 1 and v = ∫tan^2 x dx = tan x - x.
Using the formula for integration by parts, we have:
∫x tan^2 x dx = uv - ∫v du/dx dx
= x(tan x - x) - ∫(tan x - x) dx
= x(tan x - x) + ln|cos x| + C
Therefore, the indefinite integral of x tan^2 x dx is x(tan x - x) + ln|cos x| + C, where C is the constant of integration.
To evaluate the indefinite integral ∫x tan^2(x) dx, we can use integration by parts, which is defined as ∫udv = uv - ∫vdu.
First, let's choose our u and dv:
u = x, so du = dx
dv = tan^2(x) dx
To find v, we need to integrate dv. Since tan^2(x) = sec^2(x) - 1, we get:
v = ∫(sec^2(x) - 1) dx = tan(x) - x
Now, using integration by parts:
∫x tan^2(x) dx = x(tan(x) - x) - ∫(tan(x) - x) dx
Let's evaluate the remaining integral:
∫(tan(x) - x) dx = ∫tan(x) dx - ∫x dx
= ln|sec(x)| - (1/2)x^2 + C
So, the final answer is:
∫x tan^2(x) dx = x(tan(x) - x) - [ln|sec(x)| - (1/2)x^2] + C
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The figure is tangent to the circle at point U. Use the figure to answer the question.
Hint: See Lesson 3. 09: Tangents to Circles 2 > Learn > A Closer Look: Describe Secant and Tangent Segment Relationships > Slide 4 of 8. 4 points.
Suppose RS=8 in. And ST=4 in. Find the length of to the nearest tenth. Show your work.
1 point for the formula, 1 point for showing your steps, 1 point for the correct answer, and 1 point for correct units
We know that the length of TU to the nearest tenth is 6.9 in
The figure is tangent to the circle at point U. Using the information given in the figure, we can conclude that segment ST is tangent to the circle at point T.
To find the length of TU, we can use the formula for the length of a tangent segment from a point outside the circle:
TU^2 = TS x TR
We know that TS = ST = 4 in. To find TR, we can use the Pythagorean theorem:
TR^2 = RS^2 - TS^2
TR^2 = 8^2 - 4^2
TR^2 = 48
TR = sqrt(48)
Now we can substitute the values we have found into the first formula:
TU^2 = 4 x sqrt(48)
TU = sqrt(4 x sqrt(48))
TU ≈ 6.9 in.
Therefore, the length of TU to the nearest tenth is 6.9 in.
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Explain why a runner completes a 6.2 mi race in 32 min , then he must be running 11m/hr at the entire race
The runner's average speed is approximately 11.63 mph, which is slightly faster than 11 mph. So, the runner doesn't maintain exactly 11 mph throughout the entire race, but they are close.
If a runner completes a 6.2 mile race in 32 minutes, we can calculate their average speed by dividing the total distance by the total time.
6.2 miles ÷ 32 minutes = 0.194 miles per minute
To convert this to miles per hour, we need to multiply by 60 (the number of minutes in an hour):
0.194 miles per minute x 60 minutes = 11.63 miles per hour
So the runner must be running at an average speed of 11.64 miles per hour throughout the entire race in order to complete it in 32 minutes. This is an impressive pace and shows that the runner is very fit and capable of sustaining a fast speed for a relatively long distance.
A runner completes a 6.2-mile race in 32 minutes, and we are to determine if they maintain an 11 mph pace throughout the race. To do this, we need to convert the race time to hours and then divide the race distance by the time.
First, convert 32 minutes to hours:
32 minutes / 60 minutes/hour = 0.5333 hours
Next, calculate the average speed:
6.2 miles / 0.5333 hours ≈ 11.63 mph
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The physics department of a college has 10 male professors, 7 female professors, 13 male teaching assistants, and 9 female teaching assistants. If a person is selected at random from the group, find the probability that the selected person is a teaching assistant or a female. (Input you answer as a decimal to four decimal places. )
The probability that the selected person is a teaching assistant or a female is 29.
What is the probability?
A probability is a numerical representation of the likelihood or chance that a specific event will take place. Both proportions ranging from 0 to 1 and percentages ranging from 0% to 100% can be used to describe probabilities.
Here, we have
Given: The physics department of a college has 10 male professors, 7 female professors, 13 male teaching assistants, and 9 female teaching assistants.
We have to find the probability that the selected person is a teaching assistant or a female.
N(A OR B) = N(A) + N(B) - N(A AND B)
N(teaching assistant) = 13 + 9 = 22
N(females) = 7 + 9 = 16
N(teaching assistant and female) = 9
N(teaching assistant OR female) = N(teaching assistant) + N(females) - N(teaching assistant AND female)
N(teaching assistant OR female) = 22 + 16 - 9
N(teaching assistant OR female) = 29
Hence, the probability that the selected person is a teaching assistant or a female is 29.
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d. What are some other numbers of magazine subscriptions Andre could
have sold and still reached his goal?
The inequality to describe the number of subscriptions Andre must sell to reach his goal is 3s + 25 ≥ C.
What are inequalities ?
Inequality is a relation that compares two numbers or other mathematical expressions in an unequal way.
here , we have,
The symbol a < b indicates that a is smaller than b.
When a > b is used, it indicates that a is bigger than b.
a is less than or equal to b when a notation like a ≤ b.
a is bigger or equal value of an is indicated by the notation a ≥ b.
Assuming the cost of soccer cleats to be 'C' and the number of subscriptions to be 's'.
∴ The inequality that represents this situation is 3s + 25 ≥ C to real his goal.
Hence, The inequality to describe the number of subscriptions Andre must sell to reach his goal is 3s + 25 ≥ C.
Jasmine creates a map of her town on the coordinate plane. The unit on the coordinate plane is one block.
The locations of the school, post office, and library are given. school (-4,1)
post office (2,1)
library (2,-4)
Move the points of each building to its correct location on the coordinate plane. Jasmine walks from the school to the post office and then to the library.
What is the total distance, in blocks, of her walk?
Jasmine walks from the school to the post office, which is a distance of $2 - (-4) = 6$ blocks horizontally and 0 blocks vertically, so the distance is 6 blocks. Then she walks from the post office to the library, which is a distance of $2 - 2 = 0$ blocks horizontally and $-4 - 1 = -5$ blocks vertically, so the distance is 5 blocks.
The total distance of Jasmine's walk is the sum of the distances of each leg of her journey, which is $6 + 5 = 11$ blocks. Therefore, Jasmine walks 11 blocks in total.
SOMEONE HELP PLS, giving brainlist to anyone who answers!!!
Answer: $532,000
Step-by-step explanation:
If the company is making $140,000 and they get 20% each year, we just multiply it by 20%, or 0.20, and get $28,000. So, we would multiply that by 14, the years the company operated, and then add it to the original $140,000.
28,000 x 14 = 392,000
392,000 + 140,000 = 532,000
So, over the course of 14 years, the company made a profit of $532,000.
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2. A square with one side length represented by an
expression is shown below.
6(3x + 8) + 32 + 12x
Use the properties of operations to write three
different equivalent expressions to represent the
lengths of the other three sides of the square. One
of your expressions should contain only two terms.
We want to use properties to write expressions for the length of the other sides of the square.
Remember that the length of all the sides in a square is the same, so we only need to rewrite the above expression in two different ways.
First, we can use the distribute property in the first term:
[tex]\sf 6\times(3x + 8) + 32 + 12\times x[/tex]
[tex]\sf = 6\times3x + 6\times8 + 32 +12\times x[/tex]
[tex]= \sf 18\times x + 48 + 32 + 12\times x[/tex]
So this can be the length of one of the sides.
Now we can keep simplifying the above equation:
[tex]= \sf 18\times x + 48 + 32 + 12\times x[/tex]
To do it, we can use the distributive and associative property in the next way:
[tex]\sf 18\times x + 48 + 32 + 12\times x[/tex]
[tex]= \sf 18\times x + 12\times x + 48 + 32[/tex]
[tex]= \sf (18\times x + 12\times x) + (48 + 32)[/tex]
[tex]= \sf (18 + 12)\times x + 80[/tex]
[tex]= \sf 30\times x + 80[/tex]
This can be the expression to the other side.
Solve the following problems:
given: circle k(o), diameter us, mru=50°, mut=30°
find: m
The measure of angle M is 20°.
To solve the problem, we need to find the measure of angle M, given the information about Circle K with center O, diameter US, angle MRU = 50°, and angle MUT = 30°.
Step 1: Determine the relationship between angles MRU and MUT.
Since MRU and MUT are both inscribed angles in Circle K, they share the same intercepted arc, which is arc MU.
Step 2: Calculate the measure of arc MU.
The measure of an intercepted arc is twice the measure of the inscribed angle. Since angle MRU = 50°, the measure of arc MU will be 2 * 50° = 100°.
Step 3: Find the measure of angle M.
We know that angle MUT = 30°, and the measure of an intercepted arc is twice the measure of the inscribed angle. Therefore, the measure of arc MT = 2 * 30° = 60°. Now, since arc MU = 100°, we can determine the measure of arc MS (arc MS = arc MU - arc MT) which is 100° - 60° = 40°.
Step 4: Calculate the measure of angle M.
Finally, the measure of angle M can be found using the intercepted arc MS. Since the measure of an intercepted arc is twice the measure of the inscribed angle, angle M = 1/2 * arc MS = 1/2 * 40° = 20°.
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the produce manager at the local pig & whistle grocery store must determine how many pounds of
bananas to order weekly. based upon past experience, the demand for bananas is expected to be 100,
150, 200, or 250 pounds with the following probabilities: 100lbs 0.20; 150lbs 0.25, 200lbs 0.35, 250lbs 0.20.
the bananas cost the store $.45 per pound and are sold for $.085 per pound. any unsold bananas at the
end of each week are sold to a local zoo for $.30 per pound. use your knowledge of decision analysis to
model and solve this problem in order to recommend how many pounds of bananas the manager should
order each week
As per the probability, the expected demand for bananas per week is 182.5 pounds.
To model this problem, we can use decision analysis, which involves identifying the possible outcomes, assigning probabilities to each outcome, and calculating the expected value of each decision.
In this case, the possible outcomes are the demand for bananas, which can be 100, 150, 200, or 250 pounds per week. The probabilities of each demand level are given as 0.20, 0.25, 0.35, and 0.20, respectively.
Let X denote the demand for bananas in pounds. Then, the expected demand for bananas, denoted as E(X), can be calculated as follows:
E(X) = 100(0.20) + 150(0.25) + 200(0.35) + 250(0.20) = 182.5
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