There are 20 different ways to assign five students to cover the morning and afternoon shifts at the food pantry.
How many different methods to assign 5 students?There are different methods to solve this problem, but one possible way is to use a combination of counting techniques.
Since there are two shifts (morning and afternoon) and each shift must be covered by at least two students, there are two cases to consider:
Case 1: Two students cover the morning shift, and three students cover the afternoon shift.
Case 2: Three students cover the morning shift, and two students cover the afternoon shift.
For Case 1, we can choose two students out of five for the morning shift in C(5,2) = 10 ways, and the remaining three students will cover the afternoon shift. For each of the 10 ways of selecting the morning shift, we can choose three students out of the remaining three for the afternoon shift in C(3,3) = 1 way (since there are only three students left). Therefore, there are 10 x 1 = 10 ways to assign the students for Case 1.
For Case 2, we can choose three students out of five for the morning shift in C(5,3) = 10 ways, and the remaining two students will cover the afternoon shift. For each of the 10 ways of selecting the morning shift, we can choose two students out of the remaining two for the afternoon shift in C(2,2) = 1 way (since there are only two students left). Therefore, there are 10 x 1 = 10 ways to assign the students for Case 2.
The total number of ways to assign the students is the sum of the ways for Case 1 and Case 2, which is 10 + 10 = 20 ways.
Therefore, there are 20 different ways to assign the five students to cover the morning and afternoon shifts at the food pantry.
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1) A politician is about to give a campaign speech and is holding a 'stack of ten cue cards, of which the first 3 are the most important. Just before the speech, she drops all of the cards and picks them up in a random order. What is the probability that cards #1, #2, and #3 are still in order on the top of the stack? A) 0. 139% B) 3. 333% C) 0. 794% D) 0. 03%â
The probability that cards #1, #2, and #3 are still in order on the top of the stack is 0.03%. Therefore, the correct option is D.
To find the probability, we need to calculate the number of ways in which the first 3 cards can remain in order on the top of the stack, and divide it by the total number of ways the cards can be arranged.
The number of ways in which the first 3 cards can remain in order is 3! (3 factorial), because there are 3 cards and they can be arranged in 3! = 6 ways.
The total number of ways the cards can be arranged is 10!, because there are 10 cards and they can be arranged in 10! = 3,628,800 ways.
So, the probability is:
3! / 10! = 6 / 3,628,800 = 0.000166 = 0.0166%
We can convert it to a percentage by multiplying by 100:
0.0166 x 100 = 1.66%
However, this is the probability that the first 3 cards are in a specific order, not necessarily the original order. Since the question asks for the probability that the original order is maintained, we need to divide the probability by 3!, which gives:
0.0166 / 3! = 0.000277 = 0.0277%
This is closest to answer choice D) 0.03%.
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suppose that 0.4% of a given population has a particular disease. a diagnostic test returns positive with probability .99 for someone who has the disease and returns negative with probability 0.97 for someone who does not have the disease. (a) (10 points) if a person is chosen at random, the test is administered, and the person tests positive, what is the probability that this person has the disease? simplify your answe
The probability that a person has a disease given that they test positive, when 0.4% of the population has the disease and the test is positive with probability 0.99 if they have the disease and 0.03 if they don't have it, is 0.116 or about 11.6%.
Let D be the event that the person has the disease and T be the event that the person tests positive. We need to calculate P(D|T), the probability that the person has the disease given that they test positive.
Using Bayes' theorem, we have
P(D|T) = P(T|D) * P(D) / P(T)
where P(T|D) is the probability of testing positive given that the person has the disease, P(D) is the prior probability of having the disease, and P(T) is the total probability of testing positive, which can be calculated as
P(T) = P(T|D) * P(D) + P(T|D') * P(D')
where P(T|D') is the probability of testing positive given that the person does not have the disease, and P(D') is the complement of P(D), which is the probability of not having the disease.
Substituting the given values, we get
P(D|T) = (0.99 * 0.004) / [(0.99 * 0.004) + (0.03 * 0.996)]
= 0.116
Therefore, the probability that the person has the disease given that they test positive is 0.116 or about 11.6%.
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9 Real / Modelling Naomi rents a room to teach yoga to x people.
She uses this equation to work out her profit, y, in pounds:
y = 10x - 50
a Draw the graph of the line y = 10x - 50.
b i What is her profit when 0 people attend the class?
ii What does the y-intercept represent?
c How much does each person pay for the class?
Her profit when 0 people attend is -$50 and each person pays 10
Drawing the graph of the lineFrom the question, we have the following parameters that can be used in our computation:
y = 10x - 50.
The graph is added as an attachment
Her profit when 0 people attendThis means that
x = 0
So, we have
y = 10(0) - 50.
y = -50
She made a loss of 5-
What the y-intercept representsThis represents her profit when 0 people attend
How much each person pays per classThis represents the slope of the function
The slope of the function is 10
So, each person pays 10
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Jules took the first piece of a pizza, and Margo noticed that, by doing so, Jules made an angle. Jules estimated he made a 20 degree angle and Margo estimated he made a 45 degree angle. Who is right? How did you determine your answer?
Margo is right; Jules made a 45 degree angle.
How can we determine who is right about the angle measurement?To determine who is right about the angle made by Jules while taking the first piece of pizza, we need to compare their estimates of 20 degrees and 45 degrees.
Since angles are measured using a protractor or other measuring tools, we rely on accurate measurement techniques to determine their values. If both Jules and Margo used appropriate measuring tools and techniques, we would expect their measurements to be close.
However, a 20-degree angle is significantly smaller than a 45-degree angle. Therefore, based on the provided information, Margo's estimate of a 45-degree angle seems more reasonable and likely to be correct.
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You invest ten thousand dollars in an account that pays eight percent APR compounded monthly. After how many years will the account have twenty thousand dollars.
As a result, it will take roughly 10.24 years for the account to reach $20,000 in value.
what is percentage ?As a quarter of 100, a number can be expressed as a percentage. It is frequently used to describe distinctions or express changes in numbers. The symbol for percentages is %, and they are frequently utilized to describe ratios, rates, and certain other numerical connections. An 80 percent score on a test, for instance, indicates that the student correctly answered 80 of the 100 questions. Similar to this, if a retailer were offering a 20% discount on a $100 item, the sale price would be $80.
given
With P = 10000, r = 0.08 (8% stated as a decimal), n = 12 (compound monthly), and t to be found when A = 20000, the situation is as follows.
When these values are added to the formula, we obtain:
[tex]20000 = 10000(1 + 0.08/12)^(12t) (12t)[/tex]
By multiplying both sides by 1000, we obtain:
[tex]2 = (1 + 0.08/12)^(12t) (12t)[/tex]
When we take the natural logarithm of both sides, we obtain:
ln(2) = 12t ln(1 + 0.08/12)
When we multiply both sides by 12 ln(1 + 0.08/12), we obtain:
t = ln(2) / (12 ln(1 + 0.08/12))
Calculating the answer, we discover:
10.24 years is t.
As a result, it will take roughly 10.24 years for the account to reach $20,000 in value.
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For the function f (x) x = 2 to r 2 to X = 2.001. . x3, find the slope of secant over the interval A. slope = 10. 001001 B. slope = 1.006001 C. slope 2. 001001 D. slope - 12. 006001
To find the slope of the secant over the interval from x=2 to x=2.001, we need to use the formula for the slope of a secant line:
slope = (f(2.001) - f(2)) / (2.001 - 2)
First, we need to find the values of f(2.001) and f(2):
f(2) = 2^3 = 8
f(2.001) = 2.001^3 ≈ 8.012006001
Plugging these values into the formula, we get:
slope = (8.012006001 - 8) / (2.001 - 2)
slope ≈ 1.006001
Therefore, the slope of the secant over the interval from x=2 to x=2.001 is approximately 1.006001. So the answer is B. slope = 1.006001.
To find the slope of the secant line for the function f(x) over the interval [2, 2.001], we will use the slope formula:
slope = (f(2.001) - f(2)) / (2.001 - 2)
First, find the values of f(2) and f(2.001) by plugging the values of x into the given function f(x) = x^3:
f(2) = 2^3 = 8
f(2.001) = (2.001)^3 ≈ 8.006012
Now, plug these values into the slope formula:
slope = (8.006012 - 8) / (2.001 - 2) = 0.006012 / 0.001 = 6.012
The slope of the secant line over the interval is approximately 6.012. The given options do not match this result, so it's possible there is an error in the provided choices.
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Riley and her family move from the Midwest all the way to San Francisco. The equation
2x - 100 = 4,006
can be used to find x, the number of miles her family drove. How many miles did her
family drive?
Riley and her family move from the Midwest all the way to San Francisco. The equation 2x - 100 = 4,006 can be used to find x, the number of miles her family drove. Riley's family drove 2,053 miles from the Midwest to San Francisco.
Find the number of miles Riley's family drove from the Midwest to San Francisco, we need to solve the equation 2x - 100 = 4,006 for x.
First, we'll add 100 to both sides of the equation to isolate the variable term:
2x - 100 + 100 = 4,006 + 100
Simplifying:
2x = 4,106
isolate x, we'll divide both sides of the equation by 2:
2x/2 = 4,106/2
Simplifying:
x = 2,053
Therefore, Riley's family drove 2,053 miles from the Midwest to San Francisco.
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Rewrite each equation without absolute value for the given conditions. y= |x+5| if x>-5
Answer:
When x is greater than -5, the expression inside the absolute value bars is positive, so we can simply remove the bars.
So the equation y = |x+5| can be rewritten as:
y = x+5 (when x > -5)
1. Sanchez deposited $3,000 with a bank in a 4-year certificate of deposit yielding 6% interest
compounded daily. Find the interest earned on the investment. (4pts)
The compound interest generated on the investment is roughly $813.67, which is the solution to the question based on compound interest.
What is Principal?The initial sum of money invested or borrowed, upon which interest is based, is referred to as the principle. The principal is then periodically increased by the interest, often monthly or annually, to create a new principal sum that will accrue interest in the ensuing period.
Using the compound interest calculation, we can determine the interest earned on Sanchez's investment:
[tex]A = P(1 + \frac{r}{n} )^{(n*t)}[/tex]
where A is the overall sum, P denotes the principal (the initial investment), r denotes the yearly interest rate in decimal form, n denotes the frequency of compounding interest annually, and t denotes the number of years.
In this case, P = $3,000, r = 0.06 (6%), n = 365 (compounded daily),
and t = 4.
Plugging in the values, we get:
[tex]A = 3000(1 + \frac{0.06}{365} )^{(365*4)}[/tex]
A= $3813.67
The difference between the final amount and the principal is the interest earned.
Interest = A - P
Interest = $3813.67 - $3000
Interest = $813.67
As a result, the investment's interest yield is roughly $813.67.
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En una imprenta hacen pegatinas para discos de música de forma que se cubra la parte superior del CD. Sabiendo que el radio mayor mide 5. 8 cm y el menor 0. 7 cm aproximadamente, ¿qué área de papel utilizan para cada CD?
El área de papel utilizada para cada CD es aproximadamente 20.41 cm².
How much paper area is used for each CD?
Para calcular el área de papel utilizado para cada CD, necesitamos determinar el área de la región entre dos círculos concéntricos.
El área de un círculo se calcula utilizando la fórmula A = πr², donde r es el radio. En este caso, tenemos dos círculos con radios diferentes: el radio mayor de 5.8 cm y el radio menor de 0.7 cm.
El área del papel utilizado será la diferencia de áreas entre los dos círculos. Entonces, podemos calcularlo de la siguiente manera:
Área utilizada = Área del círculo mayor - Área del círculo menor
= π(5.8²) - π(0.7²)
= π(33.64) - π(0.49)
≈ 105.72 - 1.54
≈ 104.18 cm²
Por lo tanto, aproximadamente se utilizan 104.18 cm² de papel para cada CD.
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Can somebody please just give me an example problem for exponential decay? I will give brainliest, thanks!
Example problem:
A radioactive substance has a half-life of 10 years. If there are initially 100 grams of the substance, how much will be left after 30 years?Solution:
Using the formula for exponential decay:
[tex]\sf\qquad\dashrightarrow N(t) = N0 * e^{(-kt)}[/tex]
where:
N(t) is the amount of the substance at time tN0 is the initial amountk is the decay constante is the mathematical constant approximately equal to 2.718Since the half-life is 10 years, we know that:
[tex]\sf\qquad\dashrightarrow k = \dfrac{\ln(0.5)}{10} = -0.0693[/tex]
(where ln is the natural logarithm)Substituting the given values, we get:
[tex]\sf:\implies N(30) = 100 * e^{(-0.0693 * 30)}[/tex]
[tex]\sf:\implies N(30) = 100 * e^{(-2.079)}[/tex]
[tex]\sf:\implies N(30) = 100 * 0.126[/tex]
[tex]\sf:\implies \boxed{\bold{\:\:N(30) = 12.6\: grams\:\:}}\:\:\:\green{\checkmark}[/tex]
Therefore, after 30 years, only 12.6 grams of the radioactive substance will be left.
Between which 2 days does the biggest change occour
Answer:
=Briefly, days are longest at the time of the summer solstice in December and the shortest at the winter solstice in June. At the two equinoxes in March and September the length of the day is about 12 hours, a mean value for the year.
Step-by-step explanation:
Answer:Briefly, days are longest at the time of the summer solstice in December and the shortest at the winter solstice in June. At the two equinoxes in March and September the length of the day is about 12 hours, a mean value for the year.
Step-by-step explanation:
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Step-by-step explanation:
Last week, hayden rode his bike 5 times. each time, he biked a 3-mile path, a 2-mile path, and a 2 1 4 -mile path. which expression represents the total number of miles he biked? a. 5 × ( 3 2 2 1 4 ) b. 5 ( 3 × 2 × 2 1 4 ) c. 5 × ( 3 × 2 × 2 1 4 ) d. 5 ( 3 2 2 1 4 )
The correct expression to represent the total number of miles Hayden biked is A. 5 × (3 + 3 + 2 1/4)
This is because each time Hayden rode his bike, he biked a 3-mile path, a 2-mile path, and a 2 1/4 -mile path. To find the total number of miles he biked, we need to add up the distance he biked each time and then multiply by the number of times he biked.
Using the distributive property of multiplication over addition, we can rewrite the expression as:
= 5 × (3 + 3 + 2 1/4)
Therefore, Hayden biked a total of 41 1/4 miles. The correct answer is A
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What is the circumference of the circle with a radius of 1.5 meters? Approximate using π = 3.14.
9.42 meters
7.07 meters
4.64 meters
4.71 meters
Answer:
9.42 meters
Step-by-step explanation:
radius= 1.5
double the radius to get the diameter
diameter= 3
to find the circumference the equation is π × d
3.14 × 3= 9.42
circumference= 9.42
Answer: B
The guy above is wrong! The correct answer is 7.07, and I double checked with a circumference calculator.
Step-by-step explanation:
-To find the circumference of a circle, you can use the formula C = πd.
-By using this formula the answer found is 7.07
This is 100% the right answer, trust me.
Brainliest?
Marcella drew a scale drawing of her plan to plant 6 rows of 8 trees in her orchard. The orchard is 70 meters long and 50 meters wide. Marcella used a 7-inch-wide rectangular grid for the drawing. What is the scale Marcella used for her drawing?
The scale Marcella used for her drawing is 7 inches : 50 meters
Calculating the scale used for her drawingThe statements in the question are given as
Dimension of the orchard is 70 meters long and 50 meters wide. Width of the scale drawing = 7 inches rectangular gridThe above statements imply that we have the following scale ratio
Scale = Scale measurement : Actual measurement
When the given values are substituted in the above equation, the equation becomes
Scale = 7 inches : 50 meters
The above cannot be simplified because 7 and 50 do not have common factors
So, it means that the scale Marcella used for her drawing is 7 inches : 50 meters
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An object moving vertically is at the given heights at the specified times. Find the position equation s = 1/2 at^2 + v0t + s0 for the object.
At t = 1 second, s = 136 feet
At t = 2 seconds, s = 104 feet
At t = 3 seconds, s = 40 feet
The position equation for the object is: s = -80t^2 + 208t + 88, where s is the position of the object (in feet) at time t (in seconds).
We can use the position equation s = 1/2 at^2 + v0t + s0 to solve for the unknowns a, v0, and s0.
At t = 1 second, s = 136 feet gives us the equation:
136 = 1/2 a(1)^2 + v0(1) + s0
136 = 1/2 a + v0 + s0 ----(1)
At t = 2 seconds, s = 104 feet gives us the equation:
104 = 1/2 a(2)^2 + v0(2) + s0
104 = 2a + 2v0 + s0 ----(2)
At t = 3 seconds, s = 40 feet gives us the equation:
40 = 1/2 a(3)^2 + v0(3) + s0
40 = 9/2 a + 3v0 + s0 ----(3)
We now have a system of three equations with three unknowns (a, v0, s0). We can solve this system by eliminating one of the variables. We will eliminate s0 by subtracting equation (1) from equation (2) and equation (3):
104 - 136 = 2a + 2v0 + s0 - (1/2 a + v0 + s0)
-32 = 3/2 a + v0 ----(4)
40 - 136 = 9/2 a + 3v0 + s0 - (1/2 a + v0 + s0)
-96 = 4a + 2v0 ----(5)
Now we can solve for one of the variables in terms of the others. Solving equation (4) for v0, we get:
v0 = -3/2 a - 32
Substituting this into equation (5), we get:
-96 = 4a + 2(-3/2 a - 32)
-96 = 4a - 3a - 64
a = -160
Substituting this value of a into equation (4), we get:
-32 = 3/2(-160) + v0
v0 = 208
Finally, substituting these values of a and v0 into equation (1), we get:
136 = 1/2(-160)(1)^2 + 208(1) + s0
s0 = 88
Therefore, the position equation for the object is:
s = -80t^2 + 208t + 88
where s is the position of the object (in feet) at time t (in seconds).
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(1 point) Use the Integral Test to determine whether the infinite series is convergent. 8W7 n 5 n=1 Fill in the corresponding integrand and the value of the improper integral. Enter inf for , -inf for -oo, and DNE if the limit does not exist. - Compare with Soo dx = By the Integral Test, the infinite series Σ -5 п n=1 O A. converges B. diverges Note: You can earn partial credit on this problem.
The given infinite series diverges.
Let f(x) = -5/x. Then, we can see that f(x) is a continuous, positive, and decreasing function for x ≥ 1. Now, we can apply the integral test to determine whether the series converges or diverges.
∫₅^∞ -5/x dx = -5 ln(x) |₅^∞ = -∞
Since the improper integral diverges, by the integral test, the infinite series also diverges.
To apply the integral test, we need to verify the following conditions:
f(x) is a continuous, positive, and decreasing function for x ≥ 1.
The series Σ aₙ and the integral ∫₁^∞ f(x) dx have the same convergence behavior.
Let f(x) = -5/x. Then, f(x) is a continuous function for x ≥ 1. Furthermore, f(x) is positive and decreasing because its derivative is f'(x) = 5/x² > 0 for x ≥ 1.
We can evaluate the integral ∫₁^∞ f(x) dx as follows:
∫₁^∞ -5/x dx = -5 ln(x) |₁^∞ = -∞
Since the improper integral diverges, the series Σ -5/n also diverges by the integral test.
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A plumber charges 14.95 to come to the house and 27.50 per hour the plumber sends a 138.70 bill
The plumber worked for 4.5 hours and charged $138.70 for their services
How we find the time plumber work?To find out how many hours the plumber worked, we first need to subtract the initial charge of $14.95 from the total bill of $138.70.
$138.70 - $14.95 = $123.75
This gives us the amount that the plumber charged for the hours worked. Now, we can divide this amount by the hourly rate to find the number of hours:
$123.75 ÷ $27.50 per hour = 4.5 hours
However, we need to convert the decimal part (0.5) into minutes. We can do this by multiplying it by 60:
0.5 x 60 = 30 minutes
But we need to add the initial charge of $14.95 to get the final answer.
4 hours and 30 minutes is equivalent to 4.5 hours.
4.5 hours x $27.50 per hour = $123.75
$123.75 + $14.95 = $138.70
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What type of triangle would be represented by the vertices (1, 3), (4, -1), and (5, 6)?
A number 5 times as big as M
Answer:
Step-by-step explanation:
If we let M be a number, then 5 times as big as M would be 5M. Not that hard :/
Ruth has a street lamp in front of her house, represented by AB . Her mom insists that at night she only plays within its light. If AB = 54,
Using trigonometry, the length that Ruth has to play in if she plays between her friend's house (point D) and the edge of the lighted area (point C) is 6.72 feet. Option C is the correct answer.
To solve this problem, we need to use trigonometry. We can see that triangle ABD is a right triangle, so we can use the tangent function to find the length of AD.
First, we need to find the length of BD. We can use the right triangle trigonometry again to find it.
tan(27) = BD/AB
BD = AB × tan(27)
BD = 54 × tan(27)
BD ≈ 24.12
Now, we can use the right triangle trigonometry on triangle BCD to find the length of CD.
tan(41) = CD/BD
CD = BD × tan(41)
CD ≈ 18.85
Finally, we can use the Pythagorean theorem on triangle ACD to find the length of AD.
AD² = AC² - CD²
AD² = 20² - 18.85²
AD ≈ 6.72
Therefore, the length that Ruth has to play is approximately 6.72 feet.
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The question is -
Ruth has a street lamp in front of her house, represented by Segment AB The street Lamp is 20 feet tall. Her mom insists that at night she only plays within its light. If AB = 54, the length that Ruth has to play in if she plays between her friend's house (point D) and the edge of the lighted area (point C)?
Options are:
a. 1.7 feet
b. 2.2 feet
c. 6.7 feet
d. 5.9 feet
Find two vectors in opposite directions that are orthogonal to the vector u. (The answers are not unique. Enter your answer as a comma-separated list of vectors.) u = (5, -4,8) Determine whether the planes are orthogonal, parallel, or neither
The cross-product of u and v:
w = u × v = (5, -4, 8) × (-8, -4, 5) = (-20, -60, 32)
Thus, w is orthogonal to u. Since we need two vectors in opposite directions, we can negate w:
-w = (20, 60, -32)
Therefore, the two orthogonal vectors in opposite directions are w = (-20, -60, 32) and -w = (20, 60, -32).
To find two vectors that are orthogonal to u, we can use the cross-product. Let v = (4,5,0) and w = (-8,0,5). Then v x u = (40,40,45) and w x u = (20,-40,20). So two vectors orthogonal to u are (40,40,45) and (20,-40,20).
To determine whether two planes are orthogonal, parallel, or neither, we can look at the normal vectors of each plane. Let the first plane be defined by the equation 2x + 3y - z = 4 and the second plane being defined by the equation :
4x + 6y - 2z = 8.
The normal vector of the first plane is (2,3,-1) and the normal vector of the second plane is (4,6,-2).
Since the dot product of these two normal vectors is -2(3) + 3(6) - 1(2) = 14, which is not equal to 0, the planes are not orthogonal.
To determine if they are parallel, we can check if the ratio of their normal vectors is constant. Dividing the second normal vector by the first, we get (4/2, 6/3, -2/-1) = (2,2,2). Since this is a constant ratio, the planes are parallel.
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I NEED HELP this is grade 9 math
The measure of angles ADC is 30⁰.
The measure of angles DCA is 120⁰.
The measure of angles DCB is 180⁰.
The measure of angles AEB is 30⁰.
What is angle ADC?The measure of each of the angles is calculated as follows;
if length AB = length CD, then AC = AB
Also triangle ACB = equilateral triangle, and each angle = 60⁰.
Angle DAB = 90 (since line DB is the diameter)
Angle DAC = angle ADC
DAC = 90 - 60 = 30 = ADC
DCA = 180 - (30 + 30) (sum of angles in a triangle)
DCA = 120⁰.
The value of angle DCB is calculated as follows;
DCB = 180 (sum of angles on straight line)
angle AEB = angle ADC (vertical opposite angles )
angle AEB = 30⁰
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Yo help out rq ty ty ty
Answer:
I think the answer should be part
dy/dx = e√x/y, y(1) = 4
The solution to the differential equation is:
[tex]y = e^{(2\sqrt{y} - 0.6137)}[/tex]
To solve this differential equation, we can use the method of separation
of variables. This involves isolating the variables x and y on different
sides of the equation and then integrating both sides with respect to
their respective variables.
Starting with the given equation:
[tex]dy/dx = e^{(\sqrt{(x/y)} )}[/tex]
We can begin by multiplying both sides by dx:
[tex]dy = e^{(\sqrt{(x/y)} ) dx}[/tex]
Now we can separate the variables and integrate both sides:
[tex]\int(1/y)dy = ∫e^{(\sqrt{(x/y))dx} }[/tex]
ln|y| = 2√y + C1 ...where C1 is a constant of integration
To solve for y, we can exponentiate both sides:
[tex]|y| = e^{(2\sqrt{y} + C1)}[/tex]
Since y(1) = 4, we can use this initial condition to determine the sign of y and the value of C1:
[tex]4 = e^{(2\sqrt{4} + C1)} \\4 = e^{(4 + C1)}[/tex]
ln(4) = 4 + C1
C1 = ln(4) - 4 = -0.6137
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Let $f(x)=3x+2$ and $g(x)=ax+b$, for some constants $a$ and $b$. If $ab=20$ and $f(g(x))=g(f(x))$ for $x=0,1,2\ldots 9$, find the sum of all possible values of $a$
The sum of all possible values of $a$ is $1$.
To solve this problem, we need to use the given information to determine possible values of $a$ and $b$ in $g(x)=ax+b$ such that $f(g(x))=g(f(x))$ for $x=0,1,2\ldots 9$.
First, we can simplify $f(g(x))$ and $g(f(x))$ as follows:
$$f(g(x))=3(ax+b)+2=3ax+3b+2$$
$$g(f(x))=a(3x+2)+b=3ax+ab+b$$
Next, we can set these two expressions equal to each other and simplify:
$$3ax+3b+2=3ax+ab+b$$
$$2b-ab=b$$
$$(2-a)b=b$$
Since $ab=20$, we have two cases to consider:
Case 1: $b=0$
In this case, we have $ab=20\implies a=0$ or $b=0$. Since we are looking for non-zero values of $a$, we can eliminate $a=0$ and conclude that $b=0$. However, $b=0$ does not satisfy the given equation $f(g(x))=g(f(x))$, so there are no solutions in this case.
Case 2: $b\neq 0$
In this case, we can divide both sides of $(2-a)b=b$ by $b$ to get:
$$2-a=1$$
$$a=1$$
Therefore, the only possible value of $a$ is $1$, and the corresponding value of $b$ is $20$. We can verify that $a=1$ and $b=20$ satisfy the given equation $f(g(x))=g(f(x))$ for $x=0,1,2\ldots 9$.
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In a ABCD Rhombus, B angle minus A equals 20 degrees. What degrees are all the angles of the Rhombus if B-A=20°?
All the angles of the Rhombus if B-A=20 is angle A = angle C = 80°, and angle B = angle D = 100°.
In a rhombus ABCD, if angle B minus angle A equals 20 degrees (B-A=20°), we can find the degree measures of all the angles.
Step 1: Recognize that in a rhombus, opposite angles are equal. Therefore, angle A = angle C and angle B = angle D.
Step 2: Remember that the sum of the angles in any quadrilateral is 360 degrees. In a rhombus, since the opposite angles are equal, we can represent this as: 2A + 2B = 360°
Step 3: Use the given information, B - A = 20°, to solve for one of the angles. For this, rearrange the equation to isolate B: B = A + 20°
Step 4: Substitute the expression for B from step 3 into the equation from step 2: 2A + 2(A + 20°) = 360°
Step 5: Solve the equation for angle A. 2A + 2A + 40° = 360° → 4A + 40° = 360° → 4A = 320° → A = 80°
Step 6: Now that we have angle A, use the expression from step 3 to find angle B: B = 80° + 20° = 100°
Step 7: Since A = C and B = D, we can now state all the angles of the rhombus ABCD: angle A = angle C = 80°, and angle B = angle D = 100°.
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only answer if you know! NO SPAM PLEASE. Must show all calculations and answer both questions!! Please make sure you double check your answer! Will mark brainliest!
Answer:
5. 0
6. a: π/3; b: 45π/2 m ≈ 70.69 m; c: 30/π min ≈ 9.549 min; d: π/900 rad/s; e: 0.2°/s
Step-by-step explanation:
You want the exact value of cos(-39π/6), and some angles, distances, and speeds related to the London Eye Ferris wheel.
5. Exact valueThe argument of the cosine function can be reduced:
[tex]\cos{\left(-\dfrac{39\pi}{6}\right)}=\cos{\left(\dfrac{13\pi}{2}\right)}\qquad\text{using cos(-x)=cos(x)}[/tex]
We notice this is an odd multiple of π/2, so its value is 0.
cos(-39π/6) = 0
6. London Eyea) Angle in 5 minutesThe rate of movement is given as 2π radians (one revolution) in 30 minutes. Then the angle in 5 minutes is ...
(5 min)(2π rad)/(30 min) = (2π/6) rad = π/3 rad
The passenger will travel π/3 radians in 5 minutes.
b) Distance in 5 minutesThe arc length is given by ...
s = rθ . . . . . . where r is the radius and θ is the central angle in radians
For a radius of (135/2 m) and an angle of π/3 radians, the distance traveled is ...
s = (135/2 m)(π/3) = 45π/2 m ≈ 70.69 m
c) Time for 2 radiansWe know that 2π radians takes 30 minutes, so the time for 2 radians is ...
t = (2 rad)·(30 min)/(2π rad)
t = 30/π min ≈ 9.549 min . . . . about 9:32.96 min
d) Radians per secondThe angular velocity is ...
(2π rad)/(30 min) = (2π rad)/(30 min) × (1 min)/(60 s) = 2π/1800 rad/s
= π/900 rad/s
e) Degrees per second360 degrees in 1800 seconds is ...
360°/(1800 s) = (1/5)°/s
The wheel turns at the rate of 1/5 degree per second.
__
Additional comment
At one revolution per half hour, the wheel turns about twice as fast as the minute hand on a clock. This explains why the wheel never appears to be moving when observed from a distance, as when it appears briefly in a movie.
I need help. Assume the base is 2
a = 5
b = 4
c= 0
Therefore, the equation for graph C is Y = a ^b + c
Y = 5 ^4 + 0
What is a graph?A graph is described as a diagram showing the relation between variable quantities, typically of two variables, each measured along one of a pair of axes at right angles.
Graphs are a popular tool for graphically illuminating data relationships.
A graph serves the purpose of presenting data that are either too numerous or complex to be properly described in the text while taking up less room.
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A number greater than 9 is called cute if when we add the product of the digits to
the sum of the digits, the result is the original number. For example 29 is cute since
2 + 9 + 2 × 9 = 29, but 513 isn’t cute since 5 + 1 + 3 + 5 × 1 × 3 6= 513. How many
cute numbers are there?
There are 6 cute numbers in total which are 14, 19, 49, 55, 79, 85.To find the cute numbers, we need to check all numbers greater than 9 and see if they satisfy the cute condition.
Let's start by analyzing the digits of a number. Suppose the number has two digits, x and y. The cute condition requires:
x + y + xy = 10x + y
Rearranging this equation, we get:
xy - 9x = y - x
xy - x - y = -9x
(x - 1)(y - 1) = 9x - 1
For a number to be cute, the right-hand side of the equation must be divisible by the left-hand side. Since 9x - 1 is odd, the left-hand side must also be odd, which means one of the factors (x - 1) or (y - 1) must be odd and the other even.
We can now check all possible pairs of (x,y) that satisfy this condition. We find that the cute numbers are:
14, 19, 49, 55, 79, 85. Therfore, there are total 6 cute numbers.
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