True. In a continuous uniform distribution, all values within a specified interval are equally likely to occur. The mean or expected value of this distribution can be found by taking the average of the upper and lower limits of the interval.
This can be represented mathematically as (a+b)/2, where a and b are the lower and upper limits of the interval. For example, if we have a continuous uniform distribution on the interval [0,10], the mean value would be (0+10)/2 = 5.
This means that on average, we would expect a randomly selected value from this distribution to be around 5. It's important to note that the mean of a continuous uniform distribution is not affected by the shape or spread of the distribution, as all values have an equal chance of occurring.
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At a ski resort, there is a 30% chance of snow for each of the next four days. What is the probability that it snows 0 days? 1 day? 2 days? 3 days? 4 days? How many snowy days should a skier expect during this time period?
The probability would be 7/12.
Here, we have,
Consider A is the event that there is snowing in first three days and B is the event that there is snowing in next four days.
According to the question,
P(A) = 1/3
P(B)= 1/4
Thus, the probability that it snows at least once during the first week of January
= snow in first three days or snow in next four days
= P(A∪B)
=P(A) + P(B) - P(A∩B)
( ∵ A and B are independent ⇒P(A∩B) = 0 )
=1/3 + 1/4
=7/12
Hence, The probability would be 7/12.
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The monthly demand function for x units of a product sold by a monopoly is p = 5,300 - dollars, and its average cost is C = 3,010 + 2x dollars. Production is limited to 100 units. Find the revenue function!
To find the revenue function, we need to multiply the price (p) by the quantity sold (x). The price function given is p = 5,300 - dollars, so we can substitute this into our revenue function as follows:
Revenue = p * x
Revenue = (5,300 - dollars) * x
We also know that production is limited to 100 units, so we need to take that into account when determining the revenue function. If x is greater than 100, then the revenue will be limited to 100 units sold. If x is less than or equal to 100, then the revenue will be based on the actual quantity sold.
To incorporate this constraint into our revenue function, we can use a piecewise function:
Revenue = { (5,300 - dollars) * 100 if x > 100
(5,300 - dollars) * x if x <= 100 }
Simplifying the piecewise function, we get:
Revenue = { 530,000 - (dollars * 100) if x > 100
(5,300 - dollars) * x if x <= 100 }
Therefore, the revenue function for this monopoly is:
Revenue = { 530,000 - 100 * dollars if x > 100
(5,300 - dollars) * x if x <= 100 }
Hi! To find the revenue function, we first need to determine the total revenue, which is the product of the price per unit (p) and the number of units sold (x). Given the demand function p = 5,300 - x dollars and the average cost function C = 3,010 + 2x dollars, we can find the revenue function as follows:
Revenue function, R(x) = p * x
R(x) = (5,300 - x) * x
By simplifying the equation, we get:
R(x) = 5,300x - x^2
So, the revenue function for this monopoly is R(x) = 5,300x - x^2.
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please help this is my chapter 7 practice test
Answer:
90
Step-by-step explanation:
Which expression are equivalent to the given expression?
By the rule of indices the given expression is equivalent to x⁻² y⁻⁵.(option B)
What is index?
In mathematics, Index in plural indices is the power or exponent which is raised to a number or a variable. For example, for the number 2⁵, 5 is the index of 2. Which gives the value 32.
Given expression is y⁻⁸ y³ x⁰ x⁻²
Here we use the law of indices.
By the law of indices it can be stated that the indices at the time of multiplication is being added up. That is zᵃ × zᵇ = zᵃ⁺ᵇ.
Using this concept of indices the given expression can be rewritten as
y⁻⁸⁺³ x⁰⁻²
= y ⁻⁵ x⁻²
= x⁻² y⁻⁵
Hence, the given expression is equivalent to x⁻² y⁻⁵.
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you select a marble without looking and then put it back if you do this 90 times what is the best predictions that you will pick an orange or a pink marble?
If you select a marble without looking and then put it back if you do this 90 times, the probability of selecting an orange or a pink marble on any given trial is 1/2 or 0.5.
If you select a marble without looking and then put it back, the probability of selecting any particular marble on any given trial is the same for all marbles. Assuming there are only two possible outcomes - selecting an orange marble or a pink marble - the probability of selecting an orange or a pink marble on any given trial is 1/2 or 0.5.
Since the probability of selecting an orange or a pink marble on each trial is the same, the best prediction for the number of times you will select an orange or a pink marble out of 90 trials is an equal number of times for each color. Therefore, the best prediction for the number of times you will select an orange or a pink marble is 45 times each.
This is only a prediction based on probability, and the actual number of times you select an orange or a pink marble may differ from the prediction due to chance. However, over a large number of trials, the actual outcomes should approach the predicted probabilities.
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Mohal is a waiter at a restaurant. Each day he works, Mohal will make a guaranteed wage of $25, however the additional amount that Mohal earns from tips depends on the number of tables he waits on that day. From past experience, Mohal noticed that he will get about $15 in tips for each table he waits on. How much would Mohal expect to earn in a day on which he waits on 16 tables? How much would Mohal expect to make in a day when waiting on
�
t tables?
Answer:
If Mohal waits on 16 tables, he can expect to earn $25 wages + ($15 tips x 16 tables) = $265 in a day.
If Mohal waits on 1 table, he can expect to earn $25 wages + ($15 tips x 1 table) = $40 in a day.
1. This table représents Ana's check register. Her checking account had a balance of
$1,093. 12 on October 8.
Use the information in the check register to determine the balance of Ana's checking
account after the transaction on October 22.
Ana's Check Register
Date
Description
Deposit
Withdrawal
Balance
10/8
$1,093. 12
10/15
Rent
$525. 50
10/22
Paycheck
$645. 87
The balance of Ana's checking account after the transaction on October 22 is $1,213.49.
To determine the balance of Ana's checking account after the transaction on October 22, we'll follow these steps:
1. Start with the initial balance on October 8: $1,093.12
2. Subtract the withdrawal for rent on October 15: $1,093.12 - $525.50 = $567.62
3. Add the deposit from the paycheck on October 22: $567.62 + $645.87 = $1,213.49
The balance of Ana's checking account after the transaction on October 22 is $1,213.49.
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Identify if the proportion is true or false12:4=9:3
you purchase a computer for 875 plus 5% sales tax. you decide to finance it through the stores 0% program for 12 months. the terms state that you must pay
The amount of interest paid for missing the payment in the eight month would be C. $87.28
How to find the interest ?First, find the total cost of the computer including the sales price :
= 875 x ( 1 + 0. 05 )
= 875 x 1. 05
= $ 918. 75
Then, find the monthly rate of interest :
= 14. 25 / 12
= 1. 1875 %
This leads to an eight month interest rate of :
= 8 x 1. 1875 %
= 9. 5 %
The interest to be paid:
= 9. 5 % x 918. 75
= $ 87. 28
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Full question is:
You purchase a computer for $875.00 plus 5% sales tax. You decide to finance it through the store's 0% program for 12 months. The terms state that you must pay $100.00/month and that if you miss a payment, you will be assessed a late fee of $39.00 plus the interest accrued to that point at a 14.25% APR. If you miss a payment in the eighth month, how much interest will you be charged?. A. $83.13. B. $16.63. C. $87.28. D. 58.19
The school physics class has built a trebuchet (catapult) that is big enough to launch a watermelon. the math class has created the function h(t) = -16( t - 5)2 + 455 to model the height, in feet, after t seconds, of a watermelon launched into the air from a hilltop near the school the x - intercepts of this function are (-0.33 , 0) and (10.33 , 0)
the watermelon is hitting the ground at around ____ seconds
The watermelon is hitting the ground at around 10.33 seconds.
To find out when the watermelon hits the ground, we need to look for the time when the height of the watermelon is zero. This is because the watermelon will be on the ground at that point.
The x-intercepts of the function h(t) give us the times when the height is zero. So, we know that the watermelon will hit the ground at t = -0.33 seconds and t = 10.33 seconds.
However, the negative value doesn't make sense in this context, so we can ignore that solution. Therefore, the watermelon is hitting the ground at around 10.33 seconds.
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An electrical charge of 0. 000003 coulombs is 0. 06 m away from another electrical charge of 0. 0000009 coulombs. Solve this equation
The force between the two charges is 1.69 x[tex]10^-^1^5[/tex] N
How to find the equation?The force between two electric charges can be calculated using Coulomb's law, which states that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. Mathematically, it can be expressed as:
F = k * (q1 * q2) / d²
where F is the force, k is Coulomb's constant (9 x 10⁹ N m² / C²), q1 and q2 are the magnitudes of the two charges, and d is the distance between them.
Substituting the given values, we get:
F = (9 x [tex]10^9[/tex] N m² / C²) * (0.000003 C) * (0.0000009 C) / (0.06 m)²
Simplifying this expression gives:
F = 1.69 x[tex]10^-^1^5[/tex] N
Therefore, the force between the two charges is 1.69 x [tex]10^-^1^5[/tex] N.
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Find the maximum value of s=xy yz xz where x y z=21
The maximum value of s is 9261, which is obtained when x = y = z = 7.
To find the maximum value of s=xyz, we can use the AM-GM inequality, which states that the arithmetic mean of a set of non-negative numbers is greater than or equal to the geometric mean of the same set of numbers.
Mathematically, this can be represented as:[tex](1/3)(x + y + z) \geq (xyz)^(1/3)[/tex]Multiplying both sides of the inequality by[tex]3(xyz)^(1/3)[/tex],
we get: [tex](x + y + z) \geq 3(xyz)^(1/3)[/tex] Now,
we can substitute the given value of x + y + z = 21, to obtain: 21 ≥ [tex]3(xyz)^(1/3)[/tex]
Cubing both sides of the inequality, we get: [tex]21^3 \geq 27(xyz)[/tex]
Simplifying the expression, we obtain: s=[tex]xyz \leq (21^3)/27[/tex]= 9261.
The maximum value of s=xyz is obtained when x = y = z = 7, and the value of s is equal to 9261. This result is obtained using the AM-GM inequality, which is a useful tool for solving optimization problems involving non-negative numbers.
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The population of a certain bacteria is known to double every 10 hours. Assuming exponential growth, determine the time that it would take for the bacteria to triple in number
It would take approximately 20 hours for the bacteria to triple in number.
Given that the bacteria population doubles every 10 hours, we can use exponential growth to determine the time it would take for the population to triple.
Let's represent the initial population as P0 and the time it takes for the population to triple as t.
Using the concept of exponential growth, we can express the population at time t as P(t) = P0 * 2^(t/10).
Since we want the population to triple, we set P(t) = 3 * P0:
3 * P0 = P0 * 2^(t/10).
We can cancel out P0 from both sides of the equation:
3 = 2^(t/10).
To solve for t, we can take the logarithm of both sides. Using the base-2 logarithm (log2) gives us:
log2(3) = t/10.
Using a calculator, we find that log2(3) is approximately 1.585.
Now, we can solve for t:
1.585 = t/10.
Multiplying both sides of the equation by 10 gives us:
15.85 = t.
Rounding to the nearest hour, the time it would take for the bacteria population to triple is approximately 16 hours.
Therefore, the bacteria population would take approximately 20 hours to triple in number.
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In ΔGHI, h = 9. 6 cm, g = 9. 3 cm and ∠G=109°. Find all possible values of ∠H, to the nearest 10th of a degree
The two possible values for angle H in triangle GHI are approximately 93.1 degrees and 273.1 degrees, rounded to the nearest tenth of a degree
How to find possible angle in GHI triangle?To find the possible values of angle H in triangle GHI, we can use the law of cosines.
Let's label angle H as x. Then, we can use the law of cosines to solve for x:
cos(x) = (9.3² + 9.6² - 2(9.3)(9.6)cos(109))/ (2 * 9.3 * 9.6)
Simplifying this equation, we get:
cos(x) = -0.0588
To solve for x, we can take the inverse cosine of both sides:
x = cos⁻ ¹ (-0.0588)
Using a calculator, we can find that x is approximately 93.1 degrees.
However, there is another possible value for angle H. Since cosine is negative in the second and third quadrants,
We can add 180 degrees to our previous result to find the second possible value for angle H:
x = 93.1 + 180 = 273.1 degrees
So the two possible values for angle H are approximately 93.1 degrees and 273.1 degrees, rounded to the nearest tenth of a degree.
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The radius of a cylinder water tank is 6 Ft and it’s height is 11 ft what is the volume of the tank.
Answer:
1243 ft³
Step-by-step explanation:
Given the volume formula for a cylinder:
[tex]V=\pi r^{2} h[/tex]
and we know that the radius is 6 and the height is 11, we can substitute:
[tex]V=\pi 6^{2} 11[/tex]
square 6
V=π36(11)
use 3.14 for pi and multiply everything together
V=3.14(36)(11)
simplify
V=1243.44
1243.44 rounded to the nearest whole number is 1,243 ft³.
Hope this helps! :)
a group conducting a survey randomly selects adults in a certain region. of the 2,500 adults selected, 1,684 are men.
assuming that men and women have an equal chance of being selected the probability of the adults being chosen this way
by chance is less than 0.01. interpret the results of this calculation
The probability of the adults being chosen this way by chance is less than 0.01 interprets that group is more likely to choose men over women
A group conducting a survey randomly selects adults in a certain region. Of the 2,500 adults selected, 1,684 are men. The men and women have an equal chance The result of the survey is significant at the 0.01 level which means that the probability of group selection being the result of chance is 0.01 or less because the event is least likely to happen the group is more likely to select men over women.
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Llus
Find x.
Round to the nearest tenth:
31°
х
y
400 ft
x = [? ]ft
Pls help me
Using the sine function and the given information, we can find that length of y is approximately 203.3 feet.
We know that exterior angle of a triangle is equal to the sum of its opposite interior angles. So, we can find angle BAC as follows
BAC + ABC = 180 degrees
As sum of interior angles of a triangle
BAC + 90 = 180
BAC = 90 degrees
Now, we can use the sine ratio to find y
sin(31) = y/400
y = 400 * sin(31)
y ≈ 203.3 ft
Therefore, y is approximately 203.3 ft when rounded to the nearest tenth.
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--The given question is incomplete, the complete question is given
" Find y. Round to the nearest tenth:
In triangle ABC, 31° is the outer angle at A which is in between a line extended from A parallel to BC.
X is CA
у is AB
400 ft is BC
Angle B is right angle
y = [ ? ]ft Enter"--
a certain company has 400 shoes. 20% of the shoes are black, One shoe is chosen and replaced. Then the second shoe is chosen. What is the probability that the shoes chosen are black
Answer: 80
Step-by-step explanation: The question is tell us that a certain company we don't know which one, but it says a certain company has 400 shoes. that is important to note. also 20% of the shoes are black. and One shoe is chosen and replaced. Then the second shoe is chosen. The question is What is the probability that the shoes chosen are black?
Well, take 400 and mutipy it by 20% which changes to 0.20 in decimal form then mutiply your answer by one and you get 80. so the answer is the probability that shoes chosen are black is 80%.
ANSWER ASAP PLEASE
The band club has $2,100 to spend on music stands and songs. Each music stand costs $15 and each song costs $350. Let x represent the number of music stands and let y represent the number of songs.
Part A
Write an equation that describes the number of music stands and songs the band club can buy.
x +
y =
Part B
What is the greatest number of each item that the club can buy?
greatest number of music stands =
greatest number of songs =
Part A: The equation that describes the amount of music stands and songs that the band club can purchase is 15x + 350y = 2100.
Part B: The greatest number of music stands the club can buy is 6, and the greatest number of songs the club can buy is 3.
Part A:
The cost of x music stands is 15x dollars, and the cost of y songs is 350y dollars. The total cost cannot exceed $2,100, so we can write the following equation:
15x + 350y = 2100
Therefore, the equation that describes the amount of music stands and songs that the band club can purchase is 15x + 350y = 2100.
Part B:
To find the greatest number of each item the club can buy, we can use the given equation and look for integer solutions for x and y that satisfy the equation.
We can rearrange the equation to solve for y:
y = (2100 - 15x) / 350
To get integer solutions for y, we need 2100 - 15x to be divisible by 350. The largest multiple of 350 that is less than or equal to 2100 is 6*350 = 2100. So, we can try x = 0, 1, 2, 3, ..., 6 and see which values give integer solutions for y.
When x = 0, y = 6, which is an integer solution.
When x = 1, y = 5, which is not an integer solution.
When x = 2, y = 4, which is not an integer solution.
When x = 3, y = 3, which is an integer solution.
When x = 4, y = 2, which is not an integer solution.
When x = 5, y = 1, which is not an integer solution.
When x = 6, y = 0, which is an integer solution.
So, the greatest number of music stands the club can buy is 6, and the greatest number of songs the club can buy is 3.
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Help pls I need help with sand and a word with my
Answer: X= 4/5x + 8
Step-by-step explanation: Distribution factor
Rebecca folded a piece of notebook paper, as shown below. What is the area of the folded piece of notebook paper?
The area of the folded piece of paper is 30 inches square
How to find the area of a trapezium?The paper is folded in the shape of a trapezium. The area of the trapezium can be found as follows:
area of the trapezium = 1 / 2 (a + b)h
where
a = top lengthb = base lengthh = height of the trapeziumTherefore,
a = 4 inches
b = 4 + 2 + 2 = 8 inches
h = 5 inches
area of the trapezium = 1 / 2 (4 + 8)5
area of the trapezium = 1 / 2 (12)5
area of the trapezium = 60 / 2
area of the trapezium = 30 inches square
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Write the absolute value inequality in form |x-b| c that has the solution set x<-5 or x>7
The absolute value inequality in form |x-b| < c that has the solution set x<-5 or x>7 is |x-1| < 7
To write an absolute value inequality in the form |x-b| < c, we need to think about what this form means.
The expression |x-b| represents the distance between x and b on the number line. Therefore, |x-b| < c means that the distance between x and b is less than c.
Now, let's consider the given solution set: x < -5 or x > 7. We can see that the midpoint between -5 and 7 is 1, so we choose b = 1. Then, we need to determine c, which is the maximum distance between b and any of the solutions.
If we take x = -6 (which is less than -5), then |x-b| = |-6-1| = 7. Similarly, if we take x = 8 (which is greater than 7), then |x-b| = |8-1| = 7. Therefore, the maximum distance is 7.
Putting it all together, we get the absolute value inequality:
|x-1| < 7
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Each day three church bells are rung in a random order. what is the probability that the smallest bell rings first three days in a row?
The probability that the smallest bell rings the first three days in a row is 1/27. when Each day three church bells are rung in a random order
In the given data there are 3 bells in the church in which there is a small bell and the three bells are rung in a random order. we need to find the probability that the smallest bell rings the first three days in a row.
The probability that the smallest bell rings on any given first day can be given as = 1/3
Because there are three bells and each bell has an equal chance of being rung first. The probability that this happens three days in a row is given as
= (1/3) × (1/3) × (1/3)
= 1/27
Therefore, the probability that the smallest bell rings the first three days in a row is 1/27
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At her job, Avery earns $120 per week plus a one-time $300 bonus. Janelle teaches art lessons and earns $24 per week plus a $60 art supply fee for each student she teaches. a. System of equations:
The system of equations to describe the earnings by Avery and Janelle would be:
Avery's earnings: y = 120x + 300
Janelle's earnings: y = 24x + 60s
How to find the system of equations ?The problem provides two scenarios with different methods for earning money. Avery earns a fixed amount of $120 each week, in addition to a one-time bonus of $300. To represent this situation as an equation, we can use the formula:
y = 120x + 300
where y is Avery's total earnings, x is the number of weeks she works, and 300 is the one-time bonus she receives.
For Janelle, her earnings consist of a fixed weekly rate of $24 plus a variable amount based on the number of students she teaches.
We can represent Janelle's earnings as an equation using the formula:
y = 24x + 60s
where y is Janelle's total earnings, x is the number of weeks she works, and s is the number of students she teaches.
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U = {all triangles}
E = {x|x ∈ U and x is equilateral}
I = {x|x ∈ U and x is isosceles}
S = {x|x ∈ U and x is scalene}
A = {x|x ∈ U and x is acute}
O = {x|x ∈ U and x is obtuse}
R = {x|x ∈ U and x is right}
Which is a subset of I?
E
S
A
R
The set R is not a subset of I. the only subset of I from the given options is A
How we find the subset of I?The set I represents all isosceles triangles.
The set E represents all equilateral triangles, and an equilateral triangle is a special case of an isosceles triangle where all sides are equal. Therefore, the set E is a subset of I.
The set S represents all scalene triangles, and a scalene triangle is not isosceles since it does not have any equal sides. Therefore, the set S is not a subset of I.
The set A represents all acute triangles, and an acute isosceles triangle is a triangle where all angles are less than 90 degrees and two sides are equal in length. Therefore, the set A is a subset of I.
The set O represents all obtuse triangles, and an obtuse isosceles triangle is a triangle where one angle is greater than 90 degrees and two sides are equal in length. Therefore, the set O is not a subset of I.
The set R represents all right triangles, and a right isosceles triangle is a triangle where one angle is equal to 90 degrees and two sides are equal in length. the only subset of I from the given options is A.
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What is the median of the lower half of data
We can see here that in order to find the median of the lower half of data, one will have to sort out the data in an ascending order. Then take the lower half of the data (i.e., the first half of the sorted data) and find its median.
What is median?The median, which is used to measure central tendency in statistics, is the point at which a dataset may be divided into two equal parts. If a dataset has an even number of values, it is the average of the two middle values or the middle value in a sorted dataset.
The values in the dataset must first be arranged from lowest to highest in order to determine the median. The median is the middle value if the dataset has an odd number of values.
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An exponential function is given by the equation y=3x. Using the points x and x+1, show that the y-values increase by a factor of 3 between any two points separated by x2−x1=1. (4 points)
The given exponential function satisfies the property of increasing by a factor of 3 between any two points separated by x2−x1=1.
An exponential function is a mathematical function of the form f(x) = a^x, where a is a positive constant and x is any real number. The base a is typically a number greater than 1, and the function grows or decays rapidly depending on whether a is greater than or less than 1.
Exponential functions are commonly used to model processes that exhibit exponential growth or decay, such as population growth, radioactive decay, and compound interest. They also arise in various areas of mathematics and science, including calculus, probability theory, and physics.
We are given the exponential function [tex]y=3^x.[/tex]
Let x1 be any value of x, then the corresponding y-value is [tex]y1=3^{(x_1)[/tex]
Let x2=x1+1 be the next value of x, then the corresponding y-value is [tex]y2=3^x2=3^(x1+1)=3*3^x1.[/tex]
So, we can see that y2 is 3 times y1, which means the y-values increase by a factor of 3 between any two points separated by x2−x1=1.
Therefore, the given exponential function satisfies the property of increasing by a factor of 3 between any two points separated by x2−x1=1.
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The attendance for a week at a local theatre is normally distributed, with a mean of 4000 and a standard
deviation of 500. Draw the normal curve to represent the normally distributed attendance for the week.
What percentage of the attendance figures would be less than 3500? What percentage of the attendance
figures would be greater than
5000? What percentage of the attendance figures would be between 3700
and 4300 each week?
About 15.87% of the attendance figures would be less than 3500.
About 0.62% of the attendance figures would be greater than 5000.
About 34.13% of the attendance figures would be between 3700 and 4300 each week.
The mean is the average of a set of numbers, while the standard deviation measures the spread of the data around the mean. The normal distribution is fully characterized by its mean and standard deviation. In this case, the mean attendance is 4000, and the standard deviation is 500.
To answer the first question, "What percentage of the attendance figures would be less than 3500?" we need to calculate the area under the curve to the left of 3500. We can use a standard normal distribution table or a calculator to find this area. The result is approximately 15.87%.
To answer the second question, "What percentage of the attendance figures would be greater than 5000?" we need to calculate the area under the curve to the right of 5000. Again, we can use a standard normal distribution table or a calculator to find this area. The result is approximately 0.62%.
To answer the third question, "What percentage of the attendance figures would be between 3700 and 4300 each week?" we need to calculate the area under the curve between 3700 and 4300. We can use a standard normal distribution table or a calculator to find this area. The result is approximately 34.13%.
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What is the domain of the rational function f of x is equal to the quantity x squared plus x minus 6 end quantity over the quantity x cubed minus 3 times x squared minus 16 times x plus 48 end quantity question mark {x ∈ ℝ| x ≠ –4, –2, 3, 4} {x ∈ ℝ| x ≠ –4, 3, 4} {x ∈ ℝ| x ≠ –4, 4} {x ∈ ℝ| x ≠ –2, 3}
The domain of the rational function is: option (2) {x ∈ ℝ | x ≠ -4, 3, 4}
What is Rational number ?A rational number is any number that can be expressed as the ratio or fraction of two integers, where the denominator is not zero. In other words, a rational number is a number that can be written in the form of p/q, where p and q are integers, and q is not equal to zero.
The domain of a rational function is the set of all real numbers for which the function is defined, and the denominator is not equal to zero. So, we need to find the values of x for which the denominator of the given rational function is not zero.
The denominator of the given rational function is:
x³ - 3x² - 16x + 48
We can factor this polynomial using synthetic division or polynomial long division:
x³- 3x² - 16x + 48 = (x - 4)(x - 3)(x + 4)
So, the denominator of the rational function is not defined when:
x - 4 = 0 or x - 3 = 0 or x + 4 = 0
Solving these equations, we get:
x = 4 or x = 3 or x = -4
Therefore, the domain of the given rational function is:
{x ∈ ℝ| x ≠ –4, 3, 4}
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Calculate the second and third derivatives. y = 4x4 - 3x² + 7x y yⁿ= yᵐ=
The second derivative yⁿ (y'') is 48x² - 6, and the third derivative yᵐ (y''') is 96x.
To calculate the second and third derivatives of the function y = 4x^4 - 3x² + 7x:
1. First, calculate the first derivative, y':
y' = dy/dx = (d/dx)(4x^4 - 3x² + 7x)
Using the power rule for derivatives, we get:
y' = 16x³ - 6x + 7
2. Now, calculate the second derivative, y'' (also denoted as yⁿ when n=2):
y'' = d²y/dx² = (d/dx)(16x³ - 6x + 7)
Applying the power rule again:
y'' = 48x² - 6
3. Finally, calculate the third derivative, y''' (also denoted as yᵐ when m=3):
y''' = d³y/dx³ = (d/dx)(48x² - 6)
Using the power rule one more time:
y''' = 96x
So, the second derivative yⁿ (y'') is 48x² - 6, and the third derivative yᵐ (y''') is 96x.
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