Answer:
26 million
Step-by-step explanation:
8200000 / 0.32 = 25625000
25625000 rounded to nearest million = 26 million
The total number of college students is 26 million.
Given that, in a recent year, 32% of all college students were enrolled part-time and 8.2 million college students were enrolled part-time that year.
What is an equation?A mathematical equation is a formula that uses the equals sign to represent the equality of two expressions.
Let the total number of college students be x.
Now, 32% of x=8.2 million
⇒ 0.32 x=8200000
⇒ x = 8200000/3.2
⇒ x = 2562500
2562500 rounded to nearest million = 26 million
Therefore, the total number of college students is 26 million.
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The distance that a spring will stretch varies directly as the force applied to the spring. A force of 8080 pounds is needed to stretch a spring 88 inches. What force is required to stretch the spring 1919 inches? A nothing-pound force is required to stretch the spring 1919 inches.
Answer:
176 199 pounds
Step-by-step explanation:
To answer this it is first useful to find the proportionality constant.
F= kx
8080 pounds = k.88 in
k = 91.81 pound/inch
So what force is required to stretch the spring to 1919 inches?
F = kx
= 91.81 pound/inch * 1919 inches
= 176 199 pounds
It is worth noting that this force seems rather large and the spring might have long reached its elastic limit
The money multiplier _______. A. decreases if banks increase their desired reserve ratio B. is 1 if the desired reserve ratio equals the currency drain ratio C. increases if the currency drain ratio increases D. increases if banks increase their desired reserve ratio
Answer:
A. decreases if banks increase their desired reserve ratio
Step-by-step explanation:
Since, the money multiplier is the amount of money produced by banks with each dollar of reserves,
In other words,
It estimates, how an initial deposit can lead to a bigger final increase in the total money supply.
For example :
If a commercial bank gains deposits of 1 crore and this leads to a final money supply of 10 crore, the money multiplier would be 10.
That is,
[tex]\text{Money multipliers}=\frac{1}{\text{Reserve ratio}}[/tex]
[tex]\implies \text{Money multipliers}\propto \frac{1}{\text{Reserve ratio}}[/tex]
Therefore, the money multiplier decreases if banks increase their desired reserve ratio
The money multiplier decreases when banks increase their desired reserve ratio, as they lend out less money, reducing the multiplier effect.
The money multiplier is a key concept in understanding how the banking system can increase the money supply within an economy. It is defined as the quantity of money that the banking system is able to generate from each dollar of bank reserves. The question relates the behavior of the money multiplier in response to changes in the reserve ratio and the currency drain ratio.
Answering the student's query, the money multiplier decreases if banks increase their desired reserve ratio because as they keep more reserves relative to deposits, they can lend out less money, effectively reducing the multiplier effect. Conversely, when banks decrease their reserve ratio, they can lend out a larger proportion of their deposits, which leads to an increase in the money multiplier. Therefore, the correct option is A: decreases if banks increase their desired reserve ratio.
For a special game Don has two 8-sided fair dice numbered from 1 to 8 on the faces. Just as with ordinary dice, Don rolls the dice and sums the numbers that appear on the top face, getting a sum from 2 to 16. What is the sum of the possible numbers which have a probability of 1/32 of appearing
Answer
The sum of the possible numbers are {(1,2), (2,1)}
Step-by-step explanation:
Probability for rolling two dice with the eight sided dots are 1, 2, 3, 4, 5, 6, 7 and 8 dots in each die.
When two dice are rolled or thrown simultaneously, thus number of event can be [tex]8^{2}[/tex]= 64 because each die has 1 to 8 numbers on its faces. Then the possible outcomes of the sample space are shown in the pdf document below .
As Don rolls the dice and sums the number that appear on the top face, he gets a sum from 2 to 16.
Assuming; getting sum of 2
Let E[tex]_{1}[/tex] = event of getting sum of 2
E[tex]_{1}[/tex] = { (1 , 1 ) }
Therefore, Probability of getting sum of 2 will be;
P ( E[tex]_{1}[/tex] ) = [tex]\frac{Number of favorable outcome}{Total number of possible outcome}[/tex]
= [tex]\frac{1}{64}[/tex]
GETTING SUM OF TWO ( i.e { (1 , 1 ) } ) will give the probability of 1/64 of appearing. But we are looking for the probability of 1/32 of appearing. Let look at the possibility of getting sum of 3.
Assuming; getting sum of 3
Let E[tex]_{2}[/tex] = event of getting sum of 3
E[tex]_{2}[/tex] = { (1 , 2 ) (2 , 1) }
Therefore, Probability of getting sum of 3 will be;
P ( E[tex]_{2}[/tex] ) = [tex]\frac{Number of favorable outcome}{Total number of possible outcome}[/tex]
= [tex]\frac{2}{64}[/tex]
= [tex]\frac{1}{32}[/tex]
For all probability of getting the sum greater than 3 to 16 will be void because there wont be a chance for 1/32 to appear. Therefore, the sum of the possible numbers are: {(1,2), (2,1)} which have a probability of 1/32 of appearing.
I hope this comes in handy at the rightful time!
Esteban has 236 trading cards that he wants to put in storage boxes.Each box holds 18 trading cards. If he puts 18 trading cards in every box, how many trading cards will be left over
Divide the total cards by 18:
236 / 18 = 13.111
Use the whole number and multiply by 18:
13 x 18 = 234
This means 234 cards would be in storage boxes.
236 - 234 = 2 cards would be left over.
Consider the function represented by the equation 6c = 2p – 10. Write the equation in function notation, where c is the independent variable. F(c) = one-thirdp + five-thirds f(c) = 3c + 5 f(p) = one-thirdp + five-thirds f(p) = 3c + 5
Answer:
f(c) = 3c + 5
Step-by-step explanation:
c is independent variable, p is dependent variable, so p = f(c)
6c = 2p -10
3c = p - 5
p = 3c + 5
f(c) = 3c + 5
Answer:
F(c) = 3c +5
Step-by-step explanation:
6c=2p-10
If c is considered to be as the independent variable, then p is the dependent variable
So we will clear and solve for variable p
6c = 2p-10
Dividing both sides by 2. We will have
6c/2 = 2(p-5)/2
3c= p-5
Adding 5 on both sides
3c+5= p
So p =3c+5
By writing the above equation in the functional notation form
F(c) = 3c +5
What is 10x95 Hgghhjkhhhgvfddddddderhfdcfxdgyggggfffffffffffgggfvgvfgvgbbhbbbbbbbnnbnbbbvbvbbb. Hhhhhjhhhbbbbvvvvgvvv chug. Hbvv g vgggyyyhdhggshhrhehergwgegrgwgrgegrttehrhehrhrhehrhrhrhrgrehthehrhehrhhrhrrhrhrhrhrhthehhrhthrhrhrhrhrhrhrhhrrhhdhddhfhfhudhdhdhdhffhhfhfdh
Answer:
Answer is
=950
Use the value of the linear correlation coefficient to calculate the coefficient of determination. What does this tell you about the explained variation of the data about the regression line? About the unexplained variation? requals=negative 0.387
Answer:
Coefficient of determination = 0.149769
Step-by-step explanation:
We are given the following in the question:
Coefficient of correlation, r = -0.387
We have to find the coefficient of determination.
Coefficient of determination =
[tex]r^2 = (-0.387)^2 = 0.149769 = 14.98\%[/tex]
Coefficient of determination:
It tells us about the variation explained in the model.This correlation of determination tells that 14.98% of the variation is explained in the model that is we can say that the independent variable explains 14.98% changes or prediction in the dependent variable.It ranges from 0 to 1 and a correlation of determination of 1 means that the dependent variable is 100% predicted by the independent variable.Correlation of determination of 0 means that the dependent variable cannot be predicted by the independent variable.What is the length of CD in the figure below? Show your work.
Answer:
5
Step-by-step explanation:
First, notice that these two triangles are similar using AA.
Because sides BC and EC are corresponding, you can divide 24 by 8, to determine that the ratio of similitude is 3.
That means that because sides AC and DC are corresponding, 25 - 2x divided by x is 3.
25 - 2x / x = 3
25 - 2x = 3x
25 = 5x
x = 5
x is the same as side CD, so CD = 5.
Using similar triangles and proportions, we can determine that the length of CD in the figure is 10/3 units.
In order to determine the length of CD in the figure, we can use the properties of similar triangles. Since triangles ABC and CDE are similar, we can set up a proportion using their corresponding side lengths:
δ CD / δ AB = CD / AB
Then, we can substitute the given values:
4 / 12 = CD / 20
Next, we can cross multiply and solve for CD:
12 * CD = 4 * 20
CD = (4 * 20) / 12
CD = 40 / 12
CD = 10 / 3
Therefore, the length of CD is 10/3 units.
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A bowl in the shape of a hemispere is filled with water to a depth h=3 inches. The radius of the bowl is R inches. Express the radius of the bowl R as a function of the angle theta.
The radius of the bowl R as a function of the angle theta is [tex]\mathrm{R}=\frac{3}{1-\sin \theta}[/tex]
Solution:
The figure is attached below
If we consider the centre of hemisphere be A
The radius be AC and AD
According to question,
A bowl in the shape of a hemispere is filled with water to a depth h=3 inches .i.e. BC = h = 3 inches
And radius of the bowl is R inches .i.e. R = AD =AC
Now , using trigonometric identities in triangle ABD we get
[tex]\sin \theta=\frac{\text { Perpendicular }}{\text { Hypotenuse }}=\frac{A B}{A D}[/tex]
[tex]\begin{array}{l}{\sin \theta=\frac{A B}{R}} \\\\ {A B=R \sin \theta}\end{array}[/tex]
Since , AC = AB + BC
R = R Sinθ + 3
R - R Sinθ = 3
R (1 – Sinθ ) = 3
[tex]\mathrm{R}=\frac{3}{1-\sin \theta}[/tex]
Which is the required expression for the radius of the bowl R as a function of the angle theta
Amber had 3/8 of a cake left after her party she wrapped a piece that was 1/4 of the original cake for her best friend what freshener part did she have left over for herself
Answer:
Fraction of cake that was left for Amber = [tex]\frac{1}{8}[/tex]
Step-by-step explanation:
Given:
Amber had [tex]\frac{3}{8}[/tex] of a cake left after her party.
Amber wrapped a piece of cake for her friend that was [tex]\frac{1}{4}[/tex] of the original cake.
To find the fractional part of cake that was left for Amber.
Solution:
Fraction of cake left Amber had after party = [tex]\frac{3}{8}[/tex]
Fraction of cake she wrapped for her friend = [tex]\frac{1}{4}[/tex]
To find the fractional part of cake that was left for Amber we will subtract [tex]\frac{1}{4}[/tex] of the cake from [tex]\frac{3}{8}[/tex] of the cake.
∴ Fraction of cake that was left for Amber = [tex]\frac{3}{8}-\frac{1}{4}[/tex]
To subtract fractions we need to take LCD
⇒ [tex]\frac{3}{8}-\frac{1}{4}[/tex]
LCD will be =8 as its the least common multiple of 4 and 8.
To write [tex]\frac{1}{4}[/tex] as a fraction with common denominator 8 we multiply numerator and denominator with 2.
So, we have
⇒ [tex]\frac{3}{8}-\frac{1\times 2}{4\times 2}[/tex]
⇒ [tex]\frac{3}{8}-\frac{ 2}{8}[/tex]
Then we simply subtract the numerators.
⇒ [tex]\frac{3-2}{8}[/tex]
⇒ [tex]\frac{1}{8}[/tex]
∴ Fraction of cake that was left for Amber = [tex]\frac{1}{8}[/tex]
An alcohol awareness task force at a Big-Ten university sampled 200 students after the midterm to ask them whether they went bar hopping the weekend before the midterm or spent the weekend studying, and whether they did well or poorly on the midterm. The following result was obtained. (Please show the calculation process)
Answer:
a) 0.30
b) 0.90
c) 0.15
Step-by-step explanation:
When two events are independent then conditional probability take place.
A knitter wants to make a rug for a dollhouse. The length of the rug will be 2 inches more than it's width. The total area of the rug (in square inches ) based on the width W of the rug (in inches ) is given by A (w) = w (2+w) If the desired area of the rug is 15 square inches, what is the width of the rug, in inches ?
Answer:
Step-by-step explanation:
A knitter wants to make a rug for a dollhouse. The length of the rug will be 2 inches more than it's width. Let the width of the rug be represented by w. This means that the length of the rug will be w + 2
The total area of the rug (in square inches ) based on the width W of the rug (in inches ) is given by
A(w) = w (2+w)
If the desired area of the rug is 15 square inches,,the width will be determined by substituting 15 for A(w). It becomes
15 = w^2 + 2w
w^2 + 2w - 15 = 0
w^2 + 5w - 3w - 15 = 0
w(w+5) -3(w+5)
w - 3 = 0 or w+ 5 = 0
w = 3 or w = -5
w cannot be negative. So w = 3 inches
Consider the sequence:
5, 7, 11, 19, 35,....
Write an explicit definition that defines the sequence:
a_n = 2n + 3
a_n = 3n + 2
a_n = 3n^2
a_n = 2^n + 3
Answer:
a_n = 2^n + 3
Step-by-step explanation:
The first differences have a geometric progression, so the explicit definition will be an exponential function. (It cannot be modeled by a linear or quadratic function.) The above answer is the only choice that is an exponential function.
__
First differences are ...
(7-5=)2, 4, 8, 16
Answer: [tex]a_n = 2^n + 3\ \ \ \, n=1,2,3,4,5...[/tex]
Step-by-step explanation:
The given sequence = 5, 7, 11, 19, 35,....
[tex]7-5=2\\11-7=4=2^2\\19-11=8=2^3\\35-19=16=2^4[/tex]
Here , it cam be observe that the difference between the terms is not common but can be expressed as power of 2.
We can write the terms of the sequence as
[tex]2^1+3=5\\2^2+3=4+3=7\\2^3+3=8+3=11\\2^4+3=16+3=19\\2^5+3=32+3=35[/tex]
Then , the required explicit definition that defines the sequence will be
[tex]a_n = 2^n + 3\ \ \ \, n=1,2,3,4,5...[/tex]
If the median of a list of numbers is m, the first quartile of the list is the median of the numbers in the list that are less than m. What is the first quartile of the list of numbers 42, 24, 30, 22, 26, 19, 33 and 35 ?A) 33B) 28C) 27D) 24E) 23
Answer:
First quartile list of number median is 23
Step-by-step explanation:
List of numbers = 42,24, 30 , 22 , 26, 19, 33, 35
Arrange them in a sequence:
List of numbers = 19, 22, 24, 26, 30, 33, 35, 42
The median of {19, 22, 24, 26, 30, 33, 35, 42} is= 26+30/2 = 28
Numbers that are less than the median are { 19 , 22 , 24 , 26}
then
median of the list = 22+24/2 = 23
So the first quartile list of number median is 23.
The median of the first quartile list is 23.
The given set of numbers {42, 24, 30, 22, 26, 19, 33, 35} can be arranged in ascending order as follows:
{19, 22, 24, 26, 30, 33, 35, 42}.
The median of this sequence is calculated as (26 + 30) / 2, resulting in 28.
The numbers less than the median are {19, 22, 24, 26}.
Next, the median of this subset is found by taking (22 + 24) / 2, yielding 23. Therefore, the median of the first quartile list is 23.
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A company manufactures and sells video games. A survey of video game stores indicated that at a price of $66 each, the demand would be 400 games, and at a price of $36 each, the demand would be 1,300 games. If a linear relationship between price and demand exists, which of the following equations models the price-demand relationship? (Let x represent the price per video game and y represent the demand.)
Answer:
[tex]y=-30x+2380[/tex]
Step-by-step explanation:
[tex]x\rightarrow[/tex] represent price per video game.
[tex]y\rightarrow[/tex] represent demand.
The linear equation in slope intercept form can be represented as:
[tex]y=mx+b[/tex]
where [tex]m[/tex] is slope of line or rate of change of demand of game per dollar change in price and [tex]b[/tex] is the y-intercept or initial price of game.
We can construct two points using the data given.
When price was $66 each demand was 400. [tex](66,400)[/tex]
When price was $36 each demand was 1300. [tex](36,1300)[/tex]
Using the points we can find slope [tex]m[/tex] of line.
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]m=\frac{1300-400}{36-66)}[/tex]
[tex]m=\frac{900}{-30}[/tex]
[tex]m=-30[/tex]
Using point slope form of linear equation to write the equation using a given point.
[tex]y-y_1=m(x-x_1)[/tex]
Using point [tex](66,400)[/tex].
[tex]y-400=-30(x-66)[/tex]
⇒ [tex]y-400=-30x+1980[/tex] [Using distribution]
Adding 400 to both sides:
⇒ [tex]y-400+400=-30x+1980+400[/tex]
⇒[tex]y=-30x+2380[/tex]
The linear relationship between price and demand can be written as:
[tex]y=-30x+2380[/tex]
Which of the following lists is not in order from smallest to largest?
The Answer choices are ; The correct answer choice is ; 2.33, 1.87, 3.6, 7.1
10.1, 10.5, 11.2, 12.9 - False
------------------------------
2.33, 1.87, 3.6, 7.1 - Correct
---------------------------
0.5, 1.3, 2.6, 3.8 - False
-------------------------------------
4.75, 4.8, 4.92, 5 - False
Answer: 2.33, 1.87, 3.6, 7.1
Step-by-step explanation:
2.33, 1.87, 3.6, 7.1 is not arranged from smallest to the largest. Arranging the number will be 1.87, 2.33, 3.6, 7.1.
Answer:
B. 2.33, 1.87, 3.6, 7.1
Step-by-step explanation:
This option is not in order from smallest to largest.
Hope it helped!
simplify the rational expression. state any excluded values.
x^2-3x-10/x+2
Answer:
The answer to your question is x - 5 or x = 5
Step-by-step explanation:
[tex]\frac{x^{2} -3x - 10}{x + 2}[/tex]
1.- Factor the numerator
x² - 3x - 10
find 2 numbers that added equal -3 and multiply equal -10.
These numbers are -5 and + 2
(x - 5)(x + 2)
2.- Simplify
[tex]\frac{(x - 5)(x + 2)}{x + 2)}[/tex]
Delete (x +2) in both numerator and denominator
3.- Result x - 5
x = 5
If a and b are positive numbers, find the maximum value of f(x)=xa(1−x)b, 0≤x≤1 Your answer may depend on a and b.maximum value =________.
To find the maximum value of f(x) = xa(1−x)b, use calculus to find the derivative of f(x), set it equal to 0, and solve for x to find the critical points. Evaluate f(x) at the critical points and the endpoints of the interval to find the maximum value.
Explanation:To find the maximum value of f(x) = xa(1−x)b, we can use calculus. First, find the derivative of f(x) with respect to x: f'(x) = a(1 - x)b - bx(1 - x)b-1. Set the derivative to 0 and solve for x to find the critical points. The maximum value of f(x) occurs at one of these critical points or at the endpoints of the interval [0, 1]. Plug the values of x into f(x) to find the corresponding maximum values.
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To find the maximum value of f(x)=xa(1−x)b, where a and b are positive numbers and 0≤x≤1, we need to find the critical points by finding when the derivative of the function is equal to zero or does not exist. Then, we evaluate the function at the critical points and the endpoints to determine the maximum value.
Explanation:To find the maximum value of the function f(x)=xa(1−x)b, where a and b are positive numbers and 0≤x≤1, we need to determine when the function reaches its maximum. This can be done by finding the critical points, which occur when the derivative of the function is equal to zero or does not exist.
First, let's find the derivative of f(x):
f'(x) = a(1-x)^{b-1}(bx - (b-1)x -1).
To find the critical points, we set f'(x) = 0 and solve for x.
Solving the equation, we find that the critical point occurs when x = \frac{b}{2b-1}.
Next, we evaluate f(x) at the critical point x = \frac{b}{2b-1} and also at the endpoints x = 0 and x = 1. The maximum value of f(x) will be the largest value among these.
Finally, we compare the values to find the maximum value.
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P(x)P(x)P, (, x, )is a polynomial. P(x)P(x)P, (, x, )divided by (x+7)(x+7)(, x, plus, 7, )has a remainder of 555. P(x)P(x)P, (, x, )divided by (x+3)(x+3)(, x, plus, 3, )has a remainder of -4−4minus, 4. P(x)P(x)P, (, x, )divided by (x-3)(x−3)(, x, minus, 3, )has a remainder of 666. P(x)P(x)P, (, x, )divided by (x-7)(x−7)(, x, minus, 7, )has a remainder of 999. Find the following values of P(x)P(x)P, (, x, ). P(-3)=P(−3)=P, (, minus, 3, ), equals P(7)=P(7)=P, (, 7, ), equals
Answer:
P(-3)=-4
P(7) = 9
Step-by-step explanation:
Consider P(x) is a polynomial.
According to the remainder theorem, if a polynomial, P(x), is divided by a linear polynomial (x - c), then the remainder of that division will be equivalent to f(c).
Using the given information and remainder theorem we conclude,
If P(x) is divided by (x+7), then remainder is 5.
⇒ P(-7)=5
If P(x) is divided by (x+3), then remainder is -4.
⇒ P(-3)=-4
If P(x) is divided by (x-3), then remainder is 6.
⇒ P(3)=6
If P(x) is divided by (x-7), then remainder is 9.
⇒ P(7)=9
Therefore, the required values are P(-3)=-4 and P(7) = 9.
Expand the expression using the Binomial Theorem and Pascal's Triangle: (3x-1)^3
A book contains 400 pages. If their are 80 typing errors randomly distributed throughout the book, use the Poisson distribution to determine the probability that a page contains exactly 2 errors
Using the Poisson distribution to determine the probability that a page contains exactly 2 errors is 0.0163
Solution:Given that, a book contains 400 pages.
There are 80 typing errors randomly distributed throughout the book,
We have to use the Poisson distribution to determine the probability that a page contains exactly 2 errors.
The Poisson distribution formula is given as:
[tex]\text { Probability distribution }=e^{-\lambda} \frac{\lambda^{k}}{k !}[/tex]
Where, [tex]\lambda[/tex] is event rate of distribution. For observing k events.
[tex]\text { Here rate of distribution } \lambda=\frac{\text { go mistakes }}{400 \text { pages }}=\frac{1}{5}[/tex]
And, k = 2 errors.
[tex]\begin{array}{l}{\text { Then, } \mathrm{p}(2)=e^{-\frac{1}{5}} \times \frac{\frac{1}{5}}{2 !}} \\\\ {=2.7^{-\frac{1}{5}} \times \frac{\frac{1}{5^{2}}}{2 \times 1}} \\\\ {=\frac{1}{2.7^{\frac{1}{5}}} \times \frac{\frac{1}{25}}{2}}\end{array}[/tex]
[tex]\begin{array}{l}{=\frac{1}{\sqrt[5]{2.7}} \times \frac{1}{25} \times \frac{1}{2}} \\\\ {=\frac{1}{50 \sqrt[5]{2.7}}} \\\\ {=0.0163}\end{array}[/tex]
Hence, the probability is 0.0163
What is the measurement of RT?
Answer:
65
Step-by-step explanation:
The sum of 5 times the width of a rectangle and twice its length is 26 units. The difference of 15 times the width and three times the length is 6 units. Write and solve a system of equations to find the length and width of the rectangle
Answer: length = 8 units
Width = 2 units
Step-by-step explanation:
Let the length of the rectangle be represented by L
Let the width of the rectangle be represented by W
The sum of 5 times the width of a rectangle and twice its length is 26 units. This means that
5W + 2L = 26 - - - - - - - - 1
The difference of 15 times the width and three times the length is 6 units. It means that
15W - 3L = 6 - - - - - - - - 2
The system equations are equation 1 and equation 2.
Multiplying equation 1 by 3 and equation 2 by 2, it becomes
15W + 6L = 78
30W - 6L = 12
Adding both equations,
45W = 90
W = 90/45 = 2
15W - 3L = 6
15×2 - 3L = 6
-3L = 6 - 30 = - 24
L = - 24/ -3 = 8
The length of the rectangle is 8 units, and the width is 2 units. This solution satisfies both equations given.
Let's denote the width of the rectangle as ( w ) units and the length as ( l ) units.
According to the given information, we have two equations:
1. ( 5w + 2l = 26 ) (sum of 5 times the width and twice the length is 26 units)
2. ( 15w - 3l = 6 ) (difference of 15 times the width and three times the length is 6 units)
We can set up a system of equations using these two equations.
To solve this system of equations, we can use the method of substitution or elimination. Let's use the elimination method.
First, we'll multiply the second equation by 2 to eliminate the ( l ) term:
( 2 x (15w - 3l) = 2 x 6 )
This gives us:
( 30w - 6l = 12 )
Now, we have the system of equations:
1. ( 5w + 2l = 26 )
2. ( 30w - 6l = 12 )
We can now eliminate the ( l ) term by adding these equations together.
( (5w + 2l) + (30w - 6l) = 26 + 12 )
This simplifies to:
( 35w - 4l = 38 )
Now, we have one equation with only ( w ) as a variable.
Now, we can solve for ( w ):
( 35w - 4l = 38 )
( 35w = 38 + 4l )
[tex]\( w = \frac{38 + 4l}{35} \)[/tex]
Now, we can substitute this value of ( w ) into one of the original equations to solve for ( l ).
Let's use the first equation:
( 5w + 2l = 26 )
Substituting[tex]\( w = \frac{38 + 4l}{35} \)[/tex], we get:
[tex]\( 5\left(\frac{38 + 4l}{35}\right) + 2l = 26 \)[/tex]
Now, we can solve this equation for ( l ).
[tex]\( \frac{190 + 20l}{35} + 2l = 26 \)[/tex]
Multiply both sides by 35 to clear the fraction:
( 190 + 20l + 70l = 910 )
Combine like terms:
( 90l + 190 = 910 )
Subtract 190 from both sides:
( 90l = 720 )
Now, divide both sides by 90 to solve for ( l ):
[tex]\( l = \frac{720}{90} \)[/tex]
( l = 8 )
Now that we have the value of ( l ), we can substitute it back into one of the original equations to solve for ( w ). Let's use the first equation:
( 5w + 2(8) = 26 )
( 5w + 16 = 26 )
Subtract 16 from both sides:
5w = 10
Now, divide both sides by 5 to solve for ( w ):
[tex]\( w = \frac{10}{5} \)[/tex]
w = 2
So, the length of the rectangle is 8 units and the width is 2 units.
determine the intervals on which the function is increasing, decreasing, and constant
Answer:
Step-by-step explanation:
In general, wherever a function is tending from the upper left to the lower right, it is decreasing; wherever a function is tending from the lower left to the upper right it is increasing. Constant functions are horizontal lines.
Our function is tending from upper left to lower right on negative infinity to an x-value of -4. Then it runs as a horizontal line from x values of -4 to +4. Then it tends from lower left to upper right from +4 to infinity.
24. Suppose you throw two fair number cubes. What is the probability that the sum of the results of the throw is 4,5, or 6? Show your work and explain.
Answer:
0.33
Step-by-step explanation:
Two fair number cubes can be thought as dice with sides numbered from 1 to 6. The throw of two dice may result in one of the following combinations in which (d1,d2) are the results of die 1 and 2 respectively:
Ω={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}
There are 36 as many possible combinations
The sum of both quantities will produce 11 possible results
S={2,3,4,5,6,7,8,9,10,11,12}
The combinations which produce a sum of 4 are (1,3)(2,2),(3,1), 3 in total
The combinations which produce a sum of 5 are (1,4)(2,3),(3,2),(4,1) 4 in total
The combinations which produce a sum of 6 are (1,5)(2,4),(3,3),(4,2),(5,1) 5 in total
If we want to know the probability that the sum of the results of the throw is 4,5, or 6, we compute the total ways to produce them
T=3+4+5=12 combinations
The probability is finally computed as
[tex]P=\frac{T}{36}=\frac{12}{36}=\frac{1}{3}=0.33[/tex]
Final answer:
To find the probability of getting a sum of 4, 5, or 6 when two dice are rolled, all possible combinations are listed, resulting in 12 favorable outcomes. Those outcomes are then divided by the total number of possible outcomes, 36, resulting in a probability of 1/3.
Explanation:
The question asks about the probability of getting a sum of 4, 5, or 6 when throwing two fair number cubes (dice). Each die has six faces, with numbers ranging from 1 to 6. When rolling two dice, there are 36 possible outcomes (6 outcomes from the first die multiplied by 6 outcomes from the second die).
To find the probability of obtaining a sum of 4, 5, or 6, we first want to identify all the possible combinations that lead to these sums:
For a sum of 4: (1,3), (2,2), (3,1)For a sum of 5: (1,4), (2,3), (3,2), (4,1)For a sum of 6: (1,5), (2,4), (3,3), (4,2), (5,1)Altogether, there are 3 + 4 + 5 = 12 outcomes that result in either a 4, 5, or 6 as the sum. Therefore, the probability is the number of favorable outcomes (12) divided by the total number of outcomes (36), which simplifies to 1/3. Hence, the probability is 1/3.
Let production be given by P = bLαK1−α where b and α are positive and α < 1. If the cost of a unit of labor is m and the cost of a unit of capital is n, and the company can spend only p dollars as its total budget, then maximizing the production P is subject to the constraint mL + nK = p. Show that the maximum production occurs when L=αp/m and K=(1-α)p/n.
Answer:
The proof is completed below
Step-by-step explanation:
1) Definition of info given
We have the function that we want to maximize given by (1)
[tex]P(L,K)=bL^{\alpha}K^{1-\alpha}[/tex] (1)
And the constraint is given by [tex]mL+nK=p[/tex]
2) Methodology to solve the problem
On this case in order to maximize the function on equation (1) we need to calculate the partial derivates respect to L and K, since we have two variables.
Then we can use the method of Lagrange multipliers and solve a system of equations. Since that is the appropiate method when we want to maximize a function with more than 1 variable.
The final step will be obtain the values K and L that maximizes the function
3) Calculate the partial derivates
Computing the derivates respect to L and K produce this:
[tex]\frac{dP}{dL}=b\alphaL^{\alpha-1}K^{1-\alpha}[/tex]
[tex]\frac{dP}{dK}=b(1-\alpha)L^{\alpha}K^{-\alpha}[/tex]
4) Apply the method of lagrange multipliers
Using this method we have this system of equations:
[tex]\frac{dP}{dL}=\lambda m[/tex]
[tex]\frac{dP}{dK}=\lambda n[/tex]
[tex]mL+nK=p[/tex]
And replacing what we got for the partial derivates we got:
[tex]b\alphaL^{\alpha-1}K^{1-\alpha}=\lambda m[/tex] (2)
[tex]b(1-\alpha)L^{\alpha}K^{-\alpha}=\lambda n[/tex] (3)
[tex]mL+nK=p[/tex] (4)
Now we can cancel the Lagrange multiplier [tex]\lambda[/tex] with equations (2) and (3), dividing these equations:
[tex]\frac{\lambda m}{\lambda n}=\frac{b\alphaL^{\alpha-1}K^{1-\alpha}}{b(1-\alpha)L^{\alpha}K^{-\alpha}}[/tex] (4)
And simplyfing equation (4) we got:
[tex]\frac{m}{n}=\frac{\alpha K}{(1-\alpha)L}[/tex] (5)
4) Solve for L and K
We can cross multiply equation (5) and we got
[tex]\alpha Kn=m(1-\alpha)L[/tex]
And we can set up this last equation equal to 0
[tex]m(1-\alpha)L-\alpha Kn=0[/tex] (6)
Now we can set up the following system of equations:
[tex]mL+nK=p[/tex] (a)
[tex]m(1-\alpha)L-\alpha Kn=0[/tex] (b)
We can mutltiply the equation (a) by [tex]\alpha[/tex] on both sides and add the result to equation (b) and we got:
[tex]Lm=\alpha p[/tex]
And we can solve for L on this case:
[tex]L=\frac{\alpha p}{m}[/tex]
And now in order to obtain K we can replace the result obtained for L into equations (a) or (b), replacing into equation (a)
[tex]m(\frac{\alpha P}{m})+nK=p[/tex]
[tex]\alpha P +nK=P[/tex]
[tex]nK=P(1-\alpha)[/tex]
[tex]K=\frac{P(1-\alpha)}{n}[/tex]
With this we have completed the proof.
A shortstop fields a grounder at a point one-third of the way from second base to third base. How far will he have to throw the ball to make an out at first base? Give the exact answer and an approximation to two decimal places.
Answer: d = 94.87 ft
Step-by-step explanation:
In a baseball game distances between bases is equal to 90 feet
Then if a shortstop get the ball one third of second base ( in the way from second base to third base) shortstop got the ball at 30 ft from second base
Now between the above mentioned point, first base and second base, we have a right triangle. In which distance between shortstop and first base is the hypotenuse. Then
d² = (90)² + (30)² d = √ 8100 + 900 d = 94.87 ft
Final answer:
Using the Pythagorean theorem, the exact distance a shortstop must throw to make an out at first base is the square root of 11700 feet, approximately 108.17 feet.
Explanation:
The question pertains to the distance a shortstop must throw a baseball to make an out at first base. To answer this question, we need to use the geometry of a baseball diamond, which is a square with 90 feet between each base. The Pythagorean theorem can be applied to find the distance from the shortstop to first base. Specifically, when the shortstop is one-third of the way from second to third base, we treat the path from the shortstop to first base as the hypotenuse of a right-angled triangle, with the two sides being from the shortstop to second base and from second base to first base. A full side is 90 feet, so one-third of the way is 30 feet (one-third of 90), making the side from the shortstop to second base 60 feet (90 - 30). The other side is a full 90 feet.
We then calculate the hypotenuse (the throw distance) using the Pythagorean theorem (a² + b² = c²), where a is 60 feet and b is 90 feet:
a² = 60² = 3600b² = 90² = 8100c² = a² + b² = 3600 + 8100 = 11700c = √11700 ≈ 108.1665 feetThe exact distance the shortstop must throw the ball is the square root of 11700, which is an irrational number, and an approximation to two decimal places is 108.17 feet.
For x, y ∈ R we write x ∼ y if x − y is an integer. a) Show that ∼ is an equivalence relation on R. b) Show that the set [0, 1) = {x ∈ R : 0 ≤ x < 1} is a set of representatives for the set of equivalence classes. More precisely, show that the map Φ sending x ∈ [0, 1) to the equivalence class C(x) is a bijection.
Answer:
A. It is an equivalence relation on R
B. In fact, the set [0,1) is a set of representatives
Step-by-step explanation:
A. The definition of an equivalence relation demands 3 things:
The relation being reflexive (∀a∈R, a∼a)The relation being symmetric (∀a,b∈R, a∼b⇒b∼a)The relation being transitive (∀a,b,c∈R, a∼b^b∼c⇒a∼c)And the relation ∼ fills every condition.
∼ is Reflexive:
Let a ∈ R
it´s known that a-a=0 and because 0 is an integer
a∼a, ∀a ∈ R.
∼ is Reflexive by definition
∼ is Symmetric:
Let a,b ∈ R and suppose a∼b
a∼b ⇒ a-b=k, k ∈ Z
b-a=-k, -k ∈ Z
b∼a, ∀a,b ∈ R
∼ is Symmetric by definition
∼ is Transitive:
Let a,b,c ∈ R and suppose a∼b and b∼c
a-b=k and b-c=l, with k,l ∈ Z
(a-b)+(b-c)=k+l
a-c=k+l with k+l ∈ Z
a∼c, ∀a,b,c ∈ R
∼ is Transitive by definition
We´ve shown that ∼ is an equivalence relation on R.
B. Now we have to show that there´s a bijection from [0,1) to the set of all equivalence classes (C) in the relation ∼.
Let F: [0,1) ⇒ C a function that goes as follows: F(x)=[x] where [x] is the class of x.
Now we have to prove that this function F is injective (∀x,y∈[0,1), F(x)=F(y) ⇒ x=y) and surjective (∀b∈C, Exist x such that F(x)=b):
F is injective:
let x,y ∈ [0,1) and suppose F(x)=F(y)
[x]=[y]
x ∈ [y]
x-y=k, k ∈ Z
x=k+y
because x,y ∈ [0,1), then k must be 0. If it isn´t, then x ∉ [0,1) and then we would have a contradiction
x=y, ∀x,y ∈ [0,1)
F is injective by definition
F is surjective:
Let b ∈ R, let´s find x such as x ∈ [0,1) and F(x)=[b]
Let c=║b║, in other words the whole part of b (c ∈ Z)
Set r as b-c (let r be the decimal part of b)
r=b-c and r ∈ [0,1)
Let´s show that r∼b
r=b-c ⇒ c=b-r and because c ∈ Z
r∼b
[r]=[b]
F(r)=[b]
∼ is surjective
Then F maps [0,1) into C, i.e [0,1) is a set of representatives for the set of the equivalence classes.
In Jamie's class, 1/5 of the students are boys. What percent of the students in Jamie’s class are boys?
A) 1.5%
B) 5%
C) 15%
D) 20%
Answer:
The percentage of boys students in Jamie's class is 20 %
Step-by-step explanation:
Given as :
The total number of boys in the Jamie's class = [tex]\frac{1}{5}[/tex] of the total student
Let the total number of student's in the class = x
So, The number of boys = [tex]\frac{1}{5}[/tex] × x
I.e The number of boys = [tex]\dfrac{x}{5}[/tex]
So , in percentage the number of boys students in class = [tex]\dfrac{\textrm Total number of boy student}{\textrm Total number of students}[/tex] × 100
OR, % boys students = [tex]\dfrac{\dfrac{x}{5}}{x}[/tex] × 100
or, % boys students = [tex]\frac{100}{5}[/tex]
∴ % boys students = 20
Hence The percentage of boys students in Jamie's class is 20 % . Answer
Answer:
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What would be an appropriate measure to describe the diameter of a music record?
A) millimeters
B) yards
C) Inches
D) kilometers
The Answer is C) Inches