The maximum volume of such a box is determined by forming an equation that represents the condition about the width of the box and the perimeter of one of its nonsquare sides, then finding the maximum value of the volume function obtained by differentiating the function and setting it equal to zero.
Explanation:The subject of this question is mathematics related to box dimensions, more specifically the geometry dealing with volume of a box. The box in question has a square base, which means the length and width are the same, let's call this x. Because the box is wider than it is tall, we know it's height, let's call it h, is less than x.
According to the question, the width of the box and the perimeter of one of the nonsquare sides of the box sum to no more than 117 inches. The perimeter of a nonsquare side (a rectangle) is given by 2(x+h), and if we add x (the width of the box) to this, we get x + 2(x+h) which must be less than or equal to 117. Simplifying gives 3x + 2h <= 117
We are interested in the volume of the box which can be determined by multiplying the length, width, and height (V = x*x*h). This can be simplified to V = x^2 * h. To get the maximum volume, we should make h as large as possible. Substituting 3x into the original inequality for h (since 3x <= 117), we get h <= 117 - 3x. Thus, the volume V becomes V = x^2 * (117 - 3x).
To find the maximum volume, we take the derivative of the volume function (V = x^2 * (117 - 3x)) with respect to x and set it equal to zero. This will give us the value of x for which the volume is maximum. Once we have the value of x, we substitute it back into the volume function to get the maximum volume.
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The maximum volume for such a box is 152,882.5 cubic inches
We have a box with a square base, where the width w is greater than the height h. The constraint given is that the width of the box plus the perimeter of one of the non-square sides cannot exceed 117 inches.
The perimeter of one of the non-square sides is 2h + 2d, where d is the depth. Therefore, the constraint equation becomes:
[tex]\[ w + 2h + 2d \leq 117 \][/tex]
Since the base is square, w = h. Let's denote this common value as s. So, the constraint equation simplifies to:
[tex]\[ 2s + 2d \leq 117 \][/tex]
Now, we need to express the volume of the box in terms of s and d:
[tex]\[ V = s^2d \][/tex]
We want to maximize V subject to the constraint [tex]\(2s + 2d \leq 117\).[/tex]
To proceed, let's solve the constraint equation for d:
[tex]\[ d \leq \frac{117 - 2s}{2} \][/tex]
Since d must be greater than zero, we have:
[tex]\[ 0 < d \leq \frac{117 - 2s}{2} \][/tex]
Now, we substitute [tex]\(d = \frac{117 - 2s}{2}\)[/tex] into the volume equation:
[tex]\[ V(s) = s^2 \left( \frac{117 - 2s}{2} \right) \]\[ V(s) = \frac{1}{2}s^2(117 - 2s) \][/tex]
To find the maximum volume, we'll take the derivative of V(s) with respect to s, set it equal to zero to find critical points, and then check for maximum points.
[tex]\[ V'(s) = \frac{1}{2}(234s - 6s^2) \]\\Setting \(V'(s)\) equal to zero:\[ \frac{1}{2}(234s - 6s^2) = 0 \]\[ 234s - 6s^2 = 0 \]\[ 6s(39 - s) = 0 \][/tex]
This gives us two critical points: s = 0 and s = 39. Since s represents the width of the box, we discard s = 0 as it doesn't make physical sense.
Now, we need to test s = 39 to see if it corresponds to a maximum or minimum. Since the second derivative is negative, s = 39 corresponds to a maximum.
So, the maximum volume occurs when s = 39 inches.
Substitute s = 39 into the constraint equation to find d:
[tex]\[ 2(39) + 2d = 117 \]\[ 78 + 2d = 117 \]\[ 2d = 117 - 78 \]\[ 2d = 39 \]\[ d = \frac{39}{2} \]\[ d = 19.5 \][/tex]
Therefore, the maximum volume of the box is achieved when the width and height are both 39 inches, and the depth is 19.5 inches. Let's calculate the maximum volume:
[tex]\[ V_{\text{max}} = (39)^2 \times 19.5 \]\[ V_{\text{max}} = 152,882.5 \, \text{cubic inches} \][/tex]
So, the maximum volume for such a box is 152,882.5 cubic inches.
DE=6x, EF=4x, DF=30 What is EF?
Answer:
The answer to your question is EF = 12
Step-by-step explanation:
Data
DE = 6x
EF = 4x
DF = 30
Process
DE + EF = DF
6x + 4x = 30
10x = 30
x = 30 / 10
x = 3
6(3) + 4 (3) = 30
18 + 12 = 30
30 = 30
DE = 6(3) = 18
EF = 4(3) = 12
The test statistic of z equals negative 1.37 is obtained when testing the claim that p equals1 divided by 4. a. Using a significance level of alpha equals 0.01, find the critical value(s). b. Should we reject Upper H 0 or should we fail to reject Upper H 0?
Answer: a) critical value = 0.0853, b) we reject the null hypothesis.
Step-by-step explanation:
Since we have given that
z = -1.37
And the hypothesis are given below:
[tex]H_0:p=\dfrac{1}{4}=0.25\\\\H_1:p\neq 0.25[/tex]
Since α = 0.01
since critical value = 0.0853
As we can see that 0.853 < 0.25.
so, we reject the null hypothesis.
Hence, a) critical value = 0.0853, b) we reject the null hypothesis.
Sviatoslav solved the quadratic equation x^2-x-1=0 by completing the square. In the process, he came up with the equivalent equation (x+a)^2 = b, where a and b are constants. What is b?
Answer:
[tex]x=\frac{1+\sqrt5}{2}[/tex] and [tex]x=\frac{1-\sqrt5}{2}[/tex]
and b=[tex]\frac{5}{4}[/tex]
Step-by-step explanation:
We are given that a quadratic equation
[tex]x^2-x-1=0[/tex]
We have to solve the equation by completing square and find the value of b.
[tex](x)^2-2\times x\times \frac{1}{2}+(\frac{1}{2})^2-(\frac{1}{2})^2-1=0[/tex]
[tex](x-\frac{1}{2})^2-\frac{1}{4}-1=0[/tex]
[tex](x-\frac{1}{2})^2-\frac{1-4}{4}=0[/tex]
[tex](x-\frac{1}{2})-\frac{5}{4}=0[/tex]
[tex](x-\frac{1}{2})^2-(\frac{\sqrt5}{2})^2=0[/tex]
[tex](x-\frac{1}{2})^2=(\frac{\sqrt5}{2})^2[/tex]
[tex]x-\frac{1}{2}=\frac{\sqrt5}{2}[/tex] and [tex]x-\frac{1}{2}=-\frac{\sqrt5}{2}[/tex]
[tex]x=\frac{\sqrt5}{2}+\frac{1}{2}=\frac{1+\sqrt5}{2}[/tex]
And [tex]x=\frac{1}{2}-\frac{\sqrt5}{2}=\frac{1-\sqrt5}{2}[/tex]
Therefore,[tex]x=\frac{1+\sqrt5}{2}[/tex] and [tex]x=\frac{1-\sqrt5}{2}[/tex]
and b=[tex]\frac{5}{4}[/tex]
Final answer:
To complete the square for the equation x²-x-1=0, we add (1/2)² = 1/4 to both sides, resulting in the equivalent equation (x-1/2)² = 5/4, where b is 5/4 or 1.25.
Explanation:
To solve the quadratic equation x²-x-1=0 by completing the square, we need to transform the equation into the form (x+a)² = b. First, we would rearrange the equation to isolate the terms with x: x² - x = 1. The next step is completing the square by adding (1/2 ×-1)² = (1/2)² = 1/4 to both sides of the equation, resulting in x² - x + 1/4 = 1 + 1/4. Therefore, the complete square form is (x-1/2)²= 5/4 and the constant b is 5/4 or 1.25.
MAX POINTS AND BRAINLIEST
Let f(p) be the average number of days a house stays on the market before being sold for price p in $1,000s. Which statement best describes the meaning of f(275)?
Houses sell on the market for an average of $275,000 and stay on the market an average of 275 days before being sold.
Houses sell for an average of $275,000.
f(275) indicates houses stay on the market an average of 275 days before being sold.
f(275) represents the average number of days houses stay on the market before being sold for $275,000.
The problem states:
f(p) is the average number of days a house stays on the market before being sold for price p in $1,000s
So we know:
p is the price in $1000s and
f(p) is the number of days before its sold for p
This means f(275) would be the number of days before its sold for 275,000 (since p is in $1000s).
The answer is:
f(275) represents the average number of days houses stay on the market before being sold for $275,000.
The statement "f(275) represents the average number of days houses stay on the market before being sold for $275,000" best describes the meaning of f(275).
Given,
Let f(p) be the average number of days a house stays on the market before being sold for price p in $1,000s.
In the given function f(p), the variable p represents the price of a house in thousands of dollars ($1,000s), and f(p) represents the average number of days a house stays on the market before being sold for that price.
Therefore, when you evaluate f(275), it tells you the average number of days houses stay on the market before being sold for $275,000.
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The vet told Jake that his dog, Rocco, who weighed 55 pounds, needed to lose 10 pounds. Jake started walking Rocco every day and changed the amount of food he was feeding him. Rocco lost half a pound the first week. Jake wants to determine Rocco’s weight in pounds, p, after w weeks if Rocco continues to lose weight based on his vet’s advice.
The equation of the scenario is what.
The values of p must be what.
Answer:
(on Edge) 1. "The equation of the scenario is p = 55 - 0.5w"
2. "The values of p must be any whole number 45 to 55"
BTW i got number 1 from this user https://brainly.com/profile/Alyssamarie03-2838077
Sydney and Tom each count the number of steps it takes for them to walk to school. They each count a 4 digit number of steps. Total number of steps is also 4 digit. What is the greatest possible digit in the thousands place for Sydney's or Tom's steps?
Answer:
8
Step-by-step explanation:
What is the leading coefficient of this polynomial when written in standard form?
1-2x+5x^4
Answer:
5!!
Step-by-step explanation:
When writing the polynomial in standard form, we have:
5X⁴ - 2X + 1
The leading coefficient is 5, since it's part of the term with the highest degree.
Final answer:
The leading coefficient of the polynomial 1 - 2x + 5x⁴ in standard form is 5, as standard form requires terms to be in descending powers of x.
Explanation:
The question asks to identify the leading coefficient of a polynomial when written in standard form. The given polynomial is 1 - 2x + 5x⁴. The standard form of a polynomial requires the terms to be written in descending powers of x. Therefore, the standard form of the given polynomial is 5x⁴ - 2x + 1. The leading coefficient is the coefficient of the term with the highest power of x, which in this case is 5 for the term 5x⁴.
The temple at the top of the pyramid is approximately 24 meters above ground, and there are 91 steps leading up to the temple. How high above the ground would you be if you were standing on the 50th step?
Answer:
13,18meters
Step-by-step explanation:
If the temple is in a height of 24 meters and to get there there are 91 steps, each step is 24 m / 91 = 26,37cm
The 50th step then is 26, 37cm. 50=13.18meters
Find the intervals over which the function is decreasing.
• (0,1)U(1,infinity) my answer choice
•(-infinity,-1)U(-1,0)
•(-infinity,-1)
(1,infinity)
Answer:
The answer to your question is: I agree with you, the first option
Step-by-step explanation:
• (0,1) U (1,infinity) This is the right answer because there are 2 invervals in which the graph decreases, and these intervals are listed in this option.
•(-infinity,-1)U(-1,0) This option is wrong because from (-∞ , -1) the graph grows up and also from (-1, 0).
•(-infinity,-1) The graph grows up, this option is incorrect
. (1,infinity) The graph decreases but the option is incomplete.
The correct option is A which is the function decreasing over the interval (0,1) U(1, infinity).
What is a function?The expression that established the relationship between the dependent variable and independent variable is referred to as a function. In the function as the value of the independent variable varies the value of the dependent variable also varies.
Check all the options:-
(0,1) U (1, infinity) there are two intervals in which the graph drops, and these intervals are stated in this option, making it the correct response.(-infinity,-1)U(-1,0) this choice is incorrect because the graph increases from (-, -1) and likewise from (-1, 0).t(-infinity,-1) the graph increases, hence this choice is false.t(1, infinity) the graph decreases but the option is incomplete.Therefore, the correct option is A which is the function decreasing over the interval (0,1) U(1, infinity).
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given an existing function: f(x)=0.5(x-2)2+3, what transformstiins would have to be made to result in g(x)=-2(x+3)2 -1?
Answer:
vertical scaling by a factor of -4horizontal translation 5 units leftvertical translation 11 units upStep-by-step explanation:
We notice that the multiplier of the squared term in f(x) is 0.5; in g(x), it is -2, so is a factor of -4 times that in f(x).
If we scale f(x) by a factor of -4, we get ...
-4f(x) = -2(x -2)² -12
In order for the squared quantity to be x+3, we have to add 5 to the value that is squared in f(x). That is, x -2 must become x +3. We have to replace x with (x+5) to do that, so ...
(x+5) -2 = x +3
The replacement of x with x+5 amounts to a translation of 5 units to the left.
We note that the added constant after our scaling changes from +3 to -12. Instead, we want it to be -1, so we must add 11 to the scaled function. That translates it upward by 11 units.
The attached graph shows the scaled and translated function g(x):
g(x) = -4f(x +5) +11
A line segment is divided in two segments, such that the ratio of the long segment to the short segment is equivalent to the golden ratio. If the length of the entire line segment is 15 inches long, what is the length of the longer piece of the divided segment? Use variant phialmost equals1.618.
Answer:
9.271 inches.
Step-by-step explanation:
Let AC be the length of original line segment and point B divides it into two segments such that AB is longer and BC is smaller segment.
[tex]AB+BC=AC=15[/tex]
We can describe the golden ratio as:
[tex]\frac{AB}{BC}=\frac{AC}{AB}=1.618[/tex]
[tex]\frac{AC}{AB}=1.618[/tex]
[tex]\frac{15}{AB}=1.618[/tex]
[tex]\frac{15}{1.618}=AB[/tex]
[tex]9.270704=AB[/tex]
[tex]AB=9.271[/tex]
We can verify our answer as:
[tex]AB+BC=15[/tex]
[tex]9.271+BC=15[/tex]
[tex]9.271-9.271+BC=15-9.271[/tex]
[tex]BC=5.729[/tex]
[tex]\frac{AB}{BC}=1.618[/tex]
[tex]\frac{9.271}{5.729}=1.618[/tex]
[tex]1.618=1.618[/tex]
Hence proved.
Therefore, the length of the longer side would be 9.271 inches.
Can someone help me with this?
Answer:
see below
Step-by-step explanation:
3. The answers in the image below are in "interval notation". In set notation, they might be ...
increasing: {x∈ℝ: -2 < x < 0 ∪ 2 < x}
decreasing: {x∈ℝ: 0 < x < 2}
positive: {x∈ℝ: -4 ≤ x < 1.5 ∪ 3 < x}
negative: {x∈ℝ: 1.5 < x < 3}
domain: {x∈ℝ: x ≥ -4}
range: {x∈ℝ: x ≥ -3}
y-intercept: y ∈ {2}
x-intercepts: x ∈ {1.5, 3}
relative minimum: (x, y) ∈ {(2, -3)}
relative maximum: (x, y) ∈ {(0, 2)}
__
4. Answers are in the image below.
The function is increasing where its slope is positive, decreasing where its slope is negative. Where the slope changes sign, there may be a point where the slope is 0 or undefined. The function is neither increasing nor decreasing there, so those points are not part of the intervals for increasing or decreasing.
The function is positive when it is above the x-axis, negative when it is below the x-axis. The function is neither positive nor negative where it is on the x-axis or where it is undefined.
The domain is the horizontal extent of the function. Any points where the function is undefined are excluded.
The range is the vertical extent of the function, all of the y-values where the function output is defined. Here, m(2) is not defined, so y=-7 is not an output at that point. However, y=7 is an output for m(-14 2/3). Likewise, m(9) does not have an output of 0, but m(3) does, so 0 is part of the range.
X- and y-intercepts are where the graph intersects the x- or y-axis, respectively. M(x) is undefined at (9, 0), so there is no x-intercept there.
A relative minimum is any point where the y-values increase on either side of the point. For m(x), the function is undefined at its relative minima, so it cannot be said to have any.
A relative maximum is any point where the y-value decreases on either side of the point. M(x) has one at the vertex of the parabolic segment.
If a person's eye level is h meters above sea level and she can see d kilometers to the horizon, then =d3.6h . Suppose the person can see 20.7 kilometers to the horizon. What is the height of her eye level above sea level?
Answer:
.
Step-by-step explanation:
The amount of radioactive element remaining, r, in a 100mg sample after d days is represented using the equation r=100(1/2) d/5. What is the daily percent of decrease
Answer:
12.94%
Step-by-step explanation:
r = 100(1/2)^(d/5) = 100((1/2)^(1/5))^d ≈ 100(.87055)^d
The daily decrease is 1 - 0.87055 = 0.12944 ≈ 12.94%
solve for y: x=3(y-b)
Answer:
Step-by-step explanation:
x=3(y-b)
or x=3y-3b
3y=x+3b
[tex]y=\frac{x+3b}{3}[/tex]
Assume that the significance level is alpha equals 0.01. Use the given information to find the P-value and the critical value(s). With Upper H 1 : p not equals three fourths , the test statistic is zequalsnegative 1.64.
Answer: 0.1010052
Step-by-step explanation:
Given : Significance level : [tex]\alpha=0.01[/tex]
Alternative hypothesis : [tex]H_1:\ p\neq\dfrac{3}{4}[/tex]
The test statistic value : [tex]z=-1.64[/tex]
Since , the alternative hypothesis is two-tailed, so the test is a two-tailed test.
Using standard normal distribution table for z, we have
The P-value of two-tailed test will be :-
[tex]2P(z>|-1.64|)=2P(z>1.64)\\\\=2(1-P(z\leq1.64))\\\\=2(1-0.9494974)\\\\=0.1010052[/tex]
Hence, the P-value = 0.1010052
Hey there! Can you help my math assignment, please?
What is the measure of angle D?
A. 52°
B. 54°
C. 57°
D. 126°
Explain your answer!
{Full explanation required. Answers choices are given. No spam answers, please!}
Please show your work!
Thanks!
Answer:
A. 52°
Step-by-step explanation:
All interior angles of ANY quadrilateral sum up to 360°; corresponding base angles sum up to 180°, therefore the m∠A is 52°.
I am joyous to assist you anytime.
Could someone please help me with this? Thank you!
Answer:
2 .
length of ,XY= a-6
length of ,YZ =3a+11
total length of ,XZ =41
Since,Point X,Y,Z lies in same straight line XZ..
So,
XZ= XY+ YZ
41= a-6 + 3a+11
41= 4a +5
41-5 = 4a
36/4 = a
a= 9
putting value of a in lengths XY and XZ we get,
XY= 9-6
= 3
YZ = 3*9 + 11
=27 + 11
=38
Length of XY = 3
Length of YZ = 38
Answer of 3 and 4 are as similar of above solved examples..
Mr.Drysdale earned $906.25 in intrest in one year on money that he had deposited in his local bank. If the bank paid intrest rate of 6.25% how much money did mr.Drysdale deposit?
[tex]\bf ~~~~~~ \textit{Simple Interest Earned} \\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\dotfill&\$906.25\\ P=\textit{original amount deposited}\\ r=rate\to 6.25\%\to \frac{6.25}{100}\dotfill &0.0625\\ t=years\dotfill &1 \end{cases} \\\\\\ 906.25=P(0.0625)(1)\implies 906.25=0.0625P\implies \cfrac{906.25}{0.0625}=P \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill 14500=P~\hfill[/tex]
At many golf clubs, a teaching professional provides a free 10-minute lesson to new customers. A golf magazine reports that golf facilities that provide these free lessons gain, on average, $1 comma 700 in green fees, lessons, or equipment expenditures. A teaching professional believes that the average gain is not $1 comma 700.
Answer:
What is it asking?
Step-by-step explanation:
How many distinct arrangements can be formed from all the letters of "students"? Please show your work. Thanks!
A) 10,080
B) 40,320
C) 1680
D) 720
There are a total of 8 letters in student, with 6 different letters ( there are 2 s's and 2 t's).
First find the number of arrangements that can be made using 8 letters.
This is 8! which is:
8 x 7 x 6 x 5 x 4 3 x 2 x 1 = 40,320
Now there are 2 s's and 2 t's find the number of arrangements of those:
S = 2! = 2 x 1 = 2
T = 2! = 2 x 1 = 2
Now divide the total combinations by the product of the s and t's:
40,320 / (2*2)
= 40320 / 4
= 10,080
The answer is A. 10,080
Which set of ordered pairs represent functions from A to B? Explain.
A = {a, b, c} and B = {0, 1, 2, 3}
a. {(a, 1), (c, 2), (c, 3), (b, 3)}
b. {(a, 1), (b, 2), (c, 3)}
c. {(1, a), (0, a), (2, c), (3, b)}
Answer:
c. {(1, a), (0, a), (2, c), (3, b)}
Step-by-step explanation:
The soccer team collected $800 at a car wash fundraiser. They charged $5.00 for small vehicles and $10.00 for larger vehicles. The amount collected can be modeled by the equation , where x represents the number of small vehicles and y represents the number of larger vehicles. If the number of larger vehicles washed was 50, how many small vehicles were washed in total?
a.55
b.60
c.130
d.150
Answer:
b. 60
Step-by-step explanation:
We have been given a formula [tex]5x+10y=800[/tex], where x represents the number of small vehicles and y represents the number of larger vehicles.
To find the number of small vehicles, we will substitute [tex]y=50[/tex] in the given equation as:
[tex]5x+10(50)=800[/tex]
[tex]5x+500=800[/tex]
[tex]5x+500-500=800-500[/tex]
[tex]5x=300[/tex]
[tex]\frac{5x}{5}=\frac{300}{5}[/tex]
[tex]x=60[/tex]
Therefore, 60 small vehicles were washed in total and option 'b' is the correct choice.
A kayaker travels X miles per hour down stream for three hours. On the for our return trip, the kayak or travels 1 mph slower. How far did the kayak or travel in total?
Consider this bag of marbles. What is the probability of drawing a green marble versus the ODDs of drawing a green marble? What is the difference in these two things? Make sure you show work, answer all questions and write in complete sentences.
Answer:
Probability 50%
Odds 5:5
Step-by-step explanation:
Probability is calculated as favorable cases divided by total cases.
While odds are calculated as favorable cases divided by (total cases - favorable cases)
Favorable cases (green) : 5
Total cases(green, red and blue): 10
Probability= 5/10 * 100%=50%
Odds = 5:(10-5) = 5:5
Answer:
it is 50%
Step-by-step explanation:
A teacher divides the students into three groups for project each group has the same number of students in the total of number students prime or composite
Answer:
The total number of students is composite
Step-by-step explanation:
* Lets explain how to solve the problem
- A teacher divides the students into three groups for project
- Each group has the same number of students
- We need to know the total number of students is prime number or
composite number
∵ The least number of any group is 2 students
∵ There are three groups
∵ The number of students in each group equal
∴ The least number of the students is 6
∵ Any number of students must be multiple of 3 because it will
divide by 3
∵ Any prime number has only two factors 1 and itself
∵ Any multiple of 3 (except 3) has more than two factors
∵ The class can not be 3 students because when divided to 3
groups each group will have 1 students and the least number of
students in a group is 2
∴ The number of the students in the class must be composite
Final answer:
The question deals with evenly dividing students into three groups for a group assignment, indicating that the total number of students is composite, as it is divisible by three.
Explanation:
The question involves dividing a number of students into groups for a group assignment. A key concept here is understanding whether the total number of students is a prime or composite number.
If the students were evenly divided into three groups, this implies that the total number must be divisible by three, making it a composite number.
Examples of group work dynamics include cases where students divide work equally but one ends up doing the most of it, evenly distributing tasks, or each taking on different aspects of a complex problem.
Last year, a women's professional organization made two small-business loans totaling $28,000 to young women beginning their own businesses. The money was lent at 7% and 14% simple interest rates. If the annual income the organization received from these loans was $3,430, what was each loan amount?
Answer:
$7,000 at a rate of 7% and $21,000 at a rate of 14%.
Step-by-step explanation:
Let x be amount invested at 7% and y be amount invested at 14%.
We have been given that a women's professional organization made two small-business loans totaling $28,000. We can represent this information in an equation as:
[tex]x+y=28,000...(1)[/tex]
The interest earned at 7% in one year would be [tex]0.07x[/tex] and interest earned at 14% in one year would be [tex]0.14x[/tex].
We are also told that the organization received from these loans was $3,430. We can represent this information in an equation as:
[tex]0.07x+0.14y=3,430...(2)[/tex]
Form equation (1), we will get:
[tex]x=28,000-y[/tex]
Upon substituting this value in equation (2), we will get:
[tex]0.07(28,000-y)+0.14y=3,430[/tex]
[tex]1960-0.07y+0.14y=3,430[/tex]
[tex]1960+0.07y=3,430[/tex]
[tex]1960-1960+0.07y=3,430-1960[/tex]
[tex]0.07y=1470[/tex]
[tex]\frac{0.07y}{0.07}=\frac{1470}{0.07}[/tex]
[tex]y=21,000[/tex]
Therefore, an amount of $21,000 was invested at a rate of 14%.
[tex]x=28,000-y[/tex]
[tex]x=28,000-21,000[/tex]
[tex]x=7,000[/tex]
Therefore, an amount of $7,000 was invested at a rate of 14%.
When the height of a cylinder is 12 cm and the radius is 4 cm, the circumference of the cylinder is increasing at a rate of π 4 cm/min, and the height of the cylinder is increasing four times faster than the radius. How fast is the volume of the cylinder changing?
The volume of a cylinder is changing at a rate of 256π cm³/min when the height of the cylinder is 12 cm, the radius is 4 cm, the circumference of the cylinder is increasing at a rate of π4 cm/min, and the height of the cylinder is increasing four times faster than the radius.
Explanation:The primary equation you need for this problem is the volume of a cylinder, which is V = πr²h. Given the height h of the cylinder is increasing four times faster than the radius r, if we use dh/dt for the rate of change of the height and dr/dt for the rate of change of the radius, we have dh/dt = 4dr/dt.
Also, the rate at which the circumference of the cylinder is increasing is d(2πr)/dt=2πdr/dt = π4 cm/min. Hence we can set up an equation as 2πdr/dt = π4. Solving for dr/dt, we get dr/dt = 2 cm/min.
Substituting this into the previous equation, we find that dh/dt = 8 cm/min. Now, if we take the derivative of the volume equation with respect to time, we get dV/dt = πr²dh/dt + 2πrh*dr/dt.
With the values r = 4cm, h = 12cm, dr/dt = 2 cm/min, and dh/dt = 8 cm/min, replacing in the equation above gives us: dV/dt = π*4²*8 + 2π*4*12*2 = 256π cm³/min. So, the volume of the cylinder is changing at a rate of 256π cm³/min.
Learn more about Rates of Change here:https://brainly.com/question/31226174
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An instructor gives her class the choice to do 7 questions out of the 10 on an exam.
(a)How many choices does each student have?
(b)How many choices does a student have if he/she must answer at least 3 of the first 5 questions?
Answer:
(a) 120 choices
(b) 110 choices
Step-by-step explanation:
The number of ways in which we can select k element from a group n elements is given by:
[tex]nCk=\frac{n!}{k!(n-k)!}[/tex]
So, the number of ways in which a student can select the 7 questions from the 10 questions is calculated as:
[tex]10C7=\frac{10!}{7!(10-7)!}=120[/tex]
Then each student have 120 possible choices.
On the other hand, if a student must answer at least 3 of the first 5 questions, we have the following cases:
1. A student select 3 questions from the first 5 questions and 4 questions from the last 5 questions. It means that the number of choices is given by:
[tex](5C3)(5C4)=\frac{5!}{3!(5-3)!}*\frac{5!}{4!(5-4)!}=50[/tex]
2. A student select 4 questions from the first 5 questions and 3 questions from the last 5 questions. It means that the number of choices is given by:
[tex](5C4)(5C3)=\frac{5!}{4!(5-4)!}*\frac{5!}{3!(5-3)!}=50[/tex]
3. A student select 5 questions from the first 5 questions and 2 questions from the last 5 questions. It means that the number of choices is given by:
[tex](5C5)(5C2)=\frac{5!}{5!(5-5)!}*\frac{5!}{2!(5-2)!}=10[/tex]
So, if a student must answer at least 3 of the first 5 questions, he/she have 110 choices. It is calculated as:
50 + 50 + 10 = 110
Which of these numbers are divisible by 9? 7,625 4,932 9,180 2,969 9,180 and 4,932 2,969, 9,180, and 7,625 9,180 and 7,625 2,969 and 4,932
Answer:
The answer to your question is below:
Step-by-step explanation:
A number is divisible by 9 if the sum of all the individual digits is evenly divisible by 9.
7,625 example 7 + 6 + 2 +5 = 20 it's not divisible by 9
4,932 4 + 9 + 3 + 2 = 19 it's not divisible by 9
9,180 18 it is divisible
2,969 26 it's not divisible by 9
9,180 18 it is divisible
4,932 18 it is divisible
2,969 26 it's not divisible by 9
Answer:
9180 is divisible by 9 .
Step-by-step explanation:
A number is divisible by 9 if the sum of all of it's digits is divisible by 9.
Here, for each of the given numbers, we will find sum of digits then check if the sum of digits is divisible by 9.
For 7,625 :
7 + 6 + 2 +5 = 20 ( not divisible by 9)
For 4,932 :
4 + 9 + 3 + 2 = 19 (not divisible by 9)
For 9,180 :
9+1+8 = 18 ( divisible by 9 )
For 2,969 :
2+9+6+9=26 (not divisible by 9)