MATHEMATICS COLLEGE

The base of an aquarium with given volume V is made of slate and the sides are made of glass. If slate costs five times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials. (Let x, y, and z be the dimensions of the aquarium. Enter your answer in terms of V.)

Answers

Answer 1

Answer:

Answer:

The dimensions of the aquarium that minimize the cost of the materials:

$x=y=\sqrt[3]{(2V)/(5)}\nz=\sqrt[3]{(25V)/(4)}$

Step-by-step explanation:

Let x, y and z be the dimensions of aquarium .

Surface area of an aquarium = xy+2yz+2xz

Volume of aquarium V= $Length * Breadth * Height=xyz$ ----A

We are given that slate costs five times as much (per unit area) as glass

So, Cost function : C=5xy+2yz+2xz

Now we will use langrage multiplier to find the dimensions of the aquarium that minimize the cost of the materials.

$\nabla C =\lambda \nabla V$

$((\partial C)/(\partial x),(\partial C)/(\partial y),(\partial C)/(\partial z))= \lambda ((\partial V)/(\partial x),(\partial V)/(\partial y),(\partial V)/(\partial z))$

$(5y+2z,5x+2z,2y+2x)=\lambda(yz,xz,xy)$

So,

$5y+2z=\lambda yz$ ----1

$5x+2z=\lambda xz$ -----2

$2y+2x=\lambda xy$ ----3

Multiply 1 ,2 and 3 by x,y and z respectively.

$5xy+2xz=\lambda xyz$ ----4

$5xy+2yz=\lambda xyz$ -----5

$2yz+2xz=\lambda xyz$ ----6

Now equate 4 and 5

5xy+2xz=5xy+2yz

x=y

Substitute y=x in 5 and 6 and equate them

$5x(x)+2(x)z=2(x)z+2xz\n5x^2=2xz\n5x=2z\n(5)/(2)x=z$

Substitute the values in A

$V = xyz = x * x * (5)/(2)xV=(5)/(2)x^3\n\sqrt[3]{(2)/(5)V}=x\nx=y=\sqrt[3]{(2)/(5)V}\nz=(5)/(2)x=(5)/(2)(\sqrt[3]{(2)/(5)})=\sqrt[3]{(25V)/(4)}$

Hence,

The dimensions of the aquarium that minimize the cost of the materials:

$x=y=\sqrt[3]{(2V)/(5)}\nz=\sqrt[3]{(25V)/(4)}$

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Evaluate (8 + x) + 62 when x = 4.
A. 38
B. 14
C. 64
O D. 40

Answers

Answer:

74 if thats not an answer 64

Step-by-step explanation:

8+4=12

12+62

Answer:

Step-by-step explanation:

(8+x)+62

(8+4)+62

12+62

Describe how the graph of y | x| -4 is like the graph of y= |x| and how it is different.

Answers

The graph y=|x|-4 is obtained from the graph y=|x| dy moving down 4 units the graph y=|x| along the y-axis (see, if x=0, then for y=|x|, y=0 and for y=|x|-4, y=-4).

These two graphs have the same form.

Answer:

The graph of y=|x|-4 is the same as y=|x|.

Step-by-step explanation:

What is the effect on the graph of the function f(x)=x when f(x) is replaced with -1/2f(x)?

Answers

Answer:

For negative sign , the graph reflects over x -axis and for 1/2 there will be a vertical compression

Step-by-step explanation:

Parent function is f(x)= x

We need to find the effect of the graph when f(x) is replaced with -1/2f(x)

When negative sign is multiplied outside f(x) like -f(x), then there will be a reflection over x-axis

When negative sign is multiplied inside f(x) like f(-x) then there will be a reflection over y-axis

When a number is multiplied outside f(x) then there will be a vertical stretch or compression

Here 1/2 is multiplied outside f(x). $(1)/(2)$ is less than 1 so there will be a vertical compression

For negative sign , the graph reflects over x -axis and for 1/2 there will be a vertical compression

Hey can you please help me posted picture of question

Answers

Shifting to right means subtracting the number from x.

So, shifting to right by 9 units means, subtracting 9 from x.

G(x) = F(x - 9)

G(x) = |x - 9|

So, option D is the correct answer

Use the value of the linear correlation coefficient r to find the percentage of the total variation that can be explained by the linear relationship between the two variables. r = 0.234 (Your answer should be a percent rounded to two decimal places as needed.)

Answers

Answer:

5.47% of the total variation between the two variables can be explained by the regression line.

Step-by-step explanation:

Given :

R value = 0.234

The R value is the correlation Coefficient ; which gives the type and level of correlation between two variables.

However, to Obtian the percentage variation which can be explained by by the regression line, we have to obtain the Coefficient of determination, R².

R² value is Obtained by taking the squared of the correlation Coefficient

Hence,

R² = 0.234² = 0.054756

R² = 0.05476 * 100% = 5.4756%

Hence, R² = 5.48%

This is interpreted as ; 5.47% of the total variation between the two variables can be explained by the regression line.

Find the domain of the function. (enter your answer using interval notation.) f(x) = x + 3 if x < −1 −2x if |x| ≤ 1 −2 if x > 1

Answers

The domain of the function is the union of all of the "if" parts of the function definition:

... (-∞, -1) ∪ [-1, 1] ∪ (1, ∞) = (-∞, ∞)

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