Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

The volume of a cube is increasing at a rate of 18 cubic meters per hour.
At a certain instant, the volume is 8 cubic meters.
What is the rate of change of the surface area of the cube at that instant (in square meters per hour)?
Choose 1 answer:
(A)
(root(3)(18))^(2)
(B) 36
(C)
(3)/(2)
(D) 24

The volume of a cube is increasing at a rate of $18$ cubic meters per hour. $\newline$ At a certain instant, the volume is $8$ cubic meters. $\newline$ What is the rate of change of the surface area of the cube at that instant (in square meters per hour)? $\newline$ Choose $1$ answer: $\newline$ (A) $(\sqrt[3]{18})^{2}$ $\newline$ (B) $36$ $\newline$ (C) $\frac{3}{2}$ $\newline$ (D) $24$

View step-by-step help

View step-by-step help

Volume of cubes and rectangular prisms: word problems

Full solution

Q. The volume of a cube is increasing at a rate of

18

cubic meters per hour.

\newline

At a certain instant, the volume is

8

cubic meters.

\newline

What is the rate of change of the surface area of the cube at that instant (in square meters per hour)?

\newline

Choose

1

answer:

\newline

(A)

(\sqrt[3]{18})^{2}

\newline

(B)

36

\newline

(C)

\frac{3}{2}

\newline

(D)

24

Calculate Cube Side Length: Volume of cube = $\text{side}^3$ . Given volume is $8$ cubic meters, so side = cube root of $8$ . $\newline$ side = $2$ meters.
Calculate Cube Surface Area: Surface area of cube = $6 \times \text{side}^2$ . So, surface area with side $2$ meters is $6 \times (2^2)$ . $\newline$ Surface area = $6 \times 4 = 24$ square meters.
Calculate Rate of Change of Volume: Rate of change of volume = $\frac{d(\text{Volume})}{dt} = 18$ cubic meters per hour. $\newline$ Since Volume = $\text{side}^3$ , take the derivative with respect to time ( $t$ ) to get $\frac{d(\text{Volume})}{dt} = 3 \cdot \text{side}^2 \cdot \frac{d(\text{side})}{dt}$ .
Calculate Rate of Change of Side Length: Solve for $\frac{d(\text{side})}{dt}$ : $18 = 3 \times (2^2) \times \frac{d(\text{side})}{dt}$ . $\newline$ $\frac{d(\text{side})}{dt} = \frac{18}{(3 \times 4)} = \frac{18}{12} = 1.5$ meters per hour.
Calculate Rate of Change of Surface Area: Rate of change of surface area = $\frac{d(\text{Surface area})}{dt} = 6 \times 2 \times \text{side} \times \frac{d(\text{side})}{dt}$ . Plug in $\frac{d(\text{side})}{dt} = 1.5$ meters per hour to get $\frac{d(\text{Surface area})}{dt} = 6 \times 2 \times 2 \times 1.5$ .

More problems from Volume of cubes and rectangular prisms: word problems

Question

Chinedu was given this problem:

\newline

A person stands at a distance of

18

meters east of an intersection and watches a car driving away from the intersection to the north at

3

meters per second. The function

y(t)

gives the car's distance, in meters, north of the intersection at time

t

. At a certain instant

t_{0}

\theta\left(t_{0}\right)

between the line of sight from the person to the car and the line of sight from the person to the intersection is

0

7

radians. What is the rate of change of the angle

\theta(t)

at that instant?

\newline

Which equation should Chinedu use to solve the problem?

\newline

1

\newline

\tan [\theta(t)]=\frac{y(t)}{18}

\newline

\theta(t)=\frac{18 \cdot y(t)}{2}

\newline

\theta(t)+18+y(t)=180

\newline

[\theta(t)]^{2}=18^{2}+[y(t)]^{2}

Posted 3 months ago

Question

Steve was given this problem:

\newline

r(t)

of a circle is increasing at a rate of

3

centimeters per second. At a certain instant

t_{0}

, the radius is

8

centimeters. What is the rate of change of the area

A(t)

of the circle at that instant?

\newline

Which equation should Steve use to solve the problem?

\newline

1

\newline

A(t)=2 \pi \cdot r(t)

\newline

A(t)=\pi[r(t)]^{2}

\newline

A(t)=4 \pi[r(t)]^{2}

\newline

A(t)=\frac{4}{3} \pi[r(t)]^{3}

Posted 3 months ago

Question

Benjamin invests

\$ 400

in a savings account that earns

5 \%

interest each year.

\newline

Which expression gives the balance in the account after

3

\newline

1

\newline

400 \cdot 0.05 \cdot 0.05 \cdot 0.05

\newline

400+(1+0.05)(1+0.05)(1+0.05)

\newline

400+0.05 \cdot 0.05 \cdot 0.05

\newline

400(1+0.05)(1+0.05)(1+0.05)

Posted 3 months ago

Question

\newline

T chart of values for the Exponential equation. Round to the nearest

10^{th}

\newline

When you have checked your quiz and know that your T-Chart is correct, graph this Exponential equation and upload it (along with the others) to the Absolute Value and Exponential Graph Upload Assignment for points

\newline

y=-3^{x}-4

\newline

\newline

x

\newline

y

\newline

\newline

0

\newline

\newline

1

\newline

\newline

2

\newline

\newline

3

\newline

\newline

-1

\newline

\newline

-2

Posted 3 months ago

Question

\newline

-\frac{1}{1} \text{ Points}

\newline

\newline

8

7

025

\newline

Use the disk method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the

\newline

\newline

y=\frac{1}{\sqrt{1+x^{2}}},\quad y=0,\quad x=-1,\quad x=1

\newline

\newline

-\frac{1}{1} \text{ Points}

\newline

\newline

8

7

027

Posted 3 months ago

Question

A rectangular prism has a height of

7\,\text{in}

and the base with dimensions of

8\,\text{in}

6\,\text{in}

. What is the lateral area, in square inches?

Posted 3 months ago

Question

3

6

\newline

\newline

PREVIOUS ANSWERS

\newline

10

3

1

071

\newline

\newline

The total revenue

R

earned (in thousands of dollars) from manufacturing handheld video games is given by

R(p)=-25p^{2}+1700p

p

is the price per unit

\newline

(a) Find the revenues when the prices per unit are

\$20

\$25

\$30

\newline

R(\$20)\quad\$24000

\newline

\newline

R(\$25)\quad\$26875

\newline

\checkmark thousand

\newline

R(\$30)\quad\$28500

\newline

\checkmark thousand

\newline

(b) Find the unit price that yields a maximum revenue.

\newline

\newline

What is the maximum revenue?

\newline

\$\quad

\newline

\newline

Enter an exast number.

\newline

Posted 3 months ago

Question

V \text{ cm}^3

of a spherical balloon with radius

r \text{ cm}

\newline

V = \frac{4}{3}\pi r^3

\newline

\newline

\frac{dV}{dr} = 4\pi r^2

\newline

\frac{dV}{dr}

giving your answer in simplest form.

\newline

1

\newline

t

seconds, the volume of the balloon is increasing according to the differential equation

\newline

\frac{dV}{dt} = \frac{900}{(2t+3)^2}, \quad t \geq 0

\newline

V=0

t=0

\newline

(b) (i) solve this differential equation to show that

\newline

V = \frac{300 t}{2t+3}

\newline

(ii) Hence find the upper limit to the volume of the balloon.

\newline

(c) Find the radius of the balloon at

t=3

, giving your answer in

\text{cm}

3

significant figures.

\newline

5

\newline

3

Posted 3 months ago

Question

If the total surface area of a cube is

302.46\,\text{in}^2

, which best describes the length of an edge of the cube?

\newline

5\,\text{inches}

\newline

5

6\,\text{inches}

\newline

6

7\,\text{inches}

\newline

D. greater than

7\,\text{inches}

Posted 3 months ago

Question

M

, of a solution is equal to the number of moles of solute per liter of solution. If

1.0

mole of copper sulfate is used to make

5.0

liters of copper sulfate solution, what is the molarity of the solution?

\newline

1

\newline

0.20\,M

\newline

0.25\,M

\newline

1.0\,M

\newline

5.0\,M

Posted 3 months ago