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

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

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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) $24$

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Math Problems

Grade 6

Volume of cubes and rectangular prisms: word problems

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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 $\frac{d(\text{Surface area})}{dt} = 6 \times 2 \times 2 \times 1.5$ .