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The side of a cube is increasing at a rate of 2 kilometers per hour.
At a certain instant, the side is 1.5 kilometers.
What is the rate of change of the volume of the cube at that instant (in cubic kilometers per hour)?
Choose 1 answer:
(A) 2
(B)
16.5
(C) 8
(D)
13.5

The side of a cube is increasing at a rate of $2$ kilometers per hour. $\newline$ At a certain instant, the side is $1$ . $5$ kilometers. $\newline$ What is the rate of change of the volume of the cube at that instant (in cubic kilometers per hour)? $\newline$ Choose $1$ answer: $\newline$ (A) $2$ $\newline$ (B) $\mathbf{1 6 . 5}$ $\newline$ (C) $8$ $\newline$ (D) $\mathbf{1 3 . 5}$

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Calculate unit rates with fractions

Full solution

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

2

kilometers per hour.

\newline

At a certain instant, the side is

1

.

5

kilometers.

\newline

What is the rate of change of the volume of the cube at that instant (in cubic kilometers per hour)?

\newline

Choose

1

answer:

\newline

(A)

2

\newline

(B)

\mathbf{1 6 . 5}

\newline

(C)

8

\newline

(D)

\mathbf{1 3 . 5}

Volume Formula: The formula for the volume of a cube is $V = s^3$ , where $s$ is the length of a side.
Rate of Change: To find the rate of change of the volume, we need to differentiate the volume with respect to time, so $\frac{dV}{dt} = 3s^2 \cdot \frac{ds}{dt}$ .
Given Values: Given $\frac{ds}{dt}$ (rate of change of the side) is $2\,\text{km/h}$ , and $s$ (side length) is $1.5\,\text{km}$ .
Plug in Values: Plug the values into the formula: $\frac{dV}{dt} = 3 \times (1.5 \, \text{km})^2 \times (2 \, \text{km/h})$ .
Calculate Rate: Calculate the rate of change of the volume: $\frac{dV}{dt} = 3 \times 2.25 \, \text{km}^2 \times 2 \, \text{km/h}$ .
Simplify Calculation: Simplify the calculation: $\frac{dV}{dt} = 3 \times 4.5 \, \text{km}^3/\text{h} \times 2$ .
Finish Calculation: Finish the calculation: $\frac{dV}{dt} = 27 \, \text{km}^3/\text{h}.$

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