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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) 
13.5
(B) 
16.5
(C) 8
(D) 2

The side of a cube is increasing at a rate of 22 kilometers per hour.\newlineAt a certain instant, the side is 11.55 kilometers.\newlineWhat is the rate of change of the volume of the cube at that instant (in cubic kilometers per hour)?\newlineChoose 11 answer:\newline(A) 13.5 \mathbf{1 3 . 5} \newline(B) 16.5 \mathbf{1 6 . 5} \newline(C) 88\newline(D) 22

Full solution

Q. The side of a cube is increasing at a rate of 22 kilometers per hour.\newlineAt a certain instant, the side is 11.55 kilometers.\newlineWhat is the rate of change of the volume of the cube at that instant (in cubic kilometers per hour)?\newlineChoose 11 answer:\newline(A) 13.5 \mathbf{1 3 . 5} \newline(B) 16.5 \mathbf{1 6 . 5} \newline(C) 88\newline(D) 22
  1. Volume Formula: Volume of a cube formula: V=s3V = s^3, where ss is the side length.
  2. Differentiate Volume: Differentiate the volume with respect to time to find the rate of change of volume: dVdt=3s2dsdt\frac{dV}{dt} = 3s^2 \cdot \frac{ds}{dt}.
  3. Plug in Values: Plug in the values: s=1.5kms = 1.5 \, \text{km} and dsdt=2km/h\frac{ds}{dt} = 2 \, \text{km/h}.\newlinedVdt=3×(1.5)2×2\frac{dV}{dt} = 3 \times (1.5)^2 \times 2.
  4. Calculate Rate: Calculate the rate of change: dVdt=3×2.25×2\frac{dV}{dt} = 3 \times 2.25 \times 2.
  5. Simplify Calculation: Simplify the calculation: dVdt=3×4.5×2\frac{dV}{dt} = 3 \times 4.5 \times 2.
  6. Final Calculation: Final calculation: dVdt=13.5×2\frac{dV}{dt} = 13.5 \times 2.

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