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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 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) 22\newline(B) 16.5 \mathbf{1 6 . 5} \newline(C) 88\newline(D) 13.5 \mathbf{1 3 . 5}

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) 22\newline(B) 16.5 \mathbf{1 6 . 5} \newline(C) 88\newline(D) 13.5 \mathbf{1 3 . 5}
  1. Volume Formula: The formula for the volume of a cube is V=s3V = s^3, where ss is the length of a side.
  2. Rate of Change: To find the rate of change of the volume, we need to differentiate the volume with respect to time, so dVdt=3s2dsdt\frac{dV}{dt} = 3s^2 \cdot \frac{ds}{dt}.
  3. Given Values: Given dsdt\frac{ds}{dt} (rate of change of the side) is 2km/h2\,\text{km/h}, and ss (side length) is 1.5km1.5\,\text{km}.
  4. Plug in Values: Plug the values into the formula: dVdt=3×(1.5km)2×(2km/h)\frac{dV}{dt} = 3 \times (1.5 \, \text{km})^2 \times (2 \, \text{km/h}).
  5. Calculate Rate: Calculate the rate of change of the volume: dVdt=3×2.25km2×2km/h\frac{dV}{dt} = 3 \times 2.25 \, \text{km}^2 \times 2 \, \text{km/h}.
  6. Simplify Calculation: Simplify the calculation: dVdt=3×4.5km3/h×2\frac{dV}{dt} = 3 \times 4.5 \, \text{km}^3/\text{h} \times 2.
  7. Finish Calculation: Finish the calculation: dVdt=27km3/h.\frac{dV}{dt} = 27 \, \text{km}^3/\text{h}.

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