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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) (3)/(2) (B) 24 (C) 36 (D) (root(3)(18))^(2)

The volume of a cube is increasing at a rate of 1818 cubic meters per hour.\newlineAt a certain instant, the volume is 88 cubic meters.\newlineWhat is the rate of change of the surface area of the cube at that instant (in square meters per hour)?\newlineChoose 11 answer:\newline(A) 32 \frac{3}{2} \newline(B) 2424\newline(C) 3636\newline(D) (183)2 (\sqrt[3]{18})^{2}

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Q. The volume of a cube is increasing at a rate of 1818 cubic meters per hour.\newlineAt a certain instant, the volume is 88 cubic meters.\newlineWhat is the rate of change of the surface area of the cube at that instant (in square meters per hour)?\newlineChoose 11 answer:\newline(A) 32 \frac{3}{2} \newline(B) 2424\newline(C) 3636\newline(D) (183)2 (\sqrt[3]{18})^{2}
  1. Find side length: First, find the side length of the cube when the volume is 88 cubic meters. The formula for the volume of a cube is V=s3V = s^3, where ss is the side length.
  2. Calculate side length: Calculate the side length: 8=s38 = s^3. So, s=83s = \sqrt[3]{8}, which is 22 meters.
  3. Find surface area: Now, find the surface area of the cube using the side length. The formula for the surface area of a cube is SA=6s2SA = 6s^2.
  4. Calculate surface area: Calculate the surface area: SA=6×(22)=6×4=24SA = 6 \times (2^2) = 6 \times 4 = 24 square meters.
  5. Find rate of change: Next, find the rate of change of the side length as the volume increases. The formula for the volume of a cube is V=s3V = s^3, so dVdt=3s2⋅dsdt\frac{dV}{dt} = 3s^2 \cdot \frac{ds}{dt}, where dVdt\frac{dV}{dt} is the rate of change of volume and dsdt\frac{ds}{dt} is the rate of change of the side length.
  6. Calculate rate of change: Plug in the known values: 18=3×(22)×dsdt18 = 3 \times (2^2) \times \frac{ds}{dt}. So, 18=3×4×dsdt18 = 3 \times 4 \times \frac{ds}{dt}, which means dsdt=1812=1.5\frac{ds}{dt} = \frac{18}{12} = 1.5 meters per hour.
  7. Find rate of change: Finally, find the rate of change of the surface area. The formula for the surface area of a cube is SA=6s2SA = 6s^2, so dSAdt=6×2s×dsdt.\frac{dSA}{dt} = 6 \times 2s \times \frac{ds}{dt}.
  8. Calculate rate of change: Plug in the known values: dSAdt=6×2×2×1.5\frac{dSA}{dt} = 6 \times 2 \times 2 \times 1.5. So, dSAdt=6×4×1.5=36\frac{dSA}{dt} = 6 \times 4 \times 1.5 = 36 square meters per hour.
  9. Final answer: The correct answer is (C) 3636 square meters per hour.

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