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The volume of a sphere is increasing at a rate of 
25 pi cubic meters per hour.
At a certain instant, the volume is 
(32 pi)/(3) cubic meters.
What is the rate of change of the surface area of the sphere at that instant (in square meters per hour)?
Choose 1 answer:
(A) 
(32 pi)/(9)
(B) 
25 pi
(C) 
(25)/(16)
(D) 
(root(3)(25 pi))^(2)
The surface area of a sphere with radius 
r is 
4pir^(2).
The volume of 
a sphere with radius 
r is 
(4)/(3)pir^(3).

The volume of a sphere is increasing at a rate of 25π 25 \pi cubic meters per hour.\newlineAt a certain instant, the volume is 32π3 \frac{32 \pi}{3} cubic meters.\newlineWhat is the rate of change of the surface area of the sphere at that instant (in square meters per hour)?\newlineChoose 11 answer:\newline(A) 32π9 \frac{32 \pi}{9} \newline(B) 25π 25 \pi \newline(C) 2516 \frac{25}{16} \newline(D) (25π3)2 (\sqrt[3]{25 \pi})^{2} \newlineThe surface area of a sphere with radius r r is 4πr2 4 \pi r^{2} .\newlineThe volume of a a sphere with radius r r is 43πr3 \frac{4}{3} \pi r^{3}.

Full solution

Q. The volume of a sphere is increasing at a rate of 25π 25 \pi cubic meters per hour.\newlineAt a certain instant, the volume is 32π3 \frac{32 \pi}{3} cubic meters.\newlineWhat is the rate of change of the surface area of the sphere at that instant (in square meters per hour)?\newlineChoose 11 answer:\newline(A) 32π9 \frac{32 \pi}{9} \newline(B) 25π 25 \pi \newline(C) 2516 \frac{25}{16} \newline(D) (25π3)2 (\sqrt[3]{25 \pi})^{2} \newlineThe surface area of a sphere with radius r r is 4πr2 4 \pi r^{2} .\newlineThe volume of a a sphere with radius r r is 43πr3 \frac{4}{3} \pi r^{3}.
  1. Given Information: Given: The volume of a sphere is increasing at a rate of 25π25 \pi cubic meters per hour.\newlineVolume of a sphere formula: V=43πr3V = \frac{4}{3}\pi r^3.\newlineDifferentiate both sides with respect to time tt to find the rate of change of the radius.\newlinedVdt=4πr2drdt\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}.
  2. Differentiate Volume Formula: Substitute the given rate of change of volume into the differentiated volume formula.\newline25π=4πr2drdt25 \pi = 4\pi r^2 \frac{dr}{dt}.\newlineSolve for drdt\frac{dr}{dt}.\newlinedrdt=25π4πr2\frac{dr}{dt} = \frac{25 \pi}{4\pi r^2}.
  3. Substitute Rate of Change: Given: The volume at a certain instant is (32π)/3(32 \pi)/3 cubic meters.\newlineUse the volume formula to find the radius at that instant.\newline(32π)/3=(4/3)πr3(32 \pi)/3 = (4/3)\pi r^3.\newlineSolve for r3r^3.\newliner3=(32π)/(4π)r^3 = (32 \pi)/(4\pi).\newliner3=8r^3 = 8.
  4. Find Radius at Instant: Find the radius by taking the cube root of r3r^3.\newliner=cube root of 8r = \text{cube root of } 8.\newliner=2 meters.r = 2 \text{ meters}.
  5. Calculate Radius: Substitute r=2r = 2 into the expression for drdt\frac{dr}{dt}.
    drdt=25π4π22\frac{dr}{dt} = \frac{25 \pi}{4\pi \cdot 2^2}.
    drdt=25π16π\frac{dr}{dt} = \frac{25 \pi}{16\pi}.
    drdt=2516\frac{dr}{dt} = \frac{25}{16} meters per hour.
  6. Calculate Rate of Change: Surface area of a sphere formula: A=4πr2A = 4\pi r^2. Differentiate both sides with respect to time tt to find the rate of change of the surface area. dAdt=8πrdrdt\frac{dA}{dt} = 8\pi r \frac{dr}{dt}.
  7. Differentiate Surface Area Formula: Substitute r=2r = 2 and drdt=2516\frac{dr}{dt} = \frac{25}{16} into the differentiated surface area formula.dAdt=8π2(2516)\frac{dA}{dt} = 8\pi\cdot2 \cdot \left(\frac{25}{16}\right).dAdt=8π2(2516)\frac{dA}{dt} = 8\pi\cdot2 \cdot \left(\frac{25}{16}\right).dAdt=100π16\frac{dA}{dt} = \frac{100\pi}{16}.dAdt=25π4\frac{dA}{dt} = \frac{25\pi}{4}.

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