Find out the answer to "The radius of a sphere is increasing at..."

Tobias Ali

Answered question

2021-08-14

The radius of a sphere is increasing at a rate of 2 mm/s. How fast is the volume increasing when the diameter is 100 mm?

Answer & Explanation

faldduE

Skilled2021-08-15Added 109 answers

Step 1: Define an equation that relates the volume of a sphere to its radius. $V = \frac{4}{3} \cdot π \cdot r^{3}$ Step 2: Take the time-related derivative of each side (we will define time as "t") $(\frac{d}{d t}) V = (\frac{d}{d t}) (\frac{4}{3} \cdot π \cdot r^{3}) \frac{d V}{d t} = 4 π r^{2} \frac{d r}{d t}$ Step 3: The problem statement informs us that diameter is 100m, so hence r = 50mm. We are also told the radius of the sphere is increasing at a rate of 2mm/s, so therefore dr/dt = 2mm/s. We are interested in how quickly the sphere's volume is growing, or dV/dt. $\frac{d V}{d t} = 4 π (50 m m)^{2} \cdot (2 m m / s) \frac{d V}{d t} = 68.832 m m^{3} / s$

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