Question

if a snowball melts so that its surface area decreases at a rate of 1cm²/min, find the rate at which the diameter decreases when the diameter is 10cm.

Explanation:

Step1: Write the formula for the surface area of a sphere

The surface area formula of a sphere is \(A = 4\pi r^{2}\). Since \(d = 2r\) (where \(d\) is the diameter), then \(r=\frac{d}{2}\), and \(A = 4\pi(\frac{d}{2})^{2}=\pi d^{2}\).

Step2: Differentiate the surface - area formula with respect to time \(t\)

Differentiate \(A=\pi d^{2}\) with respect to \(t\) using the chain rule. \(\frac{dA}{dt}=2\pi d\frac{dd}{dt}\).

Step3: Substitute the given values

We know that \(\frac{dA}{dt}=- 1\) (negative because the area is decreasing) and \(d = 10\).
Substitute into \(\frac{dA}{dt}=2\pi d\frac{dd}{dt}\), we get \(-1=2\pi\times10\times\frac{dd}{dt}\).

Step4: Solve for \(\frac{dd}{dt}\)

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Answer:

\(\frac{-1}{20\pi}\text{ cm/min}\)