if a snowball melts so that its surface area decreases at a rate of 1cm²/min, find the rate at which the…

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})

[ \begin{align*} \frac{dd}{dt}&=\frac{-1}{20\pi}\ \end{align*} ]

Answer:

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