if a snowball melts so that its surface area decreases at a rate of 5 cm²/min, find the rate (in cm/min) at…

Explanation:

Step1: Recall surface - area formula

The surface - area formula of a sphere is $A = 4\pi r^{2}$, and since $r=\frac{d}{2}$, we can write $A = 4\pi(\frac{d}{2})^{2}=\pi d^{2}$, where $A$ is the surface area and $d$ is the diameter.

Step2: Differentiate with respect to time $t$

Differentiating both sides of $A=\pi d^{2}$ with respect to $t$ using the chain - rule, we get $\frac{dA}{dt}=2\pi d\frac{dd}{dt}$.

Step3: Substitute given values

We are given that $\frac{dA}{dt}=- 5$ (negative because the area is decreasing) and $d = 8$. Substitute these values into the equation $\frac{dA}{dt}=2\pi d\frac{dd}{dt}$: $-5=2\pi\times8\times\frac{dd}{dt}$.

Step4: Solve for $\frac{dd}{dt}$

First, simplify the right - hand side: $-5 = 16\pi\frac{dd}{dt}$. Then, solve for $\frac{dd}{dt}$: $\frac{dd}{dt}=\frac{-5}{16\pi}$. Calculate $\frac{-5}{16\pi}\approx - 0.099$.

Answer:

$-0.099$