Chapter 1.4, Problem 32E

Solution

To find: The diameter of the snowballs.

The diameter is equal to $3.6$ inches.

Given information:

Jake stores a small cache of $4$ -inch-diameter snowballs in the basement freezer, unaware that the freezer’s self-defrosting feature will cause each snowball to lose about one cubic inch of volume every $40$ days.

Calculation:

Since the snowballs initially have $4$ -inches diameter, which means their radius are $2$ inches at the start. So, the initial volume is

  $\begin{array}{c}V\left(0\right)=\frac{4}{3}\pi {\left(2\right)}^3\\ =\frac{32}{3}\pi \end{array}$

As the snowballs lose $1$ cubic inch every $40$ days, then they will lose $\frac{1}{40}$ cubic inch per day. The volume of the snowball as a function of time will be,

  $V\left(t\right)=\left(\frac{32}{3}\pi -\frac{t}{40}\right)$

After $360$ days, the volume of the snowballs will be

  $\begin{array}{c}V\left(360\right)=\left(\frac{32}{3}\pi -\frac{360}{40}\right)\\ =\frac{32}{3}\pi -9\end{array}$

The volume of the snowballs (sphere) is given by

  $V=\frac{4}{3}\pi r^3$

Now, equate the above both volumes and solve for $r$ :

  $\begin{array}{c}\frac{4}{3}\pi r^3=\frac{32}{3}\pi -9\\ 4\pi r^3=32\pi -27\\ r^3=\frac{32\pi -27}{4\pi }\\ r={\left(\frac{32\pi -27}{4\pi }\right)}^{\frac{1}{3}}\\ r={\left(\frac{32\pi -27}{4\pi }\right)}^{\frac{1}{3}}\\ r\approx 1.8\end{array}$

So, the diameter is $3.6$ inches.

Hence, the diameter is $3.6$ inches.