Chapter 12.4, Problem 17PPS

(a)

To determine

To find: the amount of ice in the icicle

Answer to Problem 17PPS

The amount of ice in the cone is approximately $1.48{\text{ft}}^3$ .

Explanation of Solution

Given:

Diameterof cone is l.5 ft

Height is 2.5 ft

Calculation:

The objective is to find the amount of ice in the cone.

Since the amount of ice is the volume of the cone, so find the volume of the cone

The volume V of a cone is given by the formula

  $V=\frac{1}{3}B\cdot h$

Where $B=\pi r^2$ the area of the circle and h isis the height the cone.

Since radius is half the diameter of a circle, so radius of the circle is $\frac{1.5}{2}\text{ft}$ that is 0.75 ft.

The area B of the circle is given

  $\begin{array}{c}B=\pi r^2\\ =\pi {\left(0.75\right)}^2\\ =0.5625\pi \\ \approx 0.5625\left(3.14\right)\left[Since\pi \approx 3.14\right]\\ \approx 1.77\end{array}$

Put $B\approx 1.77$ and h = 2.5 in the volume formula to find the volume.

Thus, the volume V of the cone is

  $\begin{array}{c}V=\frac{1}{3}B\cdot h\\ \approx \frac{1}{3}\left(1.77\right)2.5\\ =\frac{4.425}{3}\\ \approx 1.48\end{array}$

Therefore, the amount of ice in the cone is approximately $1.48{\text{ft}}^3$ .

Conclusion:

Therefore, the amount of ice in the cone is approximately $1.48{\text{ft}}^3$ .

(b)

To determine

To find: the time required to melt $1.48{\text{ft}}^3$ ice.

Answer to Problem 17PPS

Here $1.48{\text{ft}}^3$ cubic foot of ice melts in 74 min.

Explanation of Solution

Given:

Theice melts at a rate of 0.1 cubic foot every 5 minutes.

Calculation:

The objective is to find the time required to melt $1.48{\text{ft}}^3$ ice.

Use the given rate to find the required time.

Since 0.1 cubic foot of ice melt in 5 minutes, so 1 cubic foot of ice melt in $\frac{5}{0.1}\text{ min}$ that is 50 min

  $1.48\times 50=74\text{ min}$

Thus, $1.48{\text{ft}}^3$ cubic foot of ice melts in $74\text{ min}$

Conclusion:

Thus, $1.48{\text{ft}}^3$ cubic foot of ice melts in $74\text{ min}$