Chapter 14.3, Problem 58E

Glacier

Horizontal cross sections of a piece of ice that broke from a glacier are in the shape of a quarter of a circle with a radius of approximately 50 feet. The base is divided into 20 subregions, as shown in the figure. At the center of each subregion, the height of the ice is measured, yielding the following points in cylindrical coordinates.

$\begin{array}{l}\left(5,\frac{\pi }{16},7\right),\left(15,\frac{\pi }{16},8\right),\left(25,\frac{\pi }{16},10\right),\left(35,\frac{\pi }{16},12\right),\left(45,\frac{\pi }{16},9\right),\\ \left(5,\frac{3\pi }{16},9\right),\left(15,\frac{3\pi }{16},10\right),\left(25,\frac{3\pi }{16},14\right),\left(35,\frac{3\pi }{16},15\right),\left(45,\frac{3\pi }{16},10\right),\\ \left(5,\frac{5\pi }{16},9\right),\left(15,\frac{5\pi }{16},11\right),\left(25,\frac{5\pi }{16},15\right),\left(35,\frac{5\pi }{16},18\right),\left(45,\frac{5\pi }{16},14\right),\\ \left(5,\frac{7\pi }{16},5\right),\left(15,\frac{7\pi }{16},8\right),\left(25,\frac{7\pi }{16},11\right),\left(35,\frac{7\pi }{16},16\right),\left(45,\frac{7\pi }{16},12\right)\end{array}$

(a) Approximate the volume of the piece of ice.

(b) Ice weighs approximately 57 pounds per cubic foot. Approximate the weight of the piece of ice.

(e) There are 7.48 gallons of water per cubic foot.

Approximate the number of gallons of water in the piece of ice.