D1.03: Examples 6–9

Example 6

Find the area of this trapezoid, where the numbers represent feet.

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Example 7

Find the volume of a sphere with radius 3 inches. The formula is [latex]V=\frac{4}{3}\pi{{r}^{3}}[/latex].

Example 8

Graph the formula [latex]V=\frac{4}{3}\pi{{r}^{3}}[/latex] for the values of r from 0 to 6 feet and use the graph to approximate what radius will give a volume of 400 cubic feet.

Example 9

Consider the problem of Example 8. Use graphical and numerical methods to find a value for the radius that will give a volume of 400 cubic feet, correct to within 10 cubic feet.

Perhaps you have noticed that you could use algebra (involving cube roots) instead of the graph on the problem of Examples 8 and 9 to obtain a quite exact answer for the radius needed to obtain a volume of 400 cubic feet. And that’s a fine thing to do. But, in this course, we are practicing this technique that will allow you to solve problems of this kind even if the formula is so complicated that the algebra needed to “invert” the formula is harder than anything you’ve learned in your algebra courses. In fact, some formulas are so complicated that even mathematicians find these values in just the same way you’re learning here. Using graphs in this way is a very powerful and important tool in working on technical problems!