Example 6

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

Answer: The formula is [latex]A=\frac{1}{2}h\left({{b}_{1}}+{{b}_{2}}\right)[/latex]where h is the height, [latex]{{b}_{1}}[/latex] and [latex]{{b}_{2}}[/latex] are the lengths of the two bases, and A is the area. Notice that the subscripts are used because both these sides are called bases, so b is a reasonable letter to use, but there are two different ones. We use subscripts on the b’s to distinguish between the two different bases.

Example 7

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

Answer: [latex]V=\frac{4}{3}\pi{{r}^{3}}=\frac{4}{3}\pi{{\left(3\right)}^{3}}=\frac{4}{3}\pi\cdot27=113.097[/latex]. So the volume of this sphere is 113.097 cubic inches.

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.

Answer: Notice that r is the input value and V is the output value.

r	V
0	0
1	4.188787
2	33.51029
3	113.0972
4	268.0823
5	523.5983
6	904.7779

Using the graph, we follow the line for 400 on the vertical axis across to the graph and then down to see the corresponding r value, which is approximately [latex]r=4.5[/latex] feet. We can check this by plugging in and finding that [latex]V=\frac{4}{3}\pi{{(4.5)}^{3}}=381.7[/latex] If we wanted to get a better estimate, we would try a somewhat higher value for r.

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.

Answer: The graph tells us that a good estimate for the radius is 4.5 feet. Next we check that by plugging in and finding that   [latex]V=\frac{4}{3}\pi{{(4.5)}^{3}}=381.7[/latex]cubic feet.   That’s a bit too low, so let’s try a slightly larger value for the radius. Let’s try 4.6 feet.   [latex]V=\frac{4}{3}\pi{{(4.6)}^{3}}=407.72[/latex] cubic feet.   That’s an adequate answer, according to the tolerance level stated in the problem.   Of course, if we wanted a more accurate answer, we could continue to work numerically by taking a value for the radius just a bit smaller than 4.6 feet and finding the volume for that.