Chapter 12, Problem 15PS

a.

To determine

To graph: the parabolic arch bounded by $y=9-x^2$ and the x -axis.

Answer to Problem 15PS

Explanation of Solution

Given information: Archimedes showed that the area of a parabolic arch is equal to $\frac{2}{3}$ of

the product of the base and the height.

Calculation:

The graph of the parabolic arch $y=9-x^2$ is given below.

  

b.

To determine

To find: the area of the parabolic arch by using the limit process.

Answer to Problem 15PS

Area of the parabolic arch=36

Explanation of Solution

Given information: Archimedes showed that the area of a parabolic arch is equal to $\frac{2}{3}$ of

the product of the base and the height .

Calculation:

By the limit process the area of parabola is ${\displaystyle \underset{-3}{\overset{3}{\int }}ydx}={\displaystyle \underset{-3}{\overset{3}{\int }}(9-x^2)dx}=54-18=36.$

c.

To determine

To find: the base and height of the arch and verify Archimedes’ formula.

Answer to Problem 15PS

Base=6.

Height=9.

Area=36.

Area obtained by limit process and Archimedes’ formula are same.

Explanation of Solution

Given information: Archimedes showed that the area of a parabolic arch is equal to $\frac{2}{3}$ of

the product of the base and the height

Calculation:

By Archimedes’ formula,

  $\begin{array}{l}Area=\frac{2}{3}\times base\times height.\\ \text{here}base=3+3=6,height=9.\\ \text{Thus},Area=\frac{2}{3}\times 6\times 9=36.\end{array}$

Thus area obtained by both methods are same .