Chapter 8, Problem 49RE

a.

To determine

To find: The equation of parabola with y-intercept $5$ and x-intercept $\pm 2$ .

Answer to Problem 49RE

The equation of the parabola is $y=5-\frac{5}{4}{x}^2$ .

Explanation of Solution

Given:

A bundit cake has a hole of radius of two inches and outer radius of six inches at the base and the cross sectional is parabolic.

Calculation:

The equation of the parabola with y-intercept $5$ and x-intercept $\pm 2$ is,

  $y=5-\frac{5}{4}{x}^2$ .

Therefore, the equation of the parabola is $y=5-\frac{5}{4}{x}^2$ .

b.

To determine

To find: The volume of the cake and revolve the region about an appropriate line to generate the bundt cake.

Answer to Problem 49RE

The volume is $\frac{320}{\pi }$ .

Explanation of Solution

Given:

A bundt cake has a hole of radius six inches at the base, the height is five inches and the cross sectional is parabolic

Calculation:

The volume can be calculated by shifting the parabola found in above part in right so that the x-intercept becomes $2$ and $6$ .

Now, the following integral makes use of this technique.

Thus,

  $\begin{array}{c}\underset{2}{\overset{6}{{\displaystyle \int }}}2\pi x\left(5-\frac{5}{4}{\left(x-4\right)}^2\right)\\ =\frac{320\pi }{3}\end{array}$

Therefore, the volume is $\frac{320}{\pi }$ .