Chapter 9.2, Problem 56E

(a)

To determine

To find: The position of the tacks in the fireplace arch.

Answer to Problem 56E

The position of the tacks in the fireplace arch is $\left(\pm 2.24,0\right)$ .

Explanation of Solution

Given information:

A fireplace arch is to be constructed in the shape of a semiellipse. The opening is to have a height of $2$ feet at the center and a width of $6$ feet along the base as shown below.

  

Calculation:

As the width of the semi ellipse is $6\text{ft}.$ , So the length of the major axis is $6\text{ft}.$ .

Use the formula for the length of major axis.

  $\begin{array}{l}2a=6\\ a=3\end{array}$

So, the value of the $a$ is equal to $3\text{ft}.$ .

As the height of the semi ellipse is $2\text{ft}.$ . So, the value of $b$ is equal to $2\text{ft}.$ .

Use the formula.

  $\begin{array}{c}{c}^2=a^2-b^2\\ c^2=3^2-2^2\\ c^2=9-4\\ c=\sqrt{5}\\ c\approx 2.24\text{ft}.\end{array}$

As per the diagram the center of the ellipse is origin and the major axis is horizontal.

The points for the foci of the ellipse are:

  $\begin{array}{c}\left(h\pm c,k\right)=\left(0\pm 2.24,0\right)\\ =\left(\pm 2.24,0\right)\end{array}$

Therefore, the position of the tacks in the fireplace arch is $\left(\pm 2.24,0\right)$ .

(b)

To determine

To find: The length of the string used in the semi ellipse fireplace arch.

Answer to Problem 56E

The length of the string is $6\text{ft}\text{.}$

Explanation of Solution

Given information:

A fireplace arch is to be constructed in the shape of a semiellipse. The opening is to have a height of $2$ feet at the center and a width of $6$ feet along the base as shown below.

  

Calculation:

The formula for the length of string is $2\sqrt{b^2+c^2}$ .

As solved in part(a), substitute $2$ for $b$ and $\sqrt{5}$ for $c$ in the formula.

  $\begin{array}{c}2\sqrt{b^2+c^2}=2\sqrt{2^2+{\left(\sqrt{5}\right)}^2}\\ =2\sqrt{4+5}\\ =2\sqrt{9}\\ =2\times 3\\ =6\end{array}$

Therefore, the length of the string is $6\text{ft}\text{.}$