Chapter 8.4, Problem 1P

Solution

To calculate : The estimated area of the given ellipse by using the rectangles of width $20$ and explain that the estimate is better or worse than the given estimate.

The area of the ellipse using the rectangles of width $20$ is $58,088\text{ square meters}$ and it is better than the estimated area of the elliptical region by using the rectangles of width $40$ .

Given information :

The equation of the ellipse is $\frac{x^2}{{200}^2}+\frac{y^2}{{100}^2}=1$ and the estimated area of the elliptical region by using the rectangles of width $40$ is $68800$ square meters.

Formula used:

Area $\left(A\right)$ of an ellipse is given by,

  $A=\pi ab$

Calculation :

If the width of the rectangles is $20$ , then the number of rectangles will be,

  $\begin{array}{l}=\frac{200}{20}\\ =10\end{array}$

The height of the first rectangle will be,

  $y_1=100$ .

Calculate the height of other nine rectangles using the formula $y=\sqrt{{100}^2-\frac{x^2}{4}}$ .

  $\begin{array}{l}{y}_2=\sqrt{{100}^2-\frac{{20}^2}{4}}=99.5\\ y_3=\sqrt{{100}^2-\frac{{40}^2}{4}}=98\\ y_4=\sqrt{{100}^2-\frac{{60}^2}{4}}=95.4\\ y_5=\sqrt{{100}^2-\frac{{80}^2}{4}}=91.6\\ y_6=\sqrt{{100}^2-\frac{{100}^2}{4}}=86.6\\ y_7=\sqrt{{100}^2-\frac{{120}^2}{4}}=80\\ y_8=\sqrt{{100}^2-\frac{{140}^2}{4}}=71.4\\ y_9=\sqrt{{100}^2-\frac{{160}^2}{4}}=60\\ y_{10}=\sqrt{{100}^2-\frac{{180}^2}{4}}=43.6\end{array}$

Calculate the total area $\left(A\right)$ of the rectangles.

  $\begin{array}{c}A=20\left(99.5+98+95.4+91.6+86.6+80+71.4+60+43.6\right)\\ =14,522\end{array}$

So, the area of the portion of ellipse in first quadrant is $14,522\text{ square meters}\text{.}$

The total area of the ellipse will be,

  $\begin{array}{l}=4\times 14,522\\ =58,088\text{ square meters}\text{.}\end{array}$

The area of the ellipse using formula is,

  $\begin{array}{l}=3.14\times 200\times 100\\ =62800\text{ square meters}\end{array}$

Therefore, the area of the ellipse using the rectangles of width $20$ is $58,088\text{ square meters}$ and it is better than the estimated area of the elliptical region by using the rectangles of width $40$ .