Chapter 8.4, Problem 3P

Solution

i.

Write an equation of the ellipse.

Equation of ellipse is $\frac{x^2}{62500}+\frac{y^2}{40000}=1$

Given: Horizontal major axis of 250 meters and a minor axis of 200 meters.

Concept Used:

The standard equation of an ellipse whose center is at origin is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.

Calculation:

Horizontal major axis $\left(a\right)=250\text{ meters}$

A minor axis $\left(b\right)=200\text{ meters}\text{.}$

Equation of ellipse:

  $\begin{array}{l}\frac{x^2}{{250}^2}+\frac{y^2}{{200}^2}=1\\ \frac{x^2}{62500}+\frac{y^2}{40000}=1\end{array}$

Conclusion:

Equation of ellipse is $\frac{x^2}{62500}+\frac{y^2}{40000}=1$

ii.

Approximate the area of ellipse.

Area of an ellipse is 157079.633 square meters.

Given: Horizontal major axis $\left(a\right)=250\text{ meters}$

A minor axis $\left(b\right)=200\text{ meters}\text{.}$

Concept Used: Area of an ellipse is $A=\pi ab$

Calculation:

Area of an ellipse:

  $\begin{array}{l}A=\pi \times 250\times 200\\ A=157079.633{\text{ m}}^2\end{array}$

Conclusion:

Area of an ellipse is 157079.633 square meters.