Question

A radio station has a broadcast area in the shape of a circle with equation x^2 + y^2 = 5, 625, where the constant represents square miles.a. Find the intercepts of the graph.b. State the radius in miles.c. What is the area of the region in which the broadcast from the station can be picked up?show each step

Answer 1

SOLUTION:

Given: Equation of a circle describing a radio station broadcast area.

$\begin{gathered} x^2+y^2=5625 \\ \text{Comparing with the general equation of a circle:} \\ (x-a)^2+(y-b)^2=r^2 \\ \text{Where (a,b) represents the centre of the circle} \\ r\text{ represents the radius of the circle} \\ \text{Therefore,} \\ The\text{ centre is the origin (0,0)} \\ \text{radius,} \\ r^2=\text{ 5625} \\ \text{Square}-\text{root both sides} \\ \sqrt[]{r^2}=\text{ }\sqrt[]{5625} \\ r=\text{ 75 miles} \end{gathered}$

To find:

A) Intercepts; x-intercept, y-intercept

$\begin{gathered} x-\text{intercept} \\ \text{the value of x when y=0} \\ x^2_{}+y^2=5625 \\ x^2=5625 \\ \text{square}-\text{root both sides} \\ \sqrt[]{x^2}=\text{ }\sqrt[]{5625} \\ x=\text{ 75 miles} \end{gathered}$$\begin{gathered} y-\text{intercept} \\ \text{the value of y when x=0} \\ x^2_{}+y^2=5625 \\ y^2=5625 \\ \text{square}-\text{root both sides} \\ \sqrt[]{y^2}=\text{ }\sqrt[]{5625} \\ y=\text{ 75 miles} \end{gathered}$

B) radius

$\begin{gathered} r^2=\text{ 5625} \\ \text{Square}-\text{root both sides} \\ \sqrt[]{r^2}=\text{ }\sqrt[]{5625} \\ r=\text{ 75 miles} \end{gathered}$

C) Area of region

The formula for area of a circle is given as:

$\begin{gathered} A=\text{ }\pi\times r^2 \\ A=\text{ }\frac{22}{7}\times75^2 \\ A=\frac{22}{7}\times5625 \\ A=\text{ 17678.57 sq miles (2 d.p)} \end{gathered}$