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Chapter 10: Q 75. (page 655)

Racetrack Design: Consult the figure. A racetrack is in the

shape of an ellipse, 100feet long and 50feet wide. What is the

width 10feet from a vertex?

Short Answer

Expert verified

The track is 30feetwide at the distance of10feetfrom the vertex.

Step by step solution

01

Step. 1 Given information.

The general form of equation for the ellipse is x2a2+y2b2=1

Let the center of the racetrack be at (0,0).

Length of the racetrack is 100feet.

The distance from the center of the racetrack to each vertex is 1002=50feet

The length of the minor axis is 50feet. So, b=502=25feet

02

Step 2. To find the width 10 feet from the vertex.

Substitute a=50,b=25in equation of the ellipse we get.

The equation for the given elliptical racetrack is x2502+y2252=1

We know that 10feetfrom the vertex is equal to 50-10=40feetfrom the center. So, put x=40we get.

402502+y2252=1y2252=1-402502y2=225y=15

The width of the track is 2y=30.

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