A racetrack is in the shape of an ellipse, 80 feet long and 40 feet wide. What is the width 10 feet from the end?
To find the width of the racetrack 10 feet from the end, we need to determine the width of the ellipse at that point.
An ellipse has two main axes: the major axis (the longest diameter) and the minor axis (the shortest diameter). In this case, since the length of the racetrack is 80 feet and the width is 40 feet, the major axis is 80 feet and the minor axis is 40 feet.
To calculate the width of the racetrack at any given point, we need to know the equation of the ellipse. The equation of an ellipse with the center at the origin (0,0) is:
x^2/a^2 + y^2/b^2 = 1
where a is the length of the semi-major axis and b is the length of the semi-minor axis.
Since the major axis is 80 feet (a = 80/2 = 40) and the minor axis is 40 feet (b = 40/2 = 20), the equation of the ellipse becomes:
x^2/40^2 + y^2/20^2 = 1
Now, we can substitute x with the distance from the center of the ellipse to the point we are interested in (10 feet from the end). Let's call this distance "d".
Using the equation, we have:
(10)^2/40^2 + y^2/20^2 = 1
Simplifying this equation, we get:
100/1600 + y^2/400 = 1
1/16 + y^2/400 = 1
y^2/400 = 1 - 1/16
y^2/400 = 15/16
Cross-multiplying, we find:
y^2 = (15/16) * 400
y^2 = 375
Taking the square root of both sides, we have:
y = √(375)
y ≈ 19.36 feet
Therefore, the width of the racetrack 10 feet from the end is approximately 19.36 feet.