A racetrack is in the shape of an ellipse, 100 feet long and 30 feet wide. What is the width 10 feet from a vertex?
To find the width 10 feet from a vertex of the ellipse, we can first find the distance from the center of the ellipse to the vertex on the long side.
The formula for the distance to the vertex on the long side of an ellipse is:
\[ a = \frac{1}{2} \times \text{length of the ellipse} \]
So, in this case:
\[ a = \frac{1}{2} \times 100 = 50 \text{ feet} \]
Next, we can find the width 10 feet from the vertex using the formula for the width of an ellipse at a certain distance from the center:
\[ b = \sqrt{a^2 - (a-10)^2} \]
Substitute the values:
\[ b = \sqrt{50^2 - (50-10)^2} \]
\[ b = \sqrt{2500 - 1600} \]
\[ b = \sqrt{900} = 30 \text{ feet} \]
Therefore, the width 10 feet from a vertex of the ellipse is 30 feet.