Final answer:
Using the semi-major axis (58 ft) and a point on the ellipse (21,29), we solve for the semi-minor axis (b = 21 ft) and find the equation of the ellipse to be (b) (y = √(58² - 4x²/21)).
Step-by-step explanation:
The student is given the dimensions of a semi-elliptical tunnel and must find the equation that represents its shape. We know the height of the tunnel, which is the semi-major axis (a = 58 ft), and at a point 21 ft from the center, the clearance must be 29 ft. This point (21,29) lies on the ellipse and can be used to find the semi-minor axis (b).
We begin with the standard form of the equation of an ellipse with the center at the origin (0,0) and a horizontal major axis:
(x²/a²) + (y²/b²) = 1. Here we have a = 58 ft, so we need to find b such that when x = 21 ft, y = 29 ft.
Substituting the x and y values into the ellipse equation, we get:
(21²/b²) + (29²/58²) = 1. Simplifying further, we find b² = 21²/(1 - 29²/58²), which calculates to b² = 21²/21² = 1. Therefore, b = 21 ft.
The equation of the ellipse is then y = √(58² - x²/21). Thus, the correct equation is (b) (y = √(58² - 4x²/21))