football trophy question

Football Trophy Question

# 24 with football and hyperbola trophy

QUESTION :

Find the total height of the trophy pictured if the sides are a hyperbola with equation

and the football is an ellipse 8 cm wide by 16 cm. high.
Point P is on both the ellipse and the hyperbola.

SOLUTION :

Step 1: set x = 10 to find OM.

Step 2: set x = 3.5 (7/2) to find the coordinates of P, (point on both ellipse and hyperbola).

Step 3: Find the y-coordinate of C (0, B), the center of the ellipse.
we know that a = 8 and b = 4 for the ellipse, and since it's vertical,

the equation is: .

We know that point P is on the curve.

We solve for B (the y-value at C) using the coordinates of P.

OC = 12.49

The total height of the trophy = OM + OC + CQ = 29.39 + 12.49 + 8 = 49.88 cm.