Problem

A practice ski jump hill follows the shape of a curve with the formula given below.

The hill is flat at the top and at the end of the jump, where the skier leaves the hill and jumps into the air.

Come up with a formula for the angle the skier's skis make with the horizontal, and find how far from the top of the jump he is when this angle is the greatest.

Solution

The slope of the tangent line to the curve will let us find the angle that line makes with the horizontal. Let a typical point be called x₀. Then the slope of the tangent line to the curve at x = x₀ is found by the formula

Therefore, at any point x₀, the slope of the tangent line to the curve at the point (x₀, j(x₀)) is

The angle the skis make with respect to the horizontal is 0 for x₀ = 0 or x₀ = 90, the beginning and end of the hill. The angle is positive (the tangent of the angle is the absolute value of the above expression for the slope of the tangent line) in between. To determine when the angle is the greatest, graph the formula for the angle, which is a parabola, and identify the vertex.

The vertex of the parabola is (45, 1) so the angle is at a maximum when x₀ = 45.