Question

You are on a research boat in the ocean. You see a penguin jump out of the water. The path followed by the penguin is given by $$ h=-0.05 x^{2}+1.178 x $$ where \(h\) is the height (in feet) the penguin jumps out of the water and \(x\) is the horizontal distance (in feet) traveled by the penguin over the water. How many horizontal feet did the penguin travel over the water before reaching its maximum height?

Short answer

The penguin travelled approximately 11.78 horizontal feet before reaching its maximum height.

Step-by-step solution

Identify the values for a and b in the given equation

In the given quadratic equation \(h=-0.05 x^{2}+1.178 x\), a corresponds to -0.05 and b corresponds to 1.178.

Calculate the x-coordinate of the vertex

The x-coordinate of the vertex of a parabola given by the equation: \(ax^2 + bx + c\) is denoted by \(-b/2a\). Thus, for this problem, replace a and b with the values identified in step 1, resulting in \(-1.178 / (2*-0.05)\).

Solve for x

After performing the calculation in step 2, we get \(x \approx 11.78\).