Question

The path of a diver diving from a 10 -foot high diving board is $$h=-0.44 x^{2}+2.61 x+10$$ where \(h\) is the height of the diver above water (in feet) and \(x\) is the horizontal distance (in feet) from the end of the board. How far from the end of the board will the diver enter the water?

Short answer

The diver's distance from the end of the board when she enters the water using the given condition is the positive solution of the quadratic equation.

Step-by-step solution

Formulate the equation

The equation according to the problem is \(h=-0.44x^{2}+2.61x+10\). We are asked to solve for when the diver hits the water i.e. when the height \(h\) is 0. So, the equation becomes \(0=-0.44x^{2}+2.61x+10\).

Solve the quadratic equation

Now, we need to solve for \(x\). The quadratic formula is the appropriate tool here which is \(x=[-b ± sqrt(b^{2}-4ac)] / 2a\). In the given equation \(a = -0.44, b = 2.61, c = 10\). We plug these values into the quadratic formula giving us \(x=[-2.61 ± sqrt((2.61)^{2}-4*(-0.44)*10)] / 2*(-0.44)\).

Compute the solutions

Calculating these values, we get two solutions for \(x\). The quadratic formula will always give two solutions, one with the positive square root and one with the negative square root. However, because \(x\) represents a distance, only the positive solution is relevant to this problem.