Question

ricardo throws a stone off a bridge into a river below. the stones height (in meters above the water), ( x ) seconds after ricardo threw it, is modeled by ( w(x) = -5(x - 8)(x + 4) ) what is the maximum height that the stone will reach? (square) meters show calculator

Explanation:

Step1: Find the vertex's x - coordinate

For a quadratic function in factored form \(w(x)=a(x - r_1)(x - r_2)\), the x - coordinate of the vertex (which is the time at which the maximum height occurs for a downward - opening parabola, since \(a=- 5<0\)) is the midpoint of the roots \(r_1\) and \(r_2\).
The roots of the function \(w(x)=-5(x - 8)(x + 4)\) are found by setting \(w(x) = 0\):
\(x-8 = 0\) gives \(x = 8\); \(x + 4=0\) gives \(x=-4\).
The formula for the midpoint of two numbers \(x_1\) and \(x_2\) is \(x=\frac{x_1 + x_2}{2}\). So, \(x=\frac{8+( - 4)}{2}=\frac{8 - 4}{2}=\frac{4}{2}=2\).

Step2: Find the maximum height

Substitute \(x = 2\) into the function \(w(x)\) to find the maximum height.
\(w(2)=-5(2 - 8)(2 + 4)\)
First, calculate the values inside the parentheses: \(2-8=-6\) and \(2 + 4 = 6\).
Then, \(w(2)=-5\times(-6)\times6\)
Multiply the numbers: \(-5\times(-6)=30\), and \(30\times6 = 180\).

Answer:

180