Question

1. mark shoots a basketball 25 feet from the basket. the path of the ball is modeled by the equation h(x)= -0.2(x - 12)² + 35, where x is the horizontal distance in feet from mark and h(x) is the height of the ball in feet.

a) how far away from mark is the ball at its maximum height?

b) what is the height of the ball when x = 25?

c) did mark make the basket? (the basket is 10 feet above the ground)

Explanation:

Part (a)

Step1: Identify the vertex form

The equation \( h(x) = -0.2(x - 12)^2 + 35 \) is in vertex form \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex.

Step2: Determine the horizontal distance at maximum height

In the vertex form, the \( x \)-coordinate of the vertex \( h = 12 \). Since the coefficient of \((x - 12)^2\) is negative, the parabola opens downward, so the vertex is the maximum point. Thus, the horizontal distance from Mark at maximum height is \( x = 12 \) feet.

Step1: Substitute \( x = 25 \) into the equation

We have \( h(x) = -0.2(x - 12)^2 + 35 \). Substitute \( x = 25 \):
\( h(25) = -0.2(25 - 12)^2 + 35 \)

Step2: Calculate the value

First, calculate \( 25 - 12 = 13 \). Then, \( 13^2 = 169 \). Multiply by \(-0.2\): \( -0.2\times169 = -33.8 \). Add 35: \( -33.8 + 35 = 1.2 \).

Step1: Substitute \( x = 25 \) (distance to the basket) into the height equation

We already found in part (b) that when \( x = 25 \), \( h(25) = 1.2 \) feet. Wait, no, wait. Wait, the basket is at \( x = 25 \) (since Mark is 25 feet from the basket). Wait, no, in part (b) we calculated \( h(25) = 1.2 \)? Wait, that can't be right. Wait, let's recalculate part (b). Wait, \( 25 - 12 = 13 \), \( 13^2 = 169 \), \( -0.2\times169 = -33.8 \), \( -33.8 + 35 = 1.2 \). But the basket is 10 feet above the ground. Since \( 1.2 < 10 \), Mark did not make the basket? Wait, that seems odd. Wait, maybe I made a mistake. Wait, no, the equation is \( h(x) = -0.2(x - 12)^2 + 35 \). Let's check again. \( x = 25 \): \( (25 - 12) = 13 \), \( 13^2 = 169 \), \( 0.2\times169 = 33.8 \), so \( -33.8 + 35 = 1.2 \). So the height at \( x = 25 \) is 1.2 feet, which is less than 10 feet. So Mark did not make the basket.

Step2: Compare the height at \( x = 25 \) with the basket height

The basket height is 10 feet. Since \( h(25) = 1.2 \) feet, which is less than 10 feet, Mark did not make the basket.

Answer:

12 feet

Part (b)