What is the maximum height of the football and how far from the point of impact does this occur? Express answer exact to two decimal places.

Question

When a football is kicked, its height, h(x) in feet, can be modeled by h(x)=-0.01x²+1.18x+2, where x is the ball’s horizontal distance, in feet, from the point of impact with the kicker’s foot.

a) What is the maximum height of the football and how far from the point of impact does this occur? Express answer exact to two decimal places.

b) Assuming the ball is not blocked by another player, how far down the field will it go before hitting the ground? Round to one decimal place.

c) State the y-intercept and describe in 1-2 sentences how it relates to this scenario.

Answer 1

h(x) = -0.01x² + 1.18x + 2 is an inverted parabola so the max height will occur at the vertex. To find the x-value of the vertex of a parabola, we can use x = -b/(2a) for f(x) = ax² + bx + c.

x = -b/(2a)

b = 1.18 and a = -0.01 so x = -1.18/(2(-0.01)) = 1.18/.02 = 59.00

x = 59.00 feet is then the answer to the second question of how far from the kicker's foot does the maximum height occur.

To find the maximum height, plug in x = 59:

h(59) = -0.01(59)² + 1.18(59) + 2 = 36.81 feet high

To answer the question about how far the ball goes before hitting the ground, we would set the height equal to zero, and solve for "x":

0 = -0.01x² + 1.18x + 2

Use the quadratic equation:

x = [-b ± √(b² - 4ac)]/(2a) where a = -0.01, b = 1.18, and c = 2:

x = [-1.18 ± √(1.18² - 4(-0.01)(2))]/(2(-0.01))

x = [-1.18 ± √(1.3924 + 0.08)]/(-0.02)

x = [-1.18 ± √1.4724]/(-0.02)

x = [-1.18 ± 1.2134]/(-0.02)

So x = [-1.18 + 1.2134]/(-0.02) or [-1.18 - 1.2134]/(-0.02)

x = -1.67 or x = 119.67

We would ignore the negative answer and go with x = 119.67 feet

Regarding the y-intercept, the y-intercept occurs when x = 0

h(0) = -0.01(0)² + 1.18(0) + 2 = 2 meaning that the height of the ball is initially at 2 feet when the distance away from the kickers foot is zero. This would be the height the ball when the kicker kicks it.