Question

if a basketball player shoots a foul shot, releasing the ball at a 45 - degree angle from a position 6 feet above the floor, then the path of the ball can be modeled by the function, ( h(x)=-\frac{44x^{2}}{v^{2}}+x + 6 ), where ( h ) is the height of the ball above the floor, ( x ) is the forward distance of the ball in front of the foul line, and ( v ) is the initial velocity with which the ball is shot in feet per second. suppose a player shoots a ball with an initial velocity of 27 feet per second. answer parts (a)-(d).
(a) find the height of the ball after it has traveled 4 feet in front of the foul line.
the height of the ball is 9.03 ft. (round to two decimal places as needed.)
(b) find the height of the ball after it has traveled 9 feet in front of the foul line.
the height of the ball is ( square ) ft. (round to two decimal places as needed.)

Explanation:

Step1: Substitute values for part (a)

Given \(h(x)=-\frac{44x^{2}}{v^{2}}+x + 6\), \(v = 27\) and \(x = 4\).
Substitute into the formula: \(h(4)=-\frac{44\times4^{2}}{27^{2}}+4 + 6\).
First calculate \(\frac{44\times16}{729}=\frac{704}{729}\approx0.966\).
Then \(h(4)=- 0.966+4 + 6\).

Step2: Calculate the value for part (a)

\(h(4)=-0.966 + 10=9.034\approx9.03\) (rounded to two decimal places).

Step3: Substitute values for part (b)

For part (b), \(v = 27\) and \(x = 9\).
Substitute into the formula: \(h(9)=-\frac{44\times9^{2}}{27^{2}}+9 + 6\).
Calculate \(\frac{44\times81}{729}=\frac{3564}{729}=4.89\).
Then \(h(9)=-4.89+9 + 6\).

Step4: Calculate the value for part (b)

\(h(9)=-4.89+15 = 10.11\).

Answer:

(a) \(9.03\) ft. (b) \(10.11\) ft.