Questions: 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)=-44 x^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). (b) Find the height of the ball after it has traveled 8 feet in front of the foul line. The height of the ball is 17.86 ft. (Round to two decimal places as needed.) (c) Find additional points and graph the path of the basketball. A. B. C. D. (d) The center of the hoop is located 10 feet above the floor and 15 feet in front of the foul line. Will the ball go through the hoop? Yes No If not, with what initial velocity must the ball be shot in order for the ball to go through the hoop? Select the correct choice and, if necessary, fill in the answer box within your choice. A. The initial velocity must be feet per second. (Round to two decimal places as needed.) B. The ball will go through the hoop with the given initial velocity.

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)=-44 x^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). (b) Find the height of the ball after it has traveled 8 feet in front of the foul line. The height of the ball is 17.86 ft. (Round to two decimal places as needed.) (c) Find additional points and graph the path of the basketball. A. B. C. D. (d) The center of the hoop is located 10 feet above the floor and 15 feet in front of the foul line. Will the ball go through the hoop? Yes No If not, with what initial velocity must the ball be shot in order for the ball to go through the hoop? Select the correct choice and, if necessary, fill in the answer box within your choice. A. The initial velocity must be feet per second. (Round to two decimal places as needed.) B. The ball will go through the hoop with the given initial velocity.
Transcript text: 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, $h(x)=-\frac{44 x^{2}}{v^{2}}+x+6$, 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). (b) Find the height of the ball after it has traveled 8 feet in front of the foul line. The height of the ball is 17.86 ft . (Round to two decimal places as needed.) (c) Find additional points and graph the path of the basketball. A. B. C. D. (d) The center of the hoop is located 10 feet above the floor and 15 feet in front of the foul line. Will the ball go through the hoop? Yes No If not, with what initial velocity must the ball be shot in order for the ball to go through the hoop? Select the correct choice and, if necessary, fill in the answer box within your choice. $\square$ A. The initial velocity must be $\square$ feet per second. (Round to two decimal places as needed.) B. The ball will go through the hoop with the given initial velocity.
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Solution

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Solution Steps

Step 1: Understanding the Problem

The problem involves a basketball player shooting a ball with an initial velocity of 27 feet per second at a 45-degree angle from a position 6 feet above the floor. The height of the ball as a function of the forward distance is given by: \[ h(x) = \frac{-4x^2}{v^2} + x + 6 \] where \( v \) is the initial velocity, \( x \) is the forward distance, and \( h \) is the height above the floor.

Step 2: Solving Part (b)

To find the height of the ball after it has traveled 8 feet in front of the foul line, we substitute \( x = 8 \) and \( v = 27 \) into the function: \[ h(8) = \frac{-4(8)^2}{27^2} + 8 + 6 \] \[ h(8) = \frac{-4 \cdot 64}{729} + 8 + 6 \] \[ h(8) = \frac{-256}{729} + 8 + 6 \] \[ h(8) \approx -0.35 + 8 + 6 \] \[ h(8) \approx 13.65 \]

Step 3: Solving Part (c)

To find additional points and graph the path of the basketball, we need to evaluate the function at several values of \( x \). For example:

  • At \( x = 0 \): \[ h(0) = \frac{-4(0)^2}{27^2} + 0 + 6 = 6 \]
  • At \( x = 5 \): \[ h(5) = \frac{-4(5)^2}{27^2} + 5 + 6 \] \[ h(5) = \frac{-4 \cdot 25}{729} + 5 + 6 \] \[ h(5) = \frac{-100}{729} + 5 + 6 \] \[ h(5) \approx -0.14 + 5 + 6 \] \[ h(5) \approx 10.86 \]

Final Answer

  • The height of the ball after it has traveled 8 feet in front of the foul line is approximately 13.65 feet.
  • Additional points for graphing the path of the basketball include:
    • \( (0, 6) \)
    • \( (5, 10.86) \)
    • \( (8, 13.65) \)
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