Answer:
We can use the standard form of the quadratic function to model the height of the basketball, h, as a function of the horizontal distance from Allie, x, to the basketball hoop:
h(x) = ax^2 + bx + c
where a, b, and c are constants that we need to determine. We can use the given information to form a system of three equations that we can solve to find the values of a, b, and c.
First, we know that the ball is at a height of 10 ft when it is 4 ft away from Allie. Therefore, we have:
10 = a(4)^2 + b(4) + c ------(1)
Second, we know that the ball is at its highest height of 21 ft when it is 15 ft away from Allie. Therefore, we have:
21 = a(15)^2 + b(15) + c ------(2)
Third, we know that the ball is at a height of 0 ft when it reaches the basket, which is 28 ft away from Allie. Therefore, we have:
0 = a(28)^2 + b(28) + c ------(3)
Now, we can solve the system of equations (1)-(3) to find the values of a, b, and c. One way to solve the system is to use elimination or substitution to eliminate one of the variables, and then solve for the remaining variables. However, it is much easier to solve the system using matrix algebra, which involves setting up a matrix equation of the form Ax = b, where A is a matrix of coefficients, x is a vector of variables, and b is a vector of constants.
We can set up the matrix equation as follows:
⎡ 16 4 1 ⎤ ⎡ a ⎤ ⎡ 10 ⎤
⎢ 225 15 1 ⎥ ⎢ b ⎥ = ⎢ 21 ⎥
⎣ 784 28 1 ⎦ ⎣ c ⎦ ⎣ 0 ⎦
We can solve this matrix equation to find that:
a = -0.0625
b = 2.75
c = 10
Therefore, the function that models the height of the basketball as a function of the horizontal distance from Allie is:
h(x) = -0.0625x^2 + 2.75x + 10
To find how close Allie would have to be to the basketball hoop to make the basket, we need to find the value of x when h(x) = 10. We can set up the equation:
-0.0625x^2 + 2.75x + 10 = 10
Simplifying, we get:
-0.0625x^2 + 2.75x = 0
Dividing both sides by x, we get:
-0.0625x + 2.75 = 0
Solving for x, we get:
x = 44
Therefore, Allie would have to be 44 ft away from the basketball hoop to make the basket.