Chapter 8.5, Problem 86E

(a)

To determine

To analyse: The player will make the shot or not.

Answer to Problem 86E

Yes, the player will make the shot

Explanation of Solution

Given information:

The professional basketball player’s shot is modelled by the function. The value of x and y is measured in feet.

Formula used:

Substitution method is used

Calculation:

The player will make the shot if the point (3, 0) is on the parabola (a solution). Substitute the point & evaluate:

  $\begin{array}{l}0=-\frac{1}{20}\left(3^2-19\left(3\right)+48\right)\\ 0=-\frac{1}{20}\left(9-57+48\right)\\ 0=0\end{array}$

Yes, the player will make the shot

Conclusion:

Yes, the player will make the shot

(b)

To determine

To find: the quadratic function in the standard form that models the path of the shot.

Answer to Problem 86E

The function is:

  $f\left(x\right)=-\frac{1}{15}\left(x^2-16x+39\right)$

Explanation of Solution

Formula used:

The intercept form:

  $y=a\left(x-p\right)\left(x-q\right)$

Calculation:

Two x-intercepts are given: (3, 0) & (13, 0)

The quadratic equation, use the intercept form:

  $y=a\left(x-p\right)\left(x-q\right)$

Substitute the intercepts & simplify:

  $y=a\left(x-3\right)\left(x-13\right)$

The other point, (10, 1.4), to find the value of a:

  $\begin{array}{l}1.4=a\left(10-3\right)\left(10-13\right)\\ 1.4=-21a\\ a=-\frac{1}{15}\end{array}$

Therefore, the function is:

  $f\left(x\right)=-\frac{1}{15}\left(x-3\right)\left(x-13\right)$

Simplify,

  $f\left(x\right)=-\frac{1}{15}\left(x^2-16x+39\right)$

Conclusion:

The function is:

  $f\left(x\right)=-\frac{1}{15}\left(x^2-16x+39\right)$