Chapter 9, Problem 12CT

To determine

To calculate: The point that the skier earns.

Answer to Problem 12CT

The required points earned are 55 points.

Explanation of Solution

Given information:

The function: $h=-16t^2+28t+8$

The maximum height = 1 point per foot

Time in air = 5 points per second

Perfect landing = 25 points

Initial vertical velocity = 28 feet per second

The given function is $h=-16t^2+28t+8$ .

Subtract 8 from each side of the above equation.

  $\begin{array}{c}h-8=-16t^2+28t+8-8\\ h-8-16{\left(\frac{1.75}{2}\right)}^2=-16\left(t^2-1.75t+{\left(\frac{1.75}{2}\right)}^2\right)\\ h-20.25=-16{\left(t-0.875\right)}^2\\ h=-16{\left(t-0.875\right)}^2+20.25\end{array}$

Thus, the vertex of the above equation is $\left(0.875,20.25\right)$ .

Also, the maximum height is 20.25 feet.

So, the points earned are 20 points.

Here, the total time in air can be calculated when $h=0$ .

Rewrite the equation as:

  $\begin{array}{c}-16t^2+28t+8=0\\ 4t^2-7t-2=0\end{array}$

Simplify the above equation by factoring.

  $\begin{array}{c}4t^2-7t-2=0\\ 4t^2-8t+t-2=0\\ 4t\left(t-2\right)+1\left(t-2\right)=0\\ \left(t-2\right)\left(4t+1\right)=0\end{array}$

Solve for t .

  $\begin{array}{c}4t+1=0\\ 4t=-1\\ t=-\frac{1}{4}\end{array}$

And, $\begin{array}{c}t-2=0\\ t=2\end{array}$

But time cannot be negative.

So, the total time in air is 2 seconds.

Also, the points earned are 10 points.

Since, the skier achieves a perfect landing, so the skier also earned 25 points.

Thus, the total points earned is,

  $20+10+25=55$

Hence, the required points earned are 55 points.