Chapter 8.3, Problem 28E

a.

To determine

To find: the time when horse reaches its maximum height.

Answer to Problem 28E

  $t=0.5\text{ sec}$

Explanation of Solution

Given information:

The height (in feet) of a horse t seconds after it jumps during a steeplechase is given by the function:

  $h\left(t\right)=-16t^2+16t$   ..... (1)

Concept Used:

Maximum and Minimum value:

The y -coordinate of the vertex of the graph of $f\left(x\right)=a{x}^2+bx+c$ is the maximum value if the function when $a<0$ or the minimum value of the function when $a>0$ .

Vertex $\left(x_0,y_0\right)$ of the function $f\left(x\right)=a{x}^2+bx+c$ I given by:

  $x_0=-\frac{b}{2a}$

And

  $y_0=f\left(x_0\right)$

Calculation:

Comparing the given function $h\left(t\right)=-16t^2+16t$ with $f\left(x\right)=a{x}^2+bx+c$ , we have that

  $a=-16$ , $b=16$ And $c=0$

First find the vertex $\left(t_0,y_0\right)$ of the given parabola:

  $\begin{array}{c}{t}_0=-\frac{b}{2a}\\ =-\frac{16}{2\left(-16\right)}\\ =\frac{1}{2}\\ =0.5\end{array}$

  $\begin{array}{c}{y}_0=h\left(t_0\right)\\ =-16{\left(0.5\right)}^2+16\left(0.5\right)\\ =-4+8\\ =4\end{array}$

So, the vertex is $\left(0.5,4\right)$

Since, here $a=-16<0$

So, the maximum height is reached by the horse at $t=0.5\text{ sec}$

b.

To determine

To find:if the horse can clear a fence that is 3.5 feet high.

Answer to Problem 28E

Yes, the fence would be cleared by 0.5 feet.

Explanation of Solution

Given information:

The height (in feet) of a horse t seconds after it jumps during a steeplechase is given by the function:

  $h\left(t\right)=-16t^2+16t$   ..... (1)

Concept Used:

Maximum and Minimum value:

The y -coordinate of the vertex of the graph of $f\left(x\right)=a{x}^2+bx+c$ is the maximum value if the function when $a<0$ or the minimum value of the function when $a>0$ .

Vertex $\left(x_0,y_0\right)$ of the function $f\left(x\right)=a{x}^2+bx+c$ I given by:

  $x_0=-\frac{b}{2a}$

And

  $y_0=f\left(x_0\right)$

Calculation:

From part (a), the vertex is $\left(0.5,4\right)$

Since, here $a=-16<0$

So, the maximum height reached by the horse is the y -coordinate of the vertex.

Thus, the maximum height reached by the horse is 4 feet.

Hence, the horse can clear a fence that is 3.5 feet high and the fence would be cleared by 0.5 feet.

c.

To determine

To find:The time for which the horse is in air.

Answer to Problem 28E

The horse is in airfor 1 second.

Explanation of Solution

Given information:

The height (in feet) of a horse t seconds after it jumps during a steeplechase is given by the function:

  $h\left(t\right)=-16t^2+16t$   ..... (1)

Calculation:

The time when horse again hits the ground is found by substituting $h\left(t\right)=0$ in equation (1),

  $\begin{array}{c}h\left(t\right)=0\\ -16t^2+16t=0\\ -16t\left(t-1\right)=0\end{array}$

By the zero product property,

  $\begin{array}{c}-16t=0\\ t=0\end{array}$

Or

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

Now, $t=0$ represents the time when the horse jumped. So, the horse again hits the ground at $t=1$ second.

Hence, the horse is in airfor 1 second.