Chapter 2.2, Problem 49E

a.

To determine

What is the height of the diving board.

Answer to Problem 49E

The height of the diving board is 1 meter.

Explanation of Solution

Given information: The path of a diver is modeled by the function $f(x)=-9x^2+9x+1$

Where $f(x)$ is the height of the diver in meters above the water and x is the horizontal distance (in meters) from the end of the diving board.

  

Calculation:

  $f(x)=-9x^2+9x+1\mathrm{....................}(1)$

Height of the diving board can be calculated by putting $x=0$ into the function equation (1) because $x=0$ corresponds to distance 0 fron the end of the diving board. when $x=0$ diver still did not jump.

  $f(0)=-9\times 0^2+9\times 0+1=0+0+1=1$ .

So ,the height of the diving board is 1 meter.

b.

To determine

What is the maximum height of the diving board.

Answer to Problem 49E

Maximum height of the diving board is 13/4 meter.

Explanation of Solution

Given information: The path of a diver is modeled by the function $f(x)=-9x^2+9x+1$

Where $f(x)$ is the height of the diver in meters above the water and x is the horizontal distance (in meters) from the end of the diving board.

  

Calculation:

  $f(x)=-9x^2+9x+1\mathrm{....................}(1)$

Identify the coefficients

  $a=-9,b=9,c=1.$

Since $a=-9<0$ , parabola opens down and function has a maximum value.

To find maximum value, calculate coordinates of the vertex.

  $\begin{array}{l}x=-b/2a=-9/2\times (-9)=-9/-18=1/2.\\ $ f(1/2)=(-9)\times {(1/2)}^2+9\times (1/2)+1=(-9/4)+9/2+1=13/4$\end{array}$

Maximum height of the diving board is 13/4 meter.

c.

To determine

Describe where the diver is ascending and where the diver is descending.

Answer to Problem 49E

Diver is ascending left to the $x=1/2$ and diver is descending right to the $x=1/2$ .

Explanation of Solution

Given information: The path of a diver is modeled by the function $f(x)=-9x^2+9x+1$

Where $f(x)$ is the height of the diver in meters above the water and x is the horizontal distance (in meters) from the end of the diving board.

  

Calculation:

  $f(x)=-9x^2+9x+1\mathrm{....................}(1)$

Identify the coefficients

  $a=-9,b=9,c=1.$

Since $a=-9<0$ , parabola opens down and function has a maximum value.

To find maximum value, calculate coordinates of the vertex.

  $\begin{array}{l}x=-b/2a=-9/2\times (-9)=-9/-18=1/2.\\ $ f(1/2)=(-9)\times {(1/2)}^2+9\times (1/2)+1=(-9/4)+9/2+1=13/4$\end{array}$

Since $a=-9<0$ , parabola opens down and vertex is at $x=1/2$ so, diver is ascending left to the $x=1/2$ and diver is descending right to the $x=1/2$ .