Chapter 8.2, Problem 47PPS

(a)

To determine

To find:The height of the arch for $x=0,1,2,3\text{and}4$ and make a table.

Answer to Problem 47PPS

The required table for the values of the heights

x	y
0	0
1	9
2	12
3	9
4	0

Explanation of Solution

Given information:

The given equation is $y=-3x^2+12x$ .

The given values of $x=0,1,2,3\text{and}4$ .

Consider the given form of the equation $y=-3x^2+12x$ .

Substitute the value 0 for $x$ .

  $\begin{array}{c}y=-3x^2+12x\\ =-3\times 0^2+12\times 0\\ =0\end{array}$

Substitute the value 1 for $x$ .

  $\begin{array}{c}y=-3x^2+12x\\ =-3\times 1^2+12\times 1\\ =9\end{array}$

Substitute the value 2 for $x$ .

  $\begin{array}{c}y=-3x^2+12x\\ =-3\times 2^2+12\times 2\\ =-12+24\\ =12\end{array}$

Substitute the value 3 for $x$ .

  $\begin{array}{c}y=-3x^2+12x\\ =-3\times 3^2+12\times 3\\ =-27+36\\ =9\end{array}$

Substitute the value 4 for $x$ .

  $\begin{array}{c}y=-3x^2+12x\\ =-3\times 4^2+12\times 4\\ =-48+48\\ =0\end{array}$

Thus, the table for the values of the height is

x	y
0	0
1	9
2	12
3	9
4	0

(b)

To determine

To graph: The curve of the arch using the points from the table.

Explanation of Solution

Given information:

The table for the values of the height is

x	y
0	0
1	9
2	12
3	9
4	0

Graph:

From 47a, the coordinates of the curve are $\left(0,0\right),\left(1,9\right),\left(2,12\right),\left(3,9\right),\left(4,0\right)$ .

Put all the point on the coordinate plane.

Interpretation:

Thus, peak point of the graph is at $\left(2,12\right)$ and it meets the $x$ axis at $\left(0,0\right)\text{and}\left(4,0\right)$ .

(c)

To determine

To find: The maximum height of the doorway.

Answer to Problem 47PPS

The required maximum height of the doorway is 12.

Explanation of Solution

Given information:

The table for the values of the height is

x	y
0	0
1	9
2	12
3	9
4	0

From the graph in the solution 47b. the peak point is at $\left(2,12\right)$ .

Since, the $y$ coordinate represents the height of the doorway.

Hence, the maximum height of the doorway is 12.