We want to find the x-value of the function when y=117.
A
a
B
b
0
See solution for explanation.
C
c
54
See solution for explanation.
D
d
Approximately 4.25 feet.
Practice makes perfect
a
In order to visualize what is going on in this problem, let's start by drawing a picture of our wall.
In order to graph y=135-x^2 we will need to find points on the parabola. In the context of our problem, it does not make sense for the side length of our window to be negative, so we will only use positive x-values. Similarly, it does not make sense for the area of the wall to be negative, so we will only use positive y-values.
x
y=135-x^2
y
0
y=135-(0)^2
135
3
y=135-(3)^2
126
6
y=135-(6)^2
99
9
y=135-(9)^2
54
Let's plot these points, and then connect them to finish our parabola.
This is our graph of the function.
b
We will find a reasonable domain for our situation. As we said earlier, it wouldn't make sense for the window to have a zero or negative side length. Thus, x must be greater than zero.
y=135-x^2 where x>0
Note that the width of the wall is only 9 feet. It wouldn't make sense if our window was larger than our wall, so the window must have a side length of less than 9. Therefore, x<9.
y=135-x^2 where& x>0 and x<9
⇓
y=135-x^2 where& 0
c
The range of the function will be based off of the domain of the function.
y=135-x^2 where& 0
x
y=135-x^2
y
0
y=135-(0)^2
135
9
y=135-(9)^2
54
This tells us that the smallest y can be is 54, while the largest it can be is 135. Therefore, the range of the function is 54
d
We want to estimate x when y=117. We can do this by using the graph we made earlier. Since we are trying to find when y=117, we can draw the line y=117 on our graph, see where it intersects the function, and use it to estimate the x-value.
We can see from the graph that the function has a y-value of 117 when the x-value is about 4.25. Since x represents the side length of the window, the side length of the window is approximately 4.25 feet.
