Question

math 1215 final exam

1. two brothers wish to put a new window in a building. they want the perimeter of the window to be 80 feet. the graph shows the relation between the width, x, of the opening and the area of the opening. complete parts (a) through (c) below.

(a) what is the area of the opening if the width is 10 feet?
an opening with a width of 10 feet has an area of
(1)

(b) what is the width of the opening in order for area to be a maximum? what is the maximum area of the opening?
the area is a maximum when the width of the opening is
(2)
the maximum area is
(3)

(c) identify and interpret the intercepts.
identify and interpret the leftmost intercept. select the correct choice below and fill in the answer box to complete your choice.
(type an ordered pair.)
a. the intercept
represents when the width of the opening is 0 and the height of
the opening is 0.
b. the intercept
represents when the width of the opening is 0 and both the area
and the height of the opening are 0.
c. the intercept
represents when the width of the opening is 0 and the area of the
opening is 0.
d. the intercept
represents when the height of the opening is 0 and the width of
the opening is 0.

identify and interpret the rightmost intercept. select the correct choice below and fill in the answer box to complete your choice.
(type an ordered pair.)
a. the intercept
represents when the width of the opening is 0 and the height of
the opening is 40.
b. the intercept
represents when the height of the opening is 0 and the area of
the opening is 40.
c. the intercept
represents when the width of the opening is 40 and the area of
the opening is 0.
d. the intercept
represents when the area of the opening is 40 and the width of
the opening is 0.

Explanation:

Step1: Analyze the graph for part (a)

We need to find the area when the width \( x = 10 \) feet. From the graph, we look at the \( x \)-axis (width) and find the corresponding \( y \)-value (area) when \( x = 10 \). By observing the graph, when \( x = 10 \), the area ( \( y \)-value) is 300 square feet.

Step2: Find maximum area for part (b)

The graph is a parabola opening downward (since it has a maximum point). The vertex of the parabola gives the maximum area. From the graph, the vertex is at \( (20, 400) \). So the width ( \( x \)-value) at maximum area is 20 feet, and the maximum area ( \( y \)-value) is 400 square feet.

Step3: Analyze intercepts for part (c)

Leftmost intercept:

The leftmost intercept is at \( (0, 0) \). Let's analyze the options:

• Option C: The intercept \( (0, 0) \) represents when the width of the opening is 0 and the area of the opening is 0. This makes sense because if width \( x = 0 \), area (which is width times height) will be 0.

Rightmost intercept:

Looking at the graph, the rightmost intercept is at \( (40, 0) \). Analyzing the options:

• Option C: The intercept \( (40, 0) \) represents when the width of the opening is 40 and the area of the opening is 0. This makes sense because if the area is 0, either width or height is 0. Here, when width is 40, area is 0 (so height would be 0 as well, since area = width × height).

Answer:

(a) An opening with a width of 10 feet has an area of \(\boldsymbol{300}\) square feet.
(b) The area is a maximum when the width of the opening is \(\boldsymbol{20}\) feet. The maximum area is \(\boldsymbol{400}\) square feet.
(c) Leftmost intercept: \(\boldsymbol{C}\). The intercept \(\boldsymbol{(0, 0)}\) represents when the width of the opening is 0 and the area of the opening is 0.
Rightmost intercept: \(\boldsymbol{C}\). The intercept \(\boldsymbol{(40, 0)}\) represents when the width of the opening is 40 and the area of the opening is 0.