Question

Given is the graph representing the area of a rectangle whose perimeter is 16. Using the graph, complete the table as shown here.

(2 points)
Width Area
0 _
4 16
7 _


What is the corresponding y-value of x=0?
What is the corresponding y-value of x=7?
Remaining Attempts : 2

Answer 1

Let's analyze the problem step by step.

Given that the perimeter of the rectangle is 16, we can find the relationship between the width \( w \) and the area \( A \).

The perimeter \( P \) of a rectangle is given by the formula:

\[
P = 2 \times (L + w)
\]

where \( L \) is the length and \( w \) is the width. For a perimeter of 16:

\[
16 = 2 \times (L + w)
\]

Dividing both sides by 2, we get:

\[
L + w = 8
\]

Thus, the length can be expressed as:

\[
L = 8 - w
\]

The area \( A \) of the rectangle is given by:

\[
A = L \times w = (8 - w) \times w = 8w - w^2
\]

Now, we can complete the table using this formula for the area.

1. **For \( w = 0 \)**:

\[
A = 8(0) - (0)^2 = 0
\]

So, when the width is 0, the Area is 0.

2. **For \( w = 4 \)** (as given in the table):

\[
A = 8(4) - (4)^2 = 32 - 16 = 16
\]

3. **For \( w = 7 \)**:

\[
A = 8(7) - (7)^2 = 56 - 49 = 7
\]

We can fill in the remaining values in the table as follows:

| Width | Area |
|-------|------|
| 0 | 0 |
| 4 | 16 |
| 7 | 7 |

So, the corresponding y-values are:

- For \( x = 0 \): the y-value (Area) is 0.
- For \( x = 7 \): the y-value (Area) is 7.