*1-37. write an expression for the area of each rectangle below.
a.
rectangle with height 11 cm, width 3 cm
b.
rectangle with height 3x, width 11
*1-38. consider the equation ( y = \frac{1}{2}x - 4 ).
a. without graphing, where is the x-intercept of ( y = \frac{1}{2}x - 4 ).
b. make a table and graph ( y = \frac{1}{2}x - 4 ).
table grid and graph grid
c. using your graph from part (b), how can you find the x-intercept of ( y = \frac{1}{2}x - 4 ) from the table? explain.

Question

*1-37. write an expression for the area of each rectangle below.
  a.
    rectangle with height 11 cm, width 3 cm
  b.
    rectangle with height 3x, width 11
*1-38. consider the equation ( y = \frac{1}{2}x - 4 ).
  a. without graphing, where is the x-intercept of ( y = \frac{1}{2}x - 4 ).
  b. make a table and graph ( y = \frac{1}{2}x - 4 ).
    table grid and graph grid
  c. using your graph from part (b), how can you find the x-intercept of ( y = \frac{1}{2}x - 4 ) from the table? explain.

Accepted Answer

### 1-37a
# Explanation:
## Step1: Recall area of rectangle formula
The area of a rectangle is given by $ A = \text{length} \times \text{width} $.
## Step2: Identify length and width
For rectangle a, length is $ 11 \, \text{cm} $ and width is $ 3 \, \text{cm} $.
## Step3: Calculate area
Substitute into the formula: $ A = 11 \times 3 = 33 \, \text{cm}^2 $.
# Answer: $ 33 \, \text{cm}^2 $

### 1-37b
# Explanation:
## Step1: Recall area of rectangle formula
The area of a rectangle is $ A = \text{length} \times \text{width} $.
## Step2: Identify length and width
For rectangle b, length is $ 11 $ and width is $ 3x $.
## Step3: Calculate area
Multiply the length and width: $ A = 11 \times 3x = 33x $.
# Answer: $ 33x $

### 1-38a
# Explanation:
## Step1: Recall x-intercept definition
The x-intercept is where $ y = 0 $.
## Step2: Substitute $ y = 0 $ into the equation
Set $ y = 0 $ in $ y=\frac{1}{2}x - 4 $, so $ 0=\frac{1}{2}x - 4 $.
## Step3: Solve for $ x $
Add 4 to both sides: $ 4=\frac{1}{2}x $. Multiply both sides by 2: $ x = 8 $.
# Answer: $ x = 8 $ (the x-intercept is at $ (8, 0) $)

### 1-38b
# Explanation:
## Step1: Create a table of values
Choose values for $ x $, substitute into $ y=\frac{1}{2}x - 4 $ to find $ y $:
- When $ x = 0 $: $ y=\frac{1}{2}(0)-4=-4 $
- When $ x = 2 $: $ y=\frac{1}{2}(2)-4=1 - 4=-3 $
- When $ x = 4 $: $ y=\frac{1}{2}(4)-4=2 - 4=-2 $
- When $ x = 6 $: $ y=\frac{1}{2}(6)-4=3 - 4=-1 $
- When $ x = 8 $: $ y=\frac{1}{2}(8)-4=4 - 4=0 $
- When $ x = 10 $: $ y=\frac{1}{2}(10)-4=5 - 4=1 $

| $ x $ | $ 0 $ | $ 2 $ | $ 4 $ | $ 6 $ | $ 8 $ | $ 10 $ |
|---------|---------|---------|---------|---------|---------|----------|
| $ y $ | $ -4 $ | $ -3 $ | $ -2 $ | $ -1 $ | $ 0 $ | $ 1 $ |

## Step2: Graph the points
Plot the points $ (0, -4) $, $ (2, -3) $, $ (4, -2) $, $ (6, -1) $, $ (8, 0) $, $ (10, 1) $ on a coordinate plane and draw a straight line through them.
# Answer: Table as above, graph with points plotted and line drawn.

### 1-38c
# Explanation:
## Step1: From graph
The x-intercept is the point where the graph crosses the x - axis (where $ y = 0 $). On the graph from part (b), we look for the point with $ y = 0 $; this point has $ x = 8 $, so the x - intercept is $ (8, 0) $.

## Step2: From table
In the table, we look for the row where $ y = 0 $. When $ y = 0 $, the corresponding $ x $-value is 8, so the x - intercept is at $ x = 8 $ (or the point $ (8, 0) $).
# Answer: From the graph, find where the line crosses the x - axis (y = 0); from the table, find the x - value when y = 0 (both give x = 8, so the x - intercept is (8, 0)).

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QUESTION IMAGE

Question

Explanation:

1-37a

Step1: Recall area of rectangle formula

Step2: Identify length and width

Step3: Calculate area

Step1: Recall area of rectangle formula

Step2: Identify length and width

Step3: Calculate area

Step1: Recall x-intercept definition

Step2: Substitute \( y = 0 \) into the equation

Step3: Solve for \( x \)

Answer:

1-37b