Question

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Grade 8 Mathematics Formula Sheet

Check all of the statements that are TRUE about the scenario represented:

Months of growth

3

5

8

9

15

Length of hair (inches)

8

11

15.5

17

26

(3 points)
Responses

The equation of the line would be y = 3.5 x + 3/2
The equation of the line would be y = 3.5 x + 3/2

The equation of the line would be y = 3/2 x + 3 1/2
The equation of the line would be y = 3/2 x + 3 1/2

The y-intercept (0, 3.5) represents the amount of time it takes to grow 3 1/2 inches of hair.
The y-intercept (0, 3.5) represents the amount of time it takes to grow 3 1/2 inches of hair.

The slope 3/2 means that hair can grow 1.5 inches every month
The slope 3/2 means that hair can grow 1.5 inches every month

The y-intercept (0, 3.5) means that at the starting time, there was 3.5 inches of hair
The y-intercept (0, 3.5) means that at the starting time, there was 3.5 inches of hair

The slope 3/2 means that 2 inches of growth happens every 3 months.
The slope 3/2 means that 2 inches of growth happens every 3 months.
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Answer 1

Let's analyze the scenario based on the information given:

We have the following data for the months of growth and the corresponding hair lengths:

- 3 months: 8 inches
- 5 months: 11 inches
- 8 months: 15.5 inches
- 9 months: 17 inches
- 15 months: 26 inches

To explore the options, we can determine the slope and intercept of the line representing hair growth.

1. **Finding the slope:**
We can calculate the slope (m) of the line by taking the change in hair length divided by the change in months.

Using two points, for example (3, 8) and (15, 26):
- Change in hair length = 26 - 8 = 18 inches
- Change in time = 15 - 3 = 12 months

Thus, the slope \(m = \frac{18}{12} = 1.5\) inches per month, or \( \frac{3}{2} \) inches per month.

2. **Finding the y-intercept:**
The y-intercept can be found by using the slope-intercept form \(y = mx + b\). Let's use the slope \( \frac{3}{2} \) and one of the points to solve for b:
Using the point (3, 8):
\[
8 = \frac{3}{2}(3) + b \\
8 = \frac{9}{2} + b \\
b = 8 - \frac{9}{2} = \frac{16}{2} - \frac{9}{2} = \frac{7}{2}
\]
So, the y-intercept is \( (0, 3.5) \), which means that there were 3.5 inches of hair initially at \(t = 0\) months.

Now, let's evaluate the given statements:

1. **The equation of the line would be y = 3.5 x + 3/2.**
- **False.** The correct form based on our calculations should represent the relationship established, but this is not in standard slope-intercept form.

2. **The equation of the line would be y = 3/2 x + 3 1/2.**
- **True.** This matches the slope \( \frac{3}{2} \) and the y-intercept \( 3.5 \) (which is \( \frac{7}{2} \)).

3. **The y-intercept (0, 3.5) represents the amount of time it takes to grow 3 1/2 inches of hair.**
- **False.** The y-intercept represents the initial length of hair, not the time to grow it.

4. **The slope 3/2 means that hair can grow 1.5 inches every month.**
- **True.** This is derived from the slope of \( \frac{3}{2} \), meaning 1.5 inches per month.

5. **The y-intercept (0, 3.5) means that at the starting time, there was 3.5 inches of hair.**
- **True.** This correctly represents the initial hair length.

6. **The slope 3/2 means that 2 inches of growth happens every 3 months.**
- **True.** Since the slope 3/2 means that in 2 months it grows 3 inches, in approximately 3 months, it would grow about 2 inches.

**Final Summary of True Statements:**
- The equation of the line would be y = 3/2 x + 3 1/2.
- The slope 3/2 means that hair can grow 1.5 inches every month.
- The y-intercept (0, 3.5) means that at the starting time, there was 3.5 inches of hair.
- The slope 3/2 means that 2 inches of growth happens every 3 months.