Here are the step-by-step solutions for the problems on your worksheet.
1. Julia's Hair Growth
Goal: Find the equation for hair length ($y$) based on months passed ($x$).
*
Step 1: Identify the starting point.
The problem says when she was born (0 months), her hair was 8 cm long. This is our y-intercept ($b$). So, $b = 8$.
*
Step 2: Identify the rate of change.
Her hair grows 2 cm every month. This is our slope or rate ($m$). So, $m = 2$.
*
Step 3: Write the equation.
Using the formula $y = mx + b$:
$$y = 2x + 8$$
2. Bobby Filling a Water Tank
Goal: Create a table and an equation for the water level.
*
Step 1: Understand the rate.
The tank fills at 0.12 mL per minute.
*
Step 2: Create the Table.
We multiply the minutes by 0.12 to get the capacity.
* At 1 min: $1 \times 0.12 = 0.12$
* At 2 mins: $2 \times 0.12 = 0.24$
* At 3 mins: $3 \times 0.12 = 0.36$
* At 4 mins: $4 \times 0.12 = 0.48$
*
Step 3: Write the Equation.
Since the tank starts empty (0 mL at 0 minutes), there is no starting number added.
$$y = 0.12x$$
Table:
| Time (minutes) | Capacity (mL) |
| :--- | :--- |
| 1 | 0.12 |
| 2 | 0.24 |
| 3 | 0.36 |
| 4 | 0.48 |
3. Earth's Temperature Rise
Goal: Find the equation relating temperature ($T$) to depth ($d$).
*
Step 1: Identify the starting temperature.
At the surface (depth = 0), the temperature is 5°C. This is our starting value ($b$). So, $b = 5$.
*
Step 2: Identify the rate of change.
For every 5 meters deeper, it goes up 8°C.
Rate ($m$) = $\frac{\text{Change in Temp}}{\text{Change in Depth}} = \frac{8}{5} = 1.6$.
*
Step 3: Write the equation.
$$T = 1.6d + 5$$
4. Daniel Filling Bottles
Goal: Create a table and graph description for bottles filled over time.
*
Step 1: Understand the rate.
He fills 2 bottles every minute.
*
Step 2: Create the Table.
Multiply minutes by 2.
* 1 min $\rightarrow$ 2 bottles
* 2 mins $\rightarrow$ 4 bottles
* 3 mins $\rightarrow$ 6 bottles
* 4 mins $\rightarrow$ 8 bottles
*
Step 3: Write the Equation.
He starts with 0 bottles done.
$$y = 2x$$
*
Step 4: Graphing.
To graph this, you would plot the points from the table: $(1, 2)$, $(2, 4)$, $(3, 6)$, and $(4, 8)$. Then, draw a straight line connecting them starting from the origin $(0,0)$.
Table:
| Time (minutes) | Value (bottles) |
| :--- | :--- |
| 1 | 2 |
| 2 | 4 |
| 3 | 6 |
| 4 | 8 |
5. William Measuring Height
Goal: Find the equation for height ($h$) based on years ($t$).
*
Step 1: Identify the starting point.
When he was 5 years old, he was 5 feet tall. Let's set "age 5" as our starting time ($t=0$). So, at $t=0$, height is 5. This means $b = 5$.
*
Step 2: Identify the rate of change.
He grows 1 foot every year. So, $m = 1$.
*
Step 3: Write the equation.
$$h = 1t + 5$$ (or simply $h = t + 5$)
*(Note: In this equation, $t$ represents the number of years
after he turned 5. If $t$ represented his actual age, the equation would be different, but usually, these problems ask you to start counting from the first data point given).*
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Final Answer:
1. Equation: $y = 2x + 8$
2. Table & Equation:
Table:
(1, 0.12), (2, 0.24), (3, 0.36), (4, 0.48)
Equation: $y = 0.12x$
3. Equation: $T = 1.6d + 5$ (or $T = \frac{8}{5}d + 5$)
4. Table, Graph & Equation:
Table:
(1, 2), (2, 4), (3, 6), (4, 8)
Graph: Plot these points on a coordinate plane and connect them with a straight line starting at (0,0).
Equation: $y = 2x$
5. Equation: $h = t + 5$
*(Where $h$ is height in feet and $t$ is years after age 5)*
Parent Tip: Review the logic above to help your child master the concept of function word problems worksheet.
