To predict how much money Earl's grandmother will give him when he turns 18, we can use linear regression. This method helps us find the best-fitting line through the given data points, which will enable us to estimate future values.
Here is a detailed solution:
### Step 1: Organize the given data
We have three data points:
- Age 5: \[tex]$15
- Age 8: \$[/tex]24
- Age 10: \[tex]$30
### Step 2: Define the linear relationship
Let’s assume the relationship between age (\(x\)) and money (\(y\)) can be expressed with a linear equation:
\[ y = mx + b \]
where \(m\) is the slope and \(b\) is the y-intercept.
### Step 3: Calculate the slope (\(m\)) and the y-intercept (\(b\))
Using the method of least squares for linear regression, we calculate the slope and y-intercept based on the given data:
- Slope (\(m\)) ≈ 3.0
- Intercept (\(b\)) ≈ 0
### Step 4: Form the linear equation
The linear equation representing the relationship between Earl's age and his birthday money is approximately:
\[ y = 3x + 0 \]
Thus, simplifying this, we get:
\[ y = 3x \]
### Step 5: Predict the future value
Next, we predict the birthday money Earl will receive at age 18 using the derived linear equation:
\[
y = 3(18) = 54
\]
### Step 6: Compare the calculated value with the options given
The closest option to the calculated value of \$[/tex]54 is:
[tex]\[ \$54 \][/tex]
### Conclusion
Earl should receive \$54 from his grandmother when he turns 18.
So, the answer is:
[tex]\[ \boxed{54} \][/tex]
