Answer :
To solve this problem, we need to find the number of square feet in a garden that costs [tex]$3.50 a month to maintain. We know there's a linear relationship between the monthly cost and the area of the garden, and we're given two points on this line: a 100 square foot garden costs $[/tex]2.50, and a 450 square foot garden costs [tex]$9.50.
Here's how you can solve it step-by-step:
1. Identify the Two Known Points:
- The first known point is (100 square feet, $[/tex]2.50).
- The second known point is (450 square feet, [tex]$9.50).
2. Calculate the Slope (m) of the Line:
The formula for the slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the known values:
\[
m = \frac{9.50 - 2.50}{450 - 100} = \frac{7.00}{350} = 0.02
\]
3. Determine the y-intercept (b):
Using the slope and one of the points, we can find the y-intercept \(b\) using the equation of a line \(y = mx + b\). We'll use the point (100, $[/tex]2.50):
[tex]\[
2.50 = 0.02 \times 100 + b
\][/tex]
[tex]\[
2.50 = 2.00 + b
\][/tex]
[tex]\[
b = 2.50 - 2.00 = 0.50
\][/tex]
4. Formulate the Linear Equation:
Now we have the slope [tex]\(m = 0.02\)[/tex] and the y-intercept [tex]\(b = 0.50\)[/tex], so the linear equation will be:
[tex]\[
y = 0.02x + 0.50
\][/tex]
5. Find the Area for [tex]$3.50 Maintenance Cost:
Using the equation \(y = 0.02x + 0.50\), we need to find \(x\) when \(y = 3.50\):
\[
3.50 = 0.02x + 0.50
\]
\[
3.50 - 0.50 = 0.02x
\]
\[
3.00 = 0.02x
\]
\[
x = \frac{3.00}{0.02}
\]
\[
x = 150
\]
Based on these calculations, the garden must have 150 square feet to cost $[/tex]3.50 a month to maintain. Therefore, the answer is (E) 150.
Here's how you can solve it step-by-step:
1. Identify the Two Known Points:
- The first known point is (100 square feet, $[/tex]2.50).
- The second known point is (450 square feet, [tex]$9.50).
2. Calculate the Slope (m) of the Line:
The formula for the slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the known values:
\[
m = \frac{9.50 - 2.50}{450 - 100} = \frac{7.00}{350} = 0.02
\]
3. Determine the y-intercept (b):
Using the slope and one of the points, we can find the y-intercept \(b\) using the equation of a line \(y = mx + b\). We'll use the point (100, $[/tex]2.50):
[tex]\[
2.50 = 0.02 \times 100 + b
\][/tex]
[tex]\[
2.50 = 2.00 + b
\][/tex]
[tex]\[
b = 2.50 - 2.00 = 0.50
\][/tex]
4. Formulate the Linear Equation:
Now we have the slope [tex]\(m = 0.02\)[/tex] and the y-intercept [tex]\(b = 0.50\)[/tex], so the linear equation will be:
[tex]\[
y = 0.02x + 0.50
\][/tex]
5. Find the Area for [tex]$3.50 Maintenance Cost:
Using the equation \(y = 0.02x + 0.50\), we need to find \(x\) when \(y = 3.50\):
\[
3.50 = 0.02x + 0.50
\]
\[
3.50 - 0.50 = 0.02x
\]
\[
3.00 = 0.02x
\]
\[
x = \frac{3.00}{0.02}
\]
\[
x = 150
\]
Based on these calculations, the garden must have 150 square feet to cost $[/tex]3.50 a month to maintain. Therefore, the answer is (E) 150.
