Certainly! Let's solve the problem step-by-step:
### Part (a) How many pounds of topsoil are needed to cover a surface area of 600 ft²?
1. Identify the given information:
- A 50-pound bag of topsoil covers 12 square feet.
2. Set up a proportion to find out how many pounds of topsoil are needed to cover 600 square feet:
- If 50 pounds cover 12 square feet, then let [tex]\( x \)[/tex] be the number of pounds needed to cover 600 square feet.
- Write the proportion: [tex]\(\frac{50 \text{ pounds}}{12 \text{ ft}^2} = \frac{x \text{ pounds}}{600 \text{ ft}^2}\)[/tex]
3. Solve the proportion for [tex]\( x \)[/tex]:
[tex]\[
\frac{50}{12} = \frac{x}{600}
\][/tex]
4. Cross-multiply and solve for [tex]\( x \)[/tex]:
[tex]\[
50 \times 600 = 12 \times x
\][/tex]
[tex]\[
30000 = 12x
\][/tex]
[tex]\[
x = \frac{30000}{12}
\][/tex]
[tex]\[
x = 2500 \text{ pounds}
\][/tex]
So, 2500 pounds of topsoil are needed to cover an area of 600 ft².
### Part (b) How many bags of topsoil must be purchased to cover a surface area of 600 ft²?
1. Identify again the given information:
- One bag of topsoil covers 12 square feet.
2. Set up a proportion to find out how many bags are needed for 600 square feet:
- Let [tex]\( y \)[/tex] be the number of bags needed to cover 600 square feet.
- Write the proportion: [tex]\(\frac{1 \text{ bag}}{12 \text{ ft}^2} = \frac{y \text{ bags}}{600 \text{ ft}^2}\)[/tex]
3. Solve the proportion for [tex]\( y \)[/tex]:
[tex]\[
\frac{1}{12} = \frac{y}{600}
\][/tex]
4. Cross-multiply and solve for [tex]\( y \)[/tex]:
[tex]\[
1 \times 600 = 12 \times y
\][/tex]
[tex]\[
600 = 12y
\][/tex]
[tex]\[
y = \frac{600}{12}
\][/tex]
[tex]\[
y = 50 \text{ bags}
\][/tex]
So, 50 bags of topsoil are needed to cover an area of 600 ft².
### Summary:
- 2500 pounds of topsoil are needed to cover 600 ft².
- 50 bags of topsoil are required to cover 600 ft².
