Question
Rachel wants to put mulch around her three raised flower beds, as shown in the figure below.
Each of the three rectangular beds is 3 feet in width and 4 feet in length.
Rachel needs to fill the 13 foot by 6 foot rectangular area around the beds with mulch to a depth of 3 inches.
What volume of mulch does Rachel need to buy?
Each of the three rectangular beds is 3 feet in width and 4 feet in length.
Rachel needs to fill the 13 foot by 6 foot rectangular area around the beds with mulch to a depth of 3 inches.
What volume of mulch does Rachel need to buy?
Answers
GPT-4o mini
To find the volume of mulch Rachel needs to buy, we first determine the volume of the rectangular area surrounding the three raised flower beds.
1. **Calculate the area of the rectangular area:**
The area around the flower beds is given to be 13 feet by 6 feet.
\[
\text{Area}_{\text{outer}} = 13 \, \text{feet} \times 6 \, \text{feet} = 78 \, \text{square feet}
\]
2. **Calculate the area occupied by the three flower beds:**
Each flower bed has dimensions of 3 feet by 4 feet, and there are three of them.
\[
\text{Area}_{\text{bed}} = 3 \, \text{feet} \times 4 \, \text{feet} = 12 \, \text{square feet}
\]
Therefore, the total area occupied by the three beds is
\[
\text{Total Area}_{\text{beds}} = 3 \times 12 \, \text{square feet} = 36 \, \text{square feet}
\]
3. **Calculate the area that needs mulch:**
To find the area that requires mulch, we subtract the total area of the beds from the outer area.
\[
\text{Area}_{\text{mulch}} = \text{Area}_{\text{outer}} - \text{Total Area}_{\text{beds}} = 78 \, \text{square feet} - 36 \, \text{square feet} = 42 \, \text{square feet}
\]
4. **Convert the depth of the mulch from inches to feet:**
The depth of the mulch is given as 3 inches. To convert this into feet:
\[
\text{Depth} = \frac{3 \, \text{inches}}{12 \, \text{inches/foot}} = 0.25 \, \text{feet}
\]
5. **Calculate the volume of mulch needed:**
The volume can be calculated by multiplying the area that requires mulch by the depth of the mulch.
\[
\text{Volume}_{\text{mulch}} = \text{Area}_{\text{mulch}} \times \text{Depth} = 42 \, \text{square feet} \times 0.25 \, \text{feet} = 10.5 \, \text{cubic feet}
\]
Thus, Rachel needs to buy **10.5 cubic feet** of mulch.
1. **Calculate the area of the rectangular area:**
The area around the flower beds is given to be 13 feet by 6 feet.
\[
\text{Area}_{\text{outer}} = 13 \, \text{feet} \times 6 \, \text{feet} = 78 \, \text{square feet}
\]
2. **Calculate the area occupied by the three flower beds:**
Each flower bed has dimensions of 3 feet by 4 feet, and there are three of them.
\[
\text{Area}_{\text{bed}} = 3 \, \text{feet} \times 4 \, \text{feet} = 12 \, \text{square feet}
\]
Therefore, the total area occupied by the three beds is
\[
\text{Total Area}_{\text{beds}} = 3 \times 12 \, \text{square feet} = 36 \, \text{square feet}
\]
3. **Calculate the area that needs mulch:**
To find the area that requires mulch, we subtract the total area of the beds from the outer area.
\[
\text{Area}_{\text{mulch}} = \text{Area}_{\text{outer}} - \text{Total Area}_{\text{beds}} = 78 \, \text{square feet} - 36 \, \text{square feet} = 42 \, \text{square feet}
\]
4. **Convert the depth of the mulch from inches to feet:**
The depth of the mulch is given as 3 inches. To convert this into feet:
\[
\text{Depth} = \frac{3 \, \text{inches}}{12 \, \text{inches/foot}} = 0.25 \, \text{feet}
\]
5. **Calculate the volume of mulch needed:**
The volume can be calculated by multiplying the area that requires mulch by the depth of the mulch.
\[
\text{Volume}_{\text{mulch}} = \text{Area}_{\text{mulch}} \times \text{Depth} = 42 \, \text{square feet} \times 0.25 \, \text{feet} = 10.5 \, \text{cubic feet}
\]
Thus, Rachel needs to buy **10.5 cubic feet** of mulch.