Certainly! Let's break down the problem step-by-step:
1. Calculate the volume of dirt removed:
- Start by determining the dimensions of the section from which the dirt is removed: Length = 35 feet, Width = 67 feet, and Depth = 2 feet.
- Convert the depth from feet to yards. Since 1 yard = 3 feet:
[tex]\[
\text{Depth in yards} = \frac{2}{3} \text{ yards}
\][/tex]
- The volume of dirt removed is given by:
[tex]\[
\text{Volume (cubic yards)} = \frac{\text{Length (ft)} \times \text{Width (ft)} \times \text{Depth (yards)}}{27}
\][/tex]
Using the dimensions provided:
[tex]\[
\text{Volume cut} = \frac{35 \times 67 \times \left(\frac{2}{3}\right)}{27}
\][/tex]
Simplifying inside the parentheses first:
[tex]\[
35 \times 67 = 2345
\][/tex]
[tex]\[
2345 \times \frac{2}{3} = 1563.33
\][/tex]
[tex]\[
\text{Volume cut} = \frac{1563.33}{27} \approx 57.9 \text{ cubic yards}
\][/tex]
2. Calculate the volume required to fill the new section:
- The dimensions of the fill section are: Length = 26 feet, Width = 26 feet, Depth = 4 feet.
- Convert the depth from feet to yards:
[tex]\[
\text{Depth in yards} = \frac{4}{3} \text{ yards}
\][/tex]
- The volume of dirt needed to fill the new section is given by:
[tex]\[
\text{Volume (cubic yards)} = \frac{\text{Length (ft)} \times \text{Width (ft)} \times \text{Depth (yards)}}{27}
\][/tex]
[tex]\[
\text{Volume fill} = \frac{26 \times 26 \times \left(\frac{4}{3}\right)}{27}
\][/tex]
[tex]\[
26 \times 26 = 676
\][/tex]
[tex]\[
676 \times \frac{4}{3} \approx 902.67
\][/tex]
[tex]\[
\text{Volume fill} = \frac{902.67}{27} \approx 33.4 \text{ cubic yards}
\][/tex]
3. Calculate the remaining volume of dirt:
- Subtract the volume needed to fill the new section from the volume of dirt removed:
[tex]\[
\text{Volume left} = \text{Volume cut} - \text{Volume fill}
\][/tex]
[tex]\[
\text{Volume left} = 57.9 - 33.4 \approx 24.5 \text{ cubic yards}
\][/tex]
In conclusion, the volume of dirt left after filling the new section is approximately 24.5 cubic yards, rounded to the nearest tenth.
