Question: A farmer has farmland that is a rectangle 2 mi long and 1.5 mi wide.
He wants to completely cover the farmland in dirt.
He knows the area each truckload of dirt covers, but only in square kilometers.
(a) Find the area of the farmland in square kilometers. Do not
round intermediate computations and round your final
answer to two decimal places. Use the table of conversion
facts, as needed.
km²
(b) The farmer wants to cover his farmland with dirt. He doesn't
have any to begin with and he can't buy partial truckloads
of dirt. Each truckload of dirt covers 2.1 km². How many
whole truckloads of dirt does the farmer need to buy to
completely cover his farmland?
truckloads
(c) If each truckload of dirt costs $60.29, how much will he
need to spend on dirt? Write your answer to the nearest
cent.
$
Conversion facts for length
1 inch (in) = 2.54 centimeters (cm)
1 foot (ft) = 30.48 centimeters (cm)
1 yard (yd) 0.91 meters (m)
1 mile (mi) 1.61 kilometers (km)
Note that means "is approximately equal to".
For this problem, treat as if it were = .
1. **State the problem:**
We have a rectangular farmland with length $2$ mi and width $1.5$ mi.
We want to find its area in square kilometers, then determine how many truckloads of dirt are needed to cover it, and finally calculate the total cost.
2. **Conversion and formula:**
Area of rectangle = length \times width.
We know $1$ mile $= 1.61$ kilometers.
3. **Calculate the area in miles:**
$$\text{Area}_{mi^2} = 2 \times 1.5 = 3 \text{ mi}^2$$
4. **Convert miles to kilometers:**
Since $1$ mi $= 1.61$ km, area in km² is:
$$\text{Area}_{km^2} = (2 \times 1.61) \times (1.5 \times 1.61)$$
5. **Calculate each side in km:**
$$2 \times 1.61 = 3.22 \text{ km}$$
$$1.5 \times 1.61 = 2.415 \text{ km}$$
6. **Calculate area in km²:**
$$\text{Area}_{km^2} = 3.22 \times 2.415 = 7.7733 \text{ km}^2$$
7. **Round to two decimal places:**
$$\boxed{7.77 \text{ km}^2}$$
---
8. **Number of truckloads needed:**
Each truckload covers $2.1$ km².
9. **Calculate truckloads:**
$$\frac{7.7733}{2.1} \approx 3.70157$$
10. **Since partial truckloads cannot be bought, round up:**
$$\boxed{4 \text{ truckloads}}$$
---
11. **Calculate total cost:**
Each truckload costs $60.29$.
12. **Total cost:**
$$4 \times 60.29 = 241.16$$
13. **Final answer:**
$$\boxed{241.16}$$
---
**Summary:**
- Area of farmland: $7.77$ km²
- Truckloads needed: $4$
- Total cost: $241.16$