Question

a farmer has farmland that is a rectangle 4 km long and 2.5 km wide. he wants to completely cover the farmland in dirt. he knows the area each truckload of dirt covers, but only in square miles. (a) find the area of the farmland in square miles. do not round intermediate computations and round your final answer to two decimal places. use the table of conversion facts, as needed. (b) the farmer wants to cover his farmland with dirt. he doesnt have any to begin with and he cant buy partial truckloads of dirt. each truckload of dirt covers 0.9 mi². how many whole truckloads of dirt does the farmer need to buy to completely cover his farmland? (c) if each truckload of dirt costs $85.29, 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 =.

Explanation:

Step1: Convert length and width from km to miles

The formula for converting kilometers to miles is \(1\) mile \(\approx1.61\) kilometers.
The length \(l = 4\) km. So, in miles, \(l=\frac{4}{1.61}\) miles.
The width \(w = 2.5\) km. So, in miles, \(w=\frac{2.5}{1.61}\) miles.

Step2: Calculate the area of the rectangle

The area of a rectangle \(A=l\times w\).
Substitute \(l=\frac{4}{1.61}\) and \(w = \frac{2.5}{1.61}\) into the formula:
\(A=\frac{4\times2.5}{1.61\times1.61}=\frac{10}{2.5921}\approx3.86\) square miles.

Step3: Find the number of truckloads

Each truckload covers \(0.9\) square miles.
Number of truckloads \(n=\lceil\frac{A}{0.9}
ceil\), where \(\lceil x
ceil\) is the ceiling function (rounds up to the nearest whole number).
\(n=\lceil\frac{3.86}{0.9}
ceil=\lceil4.29
ceil = 5\).

Step4: Calculate the total cost

Each truckload costs \(\$85.29\).
Total cost \(C=5\times85.29=\$426.45\).

Answer:

(a) \(3.86\)
(b) \(5\)
(c) \(426.45\)