Chapter 11.2, Problem 80AYU

Landscaping A landscape company is hired to plant trees in three new subdivisions. The company charges the developer for each tree planted, an hourly rate to plant the trees, and a fixed delivery charge. In one subdivision it took 166 labor hours to plant 250 trees for a cost of $\$7520$. In a second subdivision it took 124 labor hours to plant 200 trees for a cost of $\$5945$. In the final subdivision it took 200 labor hours to plant 300 trees for a cost of $\$8985$. Determine the cost for each tree, the hourly labor charge, and the fixed delivery charge.

To determine

To find: A landscape company is hired to plant trees in three new subdivisions. The company charges the developer for each tree planted, an hourly rate to plant the trees, and a fixed delivery charge. In one subdivision it took 166 labor hours to plant 250 trees for a cost of $\$7520$. In a second subdivision it took 124 labor hours to plant 200 trees for a cost of $\$5945$. In the final subdivision it took 200 labor hours to plant 300 trees for a cost of $\$8985$. Determine the cost for each tree, the hourly labor charge, and the fixed delivery charge.

Answer to Problem 80AYU

Solution:

The cost for each tree is $\$5$ and hourly labor charge is $\$27$.

Explanation of Solution

Given:

A landscape company is hired to plant trees in three new subdivisions. The company charges the developer for each tree planted, an hourly rate to plant the trees, and a fixed delivery charge. In one subdivision it took 166 labor hours to plant 250 trees for a cost of $\$7520$. In a second subdivision it took 124 labor hours to plant 200 trees for a cost of $\$5945$. In the final subdivision it took 200 labor hours to plant 300 trees for a cost of $\$8985$.

Calculation:

Let the hourly labor charge be $x$ and cost for each tree be $y$.

$\begin{array}{l}166x + 250y = 7520\\ 124x + 200y = 5945\\ 200x + 300y = 8985\end{array}$

We have

$\begin{array}{l}\left[\begin{array}{cc}166& 250\\ 124& 200\\ 200& 300\end{array}|\begin{array}{c}7520\\ 5945\\ 8985\end{array}\right]\\ R_3 = 2R_3\\ R_2 = 3R_2\\ \left[\begin{array}{cc}166& 250\\ 372& 600\\ 400& 600\end{array}|\begin{array}{c}7520\\ 17835\\ 17970\end{array}\right]\\ R_3 = R_3 - R_2\\ \left[\begin{array}{cc}166& 250\\ 372& 600\\ 28& 0\end{array}|\begin{array}{c}7520\\ 17835\\ 135\end{array}\right]\end{array}$

We have

$\begin{array}{l}28x = 135\\ x = 5\end{array}$

Substituting in equation we have

$\begin{array}{l}166x + 250y = 7520\\ 166\left(5\right) + 250y = 7520\\ y = 27\end{array}$

Therefore, The cost for each tree is $\$5$ and hourly labor charge is $\$27$.