Question

A farmer wants to replace some of the fences on a farm. The wooden rail fence costs about ​$ for wood and ​$ for labor to install a fence that is 12 feet long. His son estimated he would need feet of new fence.​ However, when the farmer measured he realized he would only need new feet of fence. What is the difference in cost of the​ son's estimate versus the​ farmer's estimate with regard to how many feet of fence are​ needed?

Answer 1

Explanation:

Let's break down the cost estimation for both the son's estimate and the farmer's actual measurement.

1. Son's Estimate:

- The son estimated needing **x feet** of new fence.

- The cost for **wood** for each foot is **$x**.

- The cost for **labor** for each foot is **$y** (we don't know this value yet).

So, the son's estimated cost is **$x + $y** per foot.

2. Farmer's Actual Measurement:

- The farmer measured that he would need **z feet** of new fence.

- The cost for **wood** for each foot is still **$x**.

- The cost for **labor** for each foot is still **$y** (assuming labor costs don't change).

So, the actual cost for the farmer is **$x + $y** per foot, just like the son's estimate.

Now, let's find the difference in cost between the son's estimate and the farmer's actual measurement with regard to how many feet of fence are needed:

Son's Estimate Cost (per foot): $x + $y

Farmer's Actual Cost (per foot): $x + $y

The cost difference per foot is:

Cost Difference = Son's Estimate Cost - Farmer's Actual Cost

Cost Difference = ($x + $y) - ($x + $y)

Cost Difference = $x + $y - $x - $y

Notice that the "$x" and "$y" terms cancel each other out:

Cost Difference = ($x - $x) + ($y - $y)

Cost Difference = $0 + $0

Cost Difference = $0

So, there is **no cost difference** between the son's estimate and the farmer's actual measurement when it comes to how many feet of fence are needed. The cost per foot remains the same in both cases.