Calculating the Time to Paint a Wooden Fence - A Mathematical Approach

When faced with the task of estimating the time required to paint a fence, especially when the length of the fence varies, understanding the relationship between the length of the fence and the painting time can be quite helpful. This guide will walk you through the process of calculating the painting time for a fence of a different length based on the data given.

Problem Statement

Given that a fence of 434 feet takes 2 hours and 40 minutes to paint, we need to determine how long it would take to paint a fence that is 912 feet long. To solve this, we will use a simple mathematical approach.

Step-by-Step Solution

Step 1: Convert Time to Minutes

First, we need to convert the given painting time into minutes for easier calculation. We know that 2 hours and 40 minutes equals 160 minutes.

Step 2: Establish the Proportion

The key to solving this problem is recognizing that the painting time is directly proportional to the length of the fence. Therefore, the time taken to paint 912 feet of fence can be calculated as:

[ text{Time} frac{text{Required Length}}{text{Standard Length}} times text{Time for Standard Length} ]

Step 3: Apply the Proportion

Substituting the given values into the formula:

[ text{Time} frac{912}{434} times 160 ]

Simplifying this, we get:

[ text{Time} 2.100462963 times 160 336.0736736 text{ minutes} ]

Step 4: Convert Minutes to Hours and Minutes

To convert the time back into hours and minutes, we divide 336.0736736 by 60:

[ frac{336.0736736}{60} 5.601227893 text{ hours} ]

This is approximately 5 hours and 36.1218135 minutes, or more precisely, 5 hours 36 minutes and 7.089 seconds.

Verification

To verify our solution, let's consider the following: [ 434 times 2 868 text{ feet} ] [ 434 times 3 1302 text{ feet} ] [ 912 text{ feet is more than twice but less than thrice the length of 434 feet} ]

Therefore, the time to paint 912 feet should be more than 5 hours and 20 minutes but less than 8 hours, which aligns with our calculated result of 5 hours and 36 minutes and 7.089 seconds.

Conclusion

By applying the principles of proportionality, we can accurately estimate the time required to paint a fence of any given length. This method is useful for both practical purposes and educational activities, as it reinforces the concept of direct proportion and mathematical reasoning.