Solving a Geometry Problem to Optimize Mowing Costs
As a search engine optimization (SEO) expert, optimizing our website content to enhance its visibility and engagement is a crucial part of our role. This includes understanding and crafting informative content that addresses specific viewer needs. Here, we tackle a real-world geometry problem that involves understanding spatial reasoning and optimization, all of which can provide valuable insights for our audience.
Problem Setup
The problem at hand is as follows: A rectangular field has a width to length ratio of 3:4, and a 1-meter wide path surrounds the field. The area of this 1-meter wide path is 32 square meters. We need to find the cost of mowing the field at a rate of Rs 11 per square meter and determine if it is cost-effective to mow the field ourselves.
Formulating the Problem
Let's denote the width of the field as (3x) and the length of the field as (4x).
The area of the field is:
[ A_{field} 3x times 4x 12x^2 ]The dimensions of the entire area including the path are:
[ text{Width} 3x 2 times 1 text{ meter} 3x 2 ] [ text{Length} 4x 2 times 1 text{ meter} 4x 2 ]The area of the field including the path is:
[ A_{total} (3x 2) times (4x 2) ]The area of the path alone can be calculated by subtracting the area of the field from the total area:
[ A_{path} A_{total} - A_{field} (3x 2) times (4x 2) - 12x^2 ]Calculating and Solving the Problem
Expanding the expression for the total area:
[ A_{total} (3x 2)(4x 2) 12x^2 6x 8x 4 12x^2 14x 4 ]The area of the path:
[ A_{path} 12x^2 14x 4 - 12x^2 14x 4 ]We are given that the area of the path is 32 square meters, so we can set up the equation:
[ 14x 4 32 ]Solving for (x):
[ 14x 28 ] [ x frac{28}{14} 2 ]Dimensions of the Field
Substituting (x 2) back into the dimensions of the field:
[ text{Width of the field} 3x 3 times 2 6 text{ meters} ] [ text{Length of the field} 4x 4 times 2 8 text{ meters} ]The area of the field is:
[ A_{field} 6 times 8 48 text{ square meters} ]Mowing Costs
The cost of mowing the field at a rate of Rs 11 per square meter is:
[ text{Cost} 48 times 11 Rs , 528 ]Comparing Costs
The author mentions that one might be able to trim the field themselves on a Sunday morning and get some exercise at no cost. This suggests a personal cost comparison, where the cost of mowing professionally (Rs 528) is contrasted with the time and effort required for personal mowing. The decision could depend on the individual's time value and physical condition.
Conclusion
In conclusion, while the professional mowing costs Rs 528, it would be prudent to consider the time and effort required for personal mowing, especially for a smaller area like the backyard in Bengaluru City. This problem not only illustrates the application of geometry in real-life scenarios but also offers a practical lesson in cost-effectiveness.
Keywords:
geometry problem, mowing costs, area calculation