Problem 75 - Algebra and Trigonometry

Question

Construction Costs The town of Foxton lies 10 \(\mathrm{mi}\) north of an abandoned east-west road that runs through Grimley, as shown in the figure. The point on the abandoned road closest to Foxton is 40 \(\mathrm{mi}\) from Grimley. County officials are about to build a new road connecting the two towns. They have determined that restoring the old road would cost \(\$ 100,000\) per mile, while building a new road would cost \(\$ 200,000\) per mile. How much of the abandoned road should be used (as indicated in the figure) if the officials intend to spend exactly \(\$ 6.8\) million? Would it cost less than this amount to build a new road connecting the towns directly?

Step-by-Step Solution

Answer

Officials should restore 36 miles of the old road, and using just the new road costs more than $6.8 million.

Step 1: Identify the Variables

Let \( x \) be the distance in miles of the abandoned road that will be restored. Therefore, the remaining distance needed to connect Foxton to Grimley by a straight new road is a right triangle leg with the Pythagorean components \( 10 \ \text{mi} \) north-south and \( 40 - x \ \text{mi} \) east-west.

Step 2: Cost Equation

Formulate the cost equation. Restoring \( x \ \text{mi} \) of the abandoned road costs \( 100,000x \) dollars. Building the new road, which forms the hypotenuse of a right triangle, costs \( 200,000 \cdot \sqrt{10^2 + (40-x)^2} \) dollars.

Step 3: Total Cost Equation

The total cost is given by the sum of restoring the abandoned road and building the new road: \[ 100,000x + 200,000 \cdot \sqrt{10^2 + (40-x)^2} = 6,800,000. \]

Step 4: Simplify and Solve

Simplify and solve the equation: \[ 100,000x + 200,000 \sqrt{100 + (40 - x)^2} = 6,800,000. \]- Divide each term by 100,000 to simplify: \[ x + 2 \sqrt{100 + (40 - x)^2} = 68. \]- Isolate the square root term: \[ 2 \sqrt{100 + (40 - x)^2} = 68 - x. \]- Square both sides to eliminate the square root: \[ 4(100 + (40 - x)^2) = (68 - x)^2. \]- Solve the quadratic equation for \( x \).

Step 5: Check the Entire New Road Cost

Calculate the cost of building a new road directly from Foxton to Grimley as a straight line with distances from the right triangle: \( 10 \) north and \( 40 \) east.The straight-line distance using the Pythagorean Theorem is \( \sqrt{10^2 + 40^2} \ = \sqrt{100 + 1600} \ = \sqrt{1700} \ \text{mi}. \)The cost is \( 200,000 \times \sqrt{1700} \). Compute to check the cost.

Key Concepts

Construction Costs

When planning for infrastructure projects like roads, understanding the costs can be vital for budgeting. In scenarios like the one presented with Foxton and Grimley, different routes come with varied financial implications.
The primary cost factors:

• Restoring existing infrastructure (the abandoned road), which costs \( \\( 100,000 \) per mile.
• Constructing new infrastructure, priced at \( \\) 200,000 \) per mile.

With a fixed budget of \( \$ 6.8 \) million, officials must calculate the optimal mix of restored and new roads.
By determining the length of the road segments that need to be constructed or refurbished within this budget, they can ensure efficient resource allocation.
A detailed cost analysis helps to compare different construction options, and decide whether utilizing parts of the old road or building entirely new infrastructure is more cost-effective.

Right Triangle

In the problem, the route the new road forms with the town's position results in a right triangle. Understanding right triangles is essential here, as they allow us to calculate unknown lengths and costs.
When Foxton is 10 miles north and closest to a road running east-west, it forms the legs of a right triangle.
The east-west segment (part of the abandoned road) is \( 40 - x \) miles, where \( x \) is the length of the restored road.

• The north-south leg is 10 miles.
• The west-east leg is \( 40 - x \) miles.

Using these components, the hypotenuse, or the direct new road from Foxton to Grimley, can be calculated. Understanding the triangle's structure helps to identify the costs tied to construction routes.
With each leg easily filled into the Pythagorean Theorem, we compute the direct path's length.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental tool in solving problems involving right triangles. It lets us find the length of one side if the other two are known.
The theorem is expressed as:\[c^2 = a^2 + b^2\]In this context:

• \( a = 10 \) miles (north-south leg)
• \( b = 40 - x \) miles (remaining part of the abandoned road)
• \( c \) is the hypotenuse, the direct new road

The direct road length, \( c \), comes from solving:\[c = \sqrt{10^2 + (40-x)^2}\]This value helps calculate the total cost of building the new road, contributing to the overall cost equation.
Using the Pythagorean Theorem simplifies finding the shortest distance between points, making it easier to compute costs along different paths.