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A cable company needs to run a cable line from its main line ending at point \(P\) to point \(H\) at the corner of a house. See below. The county owns the roads marked with dotted lines in the figure, and it costs the cable company 25 per foot to run the line on poles along the county roads. The area bounded by the house and roads is a privately owned field, and the cable company must pay for an easement to run lines underground in the field. It is also more costly for the company to run lines underground than to run them on poles. The total cost to run the lines underground across the field is 52 per foot. The cable company has the choice of running the line along the roads ( 100 feet north and 125 feet west) or cutting across the field. a. Calculate the cost to run the line along the roads from P to H b. Calculate the cost to run the line directly across the field from P to H. c. Set up an equation for the cost to run the line along the road a distance of \(x\) feet from point \(P\) and then cut across the field. d. Determine whether it is less costly for the company to cut across the field. If so, at what distance from point \(P\) should the company begin laying the line through the field?

Step by step solution

01

Calculate Cost Along Roads

First, determine the total length of the cable if it runs along the roads from point \(P\) to \(H\). The path is 100 feet north and 125 feet west, resulting in a total of \(100 + 125 = 225\) feet. The cost is \(25\) per foot, so the total cost is calculated by multiplying the length by the cost per foot: \[ 225 \text{ feet} \times 25 = 5625 \text{ USD} \]

02

Calculate Cost Across Field

Next, calculate the straight-line distance from \(P\) to \(H\) across the field using the Pythagorean theorem. The north and west distances form the legs of a right triangle. Thus, the direct distance \(d\) is:\[ d = \sqrt{100^2 + 125^2} \]\[ d = \sqrt{10000 + 15625} = \sqrt{25625} \approx 160.08 \text{ feet} \]The cost of running the cable across the field is \(52\) per foot, so the total cost is:\[ 160.08 \times 52 = 8324.16 \text{ USD} \]

03

Set Up Cost Equation for Hybrid Route

Set up an equation to determine the cost of running the cable \(x\) feet along the road before cutting across the field. Along the road, the cost is \(25x\), and across the field, the remaining distance is a line segment from the endpoint on the road to point \(H\). The rest of the distance across the field can be calculated as:\[ y = \sqrt{(100-x)^2 + 125^2} \]Hence, the total cost \(C\) is:\[ C = 25x + 52y \]where:\[ y = \sqrt{(100-x)^2 + 125^2} \]

04

Determine Least Costly Option

To determine the most cost-effective method, compare the costs derived in Steps 1 and 2. Note that running directly across the field costs \(8324.16\) USD compared to \(5625\) USD along roads. Since Step 2 is not less costly, try varying \(x\) to find at what point the field route becomes the least costly. Solve for \( C < 5625 \) using the equation from Step 3. Trial and error or calculus optimization methods can flesh out the precise point \(P\) where transitioning to cutting across the field becomes optimal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pythagorean theorem

One of the fundamental concepts in this problem is the Pythagorean theorem. This theorem helps us find the straight-line distance between two points in a plane, which is known as the hypotenuse. In this exercise, the cable company needs to calculate the distance from point \(P\) to \(H\) directly across the field as if cutting through a right triangle.
To find this distance, we rely on the coordinates from the roads: moving 100 feet north and 125 feet west creates two sides of a right triangle. According to the Pythagorean theorem:

• The formula is \( a^2 + b^2 = c^2 \) where \( a \) and \( b \) are the lengths of the triangle's legs, and \( c \) is the length of the hypotenuse.
• In this problem, \(a = 100\) feet and \(b = 125\) feet. Thus, the distance \(c\) is calculated as:
  \[ c = \sqrt{100^2 + 125^2} \approx 160.08 \text{ feet} \]

Using this method, companies and engineers can effectively plan and minimize distances when spanning physical spaces.

cost analysis

Understanding cost analysis is crucial for determining the most economical option among various choices. In our exercise, two main routes for laying cable are considered: along the road and across the field. Each path has its own cost structure, allowing the cable company to make informed decisions based on financial analyses.
The cost analysis involves:

• Calculating costs for two scenarios:
  • Along the roads: Runs a distance of 225 feet (combined north and west) at $25 per foot, totaling \(5625 \text{ USD}\).
  • A direct path across the field: Runs 160.08 feet using the Pythagorean theorem's result, costing \(52\) per foot, which results in \(8324.16 \text{ USD}\).
• Identifying which plan is more cost-effective for the company.
• Evaluating hybrid cost functions, which involve partial segments of multiple methods.

This approach ensures that all variables involving distance and cost are accounted for before making strategic financial decisions.

calculus optimization

Calculus optimization is a mathematical technique used to find the maximum or minimum value of a function. In the cable installation problem, we apply this concept to minimize the costs associated with different cable pathways. Here's how it works in the optimization challenge:

• The goal is to find the point \(x\) along the road where transitioning to the field becomes the least costly option.
• We consider a hybrid cable path consisting of \(x\) feet on road (cost: \(25x\) USD) and a calculated remainder across the field.
  • The remaining field distance is \( \sqrt{(100-x)^2 + 125^2} \)
  • The total associated cost is represented by the function:
    \[ C(x) = 25x + 52 \sqrt{(100-x)^2 + 125^2} \]
• We adjust \(x\) to ascertain the minimal value of \(C(x)\) where costs are the lowest.

Through derivative calculus techniques, businesses can effectively utilize optimization to decrease expenses and improve efficiency.

hybrid cost function

A hybrid cost function arises when combining different methods and costs into a single expression. In the cable laying exercise, a hybrid cost function is crafted to account for both road and field costs.

• Here, the hybrid function integrates costs of running the cable \(x\) feet on the road and then directly across the field for the remaining distance.
  This considers different rates for each segment of the path, making it crucial to precisely express the costs related to each method.
• The cost function is:
  \[ C(x) = 25x + 52 \sqrt{(100-x)^2 + 125^2} \]
• This evaluates cost on a per-foot basis depending on which segment the total path traverses, providing adaptability to changing parameters.
  Finding the optimal \(x\) involves solving this equation so that the entire plan is financially efficient for the company.

In real-world scenarios, constructing hybrid cost functions helps achieve better financial outcomes by leveraging calculations across different terrains and costs.