Industrial costs A power plant sits next to a river where the river is 800 ft wide. Laying a new cable from the plant to a location in the city 2 mi downstream on the opposite side costs \$180 per foot across the river and \$100 per foot along the land.


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b. Generate a table of values to determine whether the least expensive location for point Q is less than 2000 ft or greater than 2000 ft from point P.

Verified step by step guidance

1. Convert all measurements to the same unit. Since the river is 800 ft wide and the city is 2 miles downstream, convert 2 miles to feet. (1 mile = 5280 feet, so 2 miles = 10560 feet).
2. Define the variables: Let x be the distance from the power plant to point Q along the riverbank. The remaining distance along the riverbank to the city is then (10560 - x) feet.
3. Calculate the cost of laying the cable across the river. The cost is \$180 per foot, so the cost across the river is 180 * 800.
4. Calculate the cost of laying the cable along the land. The cost is \$100 per foot, so the cost along the land is 100 * (10560 - x).
5. Create a table of values for different values of x (e.g., 1000, 1500, 2000, 2500, 3000) and calculate the total cost for each value by adding the cost across the river and the cost along the land. Compare these costs to determine if the least expensive location for point Q is less than or greater than 2000 ft from point P.

Key Concepts

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In this context, it is essential to determine the least expensive way to lay the cable, which requires setting up a cost function based on the distances involved and then using techniques such as derivatives to find the minimum cost.

Distance Calculation

Distance calculation is crucial for determining the lengths of cable needed for both the river crossing and the land route. The total distance can be expressed as a function of the variable distance from point P to point Q, which will help in formulating the cost function that needs to be minimized.

Cost Function

A cost function represents the total cost associated with a particular decision or scenario. In this problem, the cost function will combine the costs of laying cable across the river and along the land, allowing for the analysis of different scenarios to find the least expensive option for the cable layout.