Problem 28 - Calculus and its applications

Question

A highway passes by the small town of Las Cienegas. From Las Cienegas, the highway is 5 miles to the north and 3 miles to the east. Assume that the highway is straight as it passes through this region. The town wants to build an access road at a cost of \(\$ 250,000\) per mile to connect to the highway. What is the shortest possible distance (to three decimal places) from Las Cienegas to the highway, and what would be the minimum cost, to the nearest dollar, of constructing such a road?

Step-by-Step Solution

Answer

The shortest distance is 5.000 miles, and the minimum cost is \(\$1,250,000\).

Step 1: Understand the Layout

Visualize the problem in a coordinate system. Place Las Cienegas at a point (0,0), with the highway 5 miles north (y = 5 miles) and 3 miles east (x = 3 miles). Therefore, the equation of the highway is a straight line, parallel to the x-axis, represented as y = 0.

Step 2: Find the Shortest Distance from the Town to the Highway

Use the formula for the shortest distance from a point to a line. The line equation is y=0, with the point coordinates at (3,5). The distance from a point (x_1, y_1) to a line Ax + By + C = 0 is given by \( \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \). Here, A=0, B=1, and C=0 for y=0. So, the shortest distance is \( \frac{|3*0 + 1*5 + 0|}{\sqrt{0^2 + 1^2}} = 5 \) miles.

Step 3: Calculate the Minimum Cost of the Road Construction

Use the shortest distance, which is 5 miles, to find the cost. The cost of building the road is \( \\(250,000 \times \text{distance} \). Thus, \( \\)250,000 \times 5 = \$1,250,000 \).

Key Concepts

Shortest Distance

Finding the shortest distance from a point to a line is a classic problem in calculus optimization. When we have a point and need to determine its closest interaction with a straight line or surface, this method comes to the rescue. In our problem, we're trying to determine how close Las Cienegas can get to the highway. We use the formula for calculating the shortest distance from a point to a line:

• The line is given by the equation: \( y = 0 \) (since the highway runs parallel to the x-axis).
• The coordinates of Las Cienegas are at point (3,5).

To apply the distance formula, input the values into: \[ \text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]For our setup:

• \( A = 0, B = 1, C = 0 \) for line \( y = 0 \)
• The point we are considering is \((3,5)\)

This leads to:\[ \text{Distance} = \frac{|3*0 + 1*5 + 0|}{\sqrt{0^2 + 1^2}} = 5 \text{ miles} \]Thus, the shortest distance from Las Cienegas to the highway is 5 miles.

Coordinate Geometry

Coordinate geometry, or analytic geometry, allows us to use algebra to describe geometric ideas, like points, lines, and distances. In this scenario, the town of Las Cienegas, located at (3,5), uses coordinate geometry to find how to efficiently connect to a highway located at \( y = 0 \).

• The point \((3,5)\) signifies Las Cienegas' position relative to the x and y axes.
• The highway runs along \( y = 0 \) which is parallel to the x-axis.

The tool used here, the formula to calculate the distance from a point to a line, highlights how coordinate geometry simplifies complex spatial calculations. By converting the geographical placement of Las Cienegas and the highway into a system of equations, we are able to efficiently solve for the shortest connecting path.

Cost Calculation

The cost calculation in this exercise demonstrates practical application of calculus optimization beyond numerical complexity. We're utilizing distances calculated through coordinate geometry to find the financial cost of constructing a road. With the known cost per mile given:

• The fixed road construction cost is \( \\(250,000 \) per mile.
• The calculated shortest distance is 5 miles.

To determine the total construction cost, multiply the cost per mile by the total number of miles:\[ \text{Total Cost} = 250,000 \times 5 = \\)1,250,000 \]This approach applies real-world relevance to our mathematical findings, presenting a clear financial impact based on geometric calculations. It emphasizes how mathematical precision can directly influence economic decisions in real-world scenarios, such as urban planning and infrastructure development.