A new house is built across the river which is 212 yards downstream from the nearest telephone relay station.

Question

A new house is built across the river which is 212 yards downstream from the nearest telephone relay station. The river is 96 yards wide. In order to wire the house for phone service, wire will be laid across the river under water and along the edge of the river above ground. The cost to lay wire under water is $33 per yard and the cost to lay wire above ground is $11 per yard. How much wire should be laid ON LAND to minimize the cost?

Answer 1

The question is asking for an optimal solution to minimize costs. You can think logically and say "it's best to minimize cable under the water to minimize the cost". That would be 212 yards above the ground, right along side the river. This however would be wrong as you will see below.

Imagine, if you will, that the river is a rectangle with edges A-B-C-D (clockwise) with the width (A-B, C-D) being 96 yards and the length (A-D, B-C) is 212 yards. Say the telephone relay is at point 'B' and the house is at the opposite corner, which is 'D'. Starting from 'B', you can head towards side A-D at any angle, reaching side A-D 20%, 40%, or 80% downstream... but what is optimal? Lets do the following :

θ = angle between side A-B & the path you're taking across the river.

Max θ = arctan (212/96) = ~65.6 degrees This is standard SOH-CAH-TOA rules

X = distance downstream you reach Side A-D

Y = distance the wire is going under water

X = 96*tan(θ) This is standard SOH-CAH-TOA rules

Y = 96 / cos (θ) This is standard SOH-CAH-TOA rules

G = Distance of the wire 'On Land'

G = 212 - X = 212 - 96*tan(θ)

Cost equation :

11 * G + 33 * Y = Cost

11 * (212 - 96*tan(θ)) + 33 * (96 / cos (θ)) = Cost

2332 - (1056*tan(θ)) + (3168 / cos (θ)) = Cost

If you graph this, you'll realize that at θ = ~19.47 the cost is optimized to ~$5,318.83.

You can also solve for the angle by solving for the derivative and setting it equal to 0, which would be the angle in which the slope of the cost function is 0 and the cost is minimal. I took the cheap way out and used excel. Excel didn't give me an exact answer and it is extremely close to the exact answer. Use the derivative method to get it perfect!

Y = 96 / cos (θ) = 96 / cos (19.47) = 101.823 yards, costing $3,360.16

G = 212 - 96*tan(θ) = 212 - 96*tan(19.47) = 178.061 yards, costing $1,958.67

Total Cost is $3,360.16 + $1,958.67 = $5,318.83

Thus the answer for how many yards are above ground alongside the river is 178.061 yards...

This is cheaper than if you go straight across and down, which would cost you $5,500 total...

Answer 2

212*11=2332yds..................