A utility company is asked to provide service to a customer...

A utility company is asked to provide service to a customer whose house is located in the forest, 1,000 feet away from the road along which the power lines run. The nearest transformer is located 5,000 feet down the road. The cost to install the power line is $65 per foot along the road and $100 per foot away from the road (through the forest). (a) How far from the transformer (along the road) should the company connect the power line so as to minimize installation costs? (b) What is the least it will cost to install the power line? (c) What is the most it will cost to install the power line and under what conditions?

Transcript

00:01 The company will connect the power line at some point along the road.

00:06 So let's say here okay so then the power line would run for this distance here and then it would also run from the point to the house so that would be this distance here okay, and i'm going to call this distance x so we have a right triangle here and according to the pythagorean theorem the distance of this power line will be the square root of x squared plus 1 ,000 squared and then the power line the other stretch of power line will have a distance of 5 ,000 minus x so let's find an equation for the installation cost the installation is $65 per foot along the road so let's multiply this by the number of feet along the road, which is 5 ,000 minus x and the cost is $100 per foot away from the road.

01:20 So the number of feet away from the road is the square root of x squared plus 1 ,000 squared to find where the power line should be connected to minimize cost we want to find the critical points of the cost function this is the point where the derivative of c with respect to x is 0 okay, so differentiating i get negative 65 plus 100 times 1 half times x squared plus 1 ,000 squared to the negative 1 half power times 2 x so we'll set this equal to 0 and solve for x i'm going to simplify first we get 1 half times 2 is 1 and then we have 100 x over the square root of x squared plus 1 ,000 squared i will add 65 to both sides and i will also multiply by the denominator on both sides so 65 root x squared plus 1 ,000 squared equals 100 x and i will square both sides to get 65 squared times x squared plus 1 ,000 squared equals 100 squared x squared i'll subtract 65 x squared from both sides sorry 65 squared x squared from both sides, so i'm left with 65 squared times 1 ,000 squared is equal to 100 squared minus 65 squared times x squared okay, i'm going to divide by this coefficient on both sides...

Question

Please give Ace some feedback