Pre Calculus trigonometry

Question

A company needs to connect a physical data line to a building on an island located 1000m off the shore. The hub station that needs to be connected to the island building is located 3000m away from the shortest (1000m) point to the island. The cost to run the line on land is $2/m, and the cost to run it across the water is $6/m. One option would be to run the line directly to the island in the water the whole way. Another option would be to run 3000m across the land then turn 90º and run it directly to the island 1000m in the water. Or you could choose to Run a portion across land, then make a turn of θ° and run the rest across the water. Answer all of the following:

1) What is the cost to run directly to the island?

2) What is the cost to go across the land 3000m, and then turn and go in the water?

3) What is the cost function C(θ)= that represents the cost with respect to the angle you would turn to get other possible paths/costs?

4) What are reasonable bounds for the domain (θ)?

5) What is the angle that gives the minimum cost to run the line?

6) What is the minimum cost?

7) How much cable must be used (how many total meters) to achieve the minimum cost?

8) If the owner of the island expects to make $1,500/month for access using the new line, how many months before they recoup the cost of the project?

Answer 1

1) What is the cost to run directly to the island?

The distance could be determined with Pythagorean theorem.

sqrt(3000^2+1000^2) = 3162.278 m

Multiply by $6/m.

$18973.67

2) What is the cost to go across the land 3000m, and then turn and go in the water?

3000*2+1000*6 = $12000

3) What is the cost function C(θ)= that represents the cost with respect to the angle you would turn to get other possible paths/costs?

Let (3000-x) represent the distance traveled on land, so x represents the leg adjacent to θ in a right triangle with the other leg 1000 m and the hypotenuse as the path through water. Let h represent the length of the hypotenuse (path through water).

sin(θ) = 1000/h

h = 1000/sin(θ)

tan(θ) = 1000/x

x = 1000/tan(θ)

Distance on land = 3000-1000/tan(θ)

Distance in water = 1000/sin(θ)

C(θ) = 6000-2000/tan(θ)+6000/sin(θ)

C(θ) = 2000(3-1/tan(θ)+3/sin(θ))

4) What are reasonable bounds for the domain (θ)?

The smallest turn is atan(1000/3000) = 0.32 radians = 18.4 degrees.

The largest turn is 90 degrees.

[18.4, 90]

5) What is the angle that gives the minimum cost to run the line?

I'll assume you are using a numerical solver (like a calculator) instead of calculus to find the extrema. You typically plot the function, make sure you're in degree mode, and use the maximum finding tool. I used Maxima Computer Algebra System, but the code is kinda nasty:

find_root(derivative(3-1/tan(x)+3/sin(x),x),x,atan(1/3),%pi/2)*180/%pi,numer;

70.52877936550931 degrees

6) What is the minimum cost?

Plug the angle into the cost function.

$11657

7) How much cable must be used (how many total meters) to achieve the minimum cost?

3000-1000/tan(θ)+1000/sin(θ) = 3707 meters

8) If the owner of the island expects to make $1,500/month for access using the new line, how many months before they recoup the cost of the project?

11657/1500 = 7.77

It'll take about 8 months to recoup the cost.