Math help. Trig i think??

Question

You would like to fence in your irregularly shaped yard. After spending countless hours on Angie’s List and Facebook asking for reputable fencing companies, you have found two that you would trust:

1. Mike’s Fencing Emporium – Mike’s company charges $11 per foot of fence.
2. Brad’s Bodacious Boundaries – Brad’s company charges $6500 for the first 550 feet of fence and only $7 per foot for every foot beyond 550 feet.

Your goal in this assignment is to determine which company you should hire.

Your property lines can be determined by connecting five trees. The location of these trees are as follows, measured relative to the center of your house:

1. Tree 1: 100 feet east of your house
2. Tree 2: 80 feet east and 70 feet south of your house
3. Tree 3: 40 feet west and 110 feet south of your house
4. Tree 4: 140 feet west and 40 feet north of your house
5. Tree 5: 20 feet east and 100 feet north of your house

What is the distance from Tree 1 to Tree 2? (Round your answer to one decimal place.)  feet

Hint: Draw a line straight up from Tree 2. Draw a line to the left from Tree 1. This should form a right triangle. Use this reasoning for all distances between trees.

What is the distance from Tree 2 to Tree 3?(Round your answer to one decimal place.)  feet 

What is the distance from Tree 3 to Tree 4?(Round your answer to one decimal place.)  feet 

What is the distance from Tree 4 to Tree 5?(Round your answer to one decimal place.)  feet 

What is the distance from Tree 5 to Tree 1?(Round your answer to one decimal place.)  feet 

What is the total distance that needs to be fenced in? (Note, you should round UP no matter what. Even 12.2 feet would round up to 13 feet, since you need to purchase that 13th foot of fencing)  feet

Remember, Mike's Fencing Emporium charges $11 per foot of fencing. How much money would Mike's Fencing Emporium charge to fence your property? $

Remember, Brad's Bodacious Boundaries charges $6500 for the first 550 feet of fence, then $7 per foot for every foot beyond 550. How much money would Brad's Bodacious Boundaries charge to fence your property? $

Which company is the better deal for you, as the homeowner? 

1. Brad's Bodacious Boundaries
2. Mike's Fencing Emporium

Answer 1

I won't do the whole problem, just the first length of fencing. If you grab a sheet of graph paper and a ruler, this problem becomes much easier to work with.

Let's say North is a positive direction and East is a positive direction. South and West are negative.

For the distance between Tree 1 and Tree 2, you can write their points like this: (100ft, 0) and (80ft, -70ft)

If you draw a line from Tree 2 that goes 20 feet east (so it lines up with Tree 1) and then draw a line up from the one you just drew that goes 70 feet up (so it touches Tree 1), you have now drawn a right triangle. You can use the Pythagorean theorem to solve for the hypotenuse:

a² + b² = c²

a will be equal to 20 feet and b will be equal to 70 feet. If you solve for c, the hypotenuse, that is the distance between tree 1 and 2. It will be equal to how much fence you need to buy.

Solving, we get c = 72.8 ≈ 73 feet

If you do the same procedure of drawing right triangles that touch each pair of trees and keep solving for their hypotenuses, you will find the total amount of fence needed. You can then do the math for both Brad and Mike to find the total costs.

Answer 2

I only want to piggy back off of what was already mentioned here in the previous solution, just in case a bit more explanation was needed.

As the previous tutor said, start by putting everything on graph paper. Your house is the origin and the tree's can be written as coordinate points.

Recall that the distance formula between two points is given by the formula:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Thus, if tree 1 has the coordinates (100, 0) and tree 2 has the coordinates (80, -70) then the distance between the two tree's would be

d = √[(80 - 100)² + (-70 - 0)²] = √(400 + 4900) = √5300 ≈ 72.8 ≈ 73 (Remember to round up)

Now you are going to do this with all of the remaining trees: Tree 3 is at (-40, -110), etc.

Find the distance between tree 2 and tree 3 using the coordinates (80, -70) and (-40, -110), until you work up to the distance between tree 5 and tree 1.

Next take all of your distances, and add them up (rounded to the nearest whole number of course). Just for the sake of example, let's just pretend that the sum came out to be 1200 feet. Then

Mike's company would charge:

11(1200) = $13,200

Whereas Brad's company would charge:

$6500 + $7(1200 - 550) <---------Remember the first 550 feet is a flat rate. After that, it's $7 per foot.

6500 + 7(650)

6500 + 4550

$11,050

Therefore, in this made up example of 1200 feet, Brad's company would be cheaper. But in your problem, it could be that Mike's might be cheaper. This is what you have to figure out. Find the rest of the distances, then add them up. Don't forget to round up to the nearest whole number. Then calculate the cost of each.

Hope this helps

Mr. K