precalc hw

Question

The lengths of the sides of a triangular parcel of land are approximately 500 feet, 600 feet, and 950 feet. If land is valued at $2000 per acre (1 acre is 43,560 ft2), what is the value of the parcel of land? (Round your answer to the nearest cent.)

Answer 1

Using the Cosine Law, we will calculate the angle between the 950 feet and the 500 feet sides, we will call this angle a:

 

600^2= 950^2 +500^2 - 2(950x500)cos a

 

2(950x500)cos a= 950^2+ 500^2 - 600^2

 

2cos a= (950^2 +500^2 - 600^2)/(950x500)

 

cos a= 1.6684/2=0.8342

 

     a= 33.47 degrees

 

Now that we have this angle, we may calculate the height of this triangle, which we will call h using the sine of a

definition:

 

sin a=h/500

 

h=500 x sin a

h= 500 x sin 33.47= 500x0.55=275.8 ft

 

Then, we calculate the area of this triangle using this formula:   

 

A=(950xh)/2

 

A=(950x 275.8)/2

A= 131,005 ft2

 

To determine how many acres are in that area:

 

A=131,005/43,560= 3.007 acres

 

Therefore, at $2,000 per acre, the land value would be:  3.007x2000=$6,014.92 approx.

     

 

 

 

 

 

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Answer 2

Here are some useful steps to help you find this answer

 

1.  You need to find the length of an altitude (height) of the triangle so you can find the area.  Let 500=a, 600=b, and 950=c.  The angles across from them are A, B, and C, respectively. 

 

Draw the triangle so that c (I chose c randomly) is horizontal and angle C is the top angle of the triangle.  put angle A and B wherever youd like.

 

2.  Use the law of cosines to find either angle A or B.  To find A (again, randomly picking A), use 

a^2=b^2+c^2 - 2bc * cos A

 

500^2 = 600^2+950^2 - 2(500)(600) * cos A

 

make sure you subtract the 600^2 and 950^2 before dividing by the 2(500)(600)

 

finally use inverse cosine to get the angle. 

 

3.  Draw the altitude(height) from angle C to side c.  You now have a right triangle where you know angle A and side b (hypotenuse).  Use SOH-cah-toa to find the length of the altitude you just drew in. 

 

4.  The altitude is the height of the triangle.  From your triangle, the base is 950. Use Area=.5bh to find the area of the triangle, in square feet.

 

5.  Use a proportion to calculate how many acres this is. 

 

1/43560 = x/(your area)

 

6.  Take $2000 times x to get the cost per acre.

 

good luck and please feel free to contact me with your solution and any questions.