SOLUTION: The feild next to jj smith high school is in the shape of a parallelogram. Two consecutive sides of the field measure 340 feet and 450 feet. a) if the largest angle of the parall

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Question 84705: The feild next to jj smith high school is in the shape of a parallelogram. Two consecutive sides of the field measure 340 feet and 450 feet.
a) if the largest angle of the parallelogram measures 112.3(degrees), to the nearst tenth of a foot, what is the longest distance from one coner of the field to the opposite corner.
b)the school wants to replace the field with blacktop. If the blacktop company charges $4.29 per square foot, how much would it cost to install the blacktop?
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!

The field next to jj smith high school is in the shape of a parallelogram. Two consecutive sides of the field measure 340 feet and 450 feet.
a) if the largest angle of the parallelogram measures 112.3(degrees), to the nearst tenth of a foot, what is the longest distance from one coner of the field to the opposite corner.


Let's draw the field and its longer diagonal:



Use the law of cosines:

c² = a² + b² - 2ab·cos(C)

c² = 340² + 450² - 2(340)(450)cos(112.3°)

c² = 434213.5848
     ___________
c = Φ434213.5848

c = 658.9488484 feet

We should round that off to the nearest 10 feet
since the given sides seem to be rounded to the
nearest 10 feet.

c = 660 feet

b)the school wants to replace the field with blacktop. If the blacktop company charges $4.29 per square foot, how much would it cost to install the blacktop?

To do this we must find the area.  The area of a parallelogram is found
by the formula

A = base Χ height

We know the base is 450 feet but we are not given the height, h, so we
redraw the figure:




Now we know that two adjacent angles of a parallelogram are
supplementary, therefore the angle at the lower right corner
is 180° - 112.3° = 67.7°



Now we draw in the height, h:



In the right triangle on the right, h is the side
opposite the 67.7° angle and the hypotenuse is the
side which is 340 feet.  Since the sine is the
opposite side divided by the hypotenuse, we have

               h
sin(67.7°) = —————
              340

let's put 1 under the left side:


 sin(67.7°)      h
———————————— = —————
     1          340

Now we can cross-multiply and get

h = 340·sin(67.7°)
h = 314.5713043

Now we can find the area of the parallelogram:

A = base Χ height

A = 450(314.5713043)

A = 141557.0869 square feet

At $4.29 per square foot

Cost = ($4.29)(141557.0869) = $607279.90 to 
the nearest penny

Edwin