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