Question

A room in the shape of a triangle has sides of length 7 yd, 8 yd, and 11 yd. If carpeting costs $17.50 a square yard and padding costs $3.25 a square yard, how much to the nearest dollar will it cost to carpet the room, assuming that there is no waste?

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Answer 1

By sketching this triangle it can be determined that this is not a right triangle, but rather an oblique-angled triangle. For the lengths of the three sides, let a = 7, b = 8, and c = 11, and let angle A be opposite side a, and angle B be opposite side b, and angle C be opposite side c.


Using the Law of Cosines:

cos(A) = (b² + c² - a²)/2bc = [(8)² + (11)² - (7)²)/2(8)(11) = 0.773

A = acos(0.773) = 39.4°


Using the Law of Sines:

sin(A)/a = sin(B)/b ==> sin(B) = (b/a)sin(A) = (8/7)sin(39.4°) = 0.725

B = asin(0.725) = 46.5°


Knowing angles A and B, angle C is:

C = 180° - A - B = 180° - 39.4° - 46.5° = 94.1°


Finally, the area is:

Area = (ab/2)sin(C) = [(7)(8)/2]sin(94.1°) = 27.928 yd²

The total cost is (27.928)($17.50 + $3.25) = $579.51

Answer 2

First, you have to calculate the area of the triangle. The formula would be


`A_(T) = √(s(s-a)(s-b)(s-c))`

where s is the semi-parameter of the triangle with sides a, b and c


`s= (a+b+c)/(2)`


`s= (7+8+11)/(2)=13`

So,


`A_(T) = √(13(13-7)(13-8)(13-11))`


`A_(T) = 27.93 \ yd^(2)`

So, the total cost of carpeting the room is,


`27.93 \ yd^(2) (17.50 \ per \ yd^(2)+3.25 \ \ per \ yd^(2))=579.52`

$579.5