Aroom 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?

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



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





So,





So, the total cost of carpeting the room is,



$579.5

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