CH. 5 Test

Question

The base of a triangle is 22 inches more than 4 times the height. If the area is 76 square inches, find the base and the height.

 

base=? in

 

height= ? in

Answer 1

 

The base of a triangle is 22 inches more than 4 times the height.

 

b = 22 + 4 h

 

Area of the triangle = 76 square inches.

 

Formula for the area of a triangle = 1/2 base x height

 

A = 1/2 b x h

 

Substitute the value of area (76) , and base expressed in terms of height (b = 22 + 4 h)

 

76 = 1/2 ( 22 + 4h)  h 

 

76 = ( 1/2 (22) + 1/2 (4h) )  h

 

76 = ( 11 + 2h ) h 

 

76 = 11h + 2h²

 

2h² + 11h - 76 = 0

 

Factor

 

(2h + 19) ( h-4) = 0

 

Solve for h in both factors

 

(2h + 19) = 0           

 

2h = -19

 

h = -19/2 --- This is not the height ince this is a negative number.

 

From the other factor:

 

h - 4 = 0

 

h = 4. 

 

 

Now substitute the value of area and height  in to the formula  A = 1/2 b x h and solve for b.

 

76 = 1/2 x b x 4

 

76 = 2 b

 

Divide both sides by 2

 

38 = b

 

 

 

Base = 38 inches.

 

Height = 4 inches

 

Check the answer by finding the area of the triangle using the height and base.

 

Answer 2

Equation 1: Base +22 inches  = 4(height)

Base = 4(height) - 22 inches

 

Equation 2: Area = 1/2 Base x height

76 square inches = 1/2 base x height

Substitute base from equation 1 into equation 2

 

76 inches square = 1/2(4height - 22 inches)x height 

76 inches square = (2 height - 11 inches) x height

76 inches square = 2 height squared - 11 height

76 inches square = height ( 2 height - 11) 

Combine equation 2 h squared + 11 h -76 square inches = 0

Factor equation:

(2h +19)(h-4) =0

take each bracket seperately

2h+19=0  h= -19/2 =-9.5

h-4=0  h = 4

Base = 22 + 4(4) = 38