Question

The height of a triangle is 4 feet more than 3 times the base. If the area is 112 ft find the base and height of the triangle

Answer 1

Answer:

$base = 8 ft\\height =28 ft$

Step-by-step explanation:

let base be x ft. then the height is 3x+4

$Area=\frac{1}{2} \times x \times (3x+4)=112\\3x^{2} +4x=224\\3x^{2} +4x-224=0\\x=8, x=-9.33 (no sol. as length can not be negative)\\ x=8\\base = 8 ft\\height =3(8)+4=28 ft$

Answer 2

Answer:

base = 8 , height = 28

Step-by-step explanation:

Let base be = b

Given:

                             height is 3times base = 3b

        Also 4 feet more than 3 times base = 3b + 4

                           Area = 112 square feet

              $area = \frac{1}{2} \times base \times height\\\\112 = \frac{1}{2} \times b \times (3b+ 4)\\\\224 = b(3b+4)\\\\224 = 3b^2 + 4b\\\\3b^2 + 4b - 224 = 0$

The quadratic equation with a = 3 , x = 4 , c = -224

Therefore ,

             $b = \frac{-x \pm \sqrt{x^2 - 4ac}}{2a}$

             $b= \frac{-4 \pm \sqrt{4^2 - (4 \times 3 \imes -224)}}{2 \times 3}\\\\b= \frac{-4 \pm \sqrt{16 + 2688}}{6}\\\\b= \frac{-4 \pm \sqrt{2704}}{6}\\\\b= \frac{-4 \pm52 }{6}\\\\b = \frac{-4 + 52}{6} , \ b = \frac{-4 -52}{6}\\\\b = \frac{48}{6} , \ b = -\frac{56}{6}$

Since base can't be negative base = 8 ft

Therefore , height = 3b + 4 = 3(8) + 4 = 24 + 4 = 28