Chapter 9: Q. 9.71 (page 911)

Find the base and height of a triangle whose base is four inches more than six times its height and has an area of 456 square inches.

Short Answer

Expert verified

The height of the triangle is 12 inches and the base is 76 inches.

Step by step solution

Step 1. Given information.

Base of a triangle is four inches more than six times its height.

The area of the triangle is 456 square inches.

Step 2. Form a quadratic equation.

The area of a triangle is:

$A = \frac{1}{2} b h$

Where, $b$ is the base of the triangle and $h$ is the height of the triangle.

Using the given information, it is clear that $b = 6 h + 4$ .

Substitute $b = 6 h + 4$ and $A = 456$ in the above formula.

$\begin{array}{rcl} 456 & = & \frac{1}{2} (6 h + 4) h \\ 0 & = & 3 h^{2} + 2 h - 456 \end{array}$

Step 3. Determine the height.

Split the middle term.

$\begin{array}{rcl} 3 h^{2} + 38 h - 36 h - 456 & = & 0 \\ h (3 h + 38) - 12 (3 h + 38) & = & 0 \\ (h - 12) (3 h + 38) & = & 0 \\ h & = & 12 and h = - \frac{38}{3} \end{array}$

Height of a triangle cannot be negative. So, $h = 12$ .

Step 4. Determine the base.

The base of the triangle is:

$\begin{array}{rcl} b & = & 6 h + 4 \\ = & 6 (12) + 4 \\ = & 72 + 4 \\ = & 76 \end{array}$

Step 5. Conclusion.

The height of the triangle is 12 inches and the base is 76 inches.

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Find the base and height of a triangle whose base is four inches more than six times its height and has an area of 456 square inches.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Form a quadratic equation.

Step 3. Determine the height.

Step 4. Determine the base.

Step 5. Conclusion.

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