Chapter 9.4, Problem 67PFA

(a)

To determine

Find the equation.

Answer to Problem 67PFA

  $x^2-30x-31=0$

Explanation of Solution

Given Information: $2x^2-16x-31$ , Base b = $2x$ cm and Height h = $x+14$

Formula Used : Area of Triangle = $\frac{1}{2}\times Base\times Height$

Calculation:

Equation $2x^2-16x-31$

Base b = $2x$ cm and Height h = $x+14$

Area of Triangle = $\frac{1}{2}\times Base\times Height$

So,

Area of Triangle = $\frac{1}{2}\times 2x\times \left(x+14\right)$

So,

Area of Triangle = $x^2+14x$

As areas are equivalent.

  $2x^2-16x-31=x^2+14x$

Now, simplify

  $x^2-30x-31=0$

(b)

To determine

Find the height.

Answer to Problem 67PFA

45

Explanation of Solution

Given Information: $x^2-30x-31=0$ , Base b = $2x$ cm and Height h = $x+14$

Formula Used : Area of Triangle = $\frac{1}{2}\times Base\times Height$

Calculation:

  $x^2-30x-31=0$

Now, make factors of $30x$

  $x^2-31x+x-31=0$

Now,

  $x\left(x-31\right)+1\left(x-31\right)=0$

Now,

  $\left(x-31\right)\left(x+1\right)=0$

Hence,

  $x=31$ or $x=-1$

Height h = $x+14$

Height $h=31+14=45$

(c)

To determine

Find the solution valid.

Answer to Problem 67PFA

No

Explanation of Solution

Given Information: $x^2-30x-31=0$ and Height h = $45$

Calculation:

  $x^2-30x-31=0$

As $x=-1$

So, Base is not negative so not valid solutions.