Chapter 9.4, Problem 65E

a.

To determine

To write the equation.

Answer to Problem 65E

Equation for amount of fencing to be used:

  $80=b+2w$

Equation for area enclosed by fencing:

  $750=b\times w$

Explanation of Solution

Given:

80feet of fencing.

Area=750 square feet.

Calculation:

Equation for amount of fencing to be used:

  $80=b+w+w$

So,

  $80=b+2w$

Equation for area enclosed by fencing:

Since it is a square:

  $750=b\times w$

b.

To determine

To solve the equations.

Answer to Problem 65E

  $w=35$ or $w=25$

  $b=50$ or $b=30$

Explanation of Solution

Given:

  $80=b+2w$

  $750=b\times w$

Calculation:

According to question:

  $80=b+2w$

Let it be equation 1.

  $750=b\times w$

Let it be equation 2.

From equation 1.

  $\frac{750}{b}=w$

Substitute the value of $w$ in equation 2.

  $80=b+2(\frac{750}{b})$

So,

  $80=b+\frac{1500}{b}$

So,

  $80=\frac{b^2+1500}{b}$

Therefore

  $80b=b^2+1500$

Converting the equation in standard form:

  $b^2-80b+1500=0$

On comparing the equation with $a{x}^2+bx+c$

  $a=1,b=-80,c=1500$

Finding the value of $b$

  $b=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Substituting the values,

  $b=\frac{-(-80)\pm \sqrt{{(-80)}^2-4(1)(1500)}}{2(1)}$

Opening brackets,

  $b=\frac{80\pm \sqrt{6400-6000}}{2}$

So,

  $b=\frac{80\pm \sqrt{400}}{2}$

Solving root,

  $b=\frac{80\pm 20}{2}$

Therefore,

  $b=\frac{80+20}{2}$ or $b=\frac{80-20}{2}$

So,

  $b=\frac{100}{2}$ or $b=\frac{60}{2}$

Therefore,

  $b=50$ or $b=30$

So, $\frac{750}{b}=w$

Substituting value,

  $w=\frac{750}{50}$

So,

  $w=25$

Also,

  $w=\frac{750}{30}$

So,

  $w=35$