# A rectangular field has an area of 1,764 m^2. The width of the field is 13 m more than the length. What is the perimeter of the field?

Jan 14, 2016

$170$ m

#### Explanation:

If a rectangle has width $W$ and length $L$, then the area $A = L \cdot W$ and the perimeter $P = 2 \cdot \left(L + W\right)$

In this case $W = L + 13$ and $A = 1764$

Plugging this expression for W into the formula for area gives us
$1764 = L \cdot \left(L + 13\right)$
${L}^{2} + 13 L - 1764 = 0$

It is hard to establish the factors of a big number like $1764$ visually so we use the quadratic formula
$L = \frac{- 13 \pm \sqrt{169 + 7056}}{2} = \frac{- 13 \pm \sqrt{7225}}{2} = \frac{- 13 \pm 85}{2}$
We can discount the negative root as the length of a real field cannot be negative. So $L = \frac{- 13 + 85}{2} = \frac{72}{2} = 36$

$L = 36$ give $W = 36 + 13 = 49$

Therefore the perimeter $P = 2 \cdot \left(36 + 49\right) = 2 \cdot 85 = 170$ m

Dec 6, 2016

$170$ $m$

#### Explanation:

Let the length of the rectangle be $l$, then the width will equal $13 + l$

color(blue)("Area of a rectangle"=l*w

Where, $l \mathmr{and} w$ are the length and width of the rectangle

So,

$\rightarrow 8 \cdot \left(13 + l\right) = 1764$

Use the distributive property color(purple)(a(b+c)=ab+ac

$\rightarrow 13 l + {l}^{2} = 1764$

$\rightarrow 13 l + {l}^{2} - 1764 = 0$

Write it in standard form

$\rightarrow {l}^{2} + 13 l - 1764 = 0$

Now, this is a quadratic equation. We solve it using the quadratic formula

color(violet)(l=(-b+-sqrt(b^2-4ac))/(2a)

Where $a , b \mathmr{and} c$ are the coefficients of the terms

Then,

$\rightarrow l = \frac{- 13 \pm \sqrt{{13}^{2} - 4 \left(1\right) \left(1764\right)}}{2 \left(1\right)}$

Solving this, we get

$\rightarrow l = \frac{- 13 \pm 85}{2}$

$\rightarrow l = \left(\frac{- 13 + 85}{2} , \frac{- 13 - 85}{2}\right)$

$\rightarrow l = \left(36 , - 49\right)$

As, length cannot be negative

The length of the rectangle is $36$ and the width is $49$

We need to find the perimeter

color(blue)("Perimeter of rectangle "=2(l+b)

$\rightarrow 2 \left(36 + 49\right)$

color(green)(rArr170 color(green)(m