Solve it to find the length and width of the plot

Question

The perimeter of a retangular plot of land is 84 metres and its area is 320 metres square. if the length of the plot is represented by x metres, from a quadratic equation in x, and solve it to find length and width of the plot.

 

Here is more information about the question.

 

 

Could you solve for this question step by step with detailed explanation?

Answer 1

I would like to show another way to write the quadratic equation without having denominators in the first step.  Using the variables and equations that Brian gave, we can simplify the equation for perimeter.

 

2(x + y) = 84   ----> perimeter

 

 

By dividing both sides of the equation by 2, we get a new form of the perimeter.

 

x + y = 42

 

 

Substituting this equation of perimeter into the equation of area, we get

 

x(42 - x) = 320   ---->  area in terms of x.

 

42x - x² = 320

 

Add x² and subtract 42x on both sides of the equation to make the left side of the equation equal to zero and get a positive x² term.  Having a positive x² term makes it easier to use FOIL when we have to factor it.  You could have also kept the negative x² term, but doing so will force you to use the quadratic formula.  We want to attempt FOIL first to save us time from performing any excessive work.

 

0 = x² - 42x + 320

 

This equation does match the one Brian has and is factorable using FOIL, as Brian has already shown here.

 

According to the solution he provided, you will have two pairs of dimensions that will work for the rectangular plot.

 

 

Pair 1:              

 

length = 32 m

width = 10 m

 

 

Pair 2:

 

length = 10 m

width = 32 m

Answer 2

Let x and y be the length and width of the rectangle.

Then 2(x + y)=84  and  xy = 320

Sub y = 320/x to the perimeter equation, we have 2(x + 320/x) = 84

The quadratic equation x² - 42x +320 = 0

(x -32)(x - 10) = 0

So x = 32 or 10.  Then y = 10 or 32.