A rectangle has an area of 636 square inches and a width which is five inches more than four times the length.

Question

Find the length and the width of the rectangle.

Length:

Width:

Answer 1

Area = length * width

A = LW

and we know A = 636

so 636 = LW

Let’s make an equation to represent the relationship between L and W.

We are told that the “width is five inches more than four times the length.” We can turn this phrase into an equation:

width —> W

is —> where the equal sign goes

five inches more than —> +5 (five will be adding onto WHATEVER COMES AFTER THIS STATEMENT)

four times the length —> 4*L

put it all together:

W = 4L + 5

We can make a system of equations with two unknowns, L and W:

636 = LW and W = 4L+5

To solve this system, we will use substitution because we have an equation with one of the variables alone on one side of the equal sign, W = 4L+5. In the first equation where W is, we will instead substitute the whole phrase (4L+5) so we can solve for L.

636 = LW

636 = L*(4L*5) distribute L into the parenthesis

636 = 4L^2 +5L

This is a quadratic equation so we need everything on the same side and a 0 on the other side - let’s subtract 636 from both sides so the left side becomes 0.

0 = 4L^2 + 5L - 636

I am going to use the quadratic formula to factor this equation to find the possible solutions for L.

L = -b + or - SQUARE ROOT OF (b^2-4ac) ALL OVER 2a

a = coefficient of squared term = 4

b = coefficient of non-squared term = 5

c = constant = -636

now use the quadratic formula

-b = -5

+ or -

Square root of(b^2-4ac)=square root of [5^2 - 4*(4)*(-636)]

= square root of 25- (-10176) = square root of 25+10176 =square root(10201)= 101

ALL OVER 2a = 2*4 = 8

Put it all together:

L = -5 + or - 101 all over 8

there are 2 possible answers —> one using the + and the other using the -. Since L is a length and objects cannot have a negative length, we can only use the answer that comes out positive.

possible answer 1: L = -5+101 all over 8 —> L = 96/8 = 12

possible answer 2: L = -5 -101 all over 8 —> L = -106/8

We can only use answer 1 because answer 2 is negative, so our length L = 12 inches. Now we can plug this into either of our original equations to find the width W. It is up to you which one you choose but I will use the one that was solved for W:

W = 4L + 5

W = 4*12 + 5

W = 48+5

W = 53

The length of the rectangle L is 12 inches and the width W is 53 inches. :)

Answer 2

L represents the length

4L + 5 represents the width

636 = 2(L + 4L + 5)

Can you solve for L and answer?