Question

In a rectangle, one side is 6 cm longer than the other. The area of the rectangle is 520cm2. Determine the longest side of the rectangle.

Answer 1

From the given informatio, we can draw the following picture:

where x denote the shorter side of the rectangle and x+6 the longest one. Since the area of the rectangle is equal to the base times the height, we have

`A=(x+6)x`

Since the area is 520 square centimeters, we have

`520=(x+6)x`

Then, by distributing the variable x into the parentheses, we have the following equation:

`x^2+6x=520`

or equivalently,

`x^2+6x-520=0`

So, we can to apply the quadratic formula:

`x=(-b\pm√(b^2-4ac))/(2a)`

coming from the quadratic polynomial:

`ax^2+bx+c=0`

By comparing this polynomial with ours, we can note that

`\begin{gathered} a=1 \\ b=6 \\ c=-520 \end{gathered}`

so, we get

`x=(-6\pm√(6^2-4(1)(-520)))/(2)`

which gives

`\begin{gathered} x=(-6\pm√(2116))/(2) \\ x=(-6\pm46)/(2) \end{gathered}`

Then, we have 2 possible solutions:

`\begin{gathered} x=(-6+46)/(2)=(40)/(2)=20 \\ and \\ x=(-6-46)/(2)=(-52)/(2)=-26 \end{gathered}`

However, since the dimension are always positive numbers, the searched side measure 20 cm. In other words, the shorter side measures 20cm and the longest one 20+6=26 cm. Therefore, the answer is: 26 centimeters.