Question

In the following exercises, solve by using methods of factoring, the square root principle, or the quadratic formula.

For Sophia’s graduation party, several tables of the same width will be arranged end to end to give a serving table with a total area of 75 square feet. The total length of the tables will be two more than three times the width. Find the length and width of the serving table so Sophia can purchase the correct size tablecloth. Round answer to the nearest tenth.

Short answer

The width of the tablecloth is$4.7ft.$and the length is$16.1ft.$

Step-by-step solution

Given information. 

The given statement is:

For Sophia’s graduation party, several tables of the same width will be arranged end to end to give a serving table with a total area of 75 square feet. The total length of the tables will be two more than three times the width. Find the length and width of the serving table so Sophia can purchase the correct size tablecloth. Round answer to the nearest tenth.

Set up an equation such that the length of the tables will be two more than three times the width. 

Let the width of the serving table be $w.$

$$\begin{gathered} Length=3\times width+2 \\ Length=3w+2 \end{gathered}$$

So, the rectangle has a width $w$and length$\left(3w+2\right).$

Use the formula of the area of the rectangle.

$$\begin{gathered} Area=length\times width \\ 75=w\left(3w+2\right) \\ 75=3w^2+2w \\ 3w^2+2w-75=0 \end{gathered}$$

Use the quadratic equation

The quadratic formula for the equation $a{x}^2+bx+c=0isx=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$

Now we compare the given equation with the standard form we get

$$\begin{gathered} a=3,b=2andc=-75. \\ w=\frac{-2\pm \sqrt{{\left(2\right)}^2-4\left(3\right)\left(-75\right)}}{2\left(3\right)} \\ w=\frac{-2\pm \sqrt{904}}{6} \\ w=-5.3,4.7 \end{gathered}$$

The width cannot be negative. So, the width is$4.7ft.$and the length is$3\left(4.7\right)+2=16.1ft.$