Question

The length of a rectangle is 23 cm less than 6 times it's width. it's area is 35 square centimeters. find the dimensions of the rectangle use the formula,area equal length times width.

Answer 1

We are given that the area of a rectangle is 42 square meters. The formula for the area of a rectangle is:

$A=wl$

Where "w" is the width and "l" is the length. We are also told that the width is 5 meters longer than one-third of the length, this can be written mathematically as:

$w=\frac{1}{3}l+5$

Replacing in the formula for the area:

$A=(\frac{1}{3}l+5)l$

Now we use the distributive property:

$A=\frac{1}{3}l^2+5l$

Now we replace the value of the area:

$42=\frac{1}{3}l^2+5l$

Now we subtract 42 to both sides:

$\frac{1}{3}l^2+5l-42=0$

Now we can multiply both sides by 3:

$l^2+15l-126=0$

Now we factor this equation, we need two numbers which sum is 15 and their product is 126, we get:

$(l+21)(l-6)=0$

Now we set each factor to zero:

$\begin{gathered} l+21=0 \\ l=-21 \end{gathered}$$\begin{gathered} l-6=0 \\ l=6 \end{gathered}$

We take the positive value and therefore, the length of the rectangle is 6 meters.

To determine the width we use the relationship given by the problem:

$w=\frac{1}{3}l+5$

Replacing l = 6:

$\begin{gathered} w=\frac{1}{3}(6)+5 \\ w=2+5 \\ w=7 \end{gathered}$

The width is 7 meters.