A small plane is flying a banner in the shape of a rectangle. The area...

Step 1
First, we will choose two numbers that have a product of 144 because it's the area of the banner, where the width of the banner is ${\displaystyle \frac{1}{4}}$ the length of the banner.
Guess: 6 feet width and 24 feet length.
Then, we will check those numbers. Is the width of the banner ${\displaystyle \frac{1}{4}}$ the length of the banner?
${\displaystyle \frac{1}{4}\times 24=\frac{1}{4}\times \frac{24}{1}=\frac{1\times 24}{4\times 1}=\frac{24}{4}=6}$
So, our guess is correct.
The dimensions of the banner are 6 feet and 24 feet.

Width is 6; Length is 24
Explanation:
Key points:
Rectangle shape
${\displaystyle Area\Rightarrow 144f{t}^2}$
Width ${\displaystyle =\frac{lenght}{4}}$
Let the area be A=144
Let the width be W
Let the length be L
Known: ${\displaystyle A=W\times L=144}$ (1)
Given: ${\displaystyle W=\frac{L}{4}}$ (2)
To determine the value of L
Substitute (2) into (1) giving
${\displaystyle \frac{L}{4}\times L=144}$
${\displaystyle \Rightarrow \frac{L^2}{4}=144}$
Multiply both sides by 4
${\displaystyle \frac{4}{4}\times L^2=144\times 4}$
${\displaystyle L^2=576}$
Taking the square root of each side
${\displaystyle L=\sqrt{576}=24}$
To determine the value of W
${\displaystyle W\times L=144}$
But ${\displaystyle L=24}$ so we have
${\displaystyle 24W=144}$
Divide both sides by 24
${\displaystyle \frac{24}{24}\times W=\frac{144}{24}}$
${\displaystyle W=6}$

Since the banner is in the shape of a rectangle, and we are asked to find the area of the banner, we are therefore asked to find the area of the rectangle.
The formula for an area of that shape is length multiplied by its width.
$A=144=L\cdot W$
$144=L\cdot W$
The question doesn't specify what the length and width are. However, it does say that the width is $\frac{1}{4}$ of the length, which means the length is 4 times larger than the width.
Let's say the width was "x".
If the length is 4 times larger than the width, then we can multiply the width, x, by 4. This will give length
$L=4\cdot x=4x$
Now we have values for both length and width:
$L=4xandW=x$
Recall that the area of the rectangle is length times width, which means
$A=4x\cdot x=4x^2$
But wait a minute, we already know what area is. It's 144.
$144=4x^2$
Simplify for x.
$\frac{144}{4}=\frac{4x^2}{4}$
$36=x^2$
$x=6$
Width $=x=6$ ft.
Length $=4x=4(6)=24ft$