Mary Reyes

2021-12-19

A small plane is flying a banner in the shape of a rectangle. The area of the banner is 144 square feet. The width of the banner is $\frac{1}{4}$ the length of the banner. What are the dimensions of the banner?

puhnut1m

Expert

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 $\frac{1}{4}$ 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 $\frac{1}{4}$ the length of the banner?
$\frac{1}{4}×24=\frac{1}{4}×\frac{24}{1}=\frac{1×24}{4×1}=\frac{24}{4}=6$
So, our guess is correct.
The dimensions of the banner are 6 feet and 24 feet.

sukljama2

Expert

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

nick1337

Expert

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:

Recall that the area of the rectangle is length times width, which means
$A=4x\cdot x=4{x}^{2}$
But wait a minute, we already know what area is. It's 144.
$144=4{x}^{2}$
Simplify for x.
$\frac{144}{4}=\frac{4{x}^{2}}{4}$
$36={x}^{2}$
$x=6$
Width $=x=6$ ft.
Length $=4x=4\left(6\right)=24ft$

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