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The length of a rectangle is twice its width, express its area, a, in terms of its perimeter, p....

Question:

The length of a rectangle is twice its width, express its area, a, in terms of its perimeter, p. then write a sentence to state how the area varies with the perimeter.

Area and perimeter of rectangle:

Let {eq}l, b {/eq} be the length and width of a rectangle respectively.

The area of rectangle is given by {eq}A = l*b {/eq}

Perimeter of a rectangle is given by {eq}p = 2(l+b) {/eq}

Answer and Explanation:

Let {eq}l, b {/eq} be teh length and width of a rectangle.

Given that {eq}l=2b {/eq}

The perimeter of a rectangle is {eq}p = 2(l+b) = 2(2b+b) = 6b {/eq}

Therefore, width {eq}b = \frac{p}{6} {/eq}

Area of a rectangle is given by {eq}\displaystyle{ A = l*b =2b*b\\ =2*\frac{p}{6}*\frac{p}{6}\\ =\frac{p^{2}}{18} } {/eq}

Therefore, area of the rectangle in terms of perimeter is {eq}A = \frac{p^{2}}{18} {/eq}

We can see that area is a quadrartic function of perimeter.


Learn more about this topic:

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Measuring the Area of a Rectangle: Formula & Examples

from Geometry: High School

Chapter 8 / Lesson 7
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