Chapter 11: Q. 89 (page 774)

Geometry: Find formulas for the length l and width w of a rectangle in terms of its area A and perimeter P.

Short Answer

Expert verified

The formula for length and width is $l = \frac{P + \sqrt{P^{2} - 16 A}}{4}$ and $w = \frac{P - \sqrt{P^{2} - 16 A}}{4}$ .

Step by step solution

Step 1. Given information

Area of the rectangle is given by $A = l w$ .

Perimeter of the rectangle is $P = 2 (l + w)$ .

So the system of equations is :

$A = l w P = 2 (l + w)$

Step 2. Solve the system of equations.

From first equation,

$\frac{A}{w} = \frac{l w}{w} \frac{A}{w} = l l = \frac{A}{w}$

Substitute $l = \frac{A}{w}$ in $P = 2 (l + w)$ .

role="math" localid="1646977874191" $P = 2 (\frac{A}{w} + w) = 2 (\frac{A + w^{2}}{w})$

Step 3. Solve the equation P=2(A+w2w) for w.

We get

$P w = 2 (A + w^{2}) P w = 2 A + 2 w^{2} 0 = 2 w^{2} - P w + 2 A 2 w^{2} - P w + 2 A = 0$

Using the quadratic formula,

$w = \frac{P \pm \sqrt{{(- P)}^{2} - 4 (2) (2 A)}}{2 (2)} = \frac{P \pm \sqrt{P^{2} - 16 A}}{4}$

Step 4. Find an equation for l .

From first equation,

$A = l w w = \frac{A}{l}$

Substitute this in second equation.

P = 2 (l + \frac{A}{l}) P = 2 (\frac{l^{2} + A}{l}) P l = 2 l^{2} + 2 A 2 l^{2} - P l + 2 A = 0

Using quadratic formula,

$l = \frac{P \pm \sqrt{P^{2} - 16 A}}{4}$

Step 5. Find the equation for l and w.

As the equation for both land w is same, so assume

$l = \frac{P + \sqrt{P^{2} - 16 A}}{4}$ and

$w = \frac{P - \sqrt{P^{2} - 16 A}}{4}$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Geometry: Find formulas for the length l and width w of a rectangle in terms of its area A and perimeter P.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solve the system of equations.

Step 3. Solve the equation P=2(A+w2w) for w.

Step 4. Find an equation for l .

Step 5. Find the equation for l and w.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Applied Mathematics

Pure Maths

Logic and Functions

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.