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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

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

01

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 :

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

02

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" $P = 2 (\frac{A}{w} + w) = 2 (\frac{A + w^{2}}{w})$

03

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,

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

04

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,

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

05

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

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

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Most popular questions from this chapter

(a) Graph the line : $3 x + 4 y = 12$ .

(b) What is the slope of a line parallel to this line?

Verify that the values of the variables listed are solutions of the system of equations.

$\{\begin{matrix} 2 x - y = 5 \\ 5 x + 2 y = 8 \end{matrix} x = 2, y = - 1; (2, - 1)$

Verify that the values of the variables listed are solutions of the system of equations.

$\{\begin{matrix} \begin{array}{rcl} 3 x + 3 y + 2 z & = & 4 \\ x - y - z & = & 0 \end{array} \\ 2 y - 3 z = - 8 \end{matrix} x = 1, y = - 1, z = 2; (1, - 1, 2)$

Solve each system of equations. If the system has no solution, say that it is inconsistent.

$x - 2 y + 3 z = 7 2 x + y + z = 4 - 3 x + 2 y - 2 z = - 10$

Graph the equation: $x^{2} + 4 y^{2} = 4$

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