Question

The area of a rectangle is 2 square feet. the perimeter of the rectangle is 9 ft. find the length, l , and width, w , of the rectangle. what is l squared plus w squared ?

Answer 1

Your problem conditions give rise to two equations in l and w.
   area = 2 ft² = lw
   perimeter = 9 ft = 2(l+w)

For l and w in feet, we can substitute for l in the first equation using
   9/2 - w = l . . . . . . .from the second equation
   2 = (9/2 -w)w . . . . substituting for l in the first equation
   w² -9/2w +2 = 0
   (w - 1/2)(w - 4) = 0
   w = 1/2 or 4
   l = 9/2-w = 4 or 1/2

The length and width of the rectangle are 4 ft and 1/2 ft.

l² + w² = (4 ft)² + (1/2 ft)²
l² + w² = 16.25 ft²

Answer 2

To solve this question:

• We apply rectangle concepts.
• We build a system for the equations of the length and width, and find them solving a quadratic equation.

Doing this, we get that:

• The length is of 4 feet, the width is of 0.5 feet, and length squared plus width squared is 16.25.

Solving a quadratic equation:

Given a second order polynomial expressed by the following equation:

$ax^{2} + bx + c, a\neq0$.

This polynomial has roots $x_{1}, x_{2}$ such that $ax^{2} + bx + c = a(x - x_{1})*(x - x_{2})$, given by the following formulas:

$x_{1} = \frac{-b + \sqrt{\Delta}}{2*a}$

$x_{2} = \frac{-b - \sqrt{\Delta}}{2*a}$

$\Delta = b^{2} - 4ac$

Rectangle:

A rectangle has two dimensions, length l and width w.

It's area is given by:

$A = lw$

It's perimeter is given by:

$P = 2l + 2w$

The area of a rectangle is 2 square feet.

This means that $A = 2$, and thus:

$lw = 2$

$w = \frac{2}{l}$

The perimeter of the rectangle is 9 ft.

This means that:

$2l + 2w = 9$

Since $w = \frac{2}{l}$

$2l + 2\frac{2}{l} = 9$

$2l + \frac{4}{l} = 9$

$\frac{2l^2 + 4}{l} = 9$

Applying cross multiplication:

$2l^2 + 4 = 9l$

$2l^2 - 9l + 4 = 0$

Solving the quadratic equation, we find the value of the dimensions.

It is a quadratic equation with $a = 2, b = -9, c = 4$. So

$\Delta = (-9)^{2} - 4(2)(4) = 49$

$l_{1} = \frac{-(-9) + \sqrt{49}}{2*2} = 4$

$l_{2} = \frac{-(-9) - \sqrt{49}}{2*2} = 0.5$

Thus, a possible answer is given by:

• Length of 4 feet, width of 0.5 feet:
• $l^2 + w^2 = 4^2 + 0.5^2 = 16 + 0.25 = 16.25$

A similar question is given at https://brainly.com/question/16943403