The area of a rectangle is 12 square inches. The length is 5 more than twice it’s width. How do you find the length and width?

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The area of a rectangle is 12 square inches. The length is 5 more than twice it’s width. How do you find the length and width?

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Answer 1 · 2018-03-12T17:47:45

Using the positive root in the quadratic equation, you find that $w = 1.5$ $w=1.5$ , which means $l = 8$ $l=8$

Explanation:

We know two equations from the problem statement. First is that the area of the rectangle is 12:

$l \cdot w = 12$ $l*w=12$

where $l$ $l$ is the length, and $w$ $w$ is the width. The other equation is the relationship between $l$ $l$ and $w$ $w$ . It states that 'The length is 5 more than twice it's width'. This would translate to:

$l = 2 w + 5$ $l=2w+5$

Now, we substitute the length to width relationship into the area equation:

$(2 w + 5) \cdot w = 12$ $(2w+5)*w=12$

If we expand the left-hand equation, and subtract 12 from both sides, we have the makings of a quadratic equation:

$2 w^{2} + 5 w - 12 = 0$ $2w^2+5w-12=0$

where:
$a = 2$ $a=2$
$b = 5$ $b=5$
$c = - 12$ $c=-12$

plug that into the quadratic equation:

$w = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \Rightarrow w = \frac{- 5 \pm \sqrt{5^{2} - 4 (2 \cdot - 12)}}{2 \cdot 2}$ $w=(-b+-sqrt(b^2-4ac))/(2a) rArr w=(-5+-sqrt(5^2-4(2*-12)))/(2*2)$

$w = \frac{- 5 \pm \sqrt{25 - (- 96)}}{4} \Rightarrow w = \frac{- 5 \pm \sqrt{121}}{4}$ $w=(-5+-sqrt(25-(-96)))/4 rArr w=(-5+-sqrt(121))/4$

$w = \frac{- 5 \pm 11}{4}$ $w=(-5+-11)/4$

we know that the width must be a positive number, so we only worry about the positive root:

$w = \frac{- 5 + 11}{4} \Rightarrow w = \frac{6}{4} \Rightarrow w = 1.5$ $w=(-5+11)/4 rArr w=6/4 rArr color(red)(w=1.5)$

now that we know the width ( $w$ $w$ ), we can solve for the length ( $l$ $l$ ):
$l = 2 w + 5 \Rightarrow l = 2 (1.5) + 5$ $l=2w+5 rArr l=2(1.5)+5$

$l = 3 + 5 \Rightarrow l = 8$ $l=3+5 rArr color(red)(l=8)$

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The area of a rectangle is 12 square inches. The length is 5 more than twice it’s width. How do you find the length and width?

1 Answer

Explanation:

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