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The area of a rectangle is 112 and perimeter is 64 what are the dimension

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Answer:

w = 28, 4 and l = 4, 28.

Explanation:

The formula for the area of a rectangle is
lw where l is the length and w is the width. Since the area of the rectangle is 112,
lw=112. The formula for the perimeter of a rectangle is
2(l+w) (simplified from
l+w+l+w). The perimeter is 64, therefore
2(l+w) = 64. You can simplify the equation by dividing both sides by 2, giving you
(2(l+w))/(2) =(64)/(2) =
l+w=32.

Now you have a system of equations:
\left \{ {{lw=112} \atop {l+w=32}} \right. . The second equation gives you
l=32-w by subtracting w from both sides.

You can then plug in the expression for l into the first equation, getting
(32-2)w=112. Using the distributive property, you can simply the equation to get
32w-w^(2) =112. By subtracting 112 on both sides, you get
32w-w^2-112=0, or
-w^2+32w-112 by switching the places of
-w^2 and 32w.

Then we can factor the equation. The only possible combination you can use to get
-w^2 is -w multiplied by w. So, we have part of the new equation: (-w )(w ) = 0. Then, we need a pair of numbers that has a sum of 32 and a product of -112. You can list out all the factors of 112: 1, 2, 4, 7, 8, 14, 16, 28, 56, and 112. To get -112, you can just pick a pair that has a product of 112, but make one of the numbers negative.

After some trial and error, we find that the pair we are looking for is 28 and -4. We are done factoring the equation to get
(-w+28)(w-4)=0. We don't put -4 in the first set of parentheses because the product of a negative number and another negative number is a positive number. Knowing that 28 is smaller than 32, we can multiply the -w with the -4 to get a positive 4w. Adding that to the 28w we have from the product of 28 and w, we get the 32w we're looking for.

Now we know that
(-w+28)=0 and
(w-4)=0 For the first equation, we subtract 28 on both sides to give us
-w=-28. Dividing both sides by -1 gives us
w=28. However, the second equation gives us
w=4 by adding 4 to both sides.

Therefore, w has two solutions: 28 and 4. Using
lw=112, we can plug in w to get
l(28)=112 or
l(4)=112. This also gives us two solutions for l: l = 4 or l = 28.

We now have our final answers: w = 28, 4 and l = 4, 28.

User Hermann Speiche
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