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A rectangle has an area of 24 inches. if all four sides lengths are integers, how many possible different perimeters could this rectangle have?

Sagot :

Answer:

Step-by-step explanation:

Area & Perimeter

The area is the product of a shape's side lengths.

Perimeter is the sum of all the side lengths of a shape.

In the case of a rectangle,

                                            A = lw

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

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Solving the Problem

We're told that the area of a rectangle is 24.

So,

                                           24 = lw.

If we want to find all the different perimeters that a rectangle with an area of 24 we just find how many different dimension combinations equal to an area of 24! (Excluding similar ones like 3 x 4 and 4 x 3).

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Why the Exclusion?

If we included all possible dimensions that give an area of 24, we'll have dimensions like 12 x 2 and 2 x 12.

If we find the perimeters of each we get

12 x 2:

                                         P = 2(12 + 2) = 28

2 x 12:

                                         P = 2(2 + 12) = 28.

They're the same so we must consider one of them, not both.

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There are

  • 24 = 8 x 3
  • 24 = 12 x 2
  • 24 = 6 x 4
  • 24 = 24 x 1

which is all the factors of 24.

Since there are 4 distinct combinations we'll have 4 different perimeters!

Verification:

8 x 3:

P = 2(8 + 3) = 22

12 x 2:

P = 28

6 x 4:

P = 2(6 + 4) = 20

24 x 1:

P = 2(24 + 1) = 50

All of the perimeters are distinct, there isn't two sets of dimensions that have perimeters equal to each other.

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