Question

the sides of a rectangle (not a square) are whole numbers. what must be there length, breadth such that the perimeter of the rectangle is numerically equal to the area???​

Answer 1

Given:

• It is a rectangle.
• The length and breadth must be whole numbers.
• The area is equal to the perimeter.

Find:

• A set of length and breadth that will satisfy the above requirement.

Recall:

Area of a rectangle = Length x Breadth

Perimeter of the rectangle = 2(Length x Breadth)

Define variables:

Let the length be L and the breadth be B.

Find the relationship between L and B:

$\text{Area} =\text{ L} \times \text{ B}$

$\text{Area} =\text{ LB}$

$\text{Perimeter} = 2 (\text{ L} + \text{ B})$

$\text{Perimeter} = 2\text{ L} + 2\text{ B}$

Given that both the area and the perimeter are equal:

$\text{LB} = 2\text{ L} + 2\text{ B}$

$\text{LB} - 2\text{ B} = 2\text{ L}$

$\text{B} (\text{L} - 2) = 2\text{ L}$

$\text{B} = \dfrac{2\text{ L} }{\text{L} - 2 }$

Now that we know the relationship between the L and B, we will use the Guess and Check method to find a pair of whole numbers that satisfy the relationship.

Rule out L = 2:

We can first rule out L = 2 as L = 2 will make the denominator = 0 and therefore the equation invalid.

Try L = 3:

We will first try L = 3 because we can see that 3 will make the denominator 1 and therefore eliminate the possibility of the right hand side being a fraction.

Substitute L  = 3 into the equation:

$\text{B} = \dfrac{2(3)}{3 - 2 }$

$\text{B} = \dfrac{6}{1 }$

$\text{B} = 6$

So we have the one pair of possible answer where Length = 3 units and Breath = 6 units.

Check the solution:

$\text{Length} = 3$

$\text{Breath} = 6$

$\text{Area} = 3 \times 6$

$\text{Area} = 18$

$\text{Perimeter} = 2(3 + 6)$

$\text{Perimeter} = 2(9)$

$\text{Perimeter} = 18$

Since both area and perimeter are 18, the condition set forth earlier is satisfy.

Answer: Length = 3 units and Breadth = 6 units

Answer 2

Answer:

Length = 3

Breadth = 6

Step-by-step explanation:

3, 6

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