Question

A rectangular sticker has an area of 6 square centimeters. Its perimeter is 10 centimeters.
What are the dimensions of the sticker?
centimeters by
centimeters

Answer 1

Let's assume that the dimensions of the rectangle are length (L) and width (W).

We know the formula for the area of a rectangle is A = L * W and the formula for the perimeter of a rectangle is P = 2L + 2W.

Given the area of the rectangle is 6 square centimeters: A = 6

And the perimeter of the rectangle is 10 centimeters: P = 10

We can solve this by creating a system of equations:

L * W = 6

2L + 2W = 10

To solve for the dimensions, we can use substitution or elimination method.

As the area of a rectangle is given by A = L * W, we can rearrange the first equation to express one variable in terms of the other:

W = 6 / L

Substitute W in the second equation:

2L + 2 (6 / L) = 10

2L + 12 / L = 10

Multiply through by L:

2L^2 - 10L + 12 = 0

Now we can solve for L by factoring:

2L^2 - 10L + 12 = 0

2(L^2 - 5L + 6) = 0

2(L - 3)(L - 2) = 0

This gives us two possible values for L:

L = 3 or L = 2

If L = 3:

W = 6 / 3 = 2

The dimensions are 3 centimeters by 2 centimeters.

If L = 2:

W = 6 / 2 = 3

The dimensions are 2 centimeters by 3 centimeters.

Therefore, the dimensions of the sticker are either 3 centimeters by 2 centimeters or 2 centimeters by 3 centimeters.

Answer 2

Answer:

3 centimeters by 2 centimeters

Step-by-step explanation:

The area of a rectangle is the product of its width (w) and length (l). Given that the area of the rectangular sticker is 6 square centimeters, then:

$wl = 6$

The perimeter of a rectangle is twice the sum of its width (w) and length (l). Given that the perimeter is 10 cenitmeters, then:

$2(w + l) = 10$

Therefore, we have created a system of equations:

$\begin{cases}wl = 6\\2(w + l) = 10\end{cases}$

To solve the system of equations, being by rearranging the second equation to isolate w:

$2(w+l)=10\\\\\\\dfrac{2(w+l)}{2}=\dfrac{10}{2}\\\\\\w+l=5\\\\\\w+l-l=5-l\\\\\\w=5-l$

Now, substitute $w = 5 - l$ into the first equation:

$(5-l)l=6\\\\5l-l^2=6\\\\l^2-5l+6=0\\\\l^2-3l-2l+6=0\\\\l(l-3)-2(l-3)=0\\\\(l-2)(l-3)=0$

Solve for $l$:

$\\\\l-2=0 \implies l=2\\\\l-3=0\implies l=3$

If we substitute the two values of $l$ into $w = 5 - l$, we get:

$w = 5 - 2=3\\\\w-5-3=2$

The term "length" is typically associated with the longer dimension of a rectangle, while the term "width" is associated with the shorter dimension. Therefore:

$\sf width = 2 \; centimeters$

$\sf length = 3\; centimeters$

So, the dimensions of the sticker are:

$\Large\boxed{\boxed{\textsf{3 centimeters by 2 centimeters}}}$