The area of a rectangular piece of plastic is 70 square millimeters. The perimeter is 34 millimeters. What are the dimensions of the piece of plastic?

hint: factors of 70 are 7 and 10

7 10

Let's assume the length of the rectangular piece of plastic is L millimeters and the width is W millimeters.

The formula for the area of a rectangle is: Area = Length * Width
So we can write the equation: L * W = 70

The formula for the perimeter of a rectangle is: Perimeter = 2 * (Length + Width)
So we can write the equation: 2 * (L + W) = 34

We have a system of two equations:
1) L * W = 70
2) 2 * (L + W) = 34

We can solve this system of equations using substitution or elimination method.

Let's start with the substitution method:
From equation 1, we can rearrange it to express one variable in terms of the other: W = 70 / L

Now substitute this expression for W in equation 2:
2 * (L + (70 / L)) = 34

Simplify the equation:
2L + 140 / L = 34

Multiply both sides of the equation by L to remove the fraction:
2L^2 + 140 = 34L

Rearrange the equation to form a quadratic equation:
2L^2 - 34L + 140 = 0

Now we can solve this quadratic equation to find the possible values of L. We can either factorize it or use the quadratic formula.

Using the quadratic formula:
L = (-b ± √(b^2 - 4ac)) / 2a

Here, a = 2, b = -34, and c = 140.

L = (-(-34) ± √((-34)^2 - 4(2)(140))) / (2 * 2)
L = (34 ± √(1156 - 1120)) / 4
L = (34 ± √36) / 4

Simplifying further, we get:
L = (34 ± 6) / 4

So, L can be either (34 + 6) / 4 = 40 / 4 = 10
Or, L can be (34 - 6) / 4 = 28 / 4 = 7

Now substitute these values of L back into equation 1 to find the corresponding W values:

For L = 10:
W = 70 / L = 70 / 10 = 7

For L = 7:
W = 70 / L = 70 / 7 = 10

Therefore, the dimensions of the piece of plastic can be either 10 mm by 7 mm or 7 mm by 10 mm.

To find the dimensions of the rectangular piece of plastic, we'll use the given information about its area and perimeter.

Let's assume the length of the rectangle is "L" millimeters and the width is "W" millimeters.

The formula for the area of a rectangle is: area = length x width.

Given that the area is 70 square millimeters, we can write the equation:
70 = L x W ..............(Equation 1)

The formula for the perimeter of a rectangle is: perimeter = 2(length + width).

Given that the perimeter is 34 millimeters, we can write the equation:
34 = 2(L + W) .......(Equation 2)

We have a system of two equations (Equation 1 and Equation 2) that we can solve simultaneously to find the values of L and W.

Let's rearrange Equation 2 to solve for L:
34 = 2L + 2W
2L = 34 - 2W
L = (34 - 2W) / 2
L = 17 - W ..............(Equation 3)

Substitute Equation 3 into Equation 1 to eliminate L:
70 = (17 - W) x W
70 = 17W - W^2
Rearranging the equation, we get:
W^2 - 17W + 70 = 0

Now we have a quadratic equation. We can solve it by factoring or using the quadratic formula. Factoring in this case gives us:
(W - 10)(W - 7) = 0

Setting each factor equal to zero, we can solve for W:
W - 10 = 0 => W = 10
W - 7 = 0 => W = 7

Now, substitute the values of W back into Equation 3 to find the corresponding values of L:
If W = 10, then L = 17 - 10 = 7.
If W = 7, then L = 17 - 7 = 10.

Therefore, the dimensions of the rectangular piece of plastic are 7 millimeters by 10 millimeters.