a rectangular piece of plastic has an area of 60 square millimeters. it's perimeter is 32 millimeters. what are the dimensions

Well, if the rectangular piece of plastic is a perfect rectangle, then we can call its length L and its width W.

Since the area of the rectangle is 60 square millimeters, we can write the equation as L × W = 60.

Now, let's think about the perimeter! The perimeter of a rectangle is calculated by adding up the lengths of all its sides. Since we have two lengths and two widths, we can write the equation as 2L + 2W = 32.

Now, let's solve these equations to find the dimensions:

From the first equation, we can isolate L: L = 60 ÷ W.

Substituting this into the second equation, we get: 2(60 ÷ W) + 2W = 32.

Simplifying, we have: 120 ÷ W + 2W = 32.

To make things easier, let's multiply everything by W to get rid of the denominator: 120 + 2W² = 32W.

Rearranging, we have: 2W² - 32W + 120 = 0.

Now, this is where things get a little funny. Would you believe me if I said there's a clown hiding in this equation? Don't worry, it's not a scary clown. In fact, it's quite amusing. The clown's name is "Factoring."

Factoring the quadratic equation, we find: (W - 6)(2W - 20) = 0.

So, two possible solutions for W are: W = 6 and W = 10.

Now, let's plug these values back into the first equation to find the corresponding lengths:

For W = 6, we have: L = 60 ÷ 6 = 10.

For W = 10, we have: L = 60 ÷ 10 = 6.

Therefore, the dimensions of the rectangular piece of plastic can be either 10mm by 6mm or 6mm by 10mm.

To find the dimensions of the rectangular piece of plastic, we need to use the formula for the area and perimeter of a rectangle.

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

The formula for the area of a rectangle is: Area = Length × Width
Given that the area is 60 square millimeters, we have the equation:
60 = L × W

The formula for the perimeter of a rectangle is: Perimeter = 2 × (Length + Width)
Given that the perimeter is 32 millimeters, we have the equation:
32 = 2 × (L + W)

Now we have a system of equations:
Equation 1: 60 = L × W
Equation 2: 32 = 2 × (L + W)

Solving this system of equations will give us the dimensions of the rectangular piece of plastic. Let's solve it step-by-step.

Step 1: Solve Equation 2 for L + W
32 = 2 × (L + W)
Divide both sides by 2 to isolate (L + W):
16 = L + W

Step 2: Solve Equation 1 for L in terms of W
Substitute L × W from Equation 1 into Equation 2:
16 = L + W
16 = (60 / W) + W
Multiply both sides by W to get rid of the fraction:
16W = 60 + W^2
Rearrange the equation to get it in the form of a quadratic equation:
W^2 + 16W - 60 = 0

Step 3: Solve the quadratic equation
Factor the quadratic equation:
(W - 4)(W + 15) = 0
Setting each factor equal to zero gives us two possible values for W:
W - 4 = 0 or W + 15 = 0
W = 4 or W = -15

Since we are dealing with the dimensions of a rectangle, the width cannot be negative. So we discard W = -15 as a valid solution.

Step 4: Substitute W = 4 back into Equation 1 to solve for L
60 = L × 4
Divide both sides by 4:
L = 15

Therefore, the dimensions of the rectangular piece of plastic are:
Length = 15 millimeters
Width = 4 millimeters

To find the dimensions of the rectangular piece of plastic, we can use the formulas for the area and perimeter of a rectangle.

Let's denote the length of the rectangle as "L" and the width as "W".

The area of a rectangle is given by the formula:
Area = Length x Width

And the perimeter of a rectangle is given by the formula:
Perimeter = 2 x (Length + Width)

From the given information, we have:
Area = 60 square millimeters
Perimeter = 32 millimeters

We can now set up two equations using the formulas above:

Equation 1: Area = Length x Width
60 = L x W

Equation 2: Perimeter = 2 x (Length + Width)
32 = 2 x (L + W)

Now, we have a system of two equations. We can solve it to find the values of L and W.

First, let's simplify Equation 2:
32 = 2L + 2W
16 = L + W

Now, we can use substitution method to solve the equations.

From Equation 1, we have:
L = 60 / W

Substituting this value into Equation 2:
16 = (60 / W) + W

Multiplying both sides of the equation by W to get rid of the fraction:
16W = 60 + W^2

Rearranging the equation:
W^2 + 16W - 60 = 0

This is now a quadratic equation. We can solve it by factoring or using the quadratic formula.

Factoring the equation:
(W - 4)(W + 15) = 0

Setting each factor equal to zero and solving for W:
W - 4 = 0 or W + 15 = 0

Solving for W:
W = 4 or W = -15 (Discard this solution since dimensions can't be negative)

Now, we can find the corresponding value of L using Equation 1:
L = 60 / W

Substituting W = 4 into the equation:
L = 60 / 4
L = 15

Therefore, the dimensions of the rectangular piece of plastic are:
Length = 15 millimeters
Width = 4 millimeters

width --- x

length ---- y

xy = 60
x+y = 16 ---> y = 16-x

x(16-x) = 60
16x - x^2 - 60 = 0
x^2 - 16x + 60 = 0
solve for x
(hint it factors)

btw, could you find 2 factors of 60 that add up to 16 ?