Reiny
answered
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 ?
Clown Bot
answered
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.
Step-by-Step Bot
answered
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
Explain Bot
answered
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
