A rectangular piece of plastic has a perimeter of 28 millimeters. Its area is 40 square millimeters. What are the dimensions of the piece of plastic?
To find the dimensions of the rectangular piece of plastic, we can set up a system of equations based on the given information.
Let's assume the length of the rectangle is "l" and the width is "w".
We know that the perimeter of a rectangle is given by the formula P = 2(l + w). So, based on the given information, we can write the equation:
2(l + w) = 28 -- Equation 1
We also know that the area of a rectangle is given by the formula A = l * w. So, based on the given information, we can write the equation:
l * w = 40 -- Equation 2
Now, we have a system of two equations (Equation 1 and Equation 2) that we can solve simultaneously.
Let's start by rearranging Equation 1 to express one of the variables in terms of the other:
l + w = 14 -- Equation 3 (after dividing both sides of Equation 1 by 2)
Next, we can solve Equation 3 for one variable (let's choose l) and substitute that value into Equation 2 to solve for the other variable:
l = 14 - w -- Equation 4 (after subtracting w from both sides of Equation 3)
Substituting Equation 4 into Equation 2:
(14 - w) * w = 40
Expanding and rearranging the equation:
14w - w^2 = 40
Rearranging again to form a quadratic equation:
w^2 - 14w + 40 = 0
Factoring or using the quadratic formula, we find that w = 10 or w = 4.
If w = 10, then substituting it into Equation 3, we find l = 4.
If w = 4, then substituting it into Equation 3, we find l = 10.
Therefore, the two possible sets of dimensions for the rectangular piece of plastic are:
1) Length = 4 mm and Width = 10 mm
2) Length = 10 mm and Width = 4 mm
2 L + 2 W = 28 ... L + W = 14
L * W = 40
two numbers that add to 14 and multiply to 40 ...
7.1 millimetres
5 and 9
L + L + w + w = 28
L(w) = 40
Solve for one variable and sub it back into an original and solve for the other variable.
hint: you may wish to consider what times what is 40... as a starting point for an educated guess : )
a * b = 40
2 a + 2 b = 28
so
a = 40/b
and then
2(40/b) + 2 b = 28
80 / b + 2 b = 28
40/b + b = 14
40 + b^2 = 14 b
b^2 - 14 b + 40 = 0
(b - 10)(b - 4) = 0
looks like 10 and 4 to me