A rectangular red sticker has an area of 72 square millimeters. Its perimeter is 34 millimeters. What are the dimensions of the sticker?
To find the dimensions of the rectangular red sticker, we'll solve a system of equations using the information given.
Let's assume the length of the rectangle is L and the width is W.
Given that the area of the rectangle is 72 square millimeters, we have the equation:
L * W = 72 (equation 1)
Given that the perimeter of the rectangle is 34 millimeters, we have the equation:
2L + 2W = 34 (equation 2)
Now we can solve this system of equations:
From equation 2, we can simplify it to:
L + W = 17
Rearrange this equation by subtracting W from both sides:
L = 17 - W
Substitute this equation into equation 1:
(17 - W) * W = 72
Expand and rearrange the equation:
17W - W^2 = 72
Rearrange the equation to form a quadratic equation:
W^2 - 17W + 72 = 0
Now we can solve this quadratic equation. Factoring it, we have:
(W - 8)(W - 9) = 0
Setting each factor equal to zero:
W - 8 = 0 or W - 9 = 0
Solving for W, we have two possible values for the width:
W = 8 or W = 9
If W = 8, substitute this value back into equation 1 to solve for L:
L * 8 = 72
L = 9
If W = 9, substitute this value back into equation 1 to solve for L:
L * 9 = 72
L = 8
Therefore, we have two sets of dimensions for the rectangular red sticker:
1. Length = 9 millimeters, Width = 8 millimeters
2. Length = 8 millimeters, Width = 9 millimeters
To find the dimensions of the rectangular red sticker, we need to use the given information about its area and perimeter. Let's assume the length of the rectangle is "L" and the width is "W".
We know that the formula for the area of a rectangle is calculated by multiplying its length with its width:
Area = Length * Width
Given that the area is 72 square millimeters, we can write the equation:
72 = L * W
We also know that the perimeter of a rectangle is calculated by adding the lengths of all four sides:
Perimeter = 2 * (Length + Width)
Given that the perimeter is 34 millimeters, we can write the equation:
34 = 2 * (L + W)
We now have a system of two equations with two variables (L and W):
Equation 1: 72 = L * W
Equation 2: 34 = 2 * (L + W)
To solve this system of equations, we can use substitution or elimination. In this case, we will use substitution.
Rearrange Equation 1 to solve for L:
L = 72 / W
Substitute this expression for L in Equation 2:
34 = 2 * ((72 / W) + W)
Now we can solve for W. Multiply both sides of Equation 2 by W to eliminate the fraction:
34W = 2(72 + W^2)
Expand the equation:
34W = 144 + 2W^2
Rearrange the equation and set it equal to zero:
2W^2 - 34W + 144 = 0
Now we can solve this quadratic equation for W. We can either factor it or use the quadratic formula. In this case, let's use the quadratic formula:
W = (-b ± √(b^2 - 4ac)) / (2a)
For the equation 2W^2 - 34W + 144 = 0, the values of a, b, and c are:
a = 2
b = -34
c = 144
Substituting these values into the quadratic formula:
W = (-(-34) ± √((-34)^2 - 4(2)(144))) / (2 * 2)
Simplifying:
W = (34 ± √(1156 - 1152)) / 4
W = (34 ± √4) / 4
W = (34 ± 2) / 4
Solving for W:
W1 = (34 + 2) / 4 = 36 / 4 = 9
W2 = (34 - 2) / 4 = 32 / 4 = 8
Now that we have two possible values for the width, we can plug them back into Equation 1 to find the corresponding lengths.
For W = 9:
L = 72 / 9 = 8
For W = 8:
L = 72 / 8 = 9
So, the dimensions of the rectangular red sticker can be either 9mm x 8mm or 8mm x 9mm.
L * W = 72
L = 72/W
2L + 2W = 2(72/W) + 2W = 34
Solve for W, then L.