The perimeter of a rectangle is 46m. If the width were doubled and the length were increased by 12m, the perimeter would be 82m. What are the length and width of the rectangle?
original:
2L+ 2W = 46
L+W = 23
new one:
2(L+12) + 2(2W) = 82
2L + 24 + 4W = 82
L + 2W = 29
Subtract the 1st from the 2nd
W = 6
back in the 1st ---> L=17
original rectangle is 17 by 6
check: perimeter of original = 34 + 12 = 46
new rectangle is 29 by 12
perimeter of new = 2(29) + 2(12) = 82
To solve this problem, we need to set up a system of equations based on the given information.
Let's assume the width of the rectangle is 'w' meters and the length of the rectangle is 'l' meters.
According to the first statement, the perimeter of the rectangle is 46m. Perimeter of a rectangle is given by the formula: P = 2l + 2w. So, we can write the equation as:
2l + 2w = 46 ---(Equation 1)
According to the second statement, if the width were doubled and the length were increased by 12m, the new perimeter would be 82m. In this case, the new width would be '2w' and the new length would be 'l + 12'. So, we can write the equation as:
2(l + 12) + 2(2w) = 82 ---(Equation 2)
Now, we have a system of equations. We can solve these two equations simultaneously to find the values of 'l' and 'w'.
Let's simplify Equation 2:
2l + 24 + 4w = 82
2l + 4w = 82 - 24
2l + 4w = 58 ---(Equation 3)
Now, we have two equations:
2l + 2w = 46 ---(Equation 1)
2l + 4w = 58 ---(Equation 3)
We can solve this system of equations by using the method of substitution or elimination.
Let's solve it using the method of elimination:
Multiply Equation 1 by 2 to make the coefficients of 'l' the same:
4l + 4w = 92 ---(Equation 4)
Now, subtract Equation 3 from Equation 4 to eliminate 'l':
(4l + 4w) - (2l + 4w) = 92 - 58
4l + 4w - 2l - 4w = 34
2l = 34
l = 34/2
l = 17
Now, substitute the value of 'l' (= 17) into Equation 1 to find the value of 'w':
2(17) + 2w = 46
34 + 2w = 46
2w = 46 - 34
2w = 12
w = 12/2
w = 6
Therefore, the length of the rectangle is 17m and the width is 6m.
Let's assume the length of the rectangle is "L" and the width is "W".
According to the given information, the perimeter of the original rectangle is 46m.
The formula for the perimeter of a rectangle is P = 2L + 2W, so we can write the equation as:
2L + 2W = 46 ----(1)
If the width were doubled and the length were increased by 12m, the perimeter of the new rectangle would be 82m.
We can write the equation for the new perimeter as:
2(L + 12) + 2(2W) = 82
2L + 24 + 4W = 82
2L + 4W = 58 ----(2)
To solve the system of equations (1) and (2), we can solve for either "L" or "W" in one equation and substitute it into the other equation.
Let's solve equation (1) for "L":
2L = 46 - 2W
L = 23 - W ----(3)
Substitute equation (3) into equation (2):
2(23 - W) + 4W = 58
46 - 2W + 4W = 58
2W = 12
W = 6
Now that we have the value of "W", we can substitute it into equation (3) to find "L":
L = 23 - 6
L = 17
Therefore, the length of the rectangle is 17m and the width is 6m.