If the length of a rectangle is decreased by 4 cm and the width is increased by 5 cm, the result will be a square, the area of which will be 40 cm2 greater than the area of the rectangle. Find the area of the rectangle.
Let's assume the original length of the rectangle is L and the original width is W.
According to the given information, the length is decreased by 4 cm, so the new length will be L - 4.
The width is increased by 5 cm, so the new width will be W + 5.
We know that the result of these changes is a square, so the new length and new width are equal. Therefore:
L - 4 = W + 5 -----(1)
We are also given that the area of the square is 40 cm2 greater than the area of the rectangle.
The area of the rectangle is L * W, and the area of the square is (L - 4)(W + 5).
Equating the areas, we have:
(L - 4)(W + 5) = L * W + 40 -----(2)
Now we have two equations (1) and (2) with two unknowns. Let's solve them simultaneously.
Expanding equation (2), we get:
LW + 5L - 4W - 20 = LW + 40
Simplifying, we have:
5L - 4W - 20 = 40
Adding 40 to both sides, we get:
5L - 4W = 60 -----(3)
Now we have a system of equations (1) and (3). Let's solve them simultaneously.
From equation (1), we have:
L - 4 = W + 5
Rearranging, we get:
L - W = 9 -----(4)
Adding equations (3) and (4), we have:
5L - 4W + L - W = 60 + 9
Simplifying, we get:
6L - 5W = 69
Now we can solve this equation for L or W.
Let's multiply equation (4) by 5 and equation (1) by 6 to simplify the equations:
5L - 5W = 45 -----(5)
6L - 6W = 54 -----(6)
Subtracting equation (5) from equation (6), we have:
(6L - 6W) - (5L - 5W) = 54 - 45
Simplifying, we get:
L = 9
Now we can substitute the value of L back into equation (4) to find W:
9 - W = 9
Simplifying, we get:
W = 0
But a width of 0 does not make sense in this context, so there must be an error in the problem statement or the calculations. Please double-check the problem information or the calculations.
To solve this problem, let's start by assuming that the original length and width of the rectangle are L and W, respectively.
According to the given information, if we decrease the length by 4 cm, the new length becomes L - 4 cm, and if we increase the width by 5 cm, the new width becomes W + 5 cm.
Now, the area of the original rectangle is A = L * W, and the area of the square is given as 40 cm² greater than the area of the rectangle. So, the area of the square is (L - 4) * (W + 5) + 40 cm².
Since the result is a square, the length and width of the resulting square are the same. Therefore, we can set the new length and width of the rectangle as (L - 4) and (W + 5), respectively.
Now, we have two equations:
Equation 1: A = L * W
Equation 2: (L - 4) * (W + 5) + 40 = (L - 4) * (W + 5)
Let's solve these equations to find the area of the rectangle:
Expanding Equation 2:
LW - 4W + 5L - 20 + 40 = LW - 4L + 5W + 20
LW - 4W + 5L - 4L + 5W = 60
Combine like terms:
LW - 4L - 4W + 5L + 5W = 60
LW + L + W = 60
Now let's substitute Equation 1 into the new equation:
L * W + L + W = 60
Rearrange the equation:
LW + L + W - 60 = 0
Factor the equation:
(L + 6)(W + 1) = 66
Since we are looking for the area of the rectangle, which is LW, we need to find the product of two numbers that gives us 66. By trying different combinations, we can find that L + 6 = 11 and W + 1 = 6.
Solving these equations:
L + 6 = 11
L = 11 - 6
L = 5
W + 1 = 6
W = 6 - 1
W = 5
Therefore, the length of the rectangle is 5 cm and the width is 5 cm. Thus, the area of the rectangle is 5 cm * 5 cm = 25 cm².
This wasn't any help. Give me the answer
If the rectangle has length x and width y, and the square has side s, then
s = x-4 = y+5
so, y=x-9
s^2 = xy+40
(x-4)^2 = x(x-9)+40
find x, and then just finish up.
xy