Answer:
Length = 25cm; Width = 10cm.
Explanation:
My method is to use simultaneous equations.
Let x be the length of the rectangle and y be the width.
The perimeter of a rectangle is (2 x length) + (2 x width).
Substituting our values in we can say that 70 = 2x + 2y.
This is our first of two equations.
Next, we find an equation using the area of the rectangle.
The length decreases by 5, so we can call this x - 5.
The width increases by 5, so so we can call this y + 5.
The area of a rectangle is simply length x width, so in our original rectangle, the area would be xy.
However, with our new lengths the area increases by 50, allowing us to set up an equation where xy + 50 = (x - 5)(y + 5)
We now have two simultaneous equations:
1) 70 = 2x + 2y
2) xy + 50 = (x - 5)(y + 5)
We rearrange equation 1 to make y the subject.
This gives us y = 35 - x.
We can now substitute this into equation 2.
x(35 - x) + 50 = (x - 5)(35 - x + 5)
-x^2 + 35x + 50 = (x - 5)(40 - x) [expand the left and side and collect like terms in the second bracket on the RH]
-x^2 + 35x + 50 = -x^2 + 45x - 200 [expand the RH]
35x + 50 = 45x - 200 [The -x^2’s cancel out each other]
250 = 10x
x = 25cm
Substitute this into equation 1.
70 = 2(25) + 2y
70 = 50 + 2y
2y = 20
y = 10cm