a piece of rope 80 centimeters long must be cut into 2 pieces. Each piece of rope will be used to form a rectangle. Where would the rope be cut if the area of 1 rectangle must be 1/4 the area of the other rectangle
To solve this problem, we need to find the point on the original 80-centimeter rope where it should be cut so that one of the resulting pieces can form a rectangle with an area equal to one-fourth the area of the other rectangle.
Let's start by assigning variables to the lengths of the two pieces of rope that will be cut. We'll call the length of the first piece x centimeters. Since the original rope is 80 centimeters long, the length of the second piece will be (80 - x) centimeters.
Now, let's use the formula for the area of a rectangle which is length multiplied by width. In the problem, one rectangle's area is one-fourth the area of the other rectangle. Therefore, we can set up the equation:
(x)(80 - x) = (1/4)(x)(80 - x)
Now, let's simplify this equation:
80x - x^2 = 20x - (1/4)x^2
Next, multiply both sides of the equation by 4 to get rid of the fraction:
320x - 4x^2 = 80x - x^2
Moving all the terms to one side of the equation:
4x^2 - 240x = 0
Factor out x:
x(4x - 240) = 0
Now, solve for x by setting each factor equal to zero:
x = 0
4x - 240 = 0
The first solution, x = 0, doesn't make sense in this context because it would mean not cutting the rope at all. Therefore, we can ignore that solution.
Now, solve for the second factor:
4x - 240 = 0
4x = 240
x = 240/4
x = 60
So, the rope should be cut at 60 centimeters from one end, resulting in one piece measuring 60 centimeters and the other piece measuring (80 - 60) = 20 centimeters.