Grades 11-12 Video Solutions 2013-Levels 11&12 Video Solutions 2013 problem16

Video Transcription

Question number 16, consider a rectangle one of whose sides has length of 5.  The rectangle can be cut into a square and a rectangle, one of which has an area of 4.  How many such rectangles exist?  What we notice is that we have two cases, either the width of the rectangle is 5 or its length.  And so, here in case number 1, the square has length x, the rectangle has width 5 and the smaller rectangle width 5 minus x. So we can write down the corresponding areas.  The square would have an area of x squared, which can possibly be equal to 4. And the rectangle has an area of x times 5 minus x, which can also be equal to 4.  Now, in the second case, we have our square having side length 5, so its area is now 25.  And the smaller rectangle has an area of 5 times x, which can possibly be equal to 4.  So there will be as many rectangles as there are unique values of x. And so, what we have is some equations to solve. Either x squared is equal to 4, so that x is equal to 2.  Or 5x is equal to 4, so that x is equal to 4 over 5.  Or finally, we have x times quantity 5 minus x is equal to 4, and this we have to factor.  So we have x squared plus 5x minus 4 equal to 0, or equivalently, x squared minus 5x plus 4 is equal to 0, which factors into x minus 4, x minus 1.  And so, x can be either positive 4 or positive 1. And so, we have 1, 2, 3, 4 possible values of x,  and each corresponds to a unique rectangle  with the prescribed properties. So there are 4 such rectangles.

Video Summary

There are four unique rectangles that can be created when a rectangle with one side of length 5 is cut into a square and another rectangle, where one piece has an area of 4. The cases considered include different arrangements of the square and rectangle in terms of sides and constraints on areas. Solving for different values of \(x\) (where \(x\) represents possible side lengths), four solutions are found: \(x = 2\), \(x = \frac{4}{5}\), \(x = 4\), and \(x = 1\). Each value corresponds to a distinct rectangle configuration.

Keywords

rectangle

square

area

side length

solutions