Please help with these questions:
(please show how to do)
1. How many differently shaped rectangles, with positive integer dimensions, have a perimeter equal to their area?
2. Let x be any number less than one, and let y be any number greater than one.
S= x + y
P= xy
Prove that the difference between S and P must be greater than 1.
3. Prove that it is impossible to find four distinct numbers P,Q,R,S which satisfy the equation
pq+rs = ps+qr
There are only two rectangles whose sides are whole numbers and whose area and perimeter are the same. What are they?
The approach to the solution of a problem of this type, where integer answers are sought, is by treating it as a Linear Indeterminate Algebra problem or a Diophantine Problem as follows: Letting x and y be the integer sides of our rectangles, then the area = xy and the perimeter = 2(x+y). Given that xy = 2(x+y) we derive:
1--xy - 2x - 2y = 0 or
2--x = 2y/(y-2)
3--Clearly, both x and 2y/(y-2) must also be integers.
4--Set x = k from which y then becomes 2k/(k-2).
5--Assume values of k and compute x and y.
k...0...1...2...3...4...5....6....7....8....9....10.....50.....100......
x...-....1...2...3...4...5....6....7....8....9....10.....50....100......
y...-...inf..inf..6...4...3+..3....2+..2+..2+...2+...2+.....2+......
Thus 3 and 6, (or 6 and 3), and 4 and 4 are the only possible answers as y never reaches 2 as an integer.
This method is a very powerful tool for determining integer solutions to algebraic equations where there is one more unknown than there are equations. If you are at all interested in learning more about the method, let me know and I will be glad to provide you with some other examples and references
Sure! I can help you with these questions and explain how to solve them.
1. How many differently shaped rectangles, with positive integer dimensions, have a perimeter equal to their area?
To solve this question, we need to find the dimensions of the rectangles that satisfy the given condition.
Let's assume the width of the rectangle is 'w' and the length is 'l'. The perimeter of a rectangle is given by the formula:
Perimeter = 2w + 2l
The area of a rectangle is given by the formula:
Area = w * l
We are given that the perimeter is equal to the area:
2w + 2l = w * l
To simplify this equation, let's divide both sides by 2:
w + l = (w * l) / 2
Now, we need to find the positive integer values for 'w' and 'l' that satisfy this equation. One approach to solve this is by trial and error.
Start with a small value for 'w', such as 1, and try different values for 'l'. Keep on increasing 'w' and trying different 'l' values until you find a pair that satisfies the equation. Keep track of the number of different rectangles you find.
Repeat this process for different values of 'w'. As you increase 'w', you will find more rectangles that satisfy the equation.
Keep in mind that for a rectangle, length and width cannot be equal, so you need to exclude pairs where 'w' equals 'l'.
Remember to count the number of rectangles you find and keep track of them.
2. Prove that the difference between S and P must be greater than 1.
Let's start by using the given equations for 'S' and 'P':
S = x + y
P = xy
To prove that the difference between 'S' and 'P' must be greater than 1, we need to show that S - P > 1.
Substitute the given equations into the inequality:
x + y - xy > 1
Rearrange the terms to simplify the expression:
1 - xy + x + y > 1
1 + x(1 - y) + y > 1
Since both 'x' and 'y' are numbers less than and greater than 1, respectively, we know that x(1 - y) < 0. Therefore, we can simplify further:
1 + x(1 - y) + y > 1
Since 1 + x(1 - y) is less than 1, we can conclude that the difference between S and P must be greater than 1.
3. Prove that it is impossible to find four distinct numbers P, Q, R, S that satisfy the equation pq + rs = ps + qr.
To prove this statement, we can use a proof by contradiction.
Assume that there exist four distinct numbers P, Q, R, and S which satisfy the equation pq + rs = ps + qr.
Rearranging the equation, we get:
pq - ps = qr - rs
p(q - s) = r(q - s)
Since we are given that all four numbers, P, Q, R, and S, are distinct, the quantity (q - s) cannot be zero.
Now, we can divide both sides of the equation by (q - s):
p = r
This result contradicts the assumption that P, Q, R, and S are distinct. Therefore, it is impossible to find four distinct numbers that satisfy the equation pq + rs = ps + qr.
Hence, we can conclude that it is impossible to find four distinct numbers P, Q, R, and S that satisfy the equation pq + rs = ps + qr.