To repair his kitchen subfloor, John cut a rectangular piece of plywood such that width and length were two consecutive odd integers. If the product of the length and width is 99, what were the dimensions of the piece of plywood?
c'mon. You know that 99 = 9*11
So, what might the two consecutive odd integers be? ...
To find the dimensions of the plywood, we can set up a system of equations based on the given information.
Let's assume that the width is represented by the variable "x." According to the problem, the length is "two consecutive odd integers." So, the length can be represented as (x + 2).
The problem also states that the product of the length and width is 99. Therefore, we have the equation:
width * length = 99
Substituting the variables, we get:
x * (x + 2) = 99
Expanding the expression, we have:
x^2 + 2x = 99
Rearranging the equation to set it equal to zero, we get:
x^2 + 2x - 99 = 0
Now, we can solve this quadratic equation to find the value of "x" (the width). We can either factor the equation or use the quadratic formula.
If we factor the equation, we need to find two numbers whose sum is 2 and whose product is -99. After some trial and error, we find that the two numbers are +11 and -9.
Therefore, the factored form of the equation is:
(x + 11)(x - 9) = 0
Setting each factor equal to zero, we get:
x + 11 = 0 or x - 9 = 0
Solving for "x," we find:
x = -11 or x = 9
Since the width cannot be negative, we discard the solution x = -11.
Therefore, the width (x) is 9.
To find the length (x + 2), we substitute this value back into the expression:
Length = x + 2 = 9 + 2 = 11
So, the dimensions of the plywood are 9 by 11.