To repair his kitchen subfloor, John cut a rectangular piece of plywood such that the width and length were two consecutive odd integers. If the product of the length and width is 575
, what were the dimensions of the piece of plywood?
well, √575 = 23.9
So look for odd numbers close to 24.
Also, note that 24^2 = 576, so
575 = 24^2 - 1^1
To find the dimensions of the plywood, we can set up an equation based on the given information.
Let's assume the width to be x and the length to be x+2, since they are consecutive odd integers.
Given that the product of the length and width is 575, we have the equation:
x * (x+2) = 575
Now, we can solve this quadratic equation to find the value of x.
Expanding the equation:
x^2 + 2x = 575
Rearranging the equation:
x^2 + 2x - 575 = 0
Now, we can solve this quadratic equation using various methods such as factoring, completing the square, or using the quadratic formula.
Considering that the factors of 575 are 1, 5, 23, and 25, we can try to factorize the equation.
Factoring the quadratic equation:
(x + 25)(x - 23) = 0
This gives two possible solutions: x = -25 or x = 23.
Since the width cannot be negative, we discard x = -25 and consider x = 23.
Therefore, the width of the plywood is 23, and the length is x+2 = 23+2 = 25.
Hence, the dimensions of the plywood are 23 by 25.