Wen has (9x²+24xy+16y²) marbles,where x and y are positive integers.Ha arranges the marbles as a square array. a)Express,in terms of x and y,the number of marbles on each side of the array. b)When x=2 and y=5,find the number of marbles on a side of the array
+500 Hey guest!
Let's first start off with an observation:
\(9x^{2}+24xy + 16y^2 = (3x+4y)^2\)
As such,
Part A) Because it says he arranges the marble as a square array, it makes sense that the total is a square number(of which it is). Since we know that
\(9x^{2}+24xy + 16y^2 = (3x+4y)^2 \), that's how many total marbles there are. To get the number of marbles on each side of the square array, we just take the square root of given number, which gives us:
\(\sqrt{(3x+4y)^2} = 3x+4y\) marbles on each side of the array. The key takeaway here is that since the total number of marbles is a square number, that means since in a square array each side has an equal amount of marbles, the number squared must equal the total. To get the number, we just square root the total to get how many are on each side.
Part B) Since we already have the equation \(3x+4y\) as the number of marbles on each side of the square array, this problem becomes trivialized: it then becomes a simple question of substitution, where we get:
\(3*(2) + 4*(5) = 6 + 20 = 26\)marbles on each side of the array
