A room is a square with integer dimensions. it is made 3 feet wider and 2 feet deeper. the increase in area is 46 square feed. What were the dimensions of the old room?
original: x by x
area = x^2
new dimensions: (x+3) and (x+2)
new area = (x+3)(x+2) = x^2 + 5x + 6
x^2 + 5x + 6 - x^2 = 46
5x = 40
x = 8
original was 8 by 8
check:
old area = 64
new is 11 by 10
new area = 110
change in area = 110-64 = 46 , yeahh!
To solve this problem, we need to set up an equation based on the given information. Let's assume the original dimensions of the room were x feet by y feet.
The room is made 3 feet wider, so the new width would be (x + 3) feet.
The room is also made 2 feet deeper, so the new depth would be (y + 2) feet.
The increase in area can be calculated by subtracting the original area from the new area. The original area is x * y, and the new area is (x + 3) * (y + 2).
Therefore, we can set up the equation: (x + 3) * (y + 2) - x * y = 46.
Expanding this equation gives us: xy + 2x + 3y + 6 - xy = 46.
Simplifying further, we have: 2x + 3y + 6 = 46.
Subtracting 6 from both sides gives us: 2x + 3y = 40.
Since the dimensions of the room are integers, we can start by assuming values for x and solve for y.
Let's try assuming x = 1. Plugging this into the equation, we get: 2(1) + 3y = 40.
Simplifying further gives us: 2 + 3y = 40.
Subtracting 2 from both sides gives us: 3y = 38.
Dividing both sides by 3, we get: y ≈ 12.67.
Since y is not an integer, the assumption x = 1 is incorrect.
Let's try assuming x = 2. Plugging this into the equation, we get: 2(2) + 3y = 40.
Simplifying further gives us: 4 + 3y = 40.
Subtracting 4 from both sides gives us: 3y = 36.
Dividing both sides by 3, we get: y = 12.
Therefore, the original dimensions of the room were 2 feet by 12 feet.