Two rectangular pieces of wood are
similar. The ratio of the perimeters of
the two pieces is 2 : 3. If the area of the
smaller piece is 12 square inches, what
is the area of the larger piece?
please explain!
You have to set up a proportion.
2/3 = 12/x
solve for x
2x=36
x=18
To find the area of the larger piece, we need to determine the ratio of the areas of the two rectangular pieces.
First, let's find the dimensions (length and width) of the smaller piece. We know its area is 12 square inches, so let's denote its length as 'l' and its width as 'w'.
Area of the smaller piece = length * width = lw = 12
Now, let's find the dimensions of the larger piece. Since the two pieces are similar, their dimensions are proportional. Let's denote the length of the larger piece as 'L' and its width as 'W'.
The ratio of the perimeters is given as 2 : 3. The perimeter of a rectangle is calculated as 2 * (length + width), so we can write:
(2 * l) / (2 * w) = 2 / 3
Simplifying this equation gives:
l / w = 2 / 3
Since we found that lw = 12, we can substitute this into the equation:
12 / w = 2 / 3
Cross-multiplying gives:
36 = 2w
Dividing both sides by 2 gives:
w = 18
Substituting the value of w back into the equation lw = 12:
l * 18 = 12
Dividing both sides by 18 gives:
l = 12 / 18 = 2/3
So, the dimensions of the smaller piece are l = 2/3 and w = 18.
Now, let's find the dimensions of the larger piece using the same ratio:
L / W = l / w = (2/3) / 18
Cross-multiplying gives:
3L = 2W
Since we found that lw = 12, we can substitute this into the equation:
3L = 2 * 18
Dividing both sides by 2 gives:
L = 12
So, the length of the larger piece is L = 12.
Now, let's find the width of the larger piece:
w = 18
Finally, we can calculate the area of the larger piece:
Area of the larger piece = Length * Width = L * w = 12 * 18
Calculating this gives:
Area of the larger piece = 216 square inches