We want to find how many times greater the base area of the actual merry-go-round is than the base area of the similar scale model. Recall that when two figures are similar, the value of the ratio of their areas is equal to the square of the value of the ratio of their corresponding sides lengths.
A_1/A_2=(r_1/r_2)^2Here, the subscript 1 represent the actual merry-go-round and 2 indicates the model. Note that the base area is a circumference. Then, the side length is the radius of each merry-go-round. Since the radius of the scale model is given in inches, we need to convert the radius of the actual merry-go-round from feet to inches by multiplying r_1 by 12.
A_1/A_2=(12* r_1/r_2)^2
Let's substitute r_1= 10 and r_2= 6 into the above equation to calculate the ratio of the base areas.
This means that the area of the base of the actual merry-go-round is 400 times greater than that of the scale model.
Now we want to find the base area of the actual merry-go-round. Let's use the ratio of the base areas again.
A_1/A_2=(r_1/r_2)^2
We know that A_2 represents the base area of the scale model, so A_2=450 square inches. From Part A, we also know that the ratio of the areas is 400.
A_1/450=400
Here, A_1 is equal to the base area of the actual merry-go-round. Let's solve this equation for A_1.
Finally, let's convert from square inches to square feet by using the fact that 1 square foot is equal to 12*12=144 square inches.
1 ft^2/144in.^2
We will multiply 180 000 square inches by this conversion factor to convert the units of the area.
