A bakery offers a small circular cake with a diameter of 8 inches. It also offers a large circular cake with a diameter of 24 inches. Does the top of the large cake have three times the area of the small cake? If not, how much greater is its area? Explain.

A = pi * r^2

What is the area of each cake?

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To determine if the top of the large cake has three times the area of the small cake, we need to calculate the areas of both cakes and compare them.

The formula to calculate the area of a circle is A = π*r^2, where A represents the area and r represents the radius. However, we are given the diameter of the cakes, so we need to convert the diameters to radii.

The radius of the small cake can be found by dividing the diameter by 2:
Radius of small cake = Diameter of small cake / 2 = 8 inches / 2 = 4 inches

The radius of the large cake can be found the same way:
Radius of large cake = Diameter of large cake / 2 = 24 inches / 2 = 12 inches

Now, we can calculate the areas of both cakes using the formula A = π*r^2:

Area of small cake = π*(4 inches)^2
Area of small cake = π*16 square inches
Area of small cake ≈ 50.27 square inches (rounded to 2 decimal places)

Area of large cake = π*(12 inches)^2
Area of large cake = π*144 square inches
Area of large cake ≈ 452.39 square inches (rounded to 2 decimal places)

So, the top of the large cake does not have three times the area of the small cake. To determine how much greater its area is, we subtract the area of the small cake from the area of the large cake:

Difference in area = Area of large cake - Area of small cake
Difference in area = 452.39 square inches - 50.27 square inches
Difference in area ≈ 402.12 square inches (rounded to 2 decimal places)

Therefore, the area of the top of the large cake is approximately 402.12 square inches greater than the area of the small cake.