A bakery offers a small circular cake with a diameter of 6 inches. It also offers a large circular cake with a diameter of 18 inches. Complete the explanation for whether the top of the large cake has three times the area of that of the small cake.
The area of the top of the large cake is
in2 and that of the small cake is
in2. The top of the large cake has an area equal to
times that of the small cake.
area is related to the square of the diameter
three times the diameter means nine times the area
The area of a circle is calculated using the formula A = πr^2, where A is the area and r is the radius of the circle.
To find the area of the small cake, we need to find the radius first. The diameter of the small cake is given as 6 inches, so the radius would be half of that, which is 3 inches.
Using the formula for the area of a circle, we can calculate the area of the small cake:
A_small = π * (3 inches)^2
Simplifying this:
A_small = π * 9 inches^2
The area of the small cake is 9π square inches.
Next, let's find the area of the large cake. The diameter of the large cake is given as 18 inches, so the radius would be half of that, which is 9 inches.
Using the formula for the area of a circle, we can calculate the area of the large cake:
A_large = π * (9 inches)^2
Simplifying this:
A_large = π * 81 inches^2
The area of the large cake is 81π square inches.
Now, to determine whether the top of the large cake has three times the area of the small cake, we need to compare the two areas.
The ratio of the area of the large cake to the area of the small cake is:
A_large / A_small = (81π square inches) / (9π square inches)
Simplifying this:
A_large / A_small = 9
Therefore, the top of the large cake has an area equal to 9 times the area of the small cake.
To determine whether 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.
To find the area of a circular cake, we use the formula: Area = π * (radius)^2.
Given that the diameter of the small cake is 6 inches, we can calculate the radius by dividing the diameter by 2: radius = 6 inches / 2 = 3 inches.
So, the area of the small cake is: Area of small cake = π * (3 inches)^2.
Similarly, for the large cake, the diameter is 18 inches, and thus the radius is: radius = 18 inches / 2 = 9 inches.
The area of the large cake is then: Area of large cake = π * (9 inches)^2.
To compare the areas, we need to find the ratio of the area of the large cake to the area of the small cake. So, we have:
Area of large cake / Area of small cake = (π * (9 inches)^2) / (π * (3 inches)^2).
Now, we can cancel out the common factor of π and simplify the equation:
Area of large cake / Area of small cake = (9 inches)^2 / (3 inches)^2.
Simplifying further, we have:
Area of large cake / Area of small cake = (9 inches * 9 inches) / (3 inches * 3 inches).
Calculating:
Area of large cake / Area of small cake = 81 square inches / 9 square inches = 9.
Therefore, the top of the large cake has an area equal to 9 times that of the small cake. It is not three times but nine times larger.