A small silver dollar pancake served at a restaurant has a circumference of 2 pi inches. A regular pancake has a circumference of 4 pi inches. Is the area of the regular pancake twice the area of the silver dollar pancake? Explain.
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Follow a similar analysis to the one I gave you in your previous post.
e.g. what is the radius of the circle with circumference 2π ?
A small silver dollar pancake served at a restaurant has a circumference of 2 pi inches. A regular pancake has a circumference of 4 pi inches. Is the area of the regular pancake twice the area of the of the silver dollar pancake?
To determine whether the area of the regular pancake is twice the area of the silver dollar pancake, we need to compare the formulas for the area of each pancake.
The area of a circle can be calculated using the formula A = π * r^2, where A represents the area and r represents the radius.
However, since we are given the circumferences of the pancakes and not the radii, we need to find the radius first.
The circumference of a circle can be calculated using the formula C = 2 * π * r, where C represents the circumference and r represents the radius.
Given that the silver dollar pancake has a circumference of 2π inches, we can rearrange the formula to solve for the radius:
C = 2 * π * r
2π = 2 * π * r
r = 1 inch
Now, we have the radius of the silver dollar pancake, which is 1 inch.
Using the same approach for the regular pancake, we can find its radius.
Given that the regular pancake has a circumference of 4π inches, we can set up the equation:
C = 2 * π * r
4π = 2 * π * r
2r = 4
r = 2 inches
Now, we have the radius of the regular pancake, which is 2 inches.
To find the areas of the pancakes, we substitute the radii into the area formula:
Area of silver dollar pancake = π * (1)^2 = π square inches
Area of regular pancake = π * (2)^2 = 4π square inches
Comparing the two areas, we can see that the area of the regular pancake is four times the area of the silver dollar pancake, not twice the area. This is because the area of a circle is proportional to the square of its radius.
To determine if the area of the regular pancake is twice the area of the silver dollar pancake, we need to compare their areas.
Let's start by finding the radius of each pancake.
The circumference of a circle is given by the formula C = 2πr, where C is the circumference and r is the radius.
For the small silver dollar pancake, the circumference is given as 2π inches, so we can set up the equation:
2π = 2πr
Simplifying the equation, we find that r = 1 inch.
Similarly, for the regular pancake, the circumference is given as 4π inches, so we have:
4π = 2πr
Again, simplifying the equation, we find that r = 2 inches.
Now that we have the radii of the pancakes, we can calculate their areas.
The area of a circle is given by the formula A = πr², where A is the area and r is the radius.
For the small silver dollar pancake, the radius is 1 inch, so the area is:
A = π(1)² = π square inches
For the regular pancake, the radius is 2 inches, so the area is:
A = π(2)² = 4π square inches
Now, let's compare the areas. We want to determine whether the area of the regular pancake is twice the area of the silver dollar pancake.
Area of regular pancake / Area of silver dollar pancake = (4π) / (π) = 4
The ratio of the areas is 4. Therefore, the area of the regular pancake is four times the area of the silver dollar pancake, not twice.