Tina has a jar 20 cm high. She can fit 30 candies flat on the bottom of the jar. Each candy is 1 cm high. How many candies can she fit into the jar if...a) the jar is a cylinder?b) the jar is a right prism?Are the two answers the same? Why does this happen?
a) If the jar is a cylinder, we can use the formula for the volume of a cylinder: V = πr^2h, where r is the radius and h is the height.
Since the height of the jar is 20 cm, and each candy is 1 cm high, there's enough room for 20 candies stacked on top of each other. So, the total height of the candies is 20 cm.
To find the radius of the cylinder, we need to divide its volume by its height:
V = πr^2h
V/h = πr^2
r^2 = V/(πh)
r = sqrt(V/(πh))
Plugging in the numbers, we get:
r = sqrt((30 candies x 1 cm/candy)/(π x 20 cm)) ≈ 0.84 cm
Now, we can use the formula for the area of a circle to find the number of candies that can fit on the bottom of the jar:
A = πr^2
A = π x (0.84 cm)^2 ≈ 2.22 cm^2
Since each candy is also 2.22 cm^2, we can fit 30 candies on the bottom of the jar.
To find the total number of candies that can fit in the jar, we need to multiply the number of candies on the bottom by the height of the jar (20 cm):
Number of candies = 30 candies x 20 cm
Number of candies = 600 candies
b) If the jar is a right prism, we can use the formula for the volume of a prism: V = Bh, where B is the area of the base and h is the height.
Since the candies are flat on the bottom, they form a rectangle with dimensions 2.22 cm x 10 cm (since 30 candies fit on the bottom, and each candy is 2.22 cm^2). So, the area of the base is:
B = 2.22 cm x 10 cm = 22.2 cm^2
Now, we can use the same formula as before to find the number of candies that can fit in the jar:
V = Bh
V = 22.2 cm^2 x 20 cm
V = 444 candies
Are the two answers the same?
No, the two answers are different. When the jar is a cylinder, we can fit more candies because the diameter of the base is larger than when the jar is a right prism. This means that the area of the circle (which determines how many candies can fit on the bottom) is larger, so we can fit more candies overall.