Paul wants to make 150 ice cream cups, each with two scoops of chocolate ice cream and one scoop of vanilla ice cream. The chocolate ice cream is stored in a rectangular prism, and the vanilla ice cream is stored in a cylinder. We are given the dimensions of the chocolate container (14cm x 22cm x 16cm) and the vanilla container (height 18cm, diameter 16cm). The question asks how many chocolate containers and how many vanilla containers Paul needs to buy. We are not told the volume of each scoop. The scoops are spherical with diameter 3.6 cm.

1. Problem Description

Paul wants to make 150 ice cream cups, each with two scoops of chocolate ice cream and one scoop of vanilla ice cream. The chocolate ice cream is stored in a rectangular prism, and the vanilla ice cream is stored in a cylinder. We are given the dimensions of the chocolate container (14cm x 22cm x 16cm) and the vanilla container (height 18cm, diameter 16cm). The question asks how many chocolate containers and how many vanilla containers Paul needs to buy. We are not told the volume of each scoop. The scoops are spherical with diameter 3.6 cm.

2. Solution Steps

First, we calculate the volume of a single ice cream scoop. The radius, $r$, is half the diameter:

$r = 3.6/2 = 1.8$ cm.

The volume of a sphere is given by:

$V = \frac{4}{3}\pi r^3$

Plugging in the value of $r$:

$V = \frac{4}{3}\pi (1.8)^3 = \frac{4}{3}\pi (5.832) = 7.776\pi \approx 24.429$ cm$^3$

Since each ice cream cup has two chocolate scoops and one vanilla scoop, the total volume of chocolate ice cream needed for 150 cups is:

$150 \times 2 \times 24.429 = 7328.7$ cm$^3$

The total volume of vanilla ice cream needed for 150 cups is:

$150 \times 1 \times 24.429 = 3664.35$ cm$^3$

Now, let's calculate the volume of the chocolate container (rectangular prism):

$V_{chocolate} = length \times width \times height = 22 \times 16 \times 14 = 4928$ cm$^3$

To find out how many chocolate containers Paul needs, divide the total chocolate volume needed by the volume of one container:

$\frac{7328.7}{4928} \approx 1.487$

Since Paul cannot buy parts of a container, he needs to buy 2 chocolate containers.

Next, we calculate the volume of the vanilla container (cylinder):

$V_{vanilla} = \pi r^2 h$

The radius is half the diameter: $r = 16/2 = 8$ cm

$V_{vanilla} = \pi (8)^2 (18) = \pi (64)(18) = 1152\pi \approx 3619.11$ cm$^3$

To find out how many vanilla containers Paul needs, divide the total vanilla volume needed by the volume of one container:

$\frac{3664.35}{3619.11} \approx 1.012$

Since Paul cannot buy parts of a container, he needs to buy 2 vanilla containers.

3. Final Answer

Paul needs to buy 2 chocolate containers and 2 vanilla containers.