Question

A cylindrical glass 7 cm in diameter and 10 cm tall is filled with water to a height of 9 cm.
If a ball 5 cm in diameter is dropped into the class and sinks to the bottom, will the water in the glass overflow?




Perri’s Ice Cream Shoppe is about to advertise giant spherical scoops of ice cream 8 cm in diameter!
Perri wants to be sure there is enough ice cream and wonders how many scoops can be obtained from each cylindrical container of ice cream. The containers are 20 cm in diameter and 26 cm tall.
Determine the number of scoops of ice cream one container will give her?

Answer 1

Let’s tackle the first problem regarding the cylindrical glass and the ball.

### Step 1: Volume of Water in the Glass
The formula for the volume \( V \) of a cylinder is given by:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.

1. **Dimensions of the glass**:
- Diameter = 7 cm → Radius \( r = \frac{7}{2} = 3.5 \) cm
- Height of water \( h = 9 \) cm

2. **Calculate the volume of the water**:
\[
V_{\text{water}} = \pi (3.5)^2 (9) \approx \pi \times 12.25 \times 9 \approx 110.56 \text{ cm}^3
\]

### Step 2: Volume of the Ball
The formula for the volume \( V \) of a sphere is given by:
\[
V = \frac{4}{3} \pi r^3
\]

1. **Diameter of the ball = 5 cm → Radius \( r = \frac{5}{2} = 2.5 \) cm**.

2. **Calculate the volume of the ball**:
\[
V_{\text{ball}} = \frac{4}{3} \pi (2.5)^3 = \frac{4}{3} \pi \times 15.625 \approx \frac{62.83}{3} \approx 20.94 \text{ cm}^3
\]

### Step 3: Total Volume After Dropping the Ball
Now, we will find out if the water overflows when the ball is submerged.

1. **Calculate the combined volume**:
\[
V_{\text{total}} = V_{\text{water}} + V_{\text{ball}} \approx 110.56 + 20.94 \approx 131.50 \text{ cm}^3
\]

### Step 4: Volume Capacity of the Glass
We need to check the maximum volume the glass can hold.

1. **Height of the glass = 10 cm**:
\[
V_{\text{glass}} = \pi (3.5)^2 (10) \approx \pi \times 12.25 \times 10 \approx 384.69 \text{ cm}^3
\]

### Step 5: Check for Overflow
- The volume of water and the ball combined (131.50 cm³) is less than the maximum volume of the glass (384.69 cm³).
- Therefore, **the water will not overflow** when the ball is dropped into the glass.

---

## Now let's tackle the ice cream container problem.

### Step 1: Volume of One Scoop of Ice Cream
1. **Diameter of a scoop = 8 cm → Radius \( r = \frac{8}{2} = 4 \) cm**.

2. **Calculate the volume of one scoop**:
\[
V_{\text{scoop}} = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi \times 64 \approx \frac{256 \pi}{3} \approx 268.08 \text{ cm}^3
\]

### Step 2: Volume of Ice Cream Container
1. **Diameter of the container = 20 cm → Radius \( r = \frac{20}{2} = 10 \) cm**.
2. **Height of the container = 26 cm**.

3. **Calculate the volume of the container**:
\[
V_{\text{container}} = \pi (10)^2 (26) = \pi \times 100 \times 26 = 2600 \pi \approx 8171.46 \text{ cm}^3
\]

### Step 3: Calculate Number of Scoops
1. **Number of scoops**:
\[
\text{Number of scoops} = \frac{V_{\text{container}}}{V_{\text{scoop}}} \approx \frac{8171.46}{268.08} \approx 30.5
\]

Since she can't have a fraction of a scoop, **Perri can obtain 30 scoops of ice cream from one container.**