Question

Shown below is an ice-cream cone completely filled with ice-cream. Assume the top portion (visible portion of ice-cream) is a perfect hemisphere. What is the total mass of the ice cream in grams? The density of ice-cream = 0.9 grams/cubic centimeter. Consider the value of $\pi$ = 3.14.

Hint:

Factorizing common terms like $\pi R^2$ first before inserting decimal values helps avoid arithmetic errors and speeds up computation during time-constrained exams.

Answer 1

Correct Answer: 127.17

Step 1: Understanding the Question:
This question asks for the total mass of the ice cream contained within a composite solid.
The solid is composed of:
1. A right circular cone of height $H = 9\text{ cm}$ and diameter $D = 6\text{ cm}$.
2. A hemisphere of radius $R = 3\text{ cm}$ resting on top of the cone.

Step 2: Key Formula or Approach:
To find the total mass, we calculate the combined volume of the cone and the hemisphere, and then multiply this volume by the given density of the ice cream.
1. Volume of a hemisphere:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi R^3 \]
2. Volume of a cone:
\[ V_{\text{cone}} = \frac{1}{3} \pi R^2 H \]
3. Total Volume:
\[ V_{\text{total}} = V_{\text{hemisphere}} + V_{\text{cone}} \]
4. Total Mass:
\[ \text{Mass} = V_{\text{total}} \times \text{density} \]

Step 3: Detailed Explanation:
1. Calculating the Volume of the Hemisphere:
- The radius is $R = \frac{D}{2} = \frac{6}{2} = 3\text{ cm}$.
\[ V_{\text{hemisphere}} = \frac{2}{3} \times 3.14 \times 3^3 = \frac{2}{3} \times 3.14 \times 27 = 18 \times 3.14 = 56.52\text{ cm}^3 \]
2. Calculating the Volume of the Cone:
- The radius of the cone's base is equal to the hemisphere's radius, $R = 3\text{ cm}$.
- The height of the cone is $H = 9\text{ cm}$.
\[ V_{\text{cone}} = \frac{1}{3} \times 3.14 \times 3^2 \times 9 = 3 \times 3.14 \times 9 = 27 \times 3.14 = 84.78\text{ cm}^3 \]
3. Calculating the Total Volume:
\[ V_{\text{total}} = 56.52 + 84.78 = 141.30\text{ cm}^3 \]
- Alternatively, we can factorize the volume equation:
\[ V_{\text{total}} = \pi R^2 \left(\frac{2}{3}R + \frac{1}{3}H\right) = 3.14 \times 9 \times \left(2 + 3\right) = 3.14 \times 45 = 141.30\text{ cm}^3 \]
4. Calculating the Total Mass:
- Given density $\rho = 0.9\text{ g/cm}^3$:
\[ \text{Mass} = 141.30\text{ cm}^3 \times 0.9\text{ g/cm}^3 = 127.17\text{ grams} \]

Step 4: Final Answer:
The total mass of the ice cream is 127.17 grams.