The volume of a container measures 4,000 mm3

Question

and contains 50 marbles. Each marble has a radius of 2 mm. Use 3.14 as an approximation for pi. How much space in the box is unused? Round your answer to the nearest tenth.(1 point)
Responses

33.5 mm3
33.5 mm cubed

2,325.3 mm3
2,325.3 mm cubed

1,674.7 mm3
1,674.7 mm cubed

3,900 mm3

Answer 1

To find the unused space in the container, we first need to calculate the total volume occupied by the marbles and then subtract this from the total volume of the container.

1. Calculate the volume of one marble: The formula for the volume of a sphere is given by: \[ V = \frac{4}{3} \pi r^3 \] where \( r \) is the radius. Given that the radius of each marble is 2 mm, we can calculate the volume of one marble as follows:

   \[ V = \frac{4}{3} \times 3.14 \times (2)^3 \] \[ V = \frac{4}{3} \times 3.14 \times 8 \] \[ V = \frac{4}{3} \times 25.12 \] \[ V = \frac{100.48}{3} \approx 33.49 \text{ mm}^3 \]

2. Calculate the total volume occupied by 50 marbles: \[ \text{Total volume of 50 marbles} = 50 \times 33.49 \text{ mm}^3 \approx 1674.7 \text{ mm}^3 \]

3. Calculate the unused space in the container: The total volume of the container is given as 4,000 mm³. Therefore, the unused space can be calculated as follows: \[ \text{Unused space} = \text{Total volume of container} - \text{Total volume of 50 marbles} \] \[ \text{Unused space} = 4000 \text{ mm}^3 - 1674.7 \text{ mm}^3 \approx 2325.3 \text{ mm}^3 \]

Thus, the amount of unused space in the box, rounded to the nearest tenth, is 2,325.3 mm³.

Answer 2

Kairay created a sculpture by forming a hemisphere with a radius of 20 inches and then removing a hemisphere with a radius of 15 inches from it. Calculate the volume of his sculpture. Use 3.14 as an approximation for pi. Round your answer to the nearest whole number.(1 point)
Responses

9,682 in.3
9,682 inches cubed

7,065 in.3
7,065 inches cubed

19,363 in.3
19,363 inches cubed

16,747 in.3

Answer 3

To find the volume of Kairay's sculpture, we will first calculate the volume of the larger hemisphere (radius of 20 inches) and then subtract the volume of the smaller hemisphere (radius of 15 inches).

Step 1: Calculate the volume of the larger hemisphere.

The formula for the volume of a hemisphere is given by: \[ V = \frac{2}{3} \pi r^3 \]

For the larger hemisphere with a radius of \( r = 20 \) inches: \[ V_{large} = \frac{2}{3} \times 3.14 \times (20)^3 \] Calculating \( (20)^3 \): \[ (20)^3 = 8000 \]

Now substituting back into the volume formula: \[ V_{large} = \frac{2}{3} \times 3.14 \times 8000 \] \[ V_{large} = \frac{2 \times 3.14 \times 8000}{3} = \frac{50240}{3} \approx 16746.67 \text{ in}^3 \]

Step 2: Calculate the volume of the smaller hemisphere.

For the smaller hemisphere with a radius of \( r = 15 \) inches: \[ V_{small} = \frac{2}{3} \times 3.14 \times (15)^3 \] Calculating \( (15)^3 \): \[ (15)^3 = 3375 \]

Now substituting back into the volume formula: \[ V_{small} = \frac{2}{3} \times 3.14 \times 3375 \] \[ V_{small} = \frac{2 \times 3.14 \times 3375}{3} = \frac{21165}{3} \approx 7055 \text{ in}^3 \]

Step 3: Calculate the volume of the sculpture.

Now, subtract the volume of the smaller hemisphere from the volume of the larger hemisphere: \[ V_{sculpture} = V_{large} - V_{small} \] \[ V_{sculpture} \approx 16746.67 - 7055 \approx 9691.67 \text{ in}^3 \]

Step 4: Round the result to the nearest whole number.

Rounding \( 9691.67 \) to the nearest whole number, we get: \[ V_{sculpture} \approx 9692 \text{ in}^3 \]

It's important to notice that the responses given are:

• 9,682 in³
• 7,065 in³
• 19,363 in³
• 16,747 in³

The closest response to our calculation (9692 in³) is 9,682 in³.

Thus, the volume of Kairay's sculpture is approximately \( \textbf{9,682 in³} \).