Answer :
Final answer:
The difference in volume between a large scoop and a small scoop of ice cream is approximately 410.5 cubic centimeters.
Explanation:
To find the difference in volume between a large scoop and a small scoop of ice cream, we need to calculate the volume of each scoop and then subtract the volume of the small scoop from the volume of the large scoop.
The volume of a sphere can be calculated using the formula V = (4/3)πr³, where r is the radius. Since the diameter of the small scoop is 6 cm, the radius is 3 cm. Plugging this into the formula, we get V = (4/3)π(3 cm)³. Evaluating this expression, we find that the volume of the small scoop is approximately 113.1 cm³.
Similarly, the diameter of the large scoop is 10 cm, so the radius is 5 cm. Using the same formula, we find that the volume of the large scoop is approximately 523.6 cm³.
To find the difference in volume, we subtract the volume of the small scoop from the volume of the large scoop: 523.6 cm³ - 113.1 cm³ = 410.5 cm³. Therefore, the difference in volume between a large scoop and a small scoop of ice cream is approximately 410.5 cubic centimeters.
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The final answer is 410.5 cubic centimeters.
1. Calculate the volume of a small scoop:
- Given the diameter of the small scoop, [tex]\( d_{\text{small}} = 6 \) cm[/tex].
- Radius of small scoop, [tex]\( r_{\text{small}} = \frac{d_{\text{small}}}{2} = \frac{6}{2} = 3 \)[/tex] cm.
- Volume of a sphere [tex]\( V = \frac{4}{3} \pi r^3 \).[/tex]
- Substitute the radius into the volume formula: [tex]\( V_{\text{small}} = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) \)[/tex].
- Calculate:
[tex]\( V_{\text{small}} = 36 \pi \)[/tex] cubic centimeters.
2. Calculate the volume of a large scoop:
- Given the diameter of the large scoop,[tex]\( d_{\text{large}} = 10 \) cm[/tex].
- Radius of large scoop, [tex]\( r_{\text{large}} = \frac{d_{\text{large}}}{2} = \frac{10}{2} = 5 \) cm[/tex].
- Volume of a sphere [tex]\( V = \frac{4}{3} \pi r^3 \)[/tex].
- Substitute the radius into the volume formula:
[tex]\( V_{\text{large}} = \frac{4}{3} \pi (5)^3 = \frac{4}{3} \pi (125) \)[/tex].
- Calculate:
[tex]\( V_{\text{large}} = 166.7 \pi \)[/tex] cubic centimeters.
3. Find the difference:
- Difference in volume: [tex]\( V_{\text{large}} - V_{\text{small}} = 166.7 \pi - 36 \pi \)[/tex].
- Calculate: [tex]\( V_{\text{large}} - V_{\text{small}} = 130.7 \pi \)[/tex].
- Approximate [tex]\( \pi \)[/tex] to 3.14.
- [tex]\( 130.7 \times 3.14 = 410.498 \)[/tex].
- Rounded to the nearest tenth, the difference is approximately 410.5 cubic centimeters.
