Ice-cream,completely filled in a cylinder of diameter 35 cm and height 32 cm,is to be served by completely filling identical disposable cones of diameter 4 cm and height 7 cm.The maximum number of cones that can be used in this way is
- 950
1000
- 1050
1100
The Correct Option is C
Approach Solution - 1
To determine the maximum number of identical disposable cones that can be filled with ice cream from a cylindrical container, we will calculate the volumes of the cylinder and a single cone and then divide them.
Step 1: Calculate the Volume of the Cylinder
The volume of a cylinder is given by the formula:
\(V = \pi r^2 h\)
Where r is the radius and h is the height of the cylinder.
- Diameter of the cylinder = 35 cm, so radius r = 17.5 cm
- Height of the cylinder h = 32 cm
Substituting the values:
\(V_{\text{cylinder}} = \pi \times (17.5)^2 \times 32\)
Calculating further:
\(V_{\text{cylinder}} = \pi \times 306.25 \times 32 = \pi \times 9800 \text{ cm}^3\)
Step 2: Calculate the Volume of One Cone
The volume of a cone is given by the formula:
\(V = \frac{1}{3} \pi r^2 h\)
Where r is the radius and h is the height of the cone.
- Diameter of the cone = 4 cm, so radius r = 2 cm
- Height of the cone h = 7 cm
Substituting the values:
\(V_{\text{cone}} = \frac{1}{3} \pi \times (2)^2 \times 7 = \frac{1}{3} \pi \times 4 \times 7 = \frac{28}{3} \pi \text{ cm}^3\)
Step 3: Calculate the Number of Cones
To find the number of cones, divide the volume of the cylinder by the volume of one cone:
\(\text{Number of cones} = \frac{V_{\text{cylinder}}}{V_{\text{cone}}} = \frac{\pi \times 9800}{\frac{28}{3} \pi}\)
Canceling \(\pi\) from the numerator and denominator:
\(\text{Number of cones} = \frac{9800 \times 3}{28} = \frac{29400}{28} = 1050\)
Conclusion
Therefore, the maximum number of cones that can be used is 1050.
Approach Solution -2
The volume of the ice cream in the cylinder is:
\[ V_{\text{cylinder}} = \pi r^2 h = \pi \left( \frac{35}{2} \right)^2 \times 32 = \pi \times 17.5^2 \times 32 \approx 3.1416 \times 306.25 \times 32 = 3.1416 \times 9800 = 30787.36 \, \text{cm}^3 \]
The volume of one cone is:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi \left( \frac{4}{2} \right)^2 \times 7 = \frac{1}{3} \pi \times 2^2 \times 7 = \frac{1}{3} \pi \times 4 \times 7 = \frac{28\pi}{3} \approx 29.3215 \, \text{cm}^3 \]
The number of cones that can be filled is:
\[ \text{Number of cones} = \frac{V_{\text{cylinder}}}{V_{\text{cone}}} = \frac{30787.36}{29.3215} \approx 1050 \]
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