To solve this problem, we’ll calculate the initial volume of water in the cylinder and the total volume of water needed to fill the six cones. Then, we’ll find out how much water remains in the cylinder.

Step 1: Calculate the volume of water in the cylinder

The formula for the volume of a cylinder is: $V = \pi r^2 h$ where:

• $r$ is the radius of the cylinder,
• $h$ is the height of the water in the cylinder.

For this cylinder:

• Diameter = 30 cm, so $r = 15$ cm,
• Height of water $h = 30$ cm.

So, $V_{\text{cylinder}} = \pi \times (15)^2 \times 30 = 6750\pi \, \text{cm}^3$

Step 2: Calculate the volume of one cone

The volume of a cone is given by: $V = \frac{1}{3} \pi r^2 h$ For each cone:

• Diameter = 10 cm, so $r = 5$ cm,
• Height $h = 18$ cm.

Thus, $V_{\text{cone}} = \frac{1}{3} \pi \times (5)^2 \times 18 = 150\pi \, \text{cm}^3$

Step 3: Calculate the total volume for six cones

Since there are six cones: $V_{\text{total cones}} = 6 \times 150\pi = 900\pi \, \text{cm}^3$

Step 4: Determine the remaining volume of water in the cylinder

The remaining volume of water in the cylinder will be: $V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{total cones}} = 6750\pi - 900\pi = 5850\pi \, \text{cm}^3$

Step 5: Calculate the new height of water in the cylinder

Since the volume of water left in the cylinder is now $5850\pi \, \text{cm}^3$, we can use the volume formula of a cylinder to find the new height $h_{\text{new}}$: $V_{\text{remaining}} = \pi r^2 h_{\text{new}}$ $5850\pi = \pi \times (15)^2 \times h_{\text{new}}$ $5850 = 225 \times h_{\text{new}}$ $h_{\text{new}} = \frac{5850}{225} = 26 \, \text{cm}$

Final Answer

The new height of the water in the cylinder after filling the six cones is: 26 cm.

Answer: (c) 26 cm

Would you like further details or have any questions?

Here are 5 questions to expand understanding:

1. How would the solution change if the cylinder had a different radius or height?
2. What is the effect of the number of cones on the final water level in the cylinder?
3. How would the answer change if each cone had a different height or radius?
4. Can we use similar shapes to estimate the answer without calculations?
5. How is the volume formula for a cone derived?

Tip: Always check units and make sure that all measurements are consistent before starting any calculation.