To solve this problem, we need to determine the value of $h$, the height of the cylindrical part of the container, based on the information provided. Here’s the step-by-step approach:

Step 1: Understanding the Container Dimensions

1. The container consists of:
   • A cone with radius $r = 3$ m and height $4$ m.
   • A cylinder with the same radius $r = 3$ m and unknown height $h$ m.

Step 2: Volumes of the Cone and Cylinder

The volume of a cone and cylinder can be calculated using the formulas:

• Volume of a cone: $V_{\text{cone}} = \frac{1}{3} \pi r^2 h_{\text{cone}}$
• Volume of a cylinder: $V_{\text{cylinder}} = \pi r^2 h_{\text{cylinder}}$

Substitute $r = 3$ m and $h_{\text{cone}} = 4$ m:

1. Volume of the cone: $V_{\text{cone}} = \frac{1}{3} \pi (3)^2 \cdot 4 = \frac{1}{3} \pi \cdot 9 \cdot 4 = 12 \pi \text{ cubic meters}$
2. Volume of the cylinder (with height $h$): $V_{\text{cylinder}} = \pi (3)^2 \cdot h = 9\pi h \text{ cubic meters}$

Step 3: Total Volume of the Container

The total volume of the container, when full, is the sum of the volumes of the cone and the cylinder: $V_{\text{total}} = V_{\text{cone}} + V_{\text{cylinder}} = 12 \pi + 9 \pi h$

Step 4: Determine the Rate of Filling

From the problem, we know:

• After 5 hours, the depth of the grain is 6 meters above the vertex of the cone.
• This depth includes the entire 4 meters of the cone and an additional 2 meters into the cylinder.

Thus, in 5 hours, the grain has filled:

1. The entire cone volume: $12 \pi$
2. Plus the cylinder section up to a height of 2 meters: $9 \pi \cdot 2 = 18 \pi$

So, the volume filled in 5 hours is: $12 \pi + 18 \pi = 30 \pi$

Since this volume was filled in 5 hours, the rate of filling is: $\frac{30 \pi}{5} = 6 \pi \text{ cubic meters per hour}$

Step 5: Total Time to Fill the Container

The container is completely full after 9 hours, so the total volume filled in that time is: $9 \text{ hours} \times 6 \pi = 54 \pi \text{ cubic meters}$

Step 6: Setting Up the Equation

Since the total volume when the container is full is $V_{\text{total}} = 12 \pi + 9 \pi h$, we equate this to $54 \pi$: $12 \pi + 9 \pi h = 54 \pi$

Step 7: Solving for $h$

1. Divide both sides by $\pi$: $12 + 9h = 54$
2. Subtract 12 from both sides: $9h = 42$
3. Divide by 9: $h = \frac{42}{9} = \frac{14}{3}$

Final Answer

The value of $h$ is: $h = \frac{14}{3} \text{ meters}$

Would you like further details on any step or have any other questions?

Related Questions

1. How do you calculate the volume of a cone or cylinder in different units?
2. How does the filling rate affect the volume calculations for variable-depth containers?
3. What would be the impact on $h$ if the cylinder had a different radius?
4. How would this problem change if the cone's height was altered but the cylinder’s height remained constant?
5. How can you verify the filling rate through alternative methods?

Tip

When working with composite shapes, breaking down the problem by calculating individual volumes and understanding rates can simplify complex volumetric problems.