To find the total volume of the storage bin, we need to calculate the volume of both the cylindrical portion and the conical top, and then sum them up.

1. Volume of the cylinder: The formula for the volume of a cylinder is:

   $V_{\text{cylinder}} = \pi r^2 h$

   where:

   • $r = 5.6$ ft is the radius,
   • $h = 8.4$ ft is the height of the cylindrical portion.

2. Volume of the cone: The formula for the volume of a cone is:

   $V_{\text{cone}} = \frac{1}{3} \pi r^2 H_{\text{cone}}$

   where:

   • $r = 5.6$ ft is the radius (same as the cylinder),
   • $H_{\text{cone}} = H - h = 15.9 - 8.4 = 7.5$ ft is the height of the conical portion.

Now, we can calculate both volumes and sum them up:

Step 1: Calculate the volume of the cylinder

$V_{\text{cylinder}} = \pi (5.6)^2 (8.4)$

Step 2: Calculate the volume of the cone

$V_{\text{cone}} = \frac{1}{3} \pi (5.6)^2 (7.5)$

Let's compute these values.The total volume of the storage bin is approximately $1073.87 \, \text{ft}^3$.

Would you like any further details or have any questions?

Here are 5 related questions to consider:

1. How would the volume change if the radius of the bin were doubled?
2. What is the ratio of the volume of the conical portion to the cylindrical portion?
3. How does the volume formula change if the cylinder has an elliptical base instead of a circular base?
4. Can you calculate the surface area of this bin, including the cylindrical and conical surfaces?
5. How would you determine the weight of the bin if it were filled with a material of known density?

Tip: When calculating volumes of composite shapes, always treat each part separately and then sum them up for the total volume.