To solve the problems regarding the silo composed of a cylindrical and a conical section, we'll apply the formulas for volumes and the Pythagorean theorem.
Given Information
- Let's denote:
- \( r \): radius of both the cone and cylinder
- \( h_{\text{cone}} \): height of the cone
- \( h_{\text{cylinder}} \): height of the cylinder
- \( V_{\text{cylinder}} \): volume of the cylinder = \( 157.08 , \text{m}^3 \)
1.1.1 Calculating the Perpendicular Height, \( AB \), of the Cone Using Pythagoras
While the question does not provide direct lengths or locations for heights, the perpendicular height, \( AB \), of the cone can often be calculated if we know the radius and the slant height (or the dimensions of the cone).
Assuming we have a right triangle involving the height from the apex of the cone down to the base, we could interpret it as follows:
- Let \( AC \) be the radius \( r \), \( AB \) the perpendicular height, and \( BC \) be the slant height \( s \).
Using the Pythagorean theorem: \[ s^2 = r^2 + AB^2 \]
Without specific values for \( s \) or \( r \), we can't compute \( AB \) explicitly. Normally we would need additional data about the slant height.
1.1.2 Calculating the Height of the Cylinder
The volume of the cylinder \( V_{\text{cylinder}} \) is given by the equation: \[ V_{\text{cylinder}} = \pi r^2 h_{\text{cylinder}} \]
We know \( V_{\text{cylinder}} = 157.08 , \text{m}^3 \), so rearranging gives: \[ h_{\text{cylinder}} = \frac{V_{\text{cylinder}}}{\pi r^2} \]
To solve for \( h_{\text{cylinder}} \), we need the radius \( r \). Without a numeric value for \( r \), letβs denote \( r \) as a variable.
1.1.3 Calculating the Total Volume of the Silo
The total volume \( V_{\text{silo}} \) is the sum of the volumes of both the cone and cylinder: \[ V_{\text{silo}} = V_{\text{cylinder}} + V_{\text{cone}} = \pi r^2 h_{\text{cylinder}} + \frac{1}{3} \pi r^2 h_{\text{cone}} \] Substituting \( V_{\text{cylinder}} = 157.08 , \text{m}^3 \): \[ V_{\text{silo}} = 157.08 + \frac{1}{3} \pi r^2 h_{\text{cone}} \]
Again, we would need specific values for either \( r \) or \( h_{\text{cone}} \) to compute the total volume.
Conclusion
To summarize, without additional information about \( r \), \( h_{\text{cone}} \), or \( s \) to calculate \( AB \), we cannot numerically evaluate parts 1.1.1, 1.1.2 (specifically \( h_{\text{cylinder}} \)), and 1.1.3 (the total volume). To proceed, please provide the missing measurements or dimensions.