A hole in the ground in the shape of an inverted cone is 14 meters deep and has radius at the top of 19 meters. The cone is filled to the top with sawdust. The density of the sawdust depends upon the depth, x, following the formula p(x)=2.3+1.5e^-0.5X kg/m^3. Find the total mass of sawdust in the conical hole.

We are tasked with finding the total mass of sawdust in the conical hole. The density of the sawdust depends on the depth, and the hole is shaped like an inverted cone.

### Given Information:
- Depth of the cone: \( h = 14 \) meters.
- Radius at the top: \( R = 19 \) meters.
- Density function: \( \rho(x) = 2.3 + 1.5e^{-0.5x} \) kg/m\(^3\), where \( x \) is the depth (from the top) in meters.

### Steps:

1. **Volume Element (dV)**:  
   At depth \( x \), the radius of the cone decreases linearly from the top, following the relation:
   \[
   r(x) = \frac{R}{h} \cdot (h - x) = \frac{19}{14}(14 - x) = \frac{19}{14}(14 - x)
   \]
   The volume of a thin disk at depth \( x \) with thickness \( dx \) is:
   \[
   dV = \pi r(x)^2 dx = \pi \left( \frac{19}{14}(14 - x) \right)^2 dx
   \]

2. **Mass Element (dm)**:  
   The mass of this small disk is the product of the density and the volume element:
   \[
   dm = \rho(x) dV = \left( 2.3 + 1.5e^{-0.5x} \right) \cdot \pi \left( \frac{19}{14}(14 - x) \right)^2 dx
   \]

3. **Total Mass**:  
   To find the total mass, we integrate the mass element over the entire depth of the cone from \( x = 0 \) to \( x = 14 \):
   \[
   M = \int_0^{14} \left( 2.3 + 1.5e^{-0.5x} \right) \cdot \pi \left( \frac{19}{14}(14 - x) \right)^2 dx
   \]

Let me now calculate this integral to find the total mass.

The total mass of the sawdust in the conical hole is approximately 14,742 kg.

Would you like any further details or explanations on this process? 

Here are five related questions to deepen your understanding:
1. How would the result change if the cone's radius or height were altered?
2. What if the density function had a different exponential rate? How would this affect the result?
3. Can you derive the volume of the cone independently without integrating the mass?
4. How does the shape of the cone impact the mass distribution of the sawdust?
5. How would this problem change if the density increased linearly instead of exponentially?

**Tip**: When working with integrals in physical problems, setting up the correct geometric relationship and density function is key to obtaining the right mass or volume!

Learn how to calculate the total mass of sawdust in a conical hole with a depth of 14 meters and a top radius of 19 meters. This problem involves integral calculus, exponential decay, and the disk method for calculating volume. Suitable for undergraduate calculus students.

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