You've been asked to calculate the mass of a conical pile of sawdust measuring 10 meters high with a base radius of 8 meters. From experimental measurements, you've been able to determine that the density of the sawdust depends upon the depth, d, of the sawdust in meters, according to the equation _p(d) = 180 + 10d kg/m³. Find the mass of the pile. 1877.33 x kg
You've been asked to calculate the mass of a conical pile of sawdust measuring 10 meters high with a base radius of 8 meters. From experimental measurements, you've been able to determine that the density of the sawdust depends upon the depth, d, of the sawdust in meters, according to the equation _p(d) = 180 + 10d kg/m³. Find the mass of the pile. 1877.33 x kg
Related questions
Question
![### Calculating the Mass of a Conical Pile of Sawdust
**Problem Statement:**
You've been asked to calculate the mass of a conical pile of sawdust measuring 10 meters high with a base radius of 8 meters. From experimental measurements, you've been able to determine that the density of the sawdust depends upon the depth, \( d \), of the sawdust in meters, according to the equation:
\[ \rho(d) = 180 + 10d \, \text{kg/m}^3 \]
Find the mass of the pile.
**Given:**
- Height of the cone, \( h = 10 \, \text{meters} \)
- Radius of the base, \( r = 8 \, \text{meters} \)
- Density of sawdust, \( \rho(d) = 180 + 10d \, \text{kg/m}^3 \)
**Solution Explanation:**
Unfortunately, the given answer is incorrect. Here is how you would typically solve the problem:
1. **Volume of the Cone:**
\[ V = \frac{1}{3} \pi r^2 h \]
\[ V = \frac{1}{3} \pi (8)^2 (10) \]
\[ V = \frac{1}{3} \pi (64) (10) \]
\[ V = \frac{640}{3} \pi \]
2. **Density Function and Mass Calculation:**
Given that the density varies with depth \( d \), we use the density function:
\[ \rho(d) = 180 + 10d \]
Since the density varies with depth within the cone, we need to set up an integral to calculate the mass. The variable \( d \) will vary from 0 to 10 from the top to the bottom of the cone.
3. **Using Calculus to Calculate Mass:**
We use the method of slicing to find the mass. At height \( y \) from the vertex of the cone with height \( h \):
The radius \( r(y) \) at height \( y \):
\[ r(y) = r \frac{h-y}{h} \]
\[ r(y) = 8\frac{10-y}{10} \]
\[ r(y) = 8 - 0.8y \]
The area of a slice at height \( y \):
\[ A(y](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74b6836e-7075-4604-8b7e-b1441272473c%2F6aa609bf-ee4e-477e-beec-ed763c8e1c21%2F1dkrp1n_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating the Mass of a Conical Pile of Sawdust
**Problem Statement:**
You've been asked to calculate the mass of a conical pile of sawdust measuring 10 meters high with a base radius of 8 meters. From experimental measurements, you've been able to determine that the density of the sawdust depends upon the depth, \( d \), of the sawdust in meters, according to the equation:
\[ \rho(d) = 180 + 10d \, \text{kg/m}^3 \]
Find the mass of the pile.
**Given:**
- Height of the cone, \( h = 10 \, \text{meters} \)
- Radius of the base, \( r = 8 \, \text{meters} \)
- Density of sawdust, \( \rho(d) = 180 + 10d \, \text{kg/m}^3 \)
**Solution Explanation:**
Unfortunately, the given answer is incorrect. Here is how you would typically solve the problem:
1. **Volume of the Cone:**
\[ V = \frac{1}{3} \pi r^2 h \]
\[ V = \frac{1}{3} \pi (8)^2 (10) \]
\[ V = \frac{1}{3} \pi (64) (10) \]
\[ V = \frac{640}{3} \pi \]
2. **Density Function and Mass Calculation:**
Given that the density varies with depth \( d \), we use the density function:
\[ \rho(d) = 180 + 10d \]
Since the density varies with depth within the cone, we need to set up an integral to calculate the mass. The variable \( d \) will vary from 0 to 10 from the top to the bottom of the cone.
3. **Using Calculus to Calculate Mass:**
We use the method of slicing to find the mass. At height \( y \) from the vertex of the cone with height \( h \):
The radius \( r(y) \) at height \( y \):
\[ r(y) = r \frac{h-y}{h} \]
\[ r(y) = 8\frac{10-y}{10} \]
\[ r(y) = 8 - 0.8y \]
The area of a slice at height \( y \):
\[ A(y
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
