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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

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### 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
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
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