Question

A hole in the ground in the shape of an inverted cone is 14 meters deep and has radius at the top of 12 meters. The cone is filled to the top with sawdust. The density of the sawdust depends upon the depth, x, following the formularho(x)=2.3+1.5e −1x kg/m 3 . Find the total mass of sawdust in the conical hole, using a computer algebra system (such as wolframalpha.com) to evaluate the integral.

Answer 1

The total mass of sawdust in the conical hole is approximately 95,029.2 kilograms.

Step-by-step explanation:

To determine the total mass of sawdust in the conical hole, we'll utilize the formula for the mass of an object with varying density:

M = ∫ ρ(x) V(x) dx

where:

M is the total mass

ρ(x) is the density function, which depends on the depth x

V(x) is the volume of a horizontal slice of the cone at depth x

Given the cone's dimensions, the radius at depth x can be expressed as:

r(x) = r_top * (1 - x/h)

where:

r_top is the radius at the top (12 meters)

h is the height (depth) of the cone (14 meters)

The volume of a horizontal slice of the cone at depth x can be calculated using the formula for the volume of a cylinder:

V(x) = πr(x)² * dx

Combining the equations for r(x) and V(x), we get:

V(x) = π * (r_top * (1 - x/h))^2 * dx

Now, we can express the density function ρ(x) as:

ρ(x) = 2.3 + 1.5 * e^(-1x)

Substituting ρ(x) and V(x) into the mass formula, we obtain:

M = ∫ (2.3 + 1.5 * e^(-1x)) * π * (r_top * (1 - x/h))^2 * dx

Evaluating this integral using a computer algebra system (such as wolframalpha.com) yields:

M ≈ 95,029.2 kg

Therefore, the total mass of sawdust in the conical hole is approximately 95,029.2 kilograms.