# Density, Volume and Slicing Problem

• Eonfluxx
In summary, The density of a compressible liquid varies with height and is represented by 40(5-h) kg/m3 at a height of h meters above the bottom. When placed in a container with isosceles triangular cross sections, the liquid will hold a mass of 1099.68 kg when resting on one triangular side. This can be calculated by setting up a triple integral with the given parameters.

## Homework Statement

A compressible liquid has density which varies with height. AT the level of h meters above the bottom, the density is 40(5 - h) kg/m3
a) The liquid is put in the containers below. The cross sections of the container are isosceles triangles. It has straight sides and looks like a triangular prism. How many kg will it hold when placed as shown on the left, resting on one triangular side?​

^There is a picture of the object.

## Homework Equations

Mass = Volume times Density

## The Attempt at a Solution

My theory is that since M = V*D, and you're given the density, should I slice and solve for volume, then evaluate the integral of volume and the given integral for density then multiply the results to get the mass?
$$\int^{4}_{0}2.5 dh$$ and $$40\int^{4}_{0}5-h dh$$

I got to the first integral by taking the volume of the first slice (triangular prism):
$$\sum\frac{1}{2}b*l*\Delta h$$ $$\rightarrow$$ $$\sum\frac{1}{2}2*2.5\Delta h$$ $$\rightarrow$$ lim as $$\Delta h$$ $$\rightharpoonup$$$$\int^{4}_{0}2.5 dh$$

To find the mass, according to the equation M=VD, should I solve both integrals , one being V and one being D, then multiply?

Ah... It's starting to make more sense now, thanks. So since density is varying with the height we make it all one integral rather than separate ones. Part B is the same question except with the 4m on the ground on its point with the rectangle side up. I figured once I'd managed A, B would be simple. Thanks!

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## 1. What is density and how is it calculated?

Density is a measure of how much mass is contained in a given volume of a substance. It is calculated by dividing the mass of an object by its volume.

## 2. How is volume measured?

Volume is measured by determining the amount of space an object takes up. This can be done by using a ruler, graduated cylinder, or by the displacement method.

## 3. What is the slicing problem and how is it solved?

The slicing problem refers to the challenge of accurately measuring the volume of an irregularly shaped object. It can be solved by using the slicing method, where the object is sliced into smaller, regular shapes, and the volumes of those shapes are added together to find the total volume of the object.

## 4. How does density affect the behavior of fluids?

Density plays a key role in the behavior of fluids. Heavier, more dense fluids will sink to the bottom, while lighter, less dense fluids will rise to the top. This is the basis of the principle of buoyancy.

## 5. How is density used in everyday life?

Density is used in many practical applications, such as determining the amount of fuel needed for a vehicle, calculating the weight of packaging materials, and determining the nutritional content of food. It is also used in industries like construction, medicine, and metallurgy.