Mass from density A thin 10-cm rod is made of...

Question

Mass from density A thin 10-cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of g/cm. In Chapter 6, we show that the mass of the rod is the area under the density curve.
(b) Find the mass of the right half of the rod (5 ≤ x ≤ 10) .

(b) Find the mass of the right half of the rod (5 ≤ x ≤ 10) .

Graph showing density in g/cm along a 10-cm rod, with varying density values plotted against length in cm.

Accepted Answer

Hello. In this video, we are told that an 8 centimeter solid cylinder has a density that varies along its length according to the function shown in the finger below. What is the mass of the segment from X is equal to 4 to X is equal to 8, and we want to note that the density is measured in grams per centimeter. So, in order to find the mass, we have to recall that with the density curve, mass is equal to the area under the curve. Specifically underneath the density curve. And so here we are told to find the mass between X is equal to 4. And X is equal to 8. And so in order to find the area or to find the mass, we need to find the total area underneath this curve. Let's go ahead and split up the curve where the behaviors of the graph change. So if we look at the graph right here, the graph is slowly increasing from 4 to 6 and then stays constant from 6 to 8. So let's go ahead and split this graph at the value X is equal to 6, and now we are going to use geometric methods in order to find the area underneath the curve. Let's go ahead and start by finding the area between 4 and 6. Now this area is in the form of a trapezoid. The area of the trapezoid is equal to 12 multiplied by the first height plus the second height multiplied by the width of the trapezoid. Now the first height of this trapezoid has a total height of 6 units. So this first height is going to be 6, and the second height has a total height of 8. So the second height will be 8. Furthermore, the total width of this trapezoid is 2 units long, so the area in this region is going to equal to 1/2 multiplied by 6 + 8, multiplied by the width, which is 2, and now we just need to simplify. 6 + 8 is going to simplify to be 14. And so, one half multiplied by 14, multiplied by 2, is going to give us 14. So the area within this region from 4 to 6 is going to be 14. Now we are going to repeat this process by finding the area between the values of 6 and 8. Now, notice that this area is in the form of a rectangle. This rectangle has a width of 2. And has a total height of 8. And so what this means is that in order to take the area of this region, we are going to take the area of a rectangle, which is going to be 8 multiplied by 2, which simplifies to be 16. And so now in order to find the total mass of this cylinder, all we need to do is we need to take the sum of both of the areas that we calculated. So that is going to be 14 plus 16. Which is going to simplify to be 30 g, and this is going to be the mass of the cylinder between 4 and 8 centimeters. And with that being said, the overall solution to this problem is going to be C. So I hope this video helps you in understanding how to find the mass, and we will go ahead and see you all in the next video.

Key Concepts

Density Function

Integration

Definite Integral

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