Find volume of object if cross-sections are known

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In summary, the solid in question is an equilateral triangle with altitude 14, and its cross-sections perpendicular to the altitude are semicircles. To find the volume of the solid, we need to use the formula V= \int_a^{b} A(x)dx, where a=0 and b=14. The problem is finding the cross-section, which can be done by finding the length of one side of the equilateral triangle, denoted as 2R, and setting up an equation for the radius in terms of x. This allows us to find A(x) and integrate to get the volume of the solid. It may also be helpful to think of the shape formed by a general line y=mx,
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The base of a certain solid is an equilateral triangle with altitude 14. Cross-sections perpendicular to the altitude are semicircles. Find the volume of the solid, using the formula [tex]V= \int_a^{b} A(x)dx[/tex]

applied to the picture shown attached to this post (click for a better view), with the left vertex of the triangle at the origin and the given altitude along the -axis.


i figure a= 0 and b = 14 and i will be plugging it into the formula pi*r^2 right?

the problem I'm having is finding the cross-section. can someone lend me a hand?
 

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You'll be integrating from x=0 to x=14 right?

First, find the length of one side of the equilateral, let's call it 2R.

It's clear at x=0, the radius is zero and at x=14 the radius is R, and the radius varies linearly with x. So set up an equation for the radius in terms of x.
then you can find A(x) and integrate.
 
  • #3
Think of a volume of revolution formed by a general line y = mx. What sort of shape does this yield and how is it related to the shape you are given?

Thinking about this will also provide a good check as to whether your answer is correct as there is a nice formula for its volume.
 

What is the definition of cross-sections?

Cross-sections refer to the 2-dimensional shapes that are created when a 3-dimensional object is cut by a plane.

Why is finding the volume of an object important?

Knowing the volume of an object is important for various reasons, such as determining the capacity of a container, calculating the amount of material needed for a construction project, or understanding the properties of a substance.

How do you find the volume of an object if the cross-sections are known?

To find the volume of an object if the cross-sections are known, you can use the formula V = ∫A(x)dx, where A(x) represents the area of the cross-section at a specific point on the object and the integral is taken over the entire length of the object.

What types of cross-sections can be used to find the volume of an object?

Any 2-dimensional shape can be used as a cross-section to find the volume of an object, such as circles, rectangles, triangles, or irregular shapes.

Are there any limitations to finding the volume of an object using cross-sections?

Yes, there are limitations to this method, as it may not work for objects with curved or irregular shapes. In these cases, other methods such as displacement or water displacement may be used to find the volume.

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