# Volume of solid having triangle base and semicircle slice

• songoku
In summary, the conversation discusses finding the equation for the area of a slice of a solid, using the formula A(x) = 1/2πr^2. It is determined that the volume of the solid is half the volume of a cone with a base of area (3/2)^2π and a height of 2. The conversation also includes a discussion about the solution, with one person questioning if their answer is correct.
songoku
Homework Statement
Calculate the volume of solid having triangle base with vertices (0, 0) , (2, 0) and (0, 3) whose slice perpendicular to x-axis is semicircle
Relevant Equations
Volume = ##\int_p^{q} A(x) dx##
First, I tried to find the equation of line passing through (2, 0) and (0, 3) and I got ##y=3-\frac{3}{2}x##

Then I set up equation for the area of one slice, ##A(x)##
$$A(x)=\frac{1}{2} \pi r^2$$
$$=\frac{1}{2} \pi \left( \frac{1}{2}y\right)^2$$
$$=\frac{1}{2} \pi \left(\frac{3}{2}-\frac{3}{4}x \right)^2$$

Am I correct until this point? Thanks

I am not sure I got right image of the solid. Could you show me your sketch ?

anuttarasammyak said:
I am not sure I got right image of the solid. Could you show me your sketch ?

Hopefully it is clear enough

Thanks

Delta2, Lnewqban and anuttarasammyak
Thanks. Clear enough. So we observe the volume of the solid is half the volume of the cone which has base of area ##(3/2)^2\pi## and height 2. It would be beneficial for you to check answer.

$$V=\frac{1}{2}\int_0^2 \pi [\frac{3- \frac{3}{2}x}{2}]^2 dx$$

Lnewqban and songoku
anuttarasammyak said:
Thanks. Clear enough. So we observe the volume of the solid is half the volume of the cone which has base of area ##(3/2)^2\pi## and height 2. It would be beneficial for you to check answer.

$$V=\frac{1}{2}\int_0^2 \pi [\frac{3- \frac{3}{2}x}{2}]^2 dx$$
I also get the same result but the solution is:
$$V=\int_{0}^{2} \frac{1}{2} \left(\pi \left(\frac{1}{2}-\frac{x}{4}\right)^2\right)dx$$

That's why I am confused at which part I got it wrong, but I think maybe the solution is not correct

Thank you very much anuttarasammyak

Lnewqban
The answer you referred seems to set vertexes are (0,0),(2,0), and (0,1), not (0,3).

Last edited:
songoku and Lnewqban

## 1. What is the formula for finding the volume of a solid with a triangle base and semicircle slice?

The formula for finding the volume of this type of solid is V = (1/3) * b * h * (r^2 - (1/3) * h^2), where b is the base of the triangle, h is the height of the triangle, and r is the radius of the semicircle.

## 2. How do you determine the base and height of the triangle in this type of solid?

The base of the triangle is the length of one side of the triangle, and the height is the perpendicular distance from the base to the opposite vertex. These measurements can be obtained by using a ruler or measuring tape.

## 3. Can the volume of this type of solid be found using the same formula as a regular pyramid?

No, the formula for finding the volume of a regular pyramid is V = (1/3) * b * h, where b is the base and h is the height. The formula for this type of solid takes into account the curved surface of the semicircle, making it different from a regular pyramid.

## 4. Can the volume of this type of solid be found using calculus?

Yes, the volume of this solid can also be found using calculus by integrating the cross-sectional area of the solid along the height. The resulting integral will be the same as the formula mentioned in the first question.

## 5. Are there any real-life applications of this type of solid?

Yes, this type of solid can be found in architectural structures such as domes and arches. It can also be used in engineering designs for bridges and tunnels. Additionally, it can be used in the design of certain types of storage tanks and containers.

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