Volume part 1 Assignment.sagews

Volume Assignment

Question 0

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Question 1

When viewed from above, a swimming pool has the shape of a circle with radius 5 feet. If we put the circle in the x,y-plane centered at the origin, then cross sections of the pool perpendicular to the $x$-axis are squares. Find the volume of the pool.

[Hint: The cross section through the point on the circle $(x,y)$ with $y>0$ is a square with sides of length $2y$. You need the area of this square in terms of $x$, so that you can integrate with respect to $x$. To switch from $y$ to $x$, use the equation of the circle: $x^2+y^2=25$.]

[Answer: $\frac{2000}{3}$]

Question 2

Find the volume of the solid whose base is the region between the curve $\displaystyle y=2\sqrt{\sin(x)}$ and the interval $[0,\pi]$ on the $x$-axis and whose cross sections perpendicular to the $x$-axis are equilateral triangles.

[Hint: The area of an equilaterial triangle with sides of length $s$ is $A=\displaystyle\frac{\sqrt{3}}{4}s^2$].

[Answer: $2\sqrt{3}$]

Here is a picture of the base. Imagine a solid sticking out of the screen so that cutting perpendicular to the $x$-axis reveals an equilateral triangle.

Question 3

Find the volume of a right circular cone with height $h$ and circular base of radius $r$.

[Hint: Put the top point of the cone at the origin, and lay the cone sideways so the $x$-axis goes through the center of the circular base. The cross section perpendicular to the $x$-axis at $x$ is a circle with radius $y$, where $(x,y)$ is on the line through $(0,0)$ and $(h,r)$.]

[Answer: $\frac{1}{3}\pi h r^2$]

This material was developed by Aaron Tresham at the University of Hawaii at Hilo and is
licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.