Hello everyone. So I finished non-english High school and I'm trying to solve some english exercises. I understand most of them but here I came across one that I can't really understand.
Let r be a positive constant. Consider the cylinder \(\displaystyle x^2\, +\, y^2\, \leq\, r^2,\) and let C be the part of the cylinder that satisfies \(\displaystyle 0\, \leq\, z\, \leq\, y.\)
(1) Consider the cross-section of C by the plane \(\displaystyle x\, =\, t\, (-r\, \leq\, t\, \leq\, r),\) and express its area in terms of r and t.
(2) Calculate the volume of C, and express it in terms of r.
(3) Let a be the length of the arc along the base circle of C from the point (r, 0, 0) to the point \(\displaystyle (r\, \cos(\theta),\, r\, \sin(\theta),\, 0)\, (0\, \leq\, \theta\, \leq\, \pi).\) Let b be the length of the line segment from the point \(\displaystyle (r\, \cos(\theta),\, r\, \sin(\theta),\, 0)\) to the point \(\displaystyle (r\, \cos(\theta),\, r\, \sin(\theta),\, r\, \sin(\theta)).\) Express a and b in terms of r and \(\displaystyle \theta.\)
(4) Calculuate the area of the side of C with \(\displaystyle x^2\, +\, y^2\, =\, r^2,\) and express it in terms of r.
So to begin with:
I understand that r> 0
x^2+y^2 <= r^2, is a circle equation
i know the C is some part of Cylinder, but don't rly understand which one and how I should draw it.
i have no idea what z in 0<z<y can be, is it some kind of high?
what is the plane x=0?
Does arc a mean a is like a part of 2pi*r?
Also is a line segment b like a high?
Thanks in advance.
Let r be a positive constant. Consider the cylinder \(\displaystyle x^2\, +\, y^2\, \leq\, r^2,\) and let C be the part of the cylinder that satisfies \(\displaystyle 0\, \leq\, z\, \leq\, y.\)
(1) Consider the cross-section of C by the plane \(\displaystyle x\, =\, t\, (-r\, \leq\, t\, \leq\, r),\) and express its area in terms of r and t.
(2) Calculate the volume of C, and express it in terms of r.
(3) Let a be the length of the arc along the base circle of C from the point (r, 0, 0) to the point \(\displaystyle (r\, \cos(\theta),\, r\, \sin(\theta),\, 0)\, (0\, \leq\, \theta\, \leq\, \pi).\) Let b be the length of the line segment from the point \(\displaystyle (r\, \cos(\theta),\, r\, \sin(\theta),\, 0)\) to the point \(\displaystyle (r\, \cos(\theta),\, r\, \sin(\theta),\, r\, \sin(\theta)).\) Express a and b in terms of r and \(\displaystyle \theta.\)
(4) Calculuate the area of the side of C with \(\displaystyle x^2\, +\, y^2\, =\, r^2,\) and express it in terms of r.
So to begin with:
I understand that r> 0
x^2+y^2 <= r^2, is a circle equation
i know the C is some part of Cylinder, but don't rly understand which one and how I should draw it.
i have no idea what z in 0<z<y can be, is it some kind of high?
what is the plane x=0?
Does arc a mean a is like a part of 2pi*r?
Also is a line segment b like a high?
Thanks in advance.
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