What is the formula for the surface area of a right circular cylinder, $S=2 \pi r h+2 \pi r^{2},$ solved for h? $A. h=\frac{S}{4 \pi r}\ B. h=\frac{s}{2 \pi r^{2}}\ C. h=\frac{S}{2 \pi r}-r\ D. h=r-\frac{S}{2 \pi r}$

An infinitely long circular cylinder carries a uniform magnetization $\vec{M}$ parallel to its axis. Find the magnetic field (due to $\vec{M}$) inside and outside the cylinder.

The curved surface area of a solid cylinder is $660 \mathrm{~cm}^2$. The cylinder has height 10 cm. Calculate the circumference of the cylinder. Determine the radius of the cylinder. Find the area of one circular base. What is the surface area of the cylinder?

The figure shows a container in the shape of a circular cylinder topped by a hemisphere. The radii of both cylinder and hemisphere is 3 centimetres and the height of the cylinder is 7 centimetres.

Calculate:

a. the volume of the container

b. the total surface area of the container.

A wedge is cut out of a circular cylinder of radius $4$ by two planes. One plane is perpendicular to the axis of the cylinder. The other intersects the first at an angle of $30^{\circ}$ along a diameter of the cylinder. Find the volume of the wedge.

A cylinder with circular cross scction has a radius of $2.56 \mathrm{~cm}$ and a height of $56.32 \mathrm{~cm}$. What is the volume of the cylinder? Fxpress the answer to the correct number of significant figures.

A circular cylinder is to fit inside a sphere of radius 10 cm. Calculate the maximum possible volume of the cylinder. (It is probably best to take as your independent variable the height, or half the height, of the cylinder.)

A wedge is cut out of a circular cylinder of radius 4 by two planes. One plane is perpendicular to the axis of the cylinder. The other intersects the first at an angle of 30$^{\circ}$ along a diameter of the cylinder. Find the volume of the wedge.

A wedge is cut out of a circular cylinder of radius 4 by two planes. One plane is perpendicular to the axis of the cylinder. The other intersects the first at an angle of $30^{\circ}$ along a diameter of the cylinder. Find the volume of the wedge.

Solve the exercise by taking cross-sections to be parallel to the line of intersection of the two planes.

Express by repeated integrals. Do not evaluate. The volume of the solid bounded above by the parabolic cylinder $z=1-y^2$, below by the plane $2 x+3 y+z+10=0$, and on the sides by the circular cylinder $x^2+y^2-x=0$.

a. Show that the curve $\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}+(1-\cos t) \mathbf{k}, 0 \leq t \leq 2 \pi,$ is an ellipse by showing that it is the intersection of a right circular cylinder and a plane. Find equations for the cylinder and plane. b. Sketch the ellipse on the cylinder. Add to your sketch the unit tangent vectors at $t=0, \pi / 2, \pi, \text { and } 3 \pi / 2.$ c. Show that the acceleration vector always lies parallel to the plane (orthogonal to a vector normal to the plane). Thus, if you draw the acceleration as a vector attached to the ellipse, it will lie in the plane of the ellipse. Add the acceleration vectors for t = 0, $\pi / 2, \pi, \text { and } 3 \pi / 2$ to your sketch. d. Write an integral for the length of the ellipse. Do not try to evaluate the integral; it is nonelementary. e. Estimate the length of the ellipse to two decimal places.

A clockwise couple M is applied to the circular cylinder. Determine the value of M required to initiate motion for the conditions $m_{B}=3 \mathrm{kg}$, $m_{c}$ = 6kg, $\left(\mu_{s}\right)_{B}$ = 0.50, $\left(\mu_{s}\right)_{C}$ = 0.40, and r = 0.2 m. Friction between the cylinder C and the block B is negligible.

Amy said that if the radius of a circular cylinder were doubled and the height decreased by one-half, the volume of the cylinder would remain unchanged. Do you agree with Amy? Explain why or why not.

How can you find the volume of a prism or cylinder that is not a right prism or right cylinder?