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Chapter 12: Problem 41

Use a graphing device to draw a silo consisting of a cylinder with radius 3 and height 10 surmounted by a hemisphere.

Short Answer

Expert verified
Graph a cylinder (radius 3, height 10) with a hemispherical top (radius 3) on a graphing device.

Step by step solution

01

Understand the Problem

We are asked to visualize the shape of a silo that consists of two parts: a cylindrical base and a hemispherical top. The cylinder has a base radius of 3 units and a height of 10 units, while the hemisphere sits perfectly on top of the cylinder, sharing the same radius.
02

Set Coordinates for the Base Cylinder

To graph the cylinder, use a graphing tool to plot its base. The equation for the base of the cylinder in the xy-plane is a circle: \[ x^2 + y^2 = 3^2 = 9 \] This circle extends vertically from z = 0 to z = 10.
03

Draw the Vertical Cylinder

Extend vertical lines from the circle to represent the sides of the cylinder. The equation for the cylinder in 3D is: \[ x^2 + y^2 = 9,\ 0 \leq z \leq 10 \]This forms a cylinder with radius 3 and height 10.
04

Create the Hemisphere

The hemisphere is the top part of a sphere with radius 3. Its base matches the top of the cylinder at z = 10. The equation of a sphere centered at (0,0,10), is:\[ x^2 + y^2 + (z-10)^2 = 9 \]For the hemisphere, we consider the section where \( z \geq 10 \).
05

Plot the Hemisphere on the Graph

Plot the hemisphere using the sphere's equation by considering only values of z from 10 to the maximum height of the hemispheric surface, which is 13 (since the radius is 3). This gives the hemisphere from z = 10 to z = 13.
06

Combine the Cylinder and Hemisphere

Combine the two parts on your graphing tool. Ensure the top of the cylinder and the base of the hemisphere coincide perfectly at z = 10. This will complete the model of the silo.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cylinder
A cylinder is a three-dimensional shape with two parallel circular bases connected by a curved surface. Imagine a can; that's a cylinder.
  • Base Radius: In our exercise, the radius of the cylinder's base is 3 units.
  • Height: The height is the vertical distance between the bases. Here, it's 10 units.
To graph a cylinder, you start with the circular base in the xy-plane. The equation \(x^2 + y^2 = 9\) gives us this circle in a 2D plane. Then, you stretch this circle vertically from z = 0 to z = 10 to form the 3D cylinder.
This step-by-step method helps you see how the shape emerges from a simple base to a complete 3D object.
Hemisphere
A hemisphere is essentially half of a sphere. Like slicing an orange in half, with one of these halves being a hemisphere.
  • Base Radius: It shares the same radius as the base of our cylinder, which is 3 units.
  • Positioning: Our hemisphere is positioned on top of the cylinder, starting at z = 10 where the cylinder ends.
The equation of a sphere centered at (0,0,10) is \(x^2 + y^2 + (z-10)^2 = 9\). This represents our complete sphere. To make it a hemisphere, we only take the part where \(z \geq 10\), reaching up to z = 13. This way, you're looking at the upper half of the sphere forming a dome on top of the cylinder.
Coordinate system
Coordinate systems allow us to graph shapes in a 2D or 3D space and precisely determine their positions.
  • 3D Coordinates: In three-dimensional graphing, we use (x, y, z) coordinates. The x and y determine position in a plane, and z gives the height or depth.
  • Graphing Tools: Graphing tools enable us to visualize these coordinates. You input equations, like \(x^2 + y^2 = 9\) for the cylinder's base circle, to create models.
In the exercise, we place the cylinder's base circle at z = 0 and extend it to a height of 10 along the z-axis. For the hemisphere, starting at z = 10, we ensure all parts align perfectly. This way, each point in the coordinate system contributes to forming the final shape鈥攁 silo.

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Most popular questions from this chapter

\(21-32\) Use spherical coordinates. Evaluate \(\int \mathbb{J}_{H}\left(9-x^{2}-y^{2}\right) d V,\) where \(H\) is the solid hemisphere \(x^{2}+y^{2}+z^{2} \leqslant 9, z \geqslant 0\)

Set up, but do not evaluate, integral expressions for (a) the mass, (b) the center of mass, and (c) the moment of inertia about the \(z\) -axis. $$\begin{array}{l}{\text { The hemisphere } x^{2}+y^{2}+z^{2} \leqslant 1, z \geqslant 0}; \\ {\rho(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}}\end{array}$$

Evaluate the given integral by changing to polar coordinates. \(\iint_{R}(2 x-y) d A,\) where \(R\) is the region in the first quadrant enclosed by the circle \(x^{2}+y^{2}=4\) and the lines \(x=0\) and \(y=x\)

Show that $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \sqrt{x^{2}+y^{2}+z^{2}} e^{-\left(x^{2}+y^{2}+z^{2}\right)} d x d y d z=2 \pi$$ (The improper triple integral is defined as the limit of a triple integral over a solid sphere as the radius of the sphere increases indefinitely.)

Use cylindrical coordinates. $$\begin{array}{l}{\text { Evaluate } \iint_{E} z d V, \text { where } E \text { is enclosed by the paraboloid }} \\ {z=x^{2}+y^{2} \text { and the plane } z=4}\end{array}$$
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