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Magazine Chapter 18: Problem 3
Draw a sketch of the graph of the given equation and name the surface. $$ 4 x^{2}+9 y^{2}-z^{2}=36 $$
Short Answer
Expert verified
The equation represents a hyperboloid of one sheet.
Step by step solution
01
Identify the Equation Type
The given equation is in the form of a quadratic equation involving three variables. This resembles the standard form of a hyperboloid of one sheet: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \).
02
Rearrange the Equation
Rewrite the given equation to match the standard form. Start with \( 4x^2 + 9y^2 - z^2 = 36 \), and divide all terms by 36: \(\frac{4x^2}{36} + \frac{9y^2}{36} - \frac{z^2}{36} = 1\).
03
Simplify the Fractions
Simplify the equation: \(\frac{x^2}{9} + \frac{y^2}{4} - \frac{z^2}{36} = 1 \). This confirms that the surface is a hyperboloid of one sheet with \ a^2=9, \ b^2=4, \ and \ c^2=36 \.
04
Determine the Axes Lengths
The surface intercepts the x-axis, y-axis, and z-axis at: \( a = \frac{\text{Distance on x-axis}}{\text{Scaling factor}} = 3 \, b = 2 \, c = 6 \).
05
Draw the Sketch
Draw the hyperboloid by making a sketch with the correct proportions. Plot the vertices at (卤3, 0, 0), (0, 卤2, 0), and use the hyperbolic curves to link these points while extending along the z-axis up to 卤6 units.
06
Naming the Surface
Finally, name the identified surface: the given equation represents a hyperboloid of one sheet.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
graph sketching
Sketching a graph helps visualize mathematical equations. For a hyperboloid of one sheet, start by understanding its equation's form.
A hyperboloid of one sheet has a unique shape, often curving outward.
Identify the intercepts on each axis to properly draw it.
In this case, the equation \(4x^{2}+9y^{2}-z^{2}=36\) is simplified to \(\frac{x^2}{9} + \frac{y^2}{4} - \frac{z^2}{36} = 1\), showing intercepts at (卤3,0,0), (0,卤2,0), and along the z-axis extending up to 卤6 units.
Sketching involves marking these points and drawing hyperbolic curves connecting them.
The z-axis extends more than the x and y axes, making this shape distinct.
A hyperboloid of one sheet has a unique shape, often curving outward.
Identify the intercepts on each axis to properly draw it.
In this case, the equation \(4x^{2}+9y^{2}-z^{2}=36\) is simplified to \(\frac{x^2}{9} + \frac{y^2}{4} - \frac{z^2}{36} = 1\), showing intercepts at (卤3,0,0), (0,卤2,0), and along the z-axis extending up to 卤6 units.
Sketching involves marking these points and drawing hyperbolic curves connecting them.
The z-axis extends more than the x and y axes, making this shape distinct.
quadratic equations
Quadratic equations involve variables raised to the power of two.
Here, we see a quadratic equation in three variables: \(4x^{2}+9y^{2}-z^{2}=36\).
A typical quadratic equation appears as \(ax^{2} + bx + c = 0\), but in three dimensions, it gets more complex.
Rearrange and simplify equations to match standard forms for better understanding.
In our example, the equation changes to \(\frac{x^2}{9} + \frac{y^2}{4} - \frac{z^2}{36} = 1)\) after dividing by 36.
This helps to identify it as a hyperboloid of one sheet.
Here, we see a quadratic equation in three variables: \(4x^{2}+9y^{2}-z^{2}=36\).
A typical quadratic equation appears as \(ax^{2} + bx + c = 0\), but in three dimensions, it gets more complex.
Rearrange and simplify equations to match standard forms for better understanding.
In our example, the equation changes to \(\frac{x^2}{9} + \frac{y^2}{4} - \frac{z^2}{36} = 1)\) after dividing by 36.
This helps to identify it as a hyperboloid of one sheet.
three-dimensional surfaces
Three-dimensional surfaces add depth to our understanding of shapes.
These surfaces include spheres, cylinders, and hyperboloids.
A hyperboloid of one sheet curves outward in a unique shape.
To study these surfaces, we use their equations and sketch them.
For instance, \(\frac{x^2}{9} + \frac{y^2}{4} - \frac{z^2}{36} = 1\) represents a three-dimensional hyperboloid.
Visualizing three-dimensional surfaces helps in fields like physics and engineering.
These surfaces include spheres, cylinders, and hyperboloids.
A hyperboloid of one sheet curves outward in a unique shape.
To study these surfaces, we use their equations and sketch them.
For instance, \(\frac{x^2}{9} + \frac{y^2}{4} - \frac{z^2}{36} = 1\) represents a three-dimensional hyperboloid.
Visualizing three-dimensional surfaces helps in fields like physics and engineering.
standard form equations
Standard form equations simplify identifying geometric shapes.
Equations of hyperboloids of one sheet appear as: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\).
Converting the given equation to its standard form helps in visualizing it.
For instance, \(4x^{2}+9y^{2}-z^{2}=36\) converts to \(\frac{x^2}{9} + \frac{y^2}{4} - \frac{z^2}{36} = 1\) by dividing all terms by 36.
Recognizing the standard form lets us know we鈥檙e dealing with a hyperboloid.
Equations of hyperboloids of one sheet appear as: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\).
Converting the given equation to its standard form helps in visualizing it.
For instance, \(4x^{2}+9y^{2}-z^{2}=36\) converts to \(\frac{x^2}{9} + \frac{y^2}{4} - \frac{z^2}{36} = 1\) by dividing all terms by 36.
Recognizing the standard form lets us know we鈥檙e dealing with a hyperboloid.
hyperbolic structures
Hyperbolic structures have unique mathematical and practical significance.
These structures often appear in physics and engineering.
A hyperboloid of one sheet looks like two connected curved surfaces.
In math, its equation might look like \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\).
For example, \(\frac{x^2}{9} + \frac{y^2}{4} - \frac{z^2}{36} = 1\) represents a hyperboloid where values of a, b, and c determine its shape.
Such equations help design and understand structures seen in real life.
These structures often appear in physics and engineering.
A hyperboloid of one sheet looks like two connected curved surfaces.
In math, its equation might look like \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\).
For example, \(\frac{x^2}{9} + \frac{y^2}{4} - \frac{z^2}{36} = 1\) represents a hyperboloid where values of a, b, and c determine its shape.
Such equations help design and understand structures seen in real life.
