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Chapter 13: Problem 86

Identify and briefly describe the surfaces defined by the following equations. $$x^{2}+4 y^{2}=1$$

Short Answer

Expert verified
Answer: The given equation represents a horizontal ellipse, centered at the origin (0, 0), with a semi-major axis of length 1 along the x-axis and a semi-minor axis of length 1/2 along the y-axis.

Step by step solution

01

Identify the Type of Surface

The surface is defined by the equation $$x^2 + 4y^2 = 1.$$ Since there is no z component and the sum of x and y terms involves squares, the equation represents an ellipsoid.
02

Recognize an Ellipse

If we set \(z = 0\) (the constant plane), the equation becomes $$x^2 + 4y^2 = 1$$ This equation represents an ellipse in the \(xy\)-plane, with the center at the origin \((0, 0)\).
03

Determine the Axis Lengths

In general, an ellipse has the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). By comparing the given equation to the general equation, we can identify the values of \(a^2\) and \(b^2\). In our case, \(a^2 = 1\) and \(b^2 = \frac{1}{4}\). Therefore, the semi-major and semi-minor axis lengths are: \(a = 1\) and \(b = \frac{1}{2}\).
04

Describe the Surface

The surface defined by the equation $$x^2 + 4y^2 = 1$$ is a horizontal ellipse, centered at the origin \((0, 0)\), with a semi-major axis of length 1 along the x-axis and a semi-minor axis of length \(\frac{1}{2}\) along the y-axis.

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Most popular questions from this chapter

A clothing company makes a profit of \(\$ 10\) on its long-sleeved T-shirts and \(\$ 5\) on its short-sleeved T-shirts. Assuming there is a \(\$ 200\) setup cost, the profit on \(\mathrm{T}\) -shirt sales is \(z=10 x+5 y-200,\) where \(x\) is the number of long-sleeved T-shirts sold and \(y\) is the number of short-sleeved T-shirts sold. Assume \(x\) and \(y\) are nonnegative. a. Graph the plane that gives the profit using the window $$ [0,40] \times[0,40] \times[-400,400] $$ b. If \(x=20\) and \(y=10,\) is the profit positive or negative? c. Describe the values of \(x\) and \(y\) for which the company breaks even (for which the profit is zero). Mark this set on your graph.

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Let \(h\) be continuous for all real numbers. a. Find \(f_{x}\) and \(f_{y}\) when \(f(x, y)=\int_{x}^{y} h(s) d s\) b. Find \(f_{x}\) and \(f_{y}\) when \(f(x, y)=\int_{1}^{x y} h(s) d s\)

Consider the ellipse \(x^{2}+4 y^{2}=1\) in the \(x y\) -plane. a. If this ellipse is revolved about the \(x\) -axis, what is the equation of the resulting ellipsoid? b. If this ellipse is revolved about the \(y\) -axis, what is the equation of the resulting ellipsoid?
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