A homeowner wants to make an elliptical rug from a 12 -ft by 10 -ft rectangular piece of carpeting. a. What lengths of the major and minor axes would maximize the area of the new rug? b. Write an equation of the ellipse with maximum area. Use a coordinate system with the origin at the center of the rug and horizontal major axis. c. To cut the rectangular piece of carpeting, the homeowner needs to know the location of the foci. Then she will insert tacks at the foci, take a piece of string the length of the major axis, and fasten the ends to the tacks. Drawing the string tight, she'll use a piece of chalk to trace the ellipse. At what coordinates should the tacks be located? Describe the location.
A homeowner wants to make an elliptical rug from a 12 -ft by 10 -ft rectangular piece of carpeting. a. What lengths of the major and minor axes would maximize the area of the new rug? b. Write an equation of the ellipse with maximum area. Use a coordinate system with the origin at the center of the rug and horizontal major axis. c. To cut the rectangular piece of carpeting, the homeowner needs to know the location of the foci. Then she will insert tacks at the foci, take a piece of string the length of the major axis, and fasten the ends to the tacks. Drawing the string tight, she'll use a piece of chalk to trace the ellipse. At what coordinates should the tacks be located? Describe the location.
Solution Summary: The author calculates the major and minor axes that maximize the area of an elliptical rug made from 12-ft rectangular piece of carpeting.
A homeowner wants to make an elliptical rug from a
12
-ft
by
10
-ft
rectangular piece of carpeting.
a. What lengths of the major and minor axes would maximize the area of the new rug?
b. Write an equation of the ellipse with maximum area. Use a coordinate system with the origin at the center of the rug and horizontal major axis.
c. To cut the rectangular piece of carpeting, the homeowner needs to know the location of the foci. Then she will insert tacks at the foci, take a piece of string the length of the major axis, and fasten the ends to the tacks. Drawing the string tight, she'll use a piece of chalk to trace the ellipse. At what coordinates should the tacks be located?
Describe the location.
System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.
Write the given third order linear equation as an equivalent system of first order equations with initial values.
Use
Y1 = Y, Y2 = y', and y3 = y".
-
-
√ (3t¹ + 3 − t³)y" — y" + (3t² + 3)y' + (3t — 3t¹) y = 1 − 3t²
\y(3) = 1, y′(3) = −2, y″(3) = −3
(8) - (888) -
with initial values
Y
=
If you don't get this in 3 tries, you can get a hint.
Question 2
1 pts
Let A be the value of the triple integral
SSS.
(x³ y² z) dV where D is the region
D
bounded by the planes 3z + 5y = 15, 4z — 5y = 20, x = 0, x = 1, and z = 0.
Then the value of sin(3A) is
-0.003
0.496
-0.408
-0.420
0.384
-0.162
0.367
0.364
Question 1
Let A be the value of the triple integral SSS₂ (x + 22)
=
1 pts
dV where D is the
region in
0, y = 2, y = 2x, z = 0, and
the first octant bounded by the planes x
z = 1 + 2x + y. Then the value of cos(A/4) is
-0.411
0.709
0.067
-0.841
0.578
-0.913
-0.908
-0.120
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