- An ellipsoid is formed by rotating the graph of + y² = 1 about the y-axis. A right circular cone is to be inscribed within the ellipsoid having its circular base parallel to the x-axis as shown in the figure. Let P = (x, y) denote that point of the xy-plane that has a positive x-coordinate and lies on the base of the cone. Carry out the following steps to determine the dimensions of the inscribed cone that has maximal volume. (a) Express the area of the circular base of the cone in terms of x. (b) Express the height of the cone in terms of y. (c) Determine the formula for V, the volume of the cone, as a function of only the variable y. (d) Now use calculus to find the dimensions of the cone having maximal volume. (0, 1) P=(x,y) (2,0) X
- An ellipsoid is formed by rotating the graph of + y² = 1 about the y-axis. A right circular cone is to be inscribed within the ellipsoid having its circular base parallel to the x-axis as shown in the figure. Let P = (x, y) denote that point of the xy-plane that has a positive x-coordinate and lies on the base of the cone. Carry out the following steps to determine the dimensions of the inscribed cone that has maximal volume. (a) Express the area of the circular base of the cone in terms of x. (b) Express the height of the cone in terms of y. (c) Determine the formula for V, the volume of the cone, as a function of only the variable y. (d) Now use calculus to find the dimensions of the cone having maximal volume. (0, 1) P=(x,y) (2,0) X
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.3: Hyperbolas
Problem 36E
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VIEWStep 2: Part-a, Area of the circular base
VIEWStep 3: Part-b, The height of the cone
VIEWStep 4: Part-c, The formula for volume V as a function of only variable y
VIEWStep 5: Part-d, The dimensions of the cone having maximum volume
VIEWStep 6: Using second derivative test and the dimension of the cone for maximum volume
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