In other words, what do the x-values represent? ________________[+2] What are th

Question

In other words, what do the x-values represent? ________________[+2] What are the smallest and largest values of x that make sense in terms of the cone? ___________ [+2] What do the y values represent? ____________________[+2] What are the smallest and largest values ofy that make sense in terms of the cone? ____________[+2] Find the Exact Angle theta that Maximizes the Volume Refer back to section II to find your equation for the volume of the cone. Graph it in your graphing calculator, along with the scatterplot. Does the equation go through the data points? If you have done everything correctly it Use your calculator and the equation you wrote to approximate the exact angle that maximizes the volume Maximum volume ____________[2+] occurs at theta = ____________[2+]. Explore the Relationship between theta and Volume If the volume of the cone is 300 cm^3, what would the angle theta be? ____________ [+4] If the slant height was changed to 40 cm, how would this affect the volume of the cone? Write an equation to represent the radius, height, and volume. Radius = _____________ [+2] Height ___________[+2]______ Volume = _____________[+2]

Explanation / Answer

Let r be the radius of the base.
The volume is 1/3 r^2 h = 300
h = 900 / r^2

So tan(1/2 the angle) O/A = r/ h = r^2/900
The angle is
2 arctan ( r^3/ 900)
=2 arctan (Radius X Area of Base / 900)

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The volume is 1/3 r^2 h

slant height of cone = sqrt( h^2 + r^2 )

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