Calculus Questions!

Question

1) A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder.? The total volume of the solid is 2 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area. (Round your answer to three decimal places.)

 

2) Find the point on the graph of the function that is closest to the given point. f(x) = x^2 (2,1/2)? 

Answer 1

1) The volume is a sphere and a cylinder combined...which would give you the formula

V= 4/3 π r³ + π r² h = 2

 

The surface area is A= 4πr² + 2πrh.

 

Using the first equation(since the volume is given) you can solve for h....and substitute it into the SA equation.

 

h = (2 - 4/3 Π r³) / π r²     so substitute to get

 

A = 4πr² + 2πr [ (2-4/3 π r³) / π r² )]

A = 4πr² + 4/r - 8/3 π r²

A = 4/3 π r² + 4r ^-1

A' = 8/3 π r - 4r ^-2 =0      multiply by 3r² to get rid of the fractions to get

        8πr³ - 12       = 0

            r³             = 12/(8π)

            r               =   .78159 rounds to .782. you can plug in your r in the equation above to find h if necessary.

 

 

 

2) D = [ (x₂ -  x₁)² + (y₂ - y₁)² ]^1/2

 

So one point is (2, .5) and the other point is (x,x²)......substitute it into the distance formula to get

 

D = [ (x-2)² + (x² - .5)² ]^1/2

D = [ x² -4x +4 + x⁴ - 1x² + .25 ]^1/2

D = [ x⁴  -4x +4.25)^1/2

D' = 1/2 [ x⁴ -4x + 4.25] ^-1/2 * [4x³  -4]

 

So this equals zero when the numerator equals zero....so solve 4x³ - 4 = 0

 

So x = 1 so your point is (1,1).

 

Hope this helped.