A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 6 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area. (Round your answer to three decimal places.) g

x = 0.629 cm

Step-by-step explanation:

The volume of the solid  is:

V(s)  =   V(c) + V ( two hemisphere)      

And V(s)  =  6 cm³

The volume of the cylinder is   V(c)  =  π*x²*h

Let call  " x " the radius of the base f the cylinder and of course the radius of the hemispheres

The volume of the cylinder is   V(c)  =  π*x²*h

And is equal to 6 - Volume of the sphere of radius x  ( volume of two hemisphere of the same radius is just one sphere

Then V(c)  =  6  - (4/3)*π*x³

Then  V(c)  =  π*x²*h  = 6 - (4/3)*π*x³

h  =  [ 6  - (4/3)*π*x³ ] /π*x²  

The lateral area of the cylnder is :

A(l)  = 2*π*x * h    ⇒  A(l)  =  2*π*x *  [ 6  - (4/3)*π*x³ ] /π*x²  

A(l)  = 12/x  - (8/3)*π*x²

Then surface of the area of the cylinder is:

S(c)  = A(b) + A(l)     ⇒  S(c)  =  π*x²  +  12/x  - (8/3)*π*x²

And the area of a sphere is

S(sphere) = 4π*x²

Total area of the solid is:

S(s) =  π*x²  + 12/x  - (8/3)*π*x² +  4π*x²⇒ S(s) = 5*π*x²+  12/x  - (8/3)*π*x²

Taking derivatives on both sides of the equation we get

S´(s) = 10*π*x - 12/x² - (16/3)*π*x

As  10 = 30/3

S´(s) = (46/3)*π*x - 12/x²

S´(s) = 0                 (46/3)*π*x - 12/x² = 0    

46*π*x³  =  36

x³  = 0,2492

x = ∛0,2492

x = 0.629 cm

d

step-by-step explanation:

hello and welcome!

so, when we see the words "proportional" along with a table, we need to know that we're looking numbers that divide by the same thing.

looking at this table, we simply have to do a trial and error, and divide to see which numbers have the same quotient. if you look at choice a & c, they both have equal quotients for three of the numbers, with one exception, so those can't be the answer. looking at b, there is no similarity, so that can't be it. then, we see d; if you divide all the numbers in the table, you'll see that they all divide by two, which is a proportional relationship, and our answer!

also, always remember that the x will always be the number that goes into y, and somewhere behind this table, is a formula. for this specific table, the formula would be:

since our x-value is being divided by two to get the y-value. and if they were to ask for more numbers in the series, this is the formula you would use to find them.

maybe i can