A hemisphere of radius 6 sits on a horizontal plane. A cylinder stands with its axis vertical, the center of its base at the center of the sphere, and its top circular rim touching the hemisphere. Find the radius and height of the cylinder of maximum volume.
find the max radius and height.

x =  4,24 units    radius of the base of the cylinder

h = 4,24  units     height f the cylinder

Step-by-step explanation: See Annex

Let call  " x "  and  " h "  radius and height f th cylinder, then

Volume of the cylinder is :

V(c)  = π*x²*h       (1)

From annex we can see from the right triangle

L²   =  h²  +  x²

As  L  is radius of the hemisphere, and is equal to 6

(6)²  =  h²  +  x²    ⇒   h²  =  36  -  x²   ⇒  h  =  √ (36  -  x²)

Plugging the value of h  in equation (1) we get Volume as a function of x

V(c)  = π*x²*h  

V(x)  = π*x²* √ (36  -  x²)

Taking derivatives :

V´(x)  =  2*π*x*√ (36  -  x²) +(  π*x²) * (-2*x) / √ (36  -  x²)

V´(x) = 0

2*π*x*√ (36  -  x²) - 2*π*x³/ √ (36  -  x²)  =  0

2*π*x*( 36  -  x² )  -  2*π*x³  = 0

72*π*x -  2*π*x³ - 2*π*x³  = 0

18 -  x²  =  0

x = √ 18

x =  4,24 units

And h  =  √ 36 - 18

h  = √18

h = 4,24 units

step-by-step explanation:

let's assume

number of candles set =x

price of of x candles =p(x)

we are given

a candle maker prices one set of scented candles at $10 and sells an average of 200 sets each week

so, we get point as (200,10)

he finds that when he reduces the price by $1, he then sells 50 more candle sets each week

so, we can find slope

now, we can use point slope form of line

now, we can plug values

we can use point

and find b

now, we can plug it back

and we get

now, we can find revenue

we know that

revenue = price*quantity

so, we get