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

(0,-3) see attached graph

step-by-step explanation:

a. to solve the system, graph the two equations.

x = y+3 becomes y=x-3 has m=1/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move up 1 unit and 1 unit to the right. plot a point and connect.

y=-4x+3 has m =-4/1 and b=-3. start at (0,-3) on the y-axis and plot a point. then move down four units and to the right 1 unit. plot a point and connect.

the solution is the point (x,y) where the two lines intersect. by the graph below it is (0,-3).

b. to solve by substitution, substitute y = -4x-3 into x= y+3

x= (-4x-3) +3

x=-4x

5x=0

x=0

now substitute x = 0 back in to find y.

y= -4(0)-3

y=0-3

y=-3

solution is (0,-3)

c. use elimination by adding the equations together.

x-y=3

4x+y=-3

5x=0

x=0

then substitute into one equation to find y.

y=-4(0)-3

y=-3

here is the

mark brainliest pl