Question

Izuddin has a rectangular piece of zinc with a perimeter of 32 cm. He wants to use that
piece of zinc to build an open cylinder at both ends.
Find the length and the width, in cm, of the piece of zinc that makes the volume of the
cylinder is maximum. help me solve thank you. answer for length is 12 width 4​

Answer 1

Answer:

Length is 10 2/3 width is 5 1/3 cm.

Step-by-step explanation:

The 32 cm perimeter is equivalent to  2 * circumference of the cylinder + 2  * the height.

2C + 2h = 32.

Now C = 2πr, so

2*2πr + 2h = 32

4πr + 2h = 32

2h = 32 - 4πr

h = 16 - 2πr.

Volume of the cylinder = πr^2h, so

V = π(16 - 2πr) r^2

V = 16πr^2 - 2π^2r^3

Finding the derivative:

V' =   32πr - 6π^2 r^2     This = 0 for a maximum volume:

32πr - 6π^2r^2 = 0

2πr(16 - 3πr) = 0

16 - 3πr = 0

r = 16/3π.

So the circumference = 2π *  16/3π = 32/ 3 = 10 2/3 cm.

The height = 16 - 10 2/3 = 5 1/3.

Therefore the width of the zinc is 5 1/3 and the length is 10 2/3.