Question

(Calculus - Optimisation)
Attempted this twice, but I couldn't get it :(

A fuel tank is being designed to contain 200 cubic meters of gasoline. The design of the tank calls for a cylindrical part in the middle, with hemispheres at each end. The maximum length of a tank (measured from the tips of each hemisphere) that can be safely transported is 16 m. If the hemispheres are twice as expensive per unit of area as the cylindrical part, find the radius and height of the cylindrical part so that the cost of manufacturing will be minimal.

Answer 1

V = pi*r^2*h = 200 cm^3
V = pi*r^2*16 = 200 cm^3
r = 1.99

Answer 2

Now that is not the answer obviously.

Answer 3

Minimize: SA = 2πr^2 + 2πrh
h = 16

Answer 4

volume of cylinder would be (16-2r)πr^2
16-2r is the h of that part
you with me so far?

Answer 5

Where did you get the 16-2r?

Answer 6

or is that just the height for the cylinder, because i use r when i refer to the radius

Answer 7

it is the height of the cylinder not including the hemispheres at the end

Answer 8

Yes, okay I get it

Answer 9

give me 5 minutes and i will be able to explain in full

Answer 10

okay

Answer 11

so this question is all about minimizing cost right?
so we will need to create a function for cost, and then see what variable minimize that function

Answer 12

Okay

Answer 13

we are assuming that we are in a magic math world and the walls of the tank are infinitely thin, so we treat them by their area

Answer 14

as the problem tells us to

Answer 15

so first we need to be familiar with 2 equations of surface area. the first surface area of a cylinder is 2πrh. the surface are of a sphere is 4πr. we have to hemispheres so we can just combine them and think about them as a sphere. still with me?

Answer 16

i mean 2 hemispheres (not to)

Answer 17

Isn't the surface area of a cylinder 2πrh + 2πr^2
Hemisphere: 4πr^2

Answer 18

you are right about the sphere (typo on my end). but the equation for the cylinder is assuming that the cylinder has ends. here the cylinder is open ended

Answer 19

Oh, I see

Answer 20

so here is the equation for cost. we will come back to it in a minute. remember that the hemispheres will be twice as expensive so their cost per area will be multiplied by 2
cost = 2(4πr^2) + 2πrh

Answer 21

brb (2 minutes)

Answer 22

Okay, got it

Answer 23

forget what i said about h=16-2r. thats not necessarily true. h< or equal to 16-2r.
now that we have our equation for cost, lets look at out equation for the h and r with respect to the volume

Answer 24

Okay

Answer 25

the volume is 200cm^3
so the equation we get is this:
volume of hemishpheres+volume of cylinder = 200
4/3πr^3 + πr^2h = 200

Answer 26

Right, got that

Answer 27

go on ahead, i'll brb

Answer 28

so now we can isolate r or h from the equation i just wrote and plug it in to the other equation for cost

Answer 29

you're doing this in a calculus course right?

Answer 30

Yes

Answer 31

So we isolate h from the equation for volume, put that into the cost equation and then find the derivative?

Answer 32

exactly

Answer 33

keep in mind we may find multiple solutions where derivative =0. so first we need to make sure that the results of a given solution satisfy the parameters that h+2R< OR =16. another parameter would be the h and r are both positive

Answer 34

so...have worked out the derivative yet?

Answer 35

you are going to need the product and quotient rule. it might take a while

Answer 36

Okay, i understand now

Answer 37

I am just heading out to school actually, so I shall access the openstudy site from there

Answer 38

You just saved my life, thank you so much :D

Answer 39

did it work?

Answer 40

didn't have time to calculate, I will contact you when I have though

Answer 41

or you online later on?

Answer 42

*are

Answer 43

maybe, im actually in israel so im a few hours a head of where ever you are

Answer 44

you in highschool?

Answer 45

Yes

Answer 46

what time do u think youll need help?

Answer 47

I'll be on in an hour/ an hour and 15 minutes

Answer 48

i should be here. i need to go walk my dog

Answer 49

Okay, thanks