Problems on Maxima and Minima (1 Viewer)

[SunnyScience]

I'm having some Problems with 'Problems on Maxima and Minima' (see what i did there? ) I'm just not quite grasping how to do these questions effectively yet - anyone will to help me out? :/ Thanks.

4. An open tank is to be constructed with a square base of side x metres and with four rectangular sides. The tank is to have a capacity of 32 cubic meters.
(a) show that the heigh of the tank is 32/x^2 metres (did this one )
(b) Find the least area of sheet metal from which the tank can be constructed (Keep getting 60ish but it's wrong :/)


5. A magazine advertisement is to contains 50cm^2 of lettering with clear margins of 4cm each at the top and bottom and 2cm at each side. Find the overall dimensions if the total area of the advertisement is to be a minimum (can draw the diagram.)

6. The cost per hour of running a pleasure cruiser is $((V^2/40) + 10), where V is the speed in knows.
(a) for a trip D nautical miles show that cost is $[D/V((v^2/40) + 10)]
(b) What is the most economical speed for running the cruiser on this trip?

7.
(a) The sum of the radii of two circles is 100cm. If one of the circle has a radius of c cm, show that the sum of the areas of the two circles is given by
A = 2pi(x^2 -100x + 5000)
(b) find the value of x for which A is least.



Could you guys please show me how to do maybe a couple of these, so i can possibly recognise a pattern/get the hang of it? Are these types of questions considered hard for this topic section?

Thanks guys

 

[SunnyScience]

..bump

 

[SunnyScience]

no-one love me.

 

[AAEldar]

You posted this just over 30 minutes ago, if you keep demanding an answer no one will reply.

 

[Kimyia]

Lets try 4:
a) Volume = x^2 x h = 32
Rearrange to get: h = 32/(x^2)

b) A (area of sheet metal) = x^2 + 4hx
But h = 32/(x^2)
Sub in to get: A = 128/x + x^2
Derive to get: -128/(x^2) + 2x
Make first derivative equal to zero, solve to get x = 4.
Now, find the second derivative = 256/(x^3) + 2
Sub x=4 into the second derivative which gives you 6 which is greater than zero, therefore A will be a minimum - which is what you want for the least amount of sheet metal.
Now sub x=4 into your original A formula which gives you 48m^2.

EDIT: forgot to multiply the x, thanks Shadowdude.

 

[Shadowdude]

These questions are usually hard. Be patient.

For 4b, I got 48. (bah i dont know if i'm doing it wrong)

"A (area of sheet metal) = x^2 + 4hx
But h = 32/(x^2)
Sub in to get: A = 128/(x^2) + x^2"

If you sub in, shouldn't you get: A = x^2 + 4(32/x^2)x = x^2 + 128/x ?

 

[Shadowdude]

For Question 5, is the answer... 9 by 18 cm?

 

[HeroWise]