Q)

'What are the dimensions of a cylinder that has the same maximum volume as the box (1131.97 cm2), but uses the minimum amount of material to make it?'

Here are the notes I have so far:

SA=2Pi r^2 + 2 Pi r (1131.97/Pi r^2)

Now i need to use derivatives, I don't know how!

SA1=

Then these notes-
at min. gradent =0 therefore SA1=0

rearrange to find r

sub r in to find

Anyone have any clue? thanks. its my assinment

your line

SA=2Pi r^2 + 2 Pi r (1131.97/Pi r^2)

reduces to

SA=2Pi r^2 + 2263.94/r

then

SA' = 4(pi)r - 2263.94/r^2

now set this equal to zero for a minimum surface area, so....do the algebra

r^3 = 2263.94/(4pi)

= 180.15862

take the cube root to get r,

go back into 1131.97/(Pi r^2) to get the height

(I got h = 11.296 and r = 5.648
notice that would make the diameter equal to the height, mmmmhhhh?
isn't Calculus wonderful???)

you couldn't show all the working could you? its just that i don't understand how to do them, and can't do them

thanks

To find the dimensions of the cylinder that has the same maximum volume as the given box (with a surface area of 1131.97 cm^2) while using the minimum amount of material, we can follow these steps:

Step 1: Write the equation for the surface area of a cylinder in terms of its dimensions.
The surface area of a cylinder is given by the formula: SA = 2πr^2 + 2πrh, where r is the radius of the base and h is the height of the cylinder.

Step 2: Express the height (h) in terms of the radius (r) using the given volume condition.
Since we want the cylinder to have the same maximum volume as the box, we can use the formula for the volume of a cylinder: V = πr^2h. By rearranging this formula, we can solve for h: h = V / (πr^2).

Step 3: Substitute the expression for h back into the equation for the surface area (SA) to get an equation in terms of r only.
Replace h in the surface area equation (SA = 2πr^2 + 2πrh) with V / (πr^2): SA = 2πr^2 + 2πr(V / (πr^2)). Simplify this equation to obtain SA = 2πr^2 + 2V / r.

Step 4: Take the derivative of the surface area with respect to r.
To find the minimum amount of material used, we need to minimize the surface area. Taking the derivative of the surface area equation with respect to r will give us a critical point where the derivative equals zero, indicating a possible minimum.

Step 5: Set the derivative equal to zero and solve for r.
Set SA' = 0 and solve the resulting equation for r. This will give you the critical point where the surface area is potentially minimized.

Step 6: Substitute the value of r back into the equation for h to obtain the corresponding height.
After finding the value of r that satisfies SA' = 0, substitute that value back into the equation for h = V / (πr^2). This will give you the corresponding height of the cylinder.

Finally, you will have the dimensions of the cylinder (radius r and height h) that have the same maximum volume as the box while using the minimum amount of material.