Maxima and Minima Question HELP!!!! (1 Viewer)

[Bisu]

i) A cylinder is inscribed in a cone of radius 6cm and height 20cm.

Find the dimensions of the cylinder that will make the volume of the cylinder to be maximum.

ii) A cargo service operates running a ship between port A and port B, 500km apart, at a constant speed of V km/hr. For a given speed, the cost of per hour, $C, of running the ship is given by C = 9000 + 10V^2

At what speed should the ship travel so as to minimise the cost of the trip?

iii) A courier company will only accept a box for shipment if the sum of its length and girth (distance around) does not exceed 84cm. A box in the shape of a square prism is to be used to ship an article itnerstate. Find the dimensions of the largest box that can be used.

 

[marmsie]

I think I have an answer for i) (but there may be a simpler method)

To begin with, you need to find a relationship to explain how the radius of the cylinder affects the height of the cylinder. Because we know the cylinder has to be contained by a cone, then we can say that there is a linear relationship between the radius of the cylinder and its height.

To determine this relationship we need two points and the easiest way to figure them out is to consider the two extremes. When the cylinder has a radius of 6 cm then its height must be 0 cm, also when the radius is 0 cm then the height is 20 cm. Using these two points the gradient of the slope can be calculated.

m = (20 - 0) / (0 - 6) = -10 / 3

So the linear relationship is given by the equation:

h - 0 = (-10 / 3) * (r - 6)
h = (-10 / 3) r + 20

Moving onto the volume of the cylinder:

V = PI * r^2 * h

Using the relationship between h and r to eliminate one of the variables gives:

V = PI * r^2 * ( (-10 / 3) r + 20 )
= (-10 PI / 3) r^3 + 20 PI r^2

We want to maximise this volume so deriving w.r.t. the radius:

V' = -10 PI r^2 + 40 PI r

V'' = -20 PI r + 40 PI

For turning points V' = 0

0 = -10 PI r^2 + 40 PI r
0 = -r^2 + 4 r
0 = r ( 4 - r)

Therefore the turning points are r = 0 and r = 4.

Plugging these r values into V'' to determine if they are max or min points (although logic already says that r = 0 will have to be a min as V = 0 when r = 0):

r = 0:
V'' = 40 PI > 0 -> therefore a min point

r = 4:
V'' = -40 PI < 0 -> therefore a max point

Finally, the question wants the dimensions of the cylinder. So plugging r = 4 back into the equation relating h with r gives the dimensions of the cylinder to be:

r = 4
h = 20 / 3

I hope this helps and makes sense.