More Optimization Problems

This class was focused on some more optimization problems.


By following the 6 step guide, it makes solving the problems a little easier.

Find the maximum volume of a right circular cylinder that can be inscribed in a cone of altitude 12 cm, and base radius of 4 cm, if the centres of the cylinder coincide.

Step 1)
The question tells us what to optimize, which is the volume.

Step 2)
We are optimizing volume so the formula for the volume of a cylinder is:

v=pir²h

Step 3a)
Through our picture of an inscribed cylinder in a cone, we see that we have similar triangles going on. So the height of the big triangle is proportioned to the height of the smaller triangle. As will the base of the big triangle be proportioned to the base of the smaller triangle, which is the radius of the cylinder.

Step 3b)
From this we can make the equation:

4/r = 12/12-h

From here, all we have to do is isolate one of the variables but we found out that there'll be less work if we isolate h. By isolating h we get :

h=12-3r

Step 3c)
After plugging in h into the optimization formula we'll get:

v(r)=12pi r²-3pi r³

Step 4)
We found the derivative of the optimization equation which is:

^{v'(t)=3pir(8-3r)}

Step 5)
The critical numbers is when r=8/3. We also checked if this was a max by using the first derivative test.

Step 6)

^{We find the max volume by substituing 8/3 into r.}Next Question.

So we always start with drawing a diagram first. From the diagram we can see that we have a triangle. X represents the point when he starts to walk. Using pythagoras we can find the hypotenuse, which is the distance of his boat ride. Then we can make a table of the values we know for distance, rate, and time. Time can be found by doing d/r. We have yet to find the optimization equation yet. What we need to minimize is the time. With the info in our table we can make a function t. The function t(x) is just the time he took to walk and row addeded together. Now all that is needed to do is to find the derivative and find critical points.

Yeah sorry this was kind of rushed but I've yet to eat dinner and I'm really hungry. As for scribe I'll pick Shelly. I know you said you were busy but I really don't know who hasn't scribed yet so yeah.