FindThe first step to solving this problem is realizing that log properties can apply to make the problem significantly easier. Once we have fully expanded this logarithmic formula, we can take the derivative, which is then quite easy..
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The nightmare has come to pass. All of Kelley's extensive surgeries and nasal passage scrapings have (unfortunately) gone awry, and he waits in the Ear, Nose, and Throat doctor's office waiting area spewing bloodysnot into a conical paper cup at the rate of 4 in3/min. The cup is being held with the vertex down (all the better to pool the snot in, my dear). The booger catcher has a height of 5 inches and a base of 3 inches. How fast is the mucous level rising in the cup when the snot is three inches deep? (Perhaps you should draw a picture .... ewwwww). 
Calculus II (BC): Related Rates 'n' Boogers
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Solution: Ok, what do we know?
This week's solution comes from celebrity guest B. Ross, of Lake Fenton High School. I have added in a little explanation, and it appears in italics --Mr. Kelley
1) the volume of a cone is given by V = (1/3)p(r2)(h)
2) dV/dt = 4 in3/min
3) the height, h, of the cone is 5 in.
4) the base of the cone is 3 in, which makes
r = (3/2) in. (r is the radius of the cone...)So, what's dh/dt when h=3? Well...you can't differentiate directly because in the formula for a cone, more than 2 variables are changing. But...Ahah! We can use the concept of similar triangles to express the radius of the cone in terms of the height. This requires a little explanation; observe the below diagram and equation for explanation:
As the mucous flows into the cup, the radius and height of the mucous will change, so we cannot use constants for these things. However, the dimensions of the overall cup will remain the same. We can now set up the proportion 5/h = (3/2)/r, simplify, and arrive at the result below. This is the hardest step of the whole problem! r = (3h)/10 Smashing... So now we write:V = (1/3)p(3h/10)2(h) ; which can be rewritten again; Now differentiating yields...
V = (9/300)p(h3)4 = (81/100)p(dh/dt) Solve for dh/dt ...400/(81p) = dh/dt By adding a little nutmeg and baking at 350F for 20 minutes (gross out points awarded here), we see that when the height of mucous in the cup is at 3 inches, the height is changing at a rate of about 1.571 in/min. And that sounds about right... at least to me... really, I'm grasping for straws here. I would also like to point out that real calculus teachers are not human and do not bleed. Therefore, this situation can never actually happen.B. Ross, Lake Fenton H.S.
Fenton, MI
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