Well this week we worked with related rate word problems, which I must say are not fun at all. They're called related rate because somewhere in that problem you're going to have to take a derivative. There's a lot to decipher within one word problem..First you have to figure out what type of information they're giving you and what formula you're going to have to use. Then you have to figure out what exactly you're looking for (what your answer would be) based on that information.
Here are some key words in word problems:
*"a rate of 4.5 cubic feet per minute" = cubic feet indicates VOLUME
*"how fast is something rising" = rising indicates height, which means you'd be looking for dh/dt
*"radius of" = r
*"s" = hypotenuse of a triangle
*"depth" = height
*"altitude" = height
Here's an example:
Volume: All edges of a cube are expanding at a rate of 6 centimeters per second. How fast is the volume changing when each edge is a.) 2 centimeters and b.) 10 centimeters?
*Okay first let's identify what information we're given.
They tell us that it's expanding at a rate of 6 centimeters per second, so that means:
dr/dt = 6 cm/sec
Then you notice that they're looking for the volume of a cube, so you're going to need that formula which is V = x^3 (Well in this case it would be V = r^3..but it's the same thing)
Lastly they ask for how fast the volume is changing..."how fast" is the same thing as rate, and they're talking about volume so that means that they want you to find dv/dt
*So the first thing you want to do when solving this problem is take the derivative of your formula V = x^3
*So you get dv/dt = 3x^2 dx/dt
*Now part a.) wants you to find dv/dt when the edge is 2 centimeters..So all you would do is plug in 2 for x and plug in 6 for dx/dt
And you would get dv/dt = 3(2)^2(6) = 72 cm^3/sec
*Remember, it's cm^3 because volume is always cubed
*Part b.) wants dv/dt when the edge is 10 centimeters...So for this all you do is plug in 10 in place of where you plugged in 2 for part a.
So you get dv/dt = 3(10)^2(6) = 1800 cm^3/sec
And there you go
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