Rate of change question. PLEASE HELP

Question

a block of ice melts in a room held at 70 degrees fahrenheit. Let f(t) be the base area of the block and g(t) the height of the block, measured with a ruler at time t.

a.) Assume the block of ice melts completely at time T. What values would you assign to f(T) and g(T)?

b.) Give physical reasons why lim as t-->T f(t)=f(T) and limt-->T g(t)=g(T) need not both hold: what are the limits?

c.) The limiting volume of the ice blocl at time T is zero. .Write this statement as a limit formula.

d.) Using (c.), illustrate the product rule for limits

Answer 1

Hi Sam L.,

Interesting example, nice problem (compliments to your instructor!).

Let's consider what the math requested actually says. For (a), what can you say about the area of the ice bock and its height when it has COMPLETELY melted? Would you be able to find the block, at all? Then, what values would the "measured" area and height be?

For (b), you are asked to imagine what's going on as the block is about to finish melting. Questions for thought: could the block be tall, but needlelike? Or could it be modest in area, but paper-thin? Which scenario seems more likely to you, for a melting block? Only one of these possibilities seems likely to me!

For (c), you need a formula for volume of the block. How do you assemble this from area and height? (You should know, immediately!). So express this as the similar assembly of f(t) and g(t). It doesn't matter that they are functions, you can still add, subtract, multiply, divide, them (etc.) however you think is required. Hint: one of these mathematical operations is the one you need!

For (d), I guess you are being asked to point out that if both limits go to zero, the product of the limits goes to zero, i.e. 0 * 0 = 0, as a limit. But the physical situation here is an instance of this result, certainly not a proof of it. Just saying! Maybe "illustrate" means a sketch, I don't know.

-- Cheers, -- Mr. d.