Question

one winter afternoon, unbeknownst to his mom, a child brings a snowball into the house, lays it on the floor, and then goes to watch tv. let ( w = w(t) ) be the volume of dirty water that has soaked into the carpet ( t ) minutes after the snowball was deposited on the floor. explain in practical terms what the limiting value of ( w ) represents.

• the limiting value is the rate at which the snowball melts.
• the limiting value is the total time it takes for the snowball to melt completely.
• there is no limiting value for ( w(t) ).
• the limiting value is the total volume of water frozen in the snowball.

what has happened physically when this limiting value is reached?
the limiting value is reached when the snowball has (\boldsymbol{\text{—select—}}) melted.

Explanation:

First Question (Multiple Choice)

• The first option is about the rate of melting, but the limiting value of \( W(t) \) (volume of water) should be the total water from the snowball, not the rate.
• The second option is about time, but \( W(t) \) is volume, so time is irrelevant here.
• The third option is wrong because as time passes, the snowball will melt completely, so \( W(t) \) will approach the total volume of water in the snowball.
• The fourth option: The snowball is made of frozen water (ice), so when it melts completely, the volume of water \( W(t) \) will approach the total volume of water that was frozen in the snowball. So this is correct.

When the limiting value (total volume of water from the snowball) is reached, it means all the ice in the snowball has turned into water, i.e., the snowball has completely melted.

Answer:

The limiting value is the total volume of water frozen in the snowball.

Second Question (Fill - in - the - blank)