David was eating 20 grams of ice cream every thirty seconds from his ice cream cone. as the weather was very hot, the 300-gram ice cream was melting and dripping down 5 times a minute with total weight of 5 grams. he had already eaten half of it when the melting rate doubled and he had to continue eating 40 grams per 30 seconds. how long did it take him to eat whole ice cream cone ?

  about 5 minutes 6 seconds

Step-by-step explanation:

There are two ways to interpret this problem statement. "He had already eaten half of it" could be interpreted to mean that half the initial cone is gone and 150 grams remain. Of course, some of the reduction in mass is due to dripping, not due to eating.

So, the other interpretation is that David ate an amount equal to half of the ice cream that would have existed at that time, considering the reduction due to dripping. Taking the problem statement quite literally, we will use this interpretation.

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Initial rates

The initial mass of the ice cream cone is being reduced at the rate of 5 grams per minute. If the cone were not being eaten, its mass could be modeled by ...

  c(t) = 300 -5t . . . . . where c(t) is in grams and t is in minutes

David is eating at the rate of 20 grams in 1/2 minute, or 40 grams per minute, so his eating can be modeled by ...

  e(t) = 40t

The rates of dripping and eating are said to change when David has eaten half the cone, or when ...

  e(t) = (1/2)c(t)

Solving for t, we find ...

  40t = (1/2)(300 -5t)

  85t = 300

  300/85 = t = 60/17 ≈ 3.529 . . . . minutes

At this point in time, the amount of cone remaining is equal to the amount that has been eaten, ...

  e(60/17) = 2400/17 ≈ 141.18 . . . . grams

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Doubled rates

At this point in time, the rates of dripping and eating double, so the amount of cone can now be modeled by ...

  c(t) = (2400/17) -10t

and David's eating can be modeled by ...

  e(t) = 80t

Now, we're looking for the additional time it takes to finish the cone.

  e(t) = c(t)

  80t = (2400/17) -10t

  90t = 2400/17

  t = 2400/(17·90) = 80/51

So, the total time to eat the dripping ice cream cone is ...

  60/17 minutes + 80/51 minutes = 260/51 minutes

  = 5 5/51 minutes ≈ 5 minutes 6 seconds

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Comment on the alternate interpretation

If the problem is worked so the rates double after half the initial cone (150 g) is gone (due to dripping and eating), then the rates change at time 3 1/3 minutes and it takes 1 2/3 minutes for the remaining cone to disappear at the higher rates. The total time comes out an even 5 minutes.

Our objection to this interpretation is that David did not actually eat half of the cone at the point where the rates change.