Snowplow problem With snow on the ground and falling at a...

Snowplow problem With snow on the ground and falling at a constant rate, a snowplow began plowing down a long straight road at noon. The plow traveled twice as far in the first hour as it did in the second hour. At what time did the snow start falling? Assume the plowing rate is inversely proportional to the depth of the snow.

Key Concepts

Inverse Proportionality

This concept means that one quantity increases as the other decreases. In the problem, the plowing speed is inversely proportional to the depth of the snow, so as the depth increases, the speed decreases in such a way that their product remains constant.

Constant Rate of Snowfall

A constant snowfall rate implies that snow accumulates uniformly over time. This results in a snow depth that increases linearly with time, which is a crucial element in determining how the snow depth affects plowing speed over different time intervals.

Snow Accumulation and Plowing Rate Relationship

Understanding how snow accumulation affects the plowing rate is key. Since the snow accumulates at a constant rate and the plowing speed depends on the reciprocal of the snow depth, the speed of the plow is a function of time elapsed since snowfall started. This relationship helps establish a link between the time when the snow began falling and the distance covered during successive time periods.

Comparative Distance Analysis in Time Intervals

The problem compares distances covered in two consecutive time intervals. Analyzing this information through the relationships between speed, time, and snowfall depth allows one to set up equations that link the plowing performance with the accumulation of snow. Solving these equations gives insight into determining the precise moment when the snowfall began.

Transcript

00:01 Okay, here we have a problem.

00:04 We have snowfall problem.

00:06 First we're going to start by figuring out the depth of the snow.

00:09 So the depth of the snow is the initial depth plus the rate of the snowfall.

00:13 So we're going to figure that out.

00:15 So the depth of the snow is going to be d of t equals s plus the square, sorry, the integral from zero to t of r d t, which is going to be s plus r t.

00:28 So this is our d of t.

00:30 This is our depth of snow.

00:31 It's s plus rt over time.

00:34 T starts when our snow plow starts.

00:37 So to figure out the distance traveled by our snowplow, which i'm going to call l, it's going to be because what we know, the depth of the snow, which is d, is inversely proportional to the amount of the rates of the snowplow, which is p.

00:56 So to figure out the distance traveled by the snowplow, we're going to integrate from 0 to t our inverse relationship.

01:09 So 1 over d of t over for dt, which is equal to 1 over r times the natural log of s plus rt minus the natural log of s.

01:23 I skipped a step here.

01:25 You're supposed to find c here to find c.

01:27 We know that l of 0 equals 0 and you'd plug it in and then figure out that.

01:33 That the natural, c is equal to the natural log of s times 1 over r.

01:38 So we get this.

01:40 From there, we're going to move on.

01:42 We're going to look at that the fact is the distance traveled in the first hour is equal to the two distances traveled in the second hour...

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