Calc Problem

playboy

Homework Statement

Snow began to fall during the morning of February 2 and continued steadily into the afternoon.
At noon a snowplow began removing snow from a road at a constant rate. The plow traveled 6 km from noon to 1 P.M. but only 3 km from 1 P.M. to 2 P.M. When did the snow begin to fall?

Homework Equations

t = time

x(t) = distance

dx\dt = speed of the plow

b - the number of hours before noon that it began to snow

The Attempt at a Solution

I know that the speed of the plow is (6 + 3) /2 hours = 4.5km\hour

I am trying to figure out an expression for the height of the snow at time t and then use the given ingormation that the rate of removal R is constant, and thus, find the time it began to snow

Can somebody help me out please?

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D H
Staff Emeritus
The average speed of the plow isn't going to help much here, as the plow's speed is changing (and doing so constantly). This is a calculus problem.

How does the plow's instantaneous speed relate to the constant rate at which the plow removes snow from the road?

How would you interpret the phrase "Snow began to fall during the morning of February 2 and continued steadily into the afternoon" mathematically?

playboy
I am more lost than ever.

How does the plow's instantaneous speed relate to the constant rate at which the plow removes snow from the road?
I dont think the instantaneous speed has an anything to do with the problem correct?
The plow could have been going really fast at one point and perphaps at rest at another point?

How would you interpret the phrase "Snow began to fall during the morning of February 2 and continued steadily into the afternoon" mathematically?
This simply means that the snow fell at a constant rate from morning into afternoon,
we can call this rate D.

D H
Staff Emeritus
The instantaneous speed has everything to do with the problem. You are given two distances, both of which are definite integrals of the instantaneous speed.

Formulate these integrals and you will be close to having solved the problem. You will need to have an expression for the instantaneous speed to formulate these integrals.

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playboy
Distance 1, 6km, is on the interval [0,1]

Distance 2, 3 km, is on the interval [1,2]

int(0,1) 6 dt + int(1,2) 3 dt = 9

How does that sound?

D H
Staff Emeritus
Not very good. You are integrating distance. You need to integrate velocity,

$$\int_{12:00}^{13:00} v(t) dt = 6\text{\, km}$$

$$\int_{13:00}^{14:00} v(t) dt = 3\text{\, km}$$

Now you just have to come up with an expression for $v(t)$. Hint: You know that the removal rate is a constant. How does this relate to velocity and the amount of snow on the ground?

playboy
Hint: You know that the removal rate is a constant. How does this relate to velocity and the amount of snow on the ground?
The removal rate is Constant.

Therefore, the velocity of the plow is the same as the velocity of snowfall?

D H
Staff Emeritus
The removal rate is constant, but the velocity is not. Snow is falling all afternoon long, and it had a head start on the snowplow.

D H
Staff Emeritus
How does the height of the accumulated snow on the road affect the velocity at which the plow can move?

D H
Staff Emeritus
Another hint: What units would you use to reflect the constant rate at which snow is falling on the road and the constant rate at which the plow is removing snow from the road?

playboy
The removal rate is constant, but the velocity is not. Snow is falling all afternoon long, and it had a head start on the snowplow.
I'm confused between "removal rate" and "velocity" ... I thought they were equivalent.

How does the height of the accumulated snow on the road affect the velocity at which the plow can move?
The higher the hight, the smaller the velocity.

Another hint: What units would you use to reflect the constant rate at which snow is falling on the road and the constant rate at which the plow is removing snow from the road?
Units for snow falling on road --> $$m^2/hour$$ or perhaps $$km^2/hour$$
Units for snow removal --> $$m^3/hour$$ or perhaps $$km^3/hour$$

D H
Staff Emeritus
The higher the hight, the smaller the velocity.
Good. Now you're getting somewhere.

Units for snow falling on road --> $$m^2/hour$$ or perhaps $$km^2/hour$$
Try again. Think of how the weatherman expresses snowfall rates.

Units for snow removal --> $$m^3/hour$$ or perhaps $$km^3/hour$$
Good. Obviously, the snow removal rate cannot equal the snowfall rate since they have different units.

Now factor in the time-varying height of the snow and the fixed width of the plow to determine the velocity of the plow.

Kudos on attempting to learn LaTeX. Nothing beats it (and certainly not ASCII math) for making math look nice.

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playboy
Units for snow falling on road ---> $$m/hour$$

Yeah, Latex is awesome, Something I really have to master

D H
Staff Emeritus
Excellent. Now for the velocity of the plow ...

D H
Staff Emeritus
Have to take a break. Home duties are calling. You're almost there. You should get to a point where you have two expressions involving a lot of unknowns, including the two rates, the width of the plow, and time at which it started snow. A simple algebraic manipulation will eliminate all but one (the time). Solve for the time, and voila, you have the answer.

playboy
Excellent. Now for the velocity of the plow ...
:)

Have to take a break. Home duties are calling. You're almost there. You should get to a point where you have two expressions involving a lot of unknowns, including the two rates, the width of the plow, and time at which it started snow. A simple algebraic manipulation will eliminate all but one (the time). Solve for the time, and voila, you have the answer.
Thank you so much for your time! I really appreciate it,

I am still working on this problem..

playboy
1. Homework Statement

Snow began to fall during the morning of February 2 and continued steadily into the afternoon.
At noon a snowplow began removing snow from a road at a constant rate. The plow traveled 6 km from noon to 1 P.M. but only 3 km from 1 P.M. to 2 P.M. When did the snow begin to fall?

So here is what I have so far:

$$\int_{12:00}^{13:00} v(t) dt = 6\text{\, km}$$

$$\int_{13:00}^{14:00} v(t) dt = 3\text{\, km}$$

RATE OF SNOW REMOVAL is $$km^3/hour$$/hour

RATE OF SNOW FALL is $$km/hour$$/hour

Time of snowfall is $$b$$ hours BEFORE noon.

INSTANTANEOUS veolocity of the plow, $$v(t)$$ ... i still cannot find.

Now factor in the time-varying height of the snow and the fixed width of the plow to determine the velocity of the plow.

Excellent. Now for the velocity of the plow ...

Is the "time-varying height of the snow" the "rate of snowfall" $$km\hour$$ ? The snow did have a head start to fall, so should something be added?

I suppose the width of the plow could just be a constant, which I will call $$w$$

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D H
Staff Emeritus
Good. In a short span of time $\Delta t$, what is the volume of snow $\Delta v_{\text{snow}}$ removed by the plow? (Assume a constant velocity $v(t)$ over this short span of time.) How does this relate to the known rate at which the plow removes snow, and what happens in the limit $\Delta t \to 0$?

playboy
Good. In a short span of time $\Delta t$, what is the volume of snow $\Delta v_{\text{snow}}$ removed by the plow? (Assume a constant velocity $v(t)$ over this short span of time.) How does this relate to the known rate at which the plow removes snow, and what happens in the limit $\Delta t \to 0$?
Im very unsure of this...I thought it...and all I can come up with is that the plow can remove only $$x*km^3$$ of snow...

I don't really know how to fit time into it unless i turn it into a rate.

D H
Staff Emeritus
Between time $t$ and $t+\Delta t$, the plow will have moved about $v(t)\Delta t$ meters down the road. Now suppose the snow height at this time is $h(t)$. Since the plow has a width $w$, it will have removed a volume of snow $\Delta V_{\text{snow}} = w\; h(t)v(t)\Delta t$.

How can you relate this to the plow's snow removal rate, call it $\dot V_{\text{plow}}$?

playboy
Between time $t$ and $t+\Delta t$, the plow will have moved about $v(t)\Delta t$ meters down the road. Now suppose the snow height at this time is $h(t)$. Since the plow has a width $w$, it will have removed a volume of snow $\Delta V_{\text{snow}} = w\; h(t)v(t)\Delta t$.

How can you relate this to the plow's snow removal rate, call it $\dot V_{\text{plow}}$?
I find this problem more like Physics than Calculus..

THE VOLUME OF SNOW REMOVAL IS $\Delta V_{\text{snow}} = w\; h(t)v(t)\Delta t$ which makes sense since (Km) * (Km) * (Km/time) * time = km^3 which is volume!

THE RATE OF SNOW REMOVAL IS (Km^3)/hour

Is $\dot V_{\text{plow}}$ Supposed to be the removal rate or speed of the plow?

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D H
Staff Emeritus
THE RATE OF SNOW REMOVAL IS $$Km^2$$/hour
That is like saying "the speed of my car is km/hr". Those are the units in which the removal rate is expressed (and they should be in volume/time, not area/time).

Is $\dot V_{\text{plow}}$
This is the volume removal rate. To avoid some confusion, I used capital V for volume, lower case v for velocity.

playboy
That is like saying "the speed of my car is km/hr". Those are the units in which the removal rate is expressed (and they should be in volume/time, not area/time).

This is the volume removal rate. To avoid some confusion, I used capital V for volume, lower case v for velocity.

I dont know why Latex was spiting out those numbers..I had it correct in the code :yuck:

THE RATE OF SNOW REMOVAL IS (Km^3)/hour

playboy
Sorry, I am really messing this thread up as I am having problems with Latex...

So isn't the VOLUME REMOVAL RATE, $\dot V_{\text{plow}}$, = $$Km^3$$/hour?

D H
Staff Emeritus
In post #20, I gave the clues needed to relate the fixed removal rate $\dot V_{\text{plow}}$ and the velocity of the plow $v(t)$. Do you need me to spell out the answer?