# Solve Snow Plow Problem: Find Constant k

• Matthew R
In summary, the conversation discusses a snow plow problem where the plow starts at 7am and goes at a constant rate. By 8am, it has gone 4 miles and by 9am it has gone an additional 3 miles. The differential equation and initial conditions are given and the goal is to find the constant k. However, the question actually asks for the time T from when it started snowing until 7am. The conversation also mentions a sixth-order polynomial for T which does not have a root near the given answer of 2.55.
Matthew R
Thread moved from the technical PF forums so no Homework Help Template is shown
Hi guys.

I am currently stuck on the classic snow plow problem. I have the following differential equation and initial conditions:
@ 7am plow starts off to clear snow at a constant rate
By 8am, plow has gone 4mi
By 9am, plow has gone an additional 3mi
Let t=0 when it started to snow, when did it start to snow. They gave the answer as 4:27am

k(dx/dt)=1/t

It wants me to find the constant k.

Here is my work:

∫k dx= ∫dt/t
kx +C = ln (t) At this point I am assuming that x is the independent variable and t is the dependent variable.

e^(kx) +C = t
Using the first initial condition of when it starts to snow of x(0)=0, I get the C is 0, therefore

e^(kx) = t. Using the second initial condition @8, x(t+1)=4

e^(4k)= t+1.

This is the point I get stuck. I can't seem to solve for k that has a real number.

Any help would be appreciated.

Matthew R said:
kx +C = ln (t)
Right.
Matthew R said:
e^(kx) +C = t
Wrong.

Ok, looking back then the c would be e^c which would then be Ae^(kx)=t? But having initial condition of t(0)=1 would make A=0.

Matthew R said:
Ok, looking back then the c would be e^c which would then be Ae^(kx)=t? But having initial condition of t(0)=1 would make A=0.
At t=0, the snow plow was safely in its garage.

I understand that but will always result in a zero answer if A is 0.

Matthew R said:
I understand that but will always result in a zero answer if A is 0.
The equation you used to specify its motion is only valid when... what?

At t=0. So is A not a factor? Leaving e^kx=t? Then would I just plug in the next conditions and solve for k?

haruspex said:
The equation you used to specify its motion is only valid when... what?
Matthew R said:
At t=0.
No. t=0 is when it started snowing. At that time, the plow was going nowhere. The plow set off at 7 am. (We do not know when that was as a value of t.) The equation only applies while the plow is moving.

Matthew R said:
Hi guys.

I am currently stuck on the classic snow plow problem. I have the following differential equation and initial conditions:
@ 7am plow starts off to clear snow at a constant rate
By 8am, plow has gone 4mi
By 9am, plow has gone an additional 3mi
Let t=0 when it started to snow, when did it start to snow. They gave the answer as 4:27am

k(dx/dt)=1/t

It wants me to find the constant k.

Actually it doesn't; it wants you to find $T$, which is the time from when it started snowing until 7am. You do not need to know $k$ in order to find this.

However I think you may have copied the question incorrectly, because as far as I can tell you end up with a sixth-order polynomial for $T$ which doesn't have a root anywhere near the alleged answer of $T = 2.55$.

pasmith said:
sixth-order polynomial for T
Do you mean a sixth order polynomial for ek?
pasmith said:
which doesn't have a root anywhere near the alleged answer of T=2.55

## What is the snow plow problem?

The snow plow problem is a mathematical problem that involves finding the optimal speed at which a snow plow should operate in order to clear a given amount of snow from a road in the shortest amount of time.

## Why is finding the constant k important in solving the snow plow problem?

The constant k represents the relationship between the snow plow's speed and its ability to clear snow. By finding the appropriate value for k, we can determine the most efficient speed for the snow plow to operate at.

## How do you calculate the constant k?

The constant k can be calculated by dividing the distance of the road by the amount of snow that needs to be cleared and then taking the square root of that value.

## What factors can affect the value of constant k?

The value of constant k can be affected by various factors such as the type of snow, the condition of the road, the size and speed of the snow plow, and the temperature.

## How can the snow plow problem be applied in real life situations?

The snow plow problem has practical applications in the field of transportation and logistics. It can be used to determine the optimal speed for snow plows to operate at in order to efficiently clear roads during winter weather conditions.

• Calculus and Beyond Homework Help
Replies
3
Views
2K
• Calculus and Beyond Homework Help
Replies
1
Views
630
• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
7
Views
636
• Calculus and Beyond Homework Help
Replies
8
Views
945
• Calculus and Beyond Homework Help
Replies
8
Views
540
• Calculus and Beyond Homework Help
Replies
6
Views
511
• Calculus and Beyond Homework Help
Replies
7
Views
744
• Calculus and Beyond Homework Help
Replies
5
Views
465
• Calculus and Beyond Homework Help
Replies
1
Views
1K