1. Setting the problem Solving the problem
Snowplow’s problem
Ra´ul Romero Mart´ın
raul.romero@educa.madrid.org
IES Gabriel Cisneros
Departamento de Matem´aticas

2. Setting the problem Solving the problem
´Indice
1 Setting the problem
2 Solving the problem

3. Setting the problem Solving the problem
The problem
One day, it begins to snow...

4. Setting the problem Solving the problem
The problem
One day, it begins to snow...
At noon, a snowplow starts to move for doing its work.

5. Setting the problem Solving the problem
The problem
One day, it begins to snow...
At noon, a snowplow starts to move for doing its work.
In the first hour the snowplow have moved two kilometers...

6. Setting the problem Solving the problem
The problem
One day, it begins to snow...
At noon, a snowplow starts to move for doing its work.
In the first hour the snowplow have moved two kilometers...
In the second hour, there is more snow, it is more difficult to move
and so...

7. Setting the problem Solving the problem
The problem
One day, it begins to snow...
At noon, a snowplow starts to move for doing its work.
In the first hour the snowplow have moved two kilometers...
In the second hour, there is more snow, it is more difficult to move
and so... it only moves one more kilometer.

8. Setting the problem Solving the problem
The problem
One day, it begins to snow...
At noon, a snowplow starts to move for doing its work.
In the first hour the snowplow have moved two kilometers...
In the second hour, there is more snow, it is more difficult to move
and so... it only moves one more kilometer.
The question is...

9. Setting the problem Solving the problem
The problem
One day, it begins to snow...
At noon, a snowplow starts to move for doing its work.
In the first hour the snowplow have moved two kilometers...
In the second hour, there is more snow, it is more difficult to move
and so... it only moves one more kilometer.
The question is...
What time did it begin to snow?

10. Setting the problem Solving the problem
´Indice
1 Setting the problem
2 Solving the problem

11. Setting the problem Solving the problem
Let us set the following definitions:
t ≡ time since noon
t0 ≡ time since it began to snow until noon in hours
x(t) ≡ space traveled by the snowplow at time t
v(t) ≡ snowplow’s velocity at time t
h(t) ≡ snow’s height at time t
c ≡ snowplow’s velocity with 1 meter of snow

12. Setting the problem Solving the problem
Let us set the following definitions:
t ≡ time since noon
t0 ≡ time since it began to snow until noon in hours
x(t) ≡ space traveled by the snowplow at time t
v(t) ≡ snowplow’s velocity at time t
h(t) ≡ snow’s height at time t
c ≡ snowplow’s velocity with 1 meter of snow
v =
c
h

13. Setting the problem Solving the problem
Let us set the following definitions:
t ≡ time since noon
t0 ≡ time since it began to snow until noon in hours
x(t) ≡ space traveled by the snowplow at time t
v(t) ≡ snowplow’s velocity at time t
h(t) ≡ snow’s height at time t
c ≡ snowplow’s velocity with 1 meter of snow
v =
c
h
v(t) =
A
t + t0

14. Setting the problem Solving the problem
Let us set the following definitions:
t ≡ time since noon
t0 ≡ time since it began to snow until noon in hours
x(t) ≡ space traveled by the snowplow at time t
v(t) ≡ snowplow’s velocity at time t
h(t) ≡ snow’s height at time t
c ≡ snowplow’s velocity with 1 meter of snow
v =
c
h
v(t) =
A
t + t0
x (t) =
A
t + t0

15. Setting the problem Solving the problem
Let us set the following definitions:
t ≡ time since noon
t0 ≡ time since it began to snow until noon in hours
x(t) ≡ space traveled by the snowplow at time t
v(t) ≡ snowplow’s velocity at time t
h(t) ≡ snow’s height at time t
c ≡ snowplow’s velocity with 1 meter of snow
v =
c
h
v(t) =
A
t + t0
x (t) =
A
t + t0
=⇒ x(t) = A · log(t + t0) + B

16. Setting the problem Solving the problem
x(t) = A · log(t + t0) + B
and we know x(0) = 0, x(1) = 2, x(2) = 3.

17. Setting the problem Solving the problem
x(t) = A · log(t + t0) + B
and we know x(0) = 0, x(1) = 2, x(2) = 3.This leads to

18. Setting the problem Solving the problem
x(t) = A · log(t + t0) + B
and we know x(0) = 0, x(1) = 2, x(2) = 3.This leads to
0 = A · log t0 + B

19. Setting the problem Solving the problem
x(t) = A · log(t + t0) + B
and we know x(0) = 0, x(1) = 2, x(2) = 3.This leads to
0 = A · log t0 + B =⇒ B = −A · log t0.

20. Setting the problem Solving the problem
x(t) = A · log(t + t0) + B
and we know x(0) = 0, x(1) = 2, x(2) = 3.This leads to
0 = A · log t0 + B =⇒ B = −A · log t0.
2 = A · log
1 + t0
t0
.

21. Setting the problem Solving the problem
x(t) = A · log(t + t0) + B
and we know x(0) = 0, x(1) = 2, x(2) = 3.This leads to
0 = A · log t0 + B =⇒ B = −A · log t0.
2 = A · log
1 + t0
t0
.
3 = A · log
2 + t0
t0
.

22. Setting the problem Solving the problem
x(t) = A · log(t + t0) + B
and we know x(0) = 0, x(1) = 2, x(2) = 3.This leads to
0 = A · log t0 + B =⇒ B = −A · log t0.
2 = A · log
1 + t0
t0
.
3 = A · log
2 + t0
t0
.
From here we get,
3·log
1 + t0
t0
= 2·log
2 + t0
t0

23. Setting the problem Solving the problem
x(t) = A · log(t + t0) + B
and we know x(0) = 0, x(1) = 2, x(2) = 3.This leads to
0 = A · log t0 + B =⇒ B = −A · log t0.
2 = A · log
1 + t0
t0
.
3 = A · log
2 + t0
t0
.
From here we get,
3·log
1 + t0
t0
= 2·log
2 + t0
t0
=⇒ log
1 + t0
t0
3
= log
2 + t0
t0
2

24. Setting the problem Solving the problem
x(t) = A · log(t + t0) + B
and we know x(0) = 0, x(1) = 2, x(2) = 3.This leads to
0 = A · log t0 + B =⇒ B = −A · log t0.
2 = A · log
1 + t0
t0
.
3 = A · log
2 + t0
t0
.
From here we get,
3·log
1 + t0
t0
= 2·log
2 + t0
t0
=⇒ log
1 + t0
t0
3
= log
2 + t0
t0
2
And so, we arrive to the equation (1 + t0)3 = (2 + t0)2 · t0.

25. Setting the problem Solving the problem
(1 + t0)3
= (2 + t0)2
· t0

26. Setting the problem Solving the problem
(1 + t0)3
= (2 + t0)2
· t0
After some computation we obtain t0 =
√
5 − 1
2
= 0, 61803 hours.

27. Setting the problem Solving the problem
(1 + t0)3
= (2 + t0)2
· t0
After some computation we obtain t0 =
√
5 − 1
2
= 0, 61803 hours.
So, t0 = 37 minutes 5 seconds. And our answer to the problem is...

28. Setting the problem Solving the problem
(1 + t0)3
= (2 + t0)2
· t0
After some computation we obtain t0 =
√
5 − 1
2
= 0, 61803 hours.
So, t0 = 37 minutes 5 seconds. And our answer to the problem is...
It began to snow at 11:22:55