The rate of change in the number of miles s of road cleared per hour by a snowplow is inversely proportional to the depth h of snow. That is,

Question

ds/dh = k/h

Find s as a function of h given that s = 25 miles when h = 3 inches and s = 10 miles when h = 9 inches (2 <= h <= 15).

s(h) = ?

What I have done so far:

∫ds = ∫k/h dh

s = k * ln (Ch)

When s = 25 when h = 3,

25 = k*ln(3C)

When s = 10 and when h = 9

10 = k*ln(9C)

How would I continue out the problem?

Answer 1

s = k ln h + c

25 = k ln 3 + c
10 = k ln 9 + c
-----------------subtract
15 = k ln(3/9) = k ln (1/3)
etc

Answer 2

Where we split. You did nothing wrong, just made it harder :)

∫ds = ∫k/h dh

s = k [ln h + ln C ]

you said that was
k ln (C h) which is correct but
I said
k ln h + c
which is also correct, but my way made it easy

Answer 3

Ahh I see now.

Thanks for your help!