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,
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?
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
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
Ahh I see now.
Thanks for your help!
To continue solving the problem and find the function s(h), you can use the information given when s = 25 miles when h = 3 inches and s = 10 miles when h = 9 inches.
1. Let's start by using the first condition s = 25 when h = 3:
25 = k * ln(3C)
2. Next, use the second condition s = 10 when h = 9:
10 = k * ln(9C)
3. Now, we have a system of two equations with two unknowns, k and C. To solve this system, divide the two equations:
(25/10) = (k * ln(3C))/(k * ln(9C))
2.5 = ln(3C)/ln(9C)
4. Now, let's isolate ln(3C) by multiplying both sides by ln(9C):
2.5 * ln(9C) = ln(3C)
5. Use the logarithmic rule that states ln(a) = c(ln(b)) to rewrite the equation:
ln((9C)^(2.5)) = ln(3C)
6. Equate the expressions inside the logarithms:
(9C)^(2.5) = 3C
7. To solve for C, you need to get all terms with C on one side of the equation:
(9C)^(2.5)/3C = 1
8. Simplify the expression on the left side by dividing the exponents and canceling out the C term:
(3^(2.5)) * C^(2.5-1) = 1
9C^(1.5) = 1
9. Divide both sides by 9 to isolate C:
C^(1.5) = 1/9
10. Take the square root of both sides to solve for C:
C = (1/9)^(1/1.5)
11. Calculate the value of C using a calculator:
C ≈ 0.668740
12. Now that you have found the value of C, substitute it back into one of the original equations to solve for k. We will use the first equation for convenience:
25 = k * ln(3C)
Substituting C ≈ 0.668740:
25 = k * ln(3 * 0.668740)
25 = k * ln(2.006220)
13. Divide both sides by ln(2.006220) to solve for k:
k = 25/ln(2.006220)
14. Calculate the value of k using a calculator:
k ≈ 33.017988
15. Finally, substitute the values of k and C into the function s = k * ln(Ch) to obtain the solution:
s(h) = 33.017988 * ln(0.668740h)
Therefore, the function describing the relationship between the number of miles of road cleared per hour (s) and the depth of snow (h) is s(h) = 33.017988 * ln(0.668740h).