Answer:
The number of miles of road that will be cleared each hour when the depth of the snow is 11 inches as is 3.923 miles.
Step-by-step explanation:
Let X = number of miles of road cleared and h = depth of the snowfall.
Then the rat of change in the number of miles of road cleared per hour is given by the differential equation:
[tex]\frac{dX}{dh}=\frac{k}{h}[/tex]
Simplify and integrate the above differential equation as follows:
[tex]\frac{dX}{dh}=\frac{k}{h}\\dX=\frac{k}{h} dh\\\int {dX} \,=\int {\frac{k}{h} }\, dh\\X=k\ln(h)+c[/tex]
It is provided that:
X (2.3) = 21
X (9) = 13
Use these information to compute the value of k and c as follows:
[tex]21=k\ln(2.3)+c\\(-)13=(-)k\ln(9)(-)+c\\\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\\8=k(\ln(2.3)-\ln(9))\\8=k\times \ln(\frac{2.3}{9})\\8=k\times-1.364\\k=-10.912[/tex]
Now substitute the value of k to compute c as follows:
[tex]21=k\ln(2.3}+c\\c=21-(-10.912)\times ln(2.3)\\c=21+9.089\\c=30.089[/tex]
The equation of X is:
X = -10.912 ln (h) + 30.089
Compute the number of miles of road that will be cleared each hour when the depth of the snow is 11 inches as follows:
[tex]X = -10.912 \ln (h) + 30.089\\=[-10.912\times \ln (11)]+30.089\\=-26.1658+30.089\\=3.9232\\\approx3.923[/tex]
Thus, the number of miles of road that will be cleared each hour when the depth of the snow is 11 inches as is 3.923 miles.
