SOLUTION TO PROBLEM R-12

Let t=0 be the time that it starts snowing. Let t₁ be the time the snow plow begins (i.e. 7AM). Let C be the constant rate of snow fall in feet per hour. Let L be the length of the snow plow blade and let Q be the rate at which the plow clears the road in cubic feet per hour. Let x(t) be the distance, in miles, traveled by the plow.

Then $(LCt)\Delta x=Q\Delta t$, so that

$$\frac{CL}{Q}t\frac{dx}{dt}=1\qquad\mbox{ or }\qquad kdx=\frac{dt}{t}.$$

This has solution $kx=\ln(t)+B$. Now at t=t₁ we have x=0 and so $B=-\ln(t_1)$. Thus $kx=\ln(t/t_1)$. In addition, since the plow covers 2 miles by 8AM, we have $2k=\ln\big(\frac{1+t_1}{t_1}\big)$. Also $4k=\ln \big(\frac{3+t_1}{t_1}\big)$ and dividing gives

$$\ln\big(\frac{3+t_1}{t_1}\big)=\ln\big(\frac{1+t_1}{t_1}\big)^2.$$

Solving we have t₁(3+t₁)=(1+t₁)² and hence 3t₁=1+2t₁. Consequently t₁=1 and so it began to snow at 6AM.

Questions and/or comments should be directed to Judy Downey or Griff Elder

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Last modified:   Thu Apr 11 20:11:35 CDT 2002