We\u2019ll assume the snow starts at $t=0$ hours. So after $t$ hours, the depth of snow on the ground is proportional to $t$. Since the plow\u2019s speed $v$ is inversely proportional to the amount of snow on the ground, we have $v = \\tfrac{c}{t}$, where $c>0$ is a constant of proportionality. We\u2019ll assume the constant is in units of miles so that $v$ is in miles per hour.<\/p>\n

Suppose the plow starts at $t=x$ hours. During the first hour, the plow travels 2 miles. Since distance is the integral of velocity with respect to time, we have:
\n\\[
\n2\\text{ miles} = \\int_{x}^{x+1} v\\,\\mathrm{d}t = \\int_x^{x+1} \\frac{c}{t}\\,\\mathrm{d}t = c\\,\\log\\left( \\frac{x+1}{x} \\right)
\n\\]During the second hour, the plow travels 1 mile. So we also have:
\n\\[
\n1\\text{ mile} = \\int_{x+1}^{x+2} v\\,\\mathrm{d}t = \\int_{x+1}^{x+1} \\frac{c}{t}\\,\\mathrm{d}t = c\\,\\log\\left(\\frac{x+2}{x+1} \\right)
\n\\]Dividing the first equation by the second, the units and constants of proportionality cancel. Simplifying, we obtain:
\n\\[
\n2\\,\\log\\left(\\frac{x+2}{x+1}\\right) = \\log\\left( \\frac{x+1}{x} \\right)
\n\\]Exponentiating both sides to cancel out the logs,
\n\\[
\n\\left(\\frac{x+2}{x+1}\\right)^2 = \\frac{x+1}{x}
\n\\]Finally, clearing fractions and simplifying, we obtain:
\n\\[
\nx^2+x-1 = 0
\n\\]Since we must have $x\\gt 0$, we can discard the negative solution to this equation, and keep only the positive one. This leads us to:
\n\\[
\nx = \\frac{\\sqrt{5}-1}{2} \\approx 0.618034
\n\\]Incidentally, this is the inverse of the Golden Ratio<\/a>, though that\u2019s more of a coincidence than anything meaningful\u2026 If the plow started at noon, then the snow started $x$ hours before noon, which is 11:22.55 AM (11 hours, 22 min, 55 sec).\n<\/p><\/div>\n<\/body>","protected":false},"excerpt":{"rendered":"

This week\u2019s Riddler Classic is a neat calculus problem: One morning, it starts snowing. The snow falls at a constant rate, and it continues the rest of the day. At noon, a snowplow begins to clear the road. The more snow there is on the ground, the slower the plow moves. In fact, the plow\u2019s … Continue reading “When did the snow start?”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_uf_show_specific_survey":0,"_uf_disable_surveys":false,"footnotes":""},"categories":[7],"tags":[28,4,2],"class_list":["post-2801","post","type-post","status-publish","format-standard","hentry","category-riddler","tag-calculus","tag-integration","tag-riddler"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t