※ 引述《baogua (爆掛)》之銘言：
: It began to snow on a certain morning, and the snow continued to fall steadily
: throughout the day. At noon a snowplow started to clear a road at a constant
: rate in terms of the volume of snow removed per hour. The snowplow cleared
: 2 miles by 2 pm and 1 more mile by 4 pm. When did it start snowing?
抱歉我用我的方式重做 (單位用來檢查因次)
假設下雪開始的時刻為 T (單位：hr。24時制)
單位長度馬路的積雪速率為 s (單位：m^3/(mi．hr))
鏟雪速率為 C (單位：m^3/hr)
假設鏟雪機不會繞回原處清理新的積雪，
其推進距離對時間的函數為 x(t)
距離 x(12)=0 x x+dx
─┼────────────┼─┼───→
｜ 華山一條路．鏟過不回頭 ｜ ｜
─┼────────────┼─┼───→
時刻 t=12 t t+dt
假設在時刻 t~t+dt 時，鏟雪機從 x 推進到 x+dx
dt時間內鏟雪量 Cdt (m^3)
推進長度 dx(mi) = ──────── = ──────────
單位長度積雪量 (t-T)s (m^3/mi)
dx C
→ ─ = ───
dt (t-T)s
s dt
→ ─ dx = ── , 從 t=12 ~ t=t 做定積分, x(12)=0
C t-T
s t-T
→ ─ x = ㏑(──)
C 12-T
2s 14-T
╭ x(14) = 2 → ─ = ㏑(──)
代入 ｜ C 12-T
邊界 ┤
條件 ｜ 3s 16-T
╰ x(16) = 3 → ─ = ㏑(──)
C 12-T
16-T 14-T
兩式相除 → 2 ㏑(──) = 3 ㏑(──)
12-T 12-T
16-T 2 14-T 3
取 exp → (──) = (──)
12-T 12-T
2 3
令 w = 12-T > 0 → w(w+4) = (w+2)
→ w = -1+√5
→ T = 13-√5
#
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◆ From: 114.27.0.102
※ 編輯: oNeChanPhile 來自: 114.27.8.196 (10/21 17:07)