指數對數的應用問題*2

2015-01-14 7:26 am
1.放射性元素的質量會隨時間逐漸衰減,且無論從何時算起,經過相同時間後的衰變速率皆相同。若某放射性元素物質在經歷18000年後,質量只剩下原來的十六分之一,試問該放射性元素半衰期約為幾年?

2.學生考試的成績y(分)與每週研讀的時間x(小時)的學習曲線為y=10^(3x-14)/(1+10^(3x+14))*100,請問想達到60分,每周最少要花幾小時研讀?(取至整數)

回答 (1)

2015-01-14 4:22 pm
✔ 最佳答案
1.Differential Equation is dx/dt = -kx dx/x = -kdt => ∫dx/x = -∫kdtln(x/c) = -kt => x = c*e^(-kx)
Initial case: x(0) = cFinal case: x(18000) = c*e(-18000k) = c/16=> e^(18000k) = 1618000k = ln(16) = 4*ln(2)=> k = 4*ln(2)/18000 = ln(2)/4500Half life: x(T) = c*e^(-kT) = c/2=> e^(kT) = 2 => kT= ln(2) T = ln(2)/k= ln(2)*4500/ln(2)= 4500 years= ans 2.學生考試的成績y(分)與每週研讀的時間x(小時)的學習曲線為y=10^(3x-14)/(1+10^(3x+14))*100,請問想達到60分,每周最少要花幾小時研讀?(取至整數)
Set a = 10^(3x+14) => y = 100a/(1 + a)60(1 + a) = 100a => 60 = 40aa = 3/2 = 10^(3x-14) => 3x - 14 = log(3/2)x = [14 + log(3/2)] / 3= (14 + 0.47712 - 0.30103)/3= 4.725 (hr)


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