exercise:408fb18aaa

AAdmin

Jan 15'24

Exercise

You are given

[[math]]{ }_{t} q_{0}=\frac{t^{2}}{10,000}[[/math]]

for [math]0 \lt t \lt 100[/math].

Calculate [math]\stackrel{\circ}{e}_{\text {75:1010 }}[/math].

6.6
7.0
7.4
7.8
8.2

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AAdmin

Jan 15'24

Answer: E

[[math]] \begin{aligned} & e_{75: 10}=\int_{t=0}^{t=10}{ }_{t} p_{75} d t \quad \text { where }{ }_{t} p_{x}=\frac{{ }_{t+x} p_{0}}{{ }_{x} p_{0}}=\frac{1-\frac{(t+x)^{2}}{10000}}{1-\frac{x^{2}}{10000}}=\frac{10000-(t+x)^{2}}{10000-x^{2}} \text { for } 0\lt t \lt100-x \\ & =\int_{0}^{10} \frac{10000-75^{2}-150 t-t^{2}}{10000-75^{2}} d t \\ & =\frac{1}{4375} \cdot\left[4375 t-75 t^{2}-\frac{t^{3}}{3}\right]_{t=0}^{t=10}=8.21 \end{aligned} [[/math]]