Question #290948

Q2. The force of interest at any time t, (measured in years) is given by




 0.07—0.005t ,0 ≤ t < 5 0.06—0.003t, 5≤ t < 10




δ(t) =




(a) Calculate the accumulated amount at time t = 15 of $100 invested at time




0.03, t > 10 t = 0.







 0.07−0.005t, 0≤t<5 0.06 − 0.003t, 5 ≤ t < 10




δ(t) =




What is the total accumulated in value at any time t (> 0) of investments of $100 at




0.03, t ≥ 10 times 0, 4 and 6?








1
Expert's answer
2022-01-28T06:00:35-0500

a)100×A(0,5)×A(5,10)×A(10,15)A=100×e05(0.070.005t)dt×e510(0.060.003t)dt×e0.03×5=100×e0.07t0.005t2205×e0.06t0.003t22510×e0.15=100×e0.350.0625×e0.450.2925×e0.15=100×e0.2875×e0.1575×e0.15=100×1.3331×1.1706×1.1618=$181.30a)100 \times A(0,5) \times A(5,10) \times A(10,15) \\A=100\times e^{\int_0^5(0.07-0.005t)dt} \times e^{\int_5^{10}(0.06-0.003t)dt} \times e^{0.03 \times 5}\\ =100 \times e^{0.07t-\frac{0.005t^2}{2}|_0^5} \times e^{0.06t-\frac{0.003t^2}{2}|_5^{10}} \times e^{0.15}\\ =100 \times e^{0.35-0.0625} \times e^{0.45-0.2925} \times e^{0.15}\\ =100 \times e^{0.2875} \times e^{0.1575} \times e^{0.15}\\ =100 \times 1.3331 \times 1.1706 \times 1.1618\\ =\$ 181.30




at time t=0

A=100×e0.07×5×e0.06×5×e0.073×5A=100×1.4191×1.3499×1.1618A=$222.56A=100 \times e^{0.07 \times 5} \times e^{0.06 \times 5} \times e^{0.073\times 5}\\ A=100 \times 1.4191 \times 1.3499 \times 1.1618\\ A=\$ 222.56

at time t=4

=100×e45(0.070.005t)dt×e510(0.060.003t)dt×e0.03×5=100×e0.07t0.005t2245×e0.06t0.003t22510×e0.15=100×e0.28750.24×e0.450.2925×e0.15=100×e0.0475×e0.1575×e0.15=100×1.0486×1.1706×1.1618=$142.61=100\times e^{\int_4^5(0.07-0.005t)dt} \times e^{\int_5^{10}(0.06-0.003t)dt} \times e^{0.03 \times 5}\\ =100 \times e^{0.07t-\frac{0.005t^2}{2}|_4^5} \times e^{0.06t-\frac{0.003t^2}{2}|_5^{10}} \times e^{0.15}\\ =100 \times e^{0.2875-0.24} \times e^{0.45-0.2925} \times e^{0.15}\\ =100 \times e^{0.0475} \times e^{0.1575} \times e^{0.15}\\ =100 \times 1.0486 \times 1.1706 \times 1.1618\\ =\$ 142.61

at time 6

=100×e610(0.060.003t)dt×e0.03×5=100×e0.06t0.003t22610×e0.15=100×e0.450.27×e0.15=100×e0.18×e0.15=100×1.1972×1.1618=$139.09=100 \times e^{\int_6^{10}(0.06-0.003t)dt} \times e^{0.03 \times 5}\\ =100 \times e^{0.06t-\frac{0.003t^2}{2}|_6^{10}} \times e^{0.15}\\ =100 \times e^{0.45-0.27} \times e^{0.15}\\ =100 \times e^{0.18} \times e^{0.15}\\ =100 \times 1.1972 \times 1.1618\\ =\$ 139.09\\


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