Question #290641

The force of interest δ(t) is a function of time and, at any time t (measured in years),


is given by the formula:


δ(t) =





0.07 0.005t , 0 t < 5


0.06 0.003t, 5 t < 10


0.03, t > 10


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


t = 0.

1
Expert's answer
2022-01-28T04:13:11-0500

A=$100 x A(0,5) x A(5,10) x A(10,15)

=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=$181.30=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


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