You are given:i. Mr+t = ßt², t≥0ii. l = 1,000iii. lx+10Calculate 1,000ß.= 400

Answered: You are given: i. Mætt = ßt², t≥0 ii. l…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

You are given:
i. Mr+t = ßt², t≥0
ii. l = 1,000
iii. lx+10
Calculate 1,000ß.
= 400

Expert Solution

Step 1: How to solve

To calculate 1000β, we can use the information provided in the problem. We have three equations:

i. $mu subscript blank subscript left parenthesis x plus t right parenthesis end subscript equals beta t squared comma space t space greater or equal than space 0$

ii. $l subscript x equals 1000$

iii. $l subscript blank subscript left parenthesis x plus 10 right parenthesis end subscript equals 400$

Let's start by using equation ii:

$l subscript x equals 1000$

This means that the total number of lives at age x is 1000. Now, let's use equation iii:

$l subscript blank subscript left parenthesis x plus 10 right parenthesis end subscript equals 400$

This means that the total number of lives at age x+10 is 400. We can use these values to calculate the probability of surviving from age x to age x+10, denoted as $Q_{x}$ :

$Q subscript x equals l subscript blank subscript left parenthesis x plus 10 right parenthesis end subscript over l subscript x equals 400 over 1000 equals 0.4$

Now, let's use equation i with t = 10:

$mu subscript blank subscript left parenthesis x plus 10 right parenthesis end subscript equals beta cross times 10 squared equals 100 beta$

We know that the force of mortality (μ) is related to the probability of dying (q) as follows:

$mu subscript x equals negative l n left parenthesis 1 minus q subscript x right parenthesis$

Since $Q_{x}$ is the probability of surviving, $q_{x}$ is the probability of dying, and we can rearrange the formula as follows:

$q subscript x equals 1 minus e x p left parenthesis negative mu subscript x right parenthesis$

Now, we can use the value of $Q_{x}$ to calculate $q_{x}$ :

$q subscript x equals 1 minus e x p left parenthesis negative mu subscript x right parenthesis equals 1 minus e x p left parenthesis negative 100 beta right parenthesis$