ass 18

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1 / ( (x+1)(500-x) ) = A / (x+1) + B / (500-x) so

A(500-x) + B(x+1) = 1 so

A = 1 / 501 and B = 1/501.

The integrand becomes 5010 [ 1 / (501 (x+1) ) + 1 / (501 (500 - x)) ] = 10 [ 1/(x+1) + 1 / (500 - x) ].

Thus we have

t = INT(5010 ( 1 / (x+1) + 1 / (500 - x) ) , x).

Since an antiderivative of 1/(x+1) is ln | x + 1 | and an antiderivative of 1 / (500 - x) is - ln | 500 - x | we obtain

t = 10 [ ln (x+1) - ln (500-x) + c].

(for example INT (1 / (500 - x) dx) is found by first substituting u = 1 - x, giving du = -dx. The integral then becomes INT(-1/u du), giving us - ln | u | = - ln | 500 - x | ).

We are given that x = 1 yields t = 0. Substituting x = 1 into the expression t = 10 [ ln (x+1) - ln (500-x) + c], and setting the result equal to 0, we get c = 5.52, appxox.

So t = 10 [ ln (x+1) - ln (500-x) + 5.52] = 10 ln( (x+1) / (500 - x) ) + 55.2.

75% of the population of 500 is 375. Setting x = 375 we get

t = 10 ln( (375+1) / (500 - 375) ) + 55.2 = 66.2. **

DER

ALTERNATE SOLUTION STRATEGY

Using the logistic growth model and the x and t values I get:

5010 [.001996 ( ln |x+1| + ln |500 - x| ) ] = kt + C.

** Should be just t + c. You're integrating dt, not k dt. Should also be -ln|500-x|: use substitution!!! **

Simplify:

9.999 ( ln |x+1| + ln |500 - x| ) = kt + C

Exponentiate:

9.999 [ (x+1) + (500-x) ] = Ce^kt

** Invalid step here. You are assuming that ln(a+b) = ln(a) + ln(b), which isn't consistent with the laws of logarithms. ln(a) + ln(b) = ln(ab).

That 9.999 is actually exactly 10 if you use the right fractions and don't round off.

Starting with 10 ( ln |x+1| - ln |500 - x| ) = t + C you get

10 ( ln ( |x+1| / |500 - x| ) ) = t + C so

ln ( |x+1| / |500 - x| ) = (t + C) / 10 so

( |x+1| / |500 - x| ) = e^( (kt + C) / 10 ). Since 0 < x < 500, x+1 and 500-x are positive so we have

(x+1) / (500-x) = e^( (t/10 + C) ), noting that the C is an arbitrary constant so it just 'absorbs' the division by 10 and remains an arbitrary constant.

e^(t/10 + c) = e^c * e^(t/10) = A e^(t/10), A > 0, because e^c could be any positive number, which we now call A.

So (x+1) / (500-x) = A e^(-t/10). Multiply both sides by the denominator:

x+1 = A e^(-t/10) * ( 500-x). Distributive law:

x+1 = 500 A e^(-t/10) - A x e^(-t/10). Rearrange to get x on one side:

x - A x e^(-t/10) = 500 A e^(-t/10) - 1. Factor out x:

x(1-A e^(-t/10) ) = 500 A e^(-t/10) - 1. Isolate x:

x = [ 500 A e^(-t/10) - 1 ] / [(1-A e^(-t/10) ) ].

Return for a minute to (x+1) / (500-x) = A e^(-t/10) and substitute x=1, t = 0 to get

(1 + 1) / (500 - 1) = A e^0

2 / 499 = A.

Substitute A into x = [ 500 A e^(-t/10) - 1 ] / [(1-A e^(-t/10) ) ].

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