Question

A new fast-food firm predicts that the number of franchises for its products will grow at the rate dn = 9/t+1 dt where t is the number of years, 0 st s 15. If there is one franchise (n = 1) at present (t = 0), how many franchises are predicted for 15 years from now? franchises

Answer 1

The number of franchises predicted for 15 years from now is 24. To determine the number of franchises predicted for 15 years from now, we need to solve the given differential equation.

The equation dn =
`(9)/(t+1)` dt represents the rate at which the number of franchises (dn) is changing with respect to time (dt). Integrating both sides of the equation gives us the equation n = 9 ln(t+1) + C, where C is the constant of integration.

Given that there is one franchise at present (t = 0), we can substitute n = 1 and solve for C. Plugging in the values, we get 1 = 9 ln(0+1) + C, which simplifies to C = 1 - 9 ln(1) = 1.

Now, to find the number of franchises predicted for 15 years from now (t = 15), we substitute t = 15 into the equation n = 9 ln(t+1) + C. Plugging in the values, we get n = 9 ln(15+1) + 1, which simplifies to n = 24. Therefore, the predicted number of franchises for 15 years from now is 24.

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