# Applications of Differentiation

#### chrisyuen

1. Homework Statement

A vehicle tunnel company wants to raise the tunnel fees. An expert predicts that after the increase in the tunnel fees, the number of vehicles passing through the tunnel each day will drop drastically in the first week and on the t-th day after the first week, the number N(t) (in thousands) of vehicles passing through the tunnel can be modelled by N(t) = 40 / (1+be^(-rt)) (t>=0) where b and r are positive constants.

(a) Suppose that by the end of the first week after the increase in the tunnel fees, the number of vehicles passing through the tunnel each day drops to 16 thousand and by the end of the second week, the number increases to 17.4 thousand, find b and r correct to 2 decimal places.

(b) Show that N(t) is increasing.

(c) As time passes, N(t) will approach the average number Na of vehicles passing through the tunnel each day before the increase in the tunnel fees. Find Na.

(d) The expert suggests that the company should start to advertise on the day when the rate of increase of the number of cars passing through the tunnel per day is the greatest. Using the values of b and r obtained in (a),

(i) find N''(t), and
(ii) hence determine when the company should start to advertise.

((d)(ii) 20th day)

2. Homework Equations

Differentiation Rules

3. The Attempt at a Solution

I don't know how to solve the part (d)(ii) of the question.

I think I should set the N''(t) = 0 in order to get the t.

But can anyone tell me why should I set N''(t) = 0?

Thank you very much!

Related Precalculus Mathematics Homework Help News on Phys.org

#### HallsofIvy

Science Advisor
Homework Helper
1. Homework Statement

A vehicle tunnel company wants to raise the tunnel fees. An expert predicts that after the increase in the tunnel fees, the number of vehicles passing through the tunnel each day will drop drastically in the first week and on the t-th day after the first week, the number N(t) (in thousands) of vehicles passing through the tunnel can be modelled by N(t) = 40 / (1+be^(-rt)) (t>=0) where b and r are positive constants.

(a) Suppose that by the end of the first week after the increase in the tunnel fees, the number of vehicles passing through the tunnel each day drops to 16 thousand and by the end of the second week, the number increases to 17.4 thousand, find b and r correct to 2 decimal places.

(b) Show that N(t) is increasing.

(c) As time passes, N(t) will approach the average number Na of vehicles passing through the tunnel each day before the increase in the tunnel fees. Find Na.

(d) The expert suggests that the company should start to advertise on the day when the rate of increase of the number of cars passing through the tunnel per day is the greatest. Using the values of b and r obtained in (a),

(i) find N''(t), and
(ii) hence determine when the company should start to advertise.

((d)(ii) 20th day)

2. Homework Equations

Differentiation Rules

3. The Attempt at a Solution

I don't know how to solve the part (d)(ii) of the question.

I think I should set the N''(t) = 0 in order to get the t.

But can anyone tell me why should I set N''(t) = 0?

Thank you very much!
" The expert suggests that the company should start to advertise on the day when the rate of increase of the number of cars passing through the tunnel per day is the greatest."
You find an extremum of a function by setting its derivative to 0. You are trying to find when "the rate of increase" is greatest- that is, you are trying to find when N'(t) is greatest and that happens when (N')'= N"= 0.

#### chrisyuen

I got it!

Thank you very much!

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