An automobile company is ready to introduce a new line of...

An automobile company is ready to introduce a new line of hybrid cars through a national sales campaign. After test marketing the line in a carefully selected city, the marketing research department estimates that sales (in millions of dollars) will increase at the monthly rate of $$ S^{\prime}(t)=10-10 e^{-0.1 t} \quad 0 \leq t \leq 24 $$ $t$ months after the campaign has started. (A) What will be the total sales $S(t) t$ months after the beginning of the national campaign if we assume no sales at the beginning of the campaign? (B) What are the estimated total sales for the first 12 months of the campaign? (C) When will the estimated total sales reach $\$ 100$ million? Use a graphing calculator to approximate the answer to two decimal places.

Transcript

00:01 Hi there, so for this problem we are told that an automobile company is ready to introduce a new line of hybrid cars through a national sales company.

00:09 Now, the marketing research department estimates that the sales will increase at monthly rate, so we are given that that rate is equal to 10 minus 10 times the exponential of minus 0 .1 times t, where the time is defined between 0 and 24.

00:35 Now, for part a of this problem, the question is what will be the total sales t months after the beginning of the national company if we assume that no sales at the beginning of the company? so that means that this at 0 is equal to 0.

00:53 Now, to determine the number of sales for this, we just integrate the function that we are given, which in this case is 10 minus 10 times the exponential of minus 0 .1 times the time, and then we integrate this over the time.

01:07 Now, we just need to solve this integral, so then the solution of this integral is 10 times the time, then this minus will plus because this minus the integral of the exponential will give us, we just need to divide this by minus 0 .1.

01:23 So that will give us 10 divided by 0 .1 times the exponential of minus 0 .1 times the time, and this plus a constant of integration that we're going to label as c.

01:36 So let's simplify this.

01:38 So this will be then 10 times the time plus 10 divided by 0 .1 is 100.

01:45 100 times the exponential of minus 0 .1 times the time plus c.

01:51 Now, we just need to evaluate this at 0 because this should be 0, so that will be then 10 plus, well, this at 0 will give us just 0 in the first term, then we will have 100.

02:03 We know that the exponential at 0 is 1, so that will be this, then this equals to 0.

02:08 So then here for c, that will give us minus 100.

02:11 So the solution for this is just this expression right here, but c takes the value of minus 100.

02:20 So that is the solution for part a of this problem.

02:25 Now, for part b, the question is what are the estimated total sales for the first 12 months of the campaign? we already know that if we evaluate this at 0, we will obtain 0, so we just need to evaluate this at 12.

02:43 So that will be then 10 times 12, then this plus 100 times the exponential of minus 0 .1 times 12, and then this minus 100...

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