Question

After the Introduction of a new product, the percent of the public that is aware of the product is approximated by

A(t)= 10t²2^-t

where t is time in months.

(a). Calculate A'(t).

(b). Find the rate of change of the percent of the public that is aware of the product after 2 months and after 4 months and include units.

(c). In part (b), one of your answers should have been positive while the other should have been negative. what dose that tell you about how the public awareness of the product has changed?

Answer 1

A (t) 10122

to find the rate of change take derivative

A' (t) (1022-) dt

t2. 2 A' (t) 10 dt

apply product rule of derivative  

fg) g+f.g

d ) 2 (2) A' (t) 10 dt dt

A' (t)10 2t 2 10.(-In (2) 2) t2

A (t)20t2-10 ln (2)2

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.put t=2 months

A' (2)20 2 2 10ln (2) 2.22

A (2)40 10 n (2)4

A (2)10 10 In (2) 1

A (2 10 10 0.693

$A'\left(2\right)=\:10-6.93$

${\color{Red} A'\left(2\right)=\: 3.06853}$  percent of the public per month

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.put t=4 months

$A'(4)= 20\cdot 4\cdot 2^{-4}-10\ln \left(2\right)\cdot \:2^{-4}\cdot 4^2$

$A'(4)= 20\cdot 4\cdot \frac{1}{16}-10\ln \left(2\right)\cdot \:\frac{1}{16}\cdot 16$

$A'(4)= 20\cdot \frac{1}{4}-10\ln \left(2\right)\cdot \:1$

$A'(4)= 5-10\ln \left(2\right)$

A (45 10 0.693

  percent of the public per month
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A' (4) = -1.93147

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here we have${\color{Red} A'\left(2\right)=\: 3.06853}$

here positive sign indicates increasing rate

so after 2 months percent of the public aware of the product is increasing

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here we have

A' (4) = -1.93147

here negative sign indicates decreasing rate

so after 4 months percent of the public aware of the product is decreasing