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1. The normal monthly rainfalll at the Seattle-Tacoma Airport can be approximated by the model R = 3.121 +2.399sin(0.524t+1.377), where R is measured in inches and t is the time in months. Use the model to approximate the normal yearly rainfall.

I really am not sure how to solve this prolem - would anyone be able to help? IIt also gives me a hint: the interval [0,12] represents one year...

2. The profit, in thousands of dollars, of a new product over the first 6 months, 0=< t =< 6 is approximated by the model P = 5(sqrt(t) + 30). Write the integral used to find the average value of the profit function and solve (using calculator)

Mar 25, 2019

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I'm not sure  what is wanted in the first one....the average value  ???

For the second, the average value is

6

∫  5  ( sqrt (t) + 30 )   dt  /  6   =

0

5 [ (2/3)(6)^(3/2) + 30(6) ]  / 6   ≈  158.16   (thousands  of   dollars )   Mar 25, 2019
#2
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1. The normal monthly rainfalll at the Seattle-Tacoma Airport can be approximated by the model R = 3.121 +2.399sin(0.524t+1.377), where R is measured in inches and t is the time in months. Use the model to approximate the normal yearly rainfall.

Here is the approximate annual rainfall graph

the horizontal axis is Time in months and the vertical axis is Rainfall in inches.

The approximate annual rainfall is the area between the horizonal axis and the curve from t=0 to t=12

i.e.

$$yearly\;\;rainfall\\\approx \displaystyle\int_0^{12}\; 3.121+2.399sin(0.524t+1.377)\,dt\\ =\left[\;\; 3.121t-2.399*0.524cos(0.524t+1.377) \;\; \right]_0^{12}\\ = [3.121*12-2.399*0.524cos(0.524*12+1.377) ] - [3.121*0-2.399*0.524cos(0.524*0+1.377) ] \\ = [37.452-1.257076cos(4.425) ] + [1.257076cos(1.377) ] \\$$

(37.452-1.257076cos(4.425) ) + (1.257076cos(1.377) ) = 38.050411914390589152

So annual rainfall is approximately  38.05 inches.     (note that this answer has been edited, there was a small error before)

You STILL need to check my working though  Mar 25, 2019
edited by Melody  Mar 25, 2019