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A science geek brews tea at 195 ∘F, and observes that the temperature T(t) of the tea after t minutes is changing at the rate of T′(t)=−5e−0.04t ∘F/min. What is the average temperature of the tea during the first 14 minutes after being brewed? Round your answer to the nearest hundredth of a degree.

Respuesta :

Answer:

Explanation:

Given

T′(t)=[tex]5e^{-.04t}[/tex] ∘F/min

dT/dt =[tex]5e^{-.04t}[/tex]

dT =- [tex]5e^{-.04t}dt[/tex]

integrating on both sides within limit 0 to 14 minutes

T =- 5/- .04 [tex][e^{-.04\times14} - e^0][/tex]

= 125 x ( 0.57 - 1 )

= -53.6  

Change in temperature in 14 minutes = -53.6 °F

Final temperature = 195 - 53.6 = 141.4  °F

average temp = (195 + 141.4 )/2

= 168.2 °F

The average temperature of the tea during the first 14 minutes after being brewed is  168.2 °F.

  • The calculations is as follows;

[tex]T = 5/-0.04 ^({e^{-0.04\times 14-e^{0}}})\\\\ = 125 \times ( 0.57 - 1 )[/tex]

= -53.6

Now

Change in temperature in 14 minutes should be -53.6 °F

Now

Final temperature is

= 195 - 53.6

= 141.4  °F

And,

average temp = [tex](195 + 141.4 )\div 2 [/tex]

= 168.2 °F

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