Question

I'm stuck on this one as well

The function f(t) = 65 sin (pi over 5t) + 35 models the temperature of a periodic chemical reaction where t represents time in hours. What are the maximum and minimum temperatures of the reaction, and how long does the entire cycle take?

maximum: 65°; minimum: 35°; period: pi over 5 hours

maximum: 100°; minimum: −30°; period: pi over 5 hours

maximum: 65°; minimum: 35°; period: 10 hours

maximum: 100°; minimum: −30°; period: 10 hours

Answer 1

I dont really want to download anything

Answer 2

maximum of sine is when the input is \(\frac{\pi}{2}\) (you get 1 in that case, which is the largest sine can be)

Answer 3

set
\[\frac{\pi}{5}t=\frac{\pi}{2}\] and solve for \(t\)

Answer 4

where would I start?

Answer 5

ok now i have a question
what moron wrote this problem?
the question is posed with the measure in radians, but the answer is in degrees!!

Answer 6

another copy and paste for my bad math folder
hold on, we can answer in a second

Answer 7

Okay

Answer 8

oh wait, i am a moron, not the person who wrote the problem
degrees is the output, it is temperature!!

Answer 9

\[ f(t) = 65 \sin (\frac{\pi}{5}t) + 35 \] we need very little to find the maximum value

Answer 10

I just saw that too

Answer 11

the maximum value of sine is 1
the maximum value of 65 times sine is 65
the maximum value of 65 times sine plus 35 is 100

Answer 12

i will let you find the minimum on your own, starting with
the minimum value of sine is -1

Answer 13

Where would I plug it in on the equation you gave me?

Answer 14

as for the period, the period of \(\sin(bx)\) is \(\frac{2\pi}{b}\)
put \(b=\frac{\pi}{5}\) so get your answer

Answer 15

not sure what you are asking
did you understand these lines:

the maximum value of sine is 1
the maximum value of 65 times sine is 65 t
he maximum value of 65 times sine plus 35 is 100

Answer 16

I am so lost lol,sorry..math is my weak point

Answer 17

ok lets go real slow

Answer 18

the largest \(\sin(x)\) can ever be is \(1\) , it can never be bigger

Answer 19

okay

Answer 20

if the largest \(\sin(x)\) can be is \(1\), what is the largest \(65\sin(x)\) can be?

Answer 21

the largest would be 1 as well?

Answer 22

let me rewrite it like this
suppose \(\sin(x)=1\)
then what is
\[65\times \sin(x)\]?

Answer 23

it would be 35?

Answer 24

let me try again
what is
\[65\times 1\]?

Answer 25

65

Answer 26

k good

Answer 27

then finally, what is
\[65\times 1+35\]?

Answer 28

100

Answer 29

got it

Answer 30

so 100 is the maximum

Answer 31

now lets see what we know
the biggest sine can be is \(1\) so the biggest
\[f(t) = 65 \sin (\frac{\pi}{5}t) + 35\]can be is
\[65\times1+35=100\]

Answer 32

right

Answer 33

now lets find the minimum, knowing that the minimum value of sine is \(-1\)
what do you think the value of \[f(t) = 65 \sin (\frac{\pi}{5}t) + 35\] is if you replace sine by \(-1\) ?

Answer 34

t could be -30?

Answer 35

now we got it!

Answer 36

So how would I apply that to the answer options I have?

Answer 37

i forget what they were, but i hope one had "maximum is 100, minimum is -30"

Answer 38

Yes I had two of those

Answer 39

maximum: 100°; minimum: −30°; period: pi over 5 hours
maximum: 100°; minimum: −30°; period: 10 hours

Answer 40

ok so it is one of those
now you need the period

Answer 41

I think it would be the first one

Answer 42

would you like to know how to find it?

Answer 43

Yes please

Answer 44

the period of \[\sin(bx)\] is
\[\frac{2\pi}{b}\]
it is NOT \(b\)

Answer 45

therefore the period of
\[\sin(\frac{\pi}{5}t)\] is
\[2\pi\div \frac{\pi}{5}=2\pi\times \frac{5}{\pi}=10\]

Answer 46

maximum: 65°; minimum: 35°; period: 10 hours

Answer 47

hmm no

Answer 48

we already found the max and min

Answer 49

ssorry
not that one

Answer 50

maximum: 100°; minimum: −30°; period: 10 hours

Answer 51

got it

Answer 52

THANK YOU...you help me more than my online teacher

Answer 53

more importantly i hope the steps were more or less clear
on line teacher?

Answer 54

It was more clear..Thank you again!!

Answer 55

yw