I Find This Very Hard Can Somebody Check This Out

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In summary, we found the inverse of the equation y= (2e^t + 3e^-t ) / (e^t + 2e^-t), simplified it to y = \log _e \left( {\sqrt {\frac{{3 - 2t}}{{t - 2}}} } \right), and verified its equivalence with the given solution. We also solved for the exact value of sinh x in the equation cosh x = 4.5 using the formula (coshx)^2 - (sinhx)^2 = 1. Finally, we expressed the given equation (10x^2 + 14x + 3) / (x^3 + 3x^2 -
  • #1
dagg3r
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:mad: 1. Given y= (2e^t + 3e^-t ) / (e^t + 2e^-t)

find inverse of of the equation from above

i did the working out to get

y= ([2t-3] / [2-t]) ^ 1/2

do i have to apply any log laws or anything like that?

2. cosh x = 4.5 find sinh x exactly
i applied the formula
(coshx)^2 - (sinhx)^2 = 1
therefore

- (sinhx)^2 = 1 - (coshx)^2
(sinhx)^2 = -1 + (4.5)^2
(sinhx)^2 = 19.25
sinhx = sqrt 19.25

is that exactly?

3. express as partial fractions

(10x^2 + 14x + 3) / (x^3 + 3x^2 - 4)
i got
[7/(x+2)] - [5/(x+2)^2] + [3/(x-1)]

is that right?

PLEASE REPLY THANKS
 
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  • #2
sorry made an error in the 1st question meawn to be
for my solution

y= 1/2 ln ([2t-3] / [2 - t])

not this
y= ([2t-3] / [2-t]) ^ 1/2
 
  • #3
[tex]
y = \frac{{2e^t + 3e^{ - t} }}{{e^t + 2e^{ - t} }}
[/tex]

Interchange y and t(it isn't necessary to do that but I prefer to).

[tex]
t = \frac{{2e^y + 3e^{ - y} }}{{e^y + 2e^{ - y} }}
[/tex]

[tex]
t = \frac{{\frac{{2e^{2y} + 3}}{{e^y }}}}{{\frac{{e^{2y} + 2}}{{e^y }}}}
[/tex]

[tex]
t = \frac{{2e^{2y} + 3}}{{e^{2y} + 2}}
[/tex]

[tex]
te^{2y} + 2t = 2e^{2y} + 3
[/tex]

[tex]
\left( {t - 2} \right)e^{2y} = 3 - 2t
[/tex]

[tex]
e^{2y} = \frac{{3 - 2t}}{{\left( {t - 2} \right)}}
[/tex]

[tex]
y = \log _e \left( {\sqrt {\frac{{3 - 2t}}{{t - 2}}} } \right)
[/tex]

Edit: Our answers for the first one are equivalent. To convert mine to yours just multiply the numerator and the denominator of the log argument by negative one and get rid of the square root by placing a factor of (1/2) in front of the log.
 
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1. What does the phrase "I Find This Very Hard Can Somebody Check This Out" mean?

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5. What should scientists do if they are struggling with their research?

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