Life expentancy of a star (power function)

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MathBane

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The question:

The life expectancy E of a main-sequence star depends on its mass M. The relation is given by the formula below where M is solar masses and E is solar lifetimes.

\(\displaystyle E = M^{-2.5}\)

The sun is thought to be at the middle of its life, with a total life expectancy of about 10 billion years. Thus the value E = 1 corresponds to a life expectancy of 10 billion years.

(a) Does a less massive star have a longer or a shorter life expectancy than a more massive star?

(longer)


(b) Spica is a main-sequence star that is about 7.3 solar masses. What is the life expectancy of Spica? (Round your answer to three decimal places.)

.069


(c) Express using functional notation the life expectancy of a main-sequence star with mass equal to 2.5 solar mass. (Round your answers to one decimal place.)

E(2.5)

Calculate that value.

1.011


(d) Vega is a main-sequence star that is expected to live about 6.36 billion years. What is the mass of Vega? (Round your answer to two decimal places.)

____ solar masses

(So... it would be \(\displaystyle 6.36 = M^{-2.5}\), right? So I would have to get M by itself... but... how do I do that with a negative decimal exponent?)


(e) If one main-sequence star is four times as massive as another, how do their life expectancies compare? (Round your answer to two decimal places.)

The life expectancy of the larger star is [____] times that of the smaller.
 
MathBane said:
The question:

The life expectancy E of a main-sequence star depends on its mass M. The relation is given by the formula below where M is solar masses and E is solar lifetimes.

\(\displaystyle E = M^{-2.5}\)

The sun is thought to be at the middle of its life, with a total life expectancy of about 10 billion years. Thus the value E = 1 corresponds to a life expectancy of 10 billion years.

(a) Does a less massive star have a longer or a shorter life expectancy than a more massive star?

(longer) << Rethink your answer in light of what you found in part (b)


(b) Spica is a main-sequence star that is about 7.3 solar masses. What is the life expectancy of Spica? (Round your answer to three decimal places.)

.069 <<< If Spica is more massive compared to Sun - how come it's life expectancy is less than that of the Sun (0.069<1)

(c) Express using functional notation the life expectancy of a main-sequence star with mass equal to 2.5 solar mass. (Round your answers to one decimal place.)

E(2.5) <<< ???

E = (2.5)^(-2.5)


Calculate that value.

1.011<<< Incorrect - unit???


(d) Vega is a main-sequence star that is expected to live about 6.36 billion years. What is the mass of Vega? (Round your answer to two decimal places.)

____ solar masses

Your E = 6.36/10 = 0.636

0.636 = M^(-2.5)

M = (0.636)^(-1/2.5) = (0.636)^(-0.4) = 1.198442367 solar mass <<< don't forget to round


(So... it would be \(\displaystyle 6.36 = M^{-2.5}\), right? So I would have to get M by itself... but... how do I do that with a negative decimal exponent?)


(e) If one main-sequence star is four times as massive as another, how do their life expectancies compare? (Round your answer to two decimal places.)

Work with ratio
\(\displaystyle E_1 = M_1^{-2.5}\)

\(\displaystyle E_2 = M_2^{-2.5}\)

then

\(\displaystyle \frac{E_1}{E_2} = \left (\frac{M_1}{M_2}\right)^{-2.5}\)

E_1\E_2} = (4)^{-2.5} = 0.03125[/spoiler:q83haiu9]
The life expectancy of the larger star is [____] times that of the smaller.
Click to expand...
 
The life expectancy E of a main-sequence star depends on its mass M. The relation is given by the formula below where M is solar masses and E is solar lifetimes.



The sun is thought to be at the middle of its life, with a total life expectancy of about 10 billion years. Thus the value E = 1 corresponds to a life expectancy of 10 billion years.

(a) Does a less massive star have a longer or a shorter life expectancy than a more massive star?

(longer)


(b) Spica is a main-sequence star that is about 7.3 solar masses. What is the life expectancy of Spica? (Round your answer to three decimal places.)

.069


(c) Express using functional notation the life expectancy of a main-sequence star with mass equal to 2.5 solar mass. (Round your answers to one decimal place.)

E(2.5)

Calculate that value.

1.011

(e) If one main-sequence star is four times as massive as another, how do their life expectancies compare? (Round your answer to two decimal places.)
Click to expand...

Mathbane,

Beware of careless mistakes. I suspect you simply misread the first question, and on c, I think you misread the decimal position on your calculator. Don’t “shoot yourself in the foot” on problems you know how to do. :)

Also, as Subhotosh pointed out, you need to put units on your answers.

b)
E = M^(-2.5)
E = 7.3^(-2.5) = .00694
This is in tens of billions (10 times 10^9 = 10^10) of years, so
.00694(10^10) = 6.94 x 10^7 years (in scientific notation)
or
69.4 million years

c)
Since E is a function of mass, we’d write
E = f(M) = M^(-2.5)
And plugging in 2.5 solar mass, we’d write
f(2.5) = (2.5)^(-2.5) = .1
Once again, since this is in tens of billions of years,
(.1)(10^10) = 10^9 = 1 billion years

e)
On this problem, Subhotosh has given you a complete solution. You can also rephrase your answer by flipping it around and stating the life of the smaller in terms of the life of the larger:

E2/E1 = (1/4)^(-2.5) = 32

So, in a full sentence, we’d answer:

The life expectancy of the smaller star is 32 times that of the larger.
 
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