The efficiency e of a gasoline engine as a function of its compression ratio r is given by e =1− r1−γ , where γ is a constant.
Find γ for e = 0.55 and r = 7.5.
To find the value of γ given e = 0.55 and r = 7.5, we can substitute these values into the equation e = 1 - r^(1-γ) and solve for γ.
Starting with the given equation: e = 1 - r^(1-γ)
Substituting the given values, e = 0.55 and r = 7.5: 0.55 = 1 - 7.5^(1-γ)
Rearranging the equation, we get: 7.5^(1-γ) = 1 - 0.55
Now, we want to isolate the exponent (1-γ), so we take the natural logarithm (ln) of both sides:
ln(7.5^(1-γ)) = ln(1 - 0.55)
Using the property of logarithms, we can bring down the exponent:
(1-γ) * ln(7.5) = ln(1 - 0.55)
Next, we isolate (1-γ) by dividing both sides by ln(7.5):
1 - γ = ln(1 - 0.55) / ln(7.5)
Finally, solve for γ by subtracting 1 from both sides of the equation:
γ = 1 - (ln(1 - 0.55) / ln(7.5))
Calculating this expression will give you the value of γ for e = 0.55 and r = 7.5.
Online "^" is used to indicate an exponent, e.g., x^2 = x squared.
I assume r1 = r^1 = r to the first power = r.
e = 1 - r - y
Add y and subtract e from both sides.
y = 1 - r - e
Insert values and solve.