The efficiency of an internal combustion engine is given by E= (1-v/V)^0.4
Where V and v are the respective maximum and minimum volumes of air in each cylinder.
a) Show that ∂E/ ∂V > 0 and interpret your result
b) Show that ∂E/ ∂v < 0 and interpret your result
To show that ∂E/∂V > 0, we need to find the partial derivative of E with respect to V and prove that it is greater than zero.
Step 1: Calculate E.
Given: E = (1 - v/V)^0.4
Step 2: Differentiate E with respect to V.
To differentiate E with respect to V, we need to use the chain rule.
Let's first rewrite E as a power function: E = (1 - v/V)^(2/5).
Now, we can apply the chain rule:
∂E/∂V = (2/5) * (1 - v/V)^(-3/5) * (-v/V^2)
Step 3: Simplify the equation.
Simplifying the equation, we get:
∂E/∂V = (-2v/ (5V^2)) * (1 - v/V)^(-3/5)
Step 4: Interpret the result.
Since both v and V are positive values, and (1 - v/V)^(-3/5) is always positive, the negative sign in the equation does not affect the sign of the derivative. Thus, ∂E/∂V is positive.
Interpretation:
∂E/∂V > 0 means that as the maximum volume of air in each cylinder (V) increases, the efficiency of the internal combustion engine (E) increases. In other words, a larger maximum volume of air leads to a higher efficiency in the internal combustion engine.
To show that ∂E/∂v < 0, we need to find the partial derivative of E with respect to v and prove that it is less than zero.
Step 1: Calculate E.
Given: E = (1 - v/V)^0.4
Step 2: Differentiate E with respect to v.
To differentiate E with respect to v, we need to use the chain rule.
Let's rewrite E as a power function: E = (1 - v/V)^(2/5).
Now, we can apply the chain rule:
∂E/∂v = (2/5) * (1 - v/V)^(-3/5) * (-1/V)
Step 3: Simplify the equation.
Simplifying the equation, we get:
∂E/∂v = (-2/ (5V)) * (1 - v/V)^(-3/5)
Step 4: Interpret the result.
Since V is positive, the negative sign in the equation affects the sign of the derivative. Thus, ∂E/∂v is negative.
Interpretation:
∂E/∂v < 0 means that as the minimum volume of air in each cylinder (v) increases, the efficiency of the internal combustion engine (E) decreases. In other words, a larger minimum volume of air leads to a lower efficiency in the internal combustion engine.