1.The efficiency of an internal combustion engine is:
efficiency (%) = 100 [1 - (1/(x/y)^c)]
where x/y is the ratio of the uncompressed gas to the compressed gas and C is a positive constant dependent on the engine design. Find the LIMIT of the efficiency as the compression ratio approaches infinity.
MY attempt:
I set that function to lim and I distributed the 100 so mine's looks like this:
100 - (100/(x/y))^c
at this point, i am lost because i don't know where else to go in order to find the limit. help here? thanks
Don't expand it, leave it the way it stands
As x/y becomes larger, 1÷ x/y becomes small, and [1 ÷ (x/y)^c] becomes smaller even faster.
So as x/y approaches infinity, [1 ÷ (x/y)^c] approaches zero
and the limit will be 100{1-0] = 100
To find the limit of the efficiency as the compression ratio approaches infinity, we need to simplify the expression and analyze how it behaves.
Let's start by focusing on the term inside the parentheses: (x/y). As the compression ratio approaches infinity, the value of (x/y) will go to zero because the compressed gas volume will become much smaller compared to the uncompressed gas volume.
Now, let's consider the term (1/(x/y))^c. As (x/y) approaches zero, the power to which it is raised (c) becomes insignificant. This means that the value of (1/(x/y))^c approaches 1.
Therefore, the expression simplifies to:
efficiency (%) = 100 [1 - (1/(x/y))^c]
efficiency (%) = 100 [1 - 1]
efficiency (%) = 100
As the compression ratio approaches infinity, the efficiency of the internal combustion engine reaches 100% (the maximum possible value) according to this model.
To find the limit of the efficiency as the compression ratio approaches infinity, we can use the concept of limits. Let's break down the steps:
1. Start with the expression: efficiency (%) = 100 [1 - (1/(x/y)^c)].
2. To find the limit as x/y approaches infinity, simplify the expression inside the bracket by dividing both the numerator and denominator by (x/y)^c:
efficiency (%) = 100 [1 - ((x/y)^(-c))] .
3. Now, let's evaluate the limit of the expression as x/y approaches infinity. Remember that when the base (x/y) becomes larger and larger (tending to infinity), if c is positive, then (x/y)^(-c) tends to 0. This is because as the base increases, the exponent becomes more negative, making the whole expression smaller.
So, as x/y approaches infinity, the expression (x/y)^(-c) approaches 0.
4. Now, substitute the evaluated limit back into the equation:
efficiency (%) = 100 [1 - 0] = 100%.
Therefore, the limit of the efficiency as the compression ratio approaches infinity is 100%. This means that the internal combustion engine would approach complete efficiency with increasing compression ratio.