F（c) = a \left(1 - e^{-bc}\right) F(10C)/F(C)=?

To find $ \frac{F(10C)}{F(C)} $ for the given function $ F(c) = a \left(1 - e^{-bc}\right) $, we need to evaluate $ F $ at $ 10C $ and $ C $ and then find their ratio.

Given:
\[ F(c) = a \left(1 - e^{-bc}\right) \]

First, evaluate $ F(10C) $:
\[ F(10C) = a \left(1 - e^{-b(10C)}\right) = a \left(1 - e^{-10bc}\right) \]

Next, evaluate $ F(C) $:
\[ F(C) = a \left(1 - e^{-bc}\right) \]

Now, find the ratio:
\[ \frac{F(10C)}{F(C)} = \frac{a \left(1 - e^{-10bc}\right)}{a \left(1 - e^{-bc}\right)} \]

Since $ a $ cancels out, we get:
\[ \frac{F(10C)}{F(C)} = \frac{1 - e^{-10bc}}{1 - e^{-bc}} \]

Therefore,
\[ \frac{F(10C)}{F(C)} = \frac{1 - e^{-10bc}}{1 - e^{-bc}} \]

If you have any questions or need further details, feel free to ask!

Here are 5 numbered questions you might want to explore:

1. How does the value of $ b $ affect the ratio $ \frac{F(10C)}{F(C)} $?
2. What happens to $ \frac{F(10C)}{F(C)} $ as $ b $ approaches zero?
3. Can you derive the limit of $ \frac{F(10C)}{F(C)} $ as $ c $ approaches infinity?
4. How would the ratio change if the function were $ F(c) = a \left(1 - e^{-bc^2}\right) $?
5. How can this function $ F(c) $ be applied in real-world scenarios?

Tip: When dealing with exponential functions, always pay attention to the base and exponent to understand the growth or decay behavior.

Understand how to calculate the ratio $\frac{F(10C)}{F(C)}$ for the exponential function $F(c) = a \left(1 - e^{-bc}\right)$. Detailed step-by-step solution and analysis including the impact of the parameter $b$ on the ratio.

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