When a camera flash goes off, the batteries immediately...

When a camera flash goes off, the batteries immediately begin to recharge the flash's capacitor, which stores electric charge given by $$Q(t)=Q_{0}\left(1-e^{-t / a}\right)$$ (The maximum charge capacity is $Q_{0}$ and $t$ is measured inseconds.) (a) Find the inverse of this function and explain its meaning. (b) How long does it take to recharge the capacitor to 90$\%$ of capacity if $a=2 ?$

When a camera flash goes off, the batteries immediately begin to recharge the flash's capacitor, which stores electric charge given by
$$Q(t)=Q_{0}\left(1-e^{-t / a}\right)$$
(The maximum charge capacity is $Q_{0}$ and $t$ is measured inseconds.)
(a) Find the inverse of this function and explain its meaning.
(b) How long does it take to recharge the capacitor to 90$\%$ of capacity if $a=2 ?$

Key Concepts

Exponential Charging

This concept describes how the charge on a capacitor increases over time following an exponential function. It is common in circuits involving resistors and capacitors (RC circuits), where the charge builds up asymptotically to a maximum value. The rate of charging depends on parameters in the exponential expression and reflects the physical process of energy storage.

Inverse Function

An inverse function reverses the roles of inputs and outputs of the original function. In the context of the charging capacitor, finding the inverse function means expressing the time at which a given charge is reached. This is particularly useful for solving for time when the capacitor is charged to a given fraction of its total capacity.

Natural Logarithm

The natural logarithm is a mathematical tool used to invert exponential functions. When the capacitor charging equation is rearranged, taking the natural logarithm allows one to solve for time in terms of the charge, making it essential in deriving the inverse function of the exponential charging model.

Time Constant

The time constant is a key parameter in exponential processes, representing the time it takes for a process (such as capacitor charging) to progress significantly towards its final value. It determines how quickly the capacitor charges, and in cases like this, it appears as a scaling factor in the exponent that directly influences the speed of the exponential growth.

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