Question

Please help me solve this problem step by step. And this question isn’t from physics. I got this practice homework from my math class

Answer 1

Answer: To find the inverse function, we must first solve the expression for t, so let's get started.

q=k(1-e-t/a)

1- e- t/a= q/a

e- t/a= 1 -q/a

In e-t/a = In (1-q/k)

-t/a = In (1-q/k)

t= -a In (1-q/k)

Therefore, the inverse function is: t= -a In (1-q/k)

This function returns the amount of time it takes the capacitor to store a charge q.

b)To charge the capacitor to 90% capacity, q=0.9k; this is due to the fact that the maximum charge is k. Using this and the value of a, we get:

Step-by-step explanation: t= -2 In(1-0.9k/k)

-2 In(1-0.9)

t=4.6

Therefore, it takes the capacitor 4.6 seconds to charge to ninety percent of its capacity.

Answer 2

a)

To find the inverse function we need to solve the expression for t so let's do that:

`\begin{gathered} q=k(1-e^{-(t)/(a)}) \\ 1-e^{-(t)/(a)}=(q)/(k) \\ e^{-(t)/(a)}=1-(q)/(k) \\ \ln e^{-(t)/(a)}=\ln(1-(q)/(k)) \\ -(t)/(a)=\ln(1-(q)/(k)) \\ t=-a\ln(1-(q)/(k)) \end{gathered}`

Therefore, the inverse function is:

`t=-a\ln(1-(q)/(k))`

This function will tell us the time it takes the capacitor to store a charge q.

b)

To charge the capacitor to a 90 percent capacity means that q=0.9k; this comes from the fact that maximum charge is k. Plugging this and the value of a we have:

`\begin{gathered} t=-2\ln(1-(0.9k)/(k)) \\ t=-2\ln(1-0.9) \\ t=4.6 \end{gathered}`

Therefore, it takes the capacitor 4.6 seconds to charge to ninety percent of its capacity.