💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Get the answer to your homework problem.

Try Numerade Free for 7 Days

Like

Report

When a camera flash goes off, the batteries immediately begin to recharge the flash's capacitor, which stores electric charge given by$$ Q(t) = Q_0 (1 - e^{\frac{-t}{a}}) $$(The maximum charge capacity is $ Q_0 $ and $ t $ is measured in seconds.)(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 $?

a) $t=-a \ln \left[1-\frac{Q}{Q_{o}}\right]$b) $\approx 4.605$ seconds

02:55

Clarissa N.

Calculus 1 / AB

Calculus 2 / BC

Calculus 3

Chapter 1

Functions and Models

Section 5

Inverse Functions and Logarithms

Functions

Integration Techniques

Partial Derivatives

Functions of Several Variables

Johns Hopkins University

Oregon State University

Harvey Mudd College

Idaho State University

Lectures

02:56

In mathematics, a vector (…

06:36

07:47

When a camera flash goes o…

01:20

An electronic flash unit f…

02:57

The flash unit on a camera…

01:59

The duration of a photogra…

01:45

02:46

A capacitor is charged by …

04:31

Filouke $21-54$ shows a si…

(II) At $t = 0 ,$ let $Q =…

01:46

Discharging Capacitor Volt…

07:53

Some camera flashes use fl…

all right. So here we have our equation. And in this equation, T is the input, and two is the output. So for the inverse, we would switch those and Q would be the input and t would be the output. And that means the meaning of the inverse is that we're going to find time as a function of charge, whereas what we have right now is charge as a function of time. Typically, when we find an inverse, we start by switching X and Y and then solving for why, in this case, that would mean switching Q and T and then solving for T. I do want to sell for tea, but I'm going to be a little bit rebellious. And I'm not going to switch Q and T because I want T to stand for time and cute to stand for charge. And if I switch them, their meaning is going to switch. So I'm just going to isolate E. So start by dividing both sides by Q not and then subtracting one from both sides that should say minus and then multiplying both sides by negative one. That's going to give us one minus. Q. over Q not equals E to the negative. T over a and then we'll take the natural log of both sides to have natural log of one minus. Q. Over Q. Not equals negative. T over a and then to get to you by itself will multiply both sides by the opposite of a so we have t equals the opposite of a times the natural log of one minus. Q. Over. Q. Not okay, so we'll count. That is our inverse time as a function of charge. Now we want to use that to figure out how long it takes to recharge the capacitor to 90% of capacity of a equals two. So we want to find the time, and we know that Q is 90%. And we know that a is to now when we say that Q is 90% we mean it's 90% of Q. Not so Q equals 900.9 times. Q. Not now our function was t equals negative. A so negative two times the natural log of one minus que over Q Not, which would be 0.9 Q. Not over. Q. Not we can simplify that. That's t equals negative two times a natural log of one minus 10.9. Let's keep simplifying. T equals negative two times the natural log of 20.1 and let's go ahead and approximate that we can put that in a calculator.

In mathematics, a vector (from the Latin word "vehere" meaning &qu…

In mathematics, a vector (from the Latin "mover") is a geometric o…

When a camera flash goes off, the batteries immediately begin to recharge th…

An electronic flash unit for a camera is activated when a capacitor is disch…

The flash unit on a camera operates by storing charge on a capacitor and rel…

The duration of a photographic flash is related to an $R C$ time constant, w…

A capacitor is charged by a $9.0-\mathrm{V}$ battery. The charging current $…

Filouke $21-54$ shows a simplified circuit for a photographic flash unit. Th…

(II) At $t = 0 ,$ let $Q = Q _ { 0 } ,$ and $I = 0$ in an $L C$ circuit. (a)…

Discharging Capacitor Voltage Suppose that electricity is draining from a ca…

Some camera flashes use flash tubes that require a high voltage. They obtain…