A lawyer has determined that the number of people P(t) in a...

A lawyer has determined that the number of people P(t) in a city of 2.6 million people who have been exposed to a news item after t days is given by the function $P(t) = 2,600,000(1 - e^{-0.03t})$. (a) How many days after a major crime has been reported has 55% of the population heard of the crime? (Round your answer to the nearest day.) days (b) A defense lawyer knows it will be difficult to pick an unbiased jury after 80% of the population has heard of the crime. After how many days will 80% of the population have heard of the crime? (Round your answer to the nearest day.) days

Transcript

00:01 Okay, so before we go about solving this problem, let's make sure we understand the context.

00:04 So the context is after t days, we have a town population of about 1 .2 million.

00:09 The number of people who have been exposed to a news item after t days is given by this function.

00:15 Okay, so t, let's be a t here.

00:19 Okay, so first i'm going to figure out how many days after a major crime has been reported has 40 % of the population.

00:26 So 40 % of the population, my population is not changing.

00:30 40%, same thing as 0 .4 of means multiply of 1 .2 million equals my expression here.

00:47 So what i'm going to do is just now that i set up this equation, i'm just going to try to solve for t.

00:52 So first thing i'm going to do is divide by 1 .2 million.

00:59 And what's nice is they both cancel.

01:01 So i have 0 .4 equals 1 minus this whole expression here.

01:06 Okay.

01:07 What's next? well, i want to get the t term by itself.

01:10 So i'm going to subtract one from both sides.

01:13 I'm going to get negative 0 .6 equals negative e.

01:16 Okay, since i have only two terms on both sides and they're both negative, i can multiply by negative one to both sides.

01:23 And then how do i get rid of this e? right? what's the inverse of the exponential function? the logarithmic function and specifically log of the same base will cancel.

01:33 Log base e is the same thing as ln.

01:35 So i can take the ln of both.

01:37 Sides, ln and e will cancel.

01:39 So ln of 0 .6 equals negative 0 .01 times t...

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