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Probability Between 5: 00 PM and 6: 00 PM, cars arrive at Jiffy Lube at the rate of 9 cars per hour $(0.15$ car per minute). The following formula from statistics can be used to determine the probability that a car will arrive within $t$ minutes of 5: 00 PM.$$F(t)=1-e^{-0.15 t}$$(a) Determine how many minutes are needed for the probability to reach $50 \%$.(b) Determine how many minutes are needed for the probability to reach $80 \%$.

Algebra

Chapter 5

Exponential and Logarithmic Functions

Section 4

Logarithmic Functions

Functions

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we're keeping a probability function pft. And in this function, T is the number of minutes and P is the probability that the event will occur within that number of minutes within T minutes. And our task is to determine after how many minutes so t equals what for the probability Teoh equal 50% which 50% is 50 out of 100 which is 1000.5 and then we're also tasked with determining after how many minutes the probability will have gotten up to 80% which is 80 out of 100.8. So we're gonna have to do the same thing twice. Why not derive a formula t in terms of P, and then we could use that formula twice. That would be more efficient. So we're going to take our our probability formula and solve it for T. So the first thing we're gonna do is subtract one from both sides, and then we're also going to multiply both sides by negative one, and that will isolate our exponential. And then we're gonna take the natural log of both sides. So while we're at it, let's distribute this minus one negative P +11 minus p is equal to the natural log of E to the negative 0.15 t. So what do you take a longer of them oven exponential. If they both have the same base, the result you get is the exponents. Natural log is law based E. So that's the case we have here. The natural log of E to the negative 0.15 t is negative 0.15 t. So to finish solving for tea, we're gonna divide both sides by negative 0.15 So our formula then is natural log of one minus the probability divided by negative 0.15 That's how many minutes. So now we can go and easily find out the number of minutes so T is gonna equal to the natural log of one minus p, which in this case is 10.5. Divide by negative 0.1 five. And with a little help from our calculator, we've determined that that is 4.6 to minutes. So after 4.62 minutes, the probability that the event will have occurred is 50%. That will do the same thing. But this time we want to know how many minutes? For 80%. So natural log of one minus 10.8, divided by negative 0.15 can. With a little help from our calculator, we determined that it's 10.7 three minutes.

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