Question

6. The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function n = f(t) = a / (1 + be^-0.7t), where t is measured in hours. At time t = 0 the population is 20 cells and is increasing at a rate of 12 cells/hour. Find the values of a and b. According to this model, what happens to the yeast population in the long run?

          6. The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function

n = f(t) = a / (1 + be^-0.7t),

where t is measured in hours. At time t = 0 the population is 20 cells and is increasing at a rate of 12 cells/hour. Find the values of a and b. According to this model, what happens to the yeast population in the long run?

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6. The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function

n = f(t) = a / (1 + be^-0.7t),

where t is measured in hours. At time t = 0 the population is 20 cells and is increasing at a rate of 12 cells/hour. Find the values of a and b. According to this model, what happens to the yeast population in the long run?

Added by Vicenta C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Nicole Smina

12/27/2022

Step 1

So we can substitute these values into the equation to find a: 20 = a / (1 + b) Show more…

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The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function n = f(t) = a / (1 + be^-0.7t), where t is measured in hours. At time t = 0 the population is 20 cells and is increasing at a rate of 12 cells/hour. Find the values of a and b. According to this model, what happens to the yeast population in the long run?

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Transcript

00:01 Okay, so we want to find the values of a and b.

00:03 We know that when t is equal to zero, the population, which is f of t, is 20.

00:10 And the rate of increase is 12 cells an hour.

00:14 So t is an hour.

00:15 So in one hour, if the rate of increase is 12 per hour, f of t is going to be 20 plus 12, which is 32.

00:27 So then we can go ahead and plug these in.

00:29 So we know that 32 is equal to a over 1 plus b, e to the negative 0 .7 times 1.

00:38 And then 20 is a over 1 plus b ,e to the negative 0 .7 times 0.

00:47 So plugging in f of t and t.

00:51 Okay, so 32 is equal to a over 1 plus b, e to the negative 0 .7 .6.

01:01 Just simplifying the exponent, and then 20 is equal to a over 1 plus b times e to the zero power, which is 1.

01:12 So this is 1 plus b.

01:18 Okay, now simplifying.

01:19 So 32 plus 32b to the negative 0 .7 equals a, and then 20 plus 20b is equal to a.

01:35 So now we can go ahead and plug these values in or set the two equal since they both equal a, i should say.

01:43 So 32 plus 32b, e to the negative 0 .7 is a number also so we can go ahead and simplify that on our calculator.

01:52 So let's see, 0 .7, negative raise e to that, and i'm going to multiply it by 32...

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