jenreluao1321

jenreluao1321

10.05.2021 •

Mathematics

If x varies inversely as the square root of y, and x = 9 when y = 121, find x when y = 81. 7

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Ответ:

naz731

26.01.2021

a) The initial population is 125 bacteria.

b) $P(t) = 125e^{0.8t}$

c) There are 1378 cells after 3 hours.

d) The rate of growth after 3 hours is of 1102 cells an hour.

e) The population will reach 200,000 in 9.22 hours.

Step-by-step explanation:

The population of this culture can be modeled by the following formula:

$P(t) = P_{0}e^{rt}$

In which P(t) is the population after t hours, $P_{0}$ is the initial population, and r, as decimal, is the growth rate.

(a) Find the initial population.

P(2) = 600

$600 = P_{0}e^{2r}$

Also

P(8) = 75000

$75000 = P_{0}e^{8r}$

$P_{0} = \frac{75000}{e^{8r}}$

Replacing in this equation:

$600 = P_{0}e^{2r}$

$600 = \frac{75000e^{2t}}{e^{8r}}$

$e^{-6r} = 0.008$

Applying ln to both sides

-6r = -4.83

r = 0.8

The initial population is:

$P_{0} = \frac{75000}{e^{8r}} = \frac{75000}{e^{8*0.8}} = 125$

The initial population is 125 bacteria.

(b) Find an expression for the population after t hours.

$P(t) = P_{0}e^{rt}$

$P(t) = 125e^{0.8t}$

(c) Find the number of cells after 3 hours.

$P(t) = P_{0}e^{rt}$

$P(3) = 125e^{0.8*3} = 1378$

There are 1378 cells after 3 hours.

(d) Find the rate of growth after 3 hours.

This is P'(3).

$P(t) = P_{0}e^{rt}$

$P'(t) = P_{0}re^{rt}$

So

$P'(3) = 125*0.8*e^{0.8*3} = 1102$

The rate of growth after 3 hours is of 1102 cells an hour.

(e) When will the population reach 200,000?

$P(t) = P_{0}e^{rt}$

$200000 = 125e^{0.8t}$

$e^{0.8t} = 1600$

Applying ln to both sides

0.8t = 7.38

t = 9.22

The population will reach 200,000 in 9.22 hours.

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