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2 years ago

Which transformations could be performed to show that

Mathematics

1 answer:

andre [41]2 years ago

3 0

Answer: a 180 degree rotation about the origin, then a dilation by a scale factor of 1/3

Step-by-step explanation:

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A bacteria culture starts with 400 bacteria and grows at a rate proportional to its size. After 4 hours, there are 9000 bacteria

Kaylis [27]

Answer:

A) The expression for the number of bacteria is $P(t) = 400e^{0.7783t}$ .

B) After 5 hours there will be 19593 bacteria.

C) After 5.55 hours the population of bacteria will reach 30000.

Step-by-step explanation:

A) Here we have a problem with differential equations. Recall that we can interpret the rate of change of a magnitude as its derivative. So, as the rate change proportionally to the size of the population, we have

$P' = kP$

where $P$ stands for the population of bacteria.

Writing $P'$ as $\frac{dP}{dt}$ , we get

$\frac{dP}{dt} = kP$ .

Notice that this is a separable equation, so

$\frac{dP}{P} = kdt$ .

Then, integrating in both sides of the equality:

$\int\frac{dP}{P} = \int kdt$ .

We have,

$\ln P = kt+C$ .

Now, taking exponential

$P(t) = Ce^{kt}$ .

The next step is to find the value for the constant $C$ . We do this using the initial condition $P(0)=400$ . Recall that this is the initial population of bacteria. So,

$400 = P(0) = Ce^{k0}=C$ .

Hence, the expression becomes

$P(t) = 400e^{kt}$ .

Now, we find the value for $k$ . We are going to use that $P(4)=9000$ . Notice that

$9000 = 400e^{k4}$ .

Then,

$\frac{90}{4} = e^{4k}$ .

Taking logarithm

$\ln\frac{90}{4} = 4k$ , so $\frac{1}{4}\ln\frac{90}{4} = k$ .

So, $k=0.7783788273$ , and approximating to the fourth decimal place we can take $k=0.7783$ . Hence,

$P(t) = 400e^{0.7783t}$ .

B) To find the number of bacteria after 5 hours, we only need to evaluate the expression we have obtained in the previous exercise:

$P(5) =400e^{0.7783*5} = 19593.723 \approx 19593$ .

C) In this case we want to do the reverse operation: we want to find the value of t such that

$30000 = 400e^{0.7783t}$ .

This expression is equivalent to

$75 = e^{0.7783t}$ .

Now, taking logarithm we have

$\ln 75 = 0.7783t$ .

Finally,

$t = \frac{\ln 75}{0.7783} \approx 5.55$ .

So, after 5.55 hours the population of bacteria will reach 30000.

6 0

3 years ago

It is thought that not as many Americans buy presents to celebrate Valentine's Day anymore. A random sample of 40 Americans yiel

evablogger [386]

Answer:

98% confidence interval of the true proportion of all Americans who celebrate Valentine's Day

(0.3672 , 0.7328)

Step-by-step explanation:

<u><em>Explanation:</em></u>-

Given Random sample size n =40

Sample proportion

$p = \frac{x}{n} = \frac{22}{40} = 0.55$

98% confidence interval of the true proportion of all Americans who celebrate Valentine's Day

$(p-Z_{\alpha } \sqrt{\frac{pq}{n} } , p + Z_{\alpha } \sqrt{\frac{pq}{n} } )$

The Z-value Z₀.₉₈ = Z₀.₀₂ = 2.326

98% confidence interval of the true proportion of all Americans who celebrate Valentine's Day

$(0.55-2.326\sqrt{\frac{0.55 X0.45}{40} } , 0.55 + 2.326\sqrt{\frac{0.55 X0.45}{40} } )$

( 0.55 - 0.1828 , 0.55 + 0.1828)

(0.3672 , 0.7328)

6 0

3 years ago

Read 2 more answers

in a fundraiser, Mrs. Smith pledges $1 for each mile walked . if Sam walks 20 miles how much will Mrs. Smith donate? what is yhe

EleoNora [17]

Mrs. Smith donated $20
The constant proportionality is $1 because no matter what, Mrs. smith will donate $1 for each mile. So if Sam were to only walk 1 mile, mrs. smith would donate $1. Or seen as a 1:1 proportion.

5 0

2 years ago

The scatter plot with trend line below shows the data comparing wind speed and wind

natulia [17]

Answer:

Step-by-step explanation:

8 0

2 years ago

Can someone help me with this math problem please?

Alexxandr [17]

First u have to find the square root for the number and then answer all of the root question then u have to find the angels

4 0

2 years ago

Read 2 more answers