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3 years ago

Three friends have a total of 6 identical pencils, and each one has at least one pencil. In how many ways can this happen?

Mathematics

2 answers:

Lorico [155]3 years ago

7 0

Answer:

Step-by-step explanation:

events are 1+1+4=6

1+2+3=6

1+3+2=6

1+4+1=6

2+1+3=6

2+2+2=6

2+3+1=6

3+1+2=6

3+2+1=6

4+1+1=6

total number of ways=10

leva [86]3 years ago

5 0

Answer:

10 ways

Step-by-step explanation:

First we have to find three numbers that add up to 6, as each friend has at least one pencil, meaning that there are always and only three people who have pencils. The three combinations that add up to 6 are 1,2,3; 2,2,2; and 4,1,1. Now we just have to find how many ways these pencils can be distributed between the friends.

For 2,2, and 2, it is simple as there is only one way these pencils can be distributed, 2 pencils per friend, so there is one way to distribute 2,2 and 2.

For 1,2 and 3, we can use factorial to determine how many ways these groups of pencils can be distributed. Since the first person can pick between 3,2, and 1 pencil(s), they have 3 options. Multiply this by 2 options for the second person and 1 for the third, we have 3*2*1=6 ways to distribute, or 3!.

Finally for 4,1, and 1, since two of the options are the same, that means that the 4 is the only unique one, meaning that there are only 3 distributions as one of the three friends will have the 4 pencils. This means that there are 3 ways to distribute.

Now adding up all the possible ways to distribute, we get 1+6+3=10 ways to distribute the pencils.

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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.

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