The population p(t) of a culture of the bacterium Pseudomonas aeruginosa is given by ,p(t)= -1683t^2+75,000t+ 10,000 where is the time in hours since the culture was started. Determine the time the population was at its maximum. Round to the nearest hour.

Base case (n = 1):

• On the left side: 1/(1×2) = 1/2

• On the right side: 1/(1 + 1) = 1/2

Induction hypothesis: Assume the statement is true for n = k ; that is,

1/(1×2) + 1/(2×3) + … + 1/(k × (k + 1))) = k/(k + 1)

Inductive step (n = k + 1):

1/(1×2) + 1/(2×3) + … + 1/(k × (k + 1))) + 1/((k + 1) × (k + 2)))

= k/(k + 1) + 1/((k + 1) × (k + 2))

= (k × (k + 2) + 1) / ((k + 1) × (k + 2))

= (k ² + 2k + 1) / ((k + 1) × (k + 2))

= (k + 1)² / ((k + 1) × (k + 2))

= (k + 1) / (k + 2)

and this is what we wanted to show.

You're looking for a solution in the form

[tex]y(x) = \displaystyle \sum_{n=0}^\infty a_nx^n[/tex]

Differentiating, we get

[tex]y'(x) = \displaystyle \sum_{n=0}^\infty na_nx^{n-1} = \sum_{n=1}^\infty na_nx^{n-1} = \sum_{n=0}^\infty (n+1)a_{n+1}x^n[/tex]

[tex]y''(x) = \displaystyle \sum_{n=0}^\infty (n+1)na_{n+1}x^{n-1} = \sum_{n=1}^\infty (n+1)na_{n+1}x^{n-1} = \sum_{n=0}^\infty (n+2)(n+1)a_{n+2}x^n[/tex]

Substitute these for y' and y'' in the differential equation:

[tex]\displaystyle \sum_{n=0}^\infty (n+2)(n+1)a_{n+2}x^n - \sum_{n=0}^\infty (n+1)a_{n+1}x^n = 0[/tex]

[tex]\displaystyle \sum_{n=0}^\infty \bigg((n+2)(n+1)a_{n+2}-(n+1)a_{n+1}\bigg)x^n = 0[/tex]

Then the coefficients of y are given by the recurrence

[tex]\begin{cases}a_0=y(0)\\a_1=y'(0)\\a_{n+2}=\frac{a_{n+1}}{n+2}&\text{for }n\ge0\end{cases}[/tex]

or

[tex]a_n = \dfrac{a_{n-1}}n[/tex]

But we cannot assume that [tex]a_0[/tex] and [tex]a_1[/tex] depend on each other; we can only guarantee that the recurrence holds for n ≥ 1, so that

[tex]a_2=\dfrac{a_1}2 \\\\ a_3=\dfrac{a_2}3=\dfrac{a_1}{3\times2} \\\\ a_4=\dfrac{a_3}4=\dfrac{a_1}{4\times3\times2} \\\\ \vdots \\\\ a_n=\dfrac{a_1}{n!}[/tex]

So in the power series solution, we split off the constant term and we're left with

[tex]y(x) = a_0 + a_1 \displaystyle \sum_{n=1}^\infty \frac{x^n}{n!}[/tex]

so that the fundamental solutions are

[tex]y_1=1[/tex]

and

[tex]y_2=x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\dfrac{x^4}{4!}+\cdots[/tex]

Answer:

ascending order} an order of numbers from least to greatest like 1 2 3 4

descending order} an order of numbers from greatest to least like 4 3 2 1

Answer:

[tex]4\sqrt{5}[/tex]

Step-by-step explanation:

Prime factorize 20 and 45

[tex]\sqrt{20}=\sqrt{2*2*5}=2\sqrt{5}\\\\\\\sqrt{45}= \sqrt{5*3*3}=3\sqrt{5}[/tex]

[tex]\sqrt{20}-\sqrt{5}+\sqrt{45}=2\sqrt{5}-\sqrt{5}+3\sqrt{5}[/tex]

                           [tex]=(2-1+3)\sqrt{5}\\\\\\= 4\sqrt{5}\\[/tex]

The value of exponential expression ln e⁴ is 4

Expressions in math are mathematical statements that have a minimum of two terms containing numbers or variables, or both, connected by an operator in between.

Given is an exponential expression ln e⁴ we need to simplify it,

So,

we know that,

ln aⁿ = n ln a

and =

ln e = 1

so,

ln e⁴ = 4 ln e

= 4 x 1

= 4

Hence, the value of exponential expression ln e⁴ is 4

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Answer:

0.7 = 70% probability that the player will score 0 points.

Step-by-step explanation:

For each free throw, we have these following probabilities:

0.3 probability the player makes.

0.7 probability the player misses.

What is the probability that the player will score 0 points?

He is only allowed the second if he misses the first, thus, he ends with 0 points only if he misses the first.

For any free throw:

0.7 probability the player misses, so 0.7 = 70% probability that the player misses the first free throw, and 0.7 = 70% probability that the player will score 0 points.

Answer:

Step-by-step explanation:

Interior angle opposite to each side of the polygon is:

Since the circle is unit circle, its radius is 1 unit.

Let the side is a, then as per definition of sine we have:

The perimeter is:

Answer:

πr

Step-by-step explanation:

radius = r/2

so circumference = 2π(r/2)

= 2πr/2

= πr

Answer:

The answer is B which is C=2πr

Step-by-step explanation:

i just did it

Answer: the radius is perpendicular to the chord.

If a radius of a circle bisects a chord which is not diameter, then the radius is perpendicular to the chord.

Answered by Gauthmath must click thanks and mark brainliest

The radius is perpendicular to the chord.

If the radius of the circle is perpendicular to the chord of the circle, the radius bisects the chord. The two strings are congruent only if they are equidistant from the center of the circle.

No, not all strings in a circle are diameters because the diameter passes through the center of the circle. Therefore, all the diameters of a circle are also strings, not all the strings of a circle.

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Notice that the condition x ≠ 2πk for (presumably) integer k means cos(x) ≠ ±1, and in particular cos(x) ≠ 1 so that we could divide both sides by (1 - cos(x)) safely. Doing so lets us separate the variables:

(1 - cos(x)) y' - y sin(x) = 0

==>   (1 - cos(x)) y' = y sin(x)

==>   y'/y = sin(x)/(1 - cos(x))

==>   dy/y = sin(x)/(1 - cos(x)) dx

Integrate both sides and solve for y. On the right, substitute u = 1 - cos(x) and du = sin(x) dx.

∫ dy/y = ∫ sin(x)/(1 - cos(x)) dx

∫ dy/y = ∫ du/u

ln|y| = ln|u| + C

exp(ln|y|) = exp(ln|u| + C )

exp(ln|y|) = exp(ln|u|) exp(C )

y = Cu

y = C (1 - cos(x))

Step-by-step explanation:

[tex] \frac{ {c}^{2} - 4 }{c + 3} [/tex]

[tex] \frac{(c - 2)(c + 2)}{(c + 3)} [/tex]

y=1/2x-2 is perpendicular to y=2x+3

Answer:

Sin X = 12 / 37

Step-by-step explanation:

Given a right angled triangle, we are to obtain the Sin of the angle X ;

Using trigonometry, the sine of the angle X , Sin A is defined as the ratio of the angle opposite X to the hypotenus of the right angle triangle.

In the right angle triangle :

Sin X = opposite / hypotenus

Opposite = 12 ; hypotenus = 37

Sin X = 12 / 37

Answer:

3/4 or 75% or .75

Step-by-step explanation:

You would multiply 3 * .25 because there are three people getting 25% of a pie. All of the answers above are equivalent

Answer:

GHC22.5

Step-by-step explanation:

90/3=30

30=0.75

30×0.75

=22.5

Answer:

option c is the answer for your question

Answer:

3/4

Step-by-step explanation:

There are 13 hearts in a 52 deck.

52-13=39

39/52=3/4

The probability that you are not dealt a heart from the deck of cards is 3/4.

Probability determines the chances that an event would occur. The probability the event occurs is 1 and the probability that the event does not occur is 0.

The probability that you are not dealth with a heart = 1 - (number of hearts / total number of cards)

1 - 13/52 = 39/52 =  3/4

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9514 1404 393

Answer:

  A.  1/2

Step-by-step explanation:

The points on a unit circle are (cos(θ), sin(θ)), where θ is the angle from the +x axis to to the ray from the origin through that point.

If the point is (√3/2, 1/2), then sin(θ) = 1/2.

Answer:

4000

Step-by-step explanation:

from 1 - 1000 = 300

1's    = 100

10's  = 100

100's = 100

1000's = 0

300 * 10 = 3000

then add in all the 3000's (ie 3001,3002, etc ) that adds one more thousand

3000 + 1000 = 4000

Answer:

3000?

Step-by-step explanation:

Answer:

The 95% confidence interval for the proportion of all dies that pass the probe is (0.637, 0.733).

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which

z is the z-score that has a p-value of [tex]1 - \frac{\alpha}{2}[/tex].

356 dies were examined by an inspection probe and 244 of these passed the probe.

This means that [tex]n = 356, \pi = \frac{244}{356} = 0.685[/tex]

95% confidence level

So [tex]\alpha = 0.05[/tex], z is the value of Z that has a p-value of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].  

The lower limit of this interval is:

[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.685 - 1.96\sqrt{\frac{0.685*0.315}{356}} = 0.637[/tex]

The upper limit of this interval is:

[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.685 + 1.96\sqrt{\frac{0.685*0.315}{356}} = 0.733[/tex]

The 95% confidence interval for the proportion of all dies that pass the probe is (0.637, 0.733).

Your answer iss...

It is 51º

9514 1404 393

Answer:

  $11,680.58

Step-by-step explanation:

Usually, I would say copy the example, using 70,000 instead of 55,000. However, the example you show has a couple of errors in it. You need to do what it says, not follow what it did.

__

The first 48,535 is taxed at 15%, so the tax is 0.15×48535 = 7280.25.

The next (70,000 -48,535) = 21,465 is taxed at 20.5%, so the tax is ...

  0.205×21,465 = 4400.325 ≈ 4400.33

The the total tax due on $70,000 is ...

  $7280.25 +4400.33 = $11,680.58 . . . . tax due on $70,000

_____

Additional comments

The example shown has a couple of errors. The tax on the excess amount is figured at 2.05%, not 20.5%, and the 132.53 value from that is shown as 132.23.

__

Any tax table like this one can be reduced to a set of simpler formulas. Here are the formulas for the brackets shown in your tax table.

  ≤ 48535 -- income × 0.15

  ≤ 97069 -- income × 0.205 -2669.425

  ≤ 150,473 -- income × 0.26 -8008.22

  ≤ 214,368 -- income × 0.29 -12,522.41

  > 214,368 -- income × 0.33 -21,097.13

In this case, the second row of this simpler table would give the tax on $70,000 as ...

  tax = 70,000 × 0.205 -2669.425

  tax = 14350 -2669.425 = 11680.575 ≈ 11,680.58 . . . same as above

Answer:

31/10,32/10,33/10,34/10,35/10

Step-by-step explanation:

a rational number is formed when any two integers p and q are expressed in the form of p/q

to find two two sets of rational numbers BETWEEN any two numbers

a and b we need to express a and b and rational numbers....let us express 3and4 as rational numbers 3=30/10 4=40/10

the list of rational numbers between 3and4,that is, 30/10,31/10,32/10,33/10,34/10,35/10,36/10,37/10,38/10,39/10,40/10.

therefore the five rational numbers between 3 and 4 are (31/10,32/10,33/10,34/10,35/10...

I hope that helps

9514 1404 393

Answer:

  (a)  -23

Step-by-step explanation:

The sequence is arithmetic with first term 3 and common difference -2. Then the general term is ...

  an = a1 +d(n -1)

  an = 3 -2(n -1)

and the 14th term is ...

  a14 = 3 -2(14 -1) = 3 -2(13) = 3 -26

  a14 = -23

9514 1404 393

Answer:

  108.8 km/h at 84.3°

Step-by-step explanation:

The law of cosines can be used to find the resultant ground speed. In the attached diagram, the length of interest is OR. It will be found as ...

  OR² = OP² +PR² -2·OP·PR·cos(P)

  OR² = 130² +26² -2×130×26×cos(32°) ≈ 11843.19

  OR = √11843.19 ≈ 108.8

__

The angle POR can be found from the law of sines.

  sin(POR)/PR = sin(OPR)/OR

  sin(POR) = PR/OR×sin(32°) ≈ 0.12660

  ∠POR ≈ arcsin(0.12660) ≈ 7.27°

Then the bearing of the ground track of the airplane is 77° +7.27° = 84.27°.

The airplane is traveling at about 108.8 km/h on a bearing of 84.3°.

9514 1404 393

Answer:

  (a) Yes , the base is 1/e

Step-by-step explanation:

The variable is in the exponent, so this is an exponential function.

The base is the number that has the exponent. The base is (1/e).

Answer:

Step-by-step explanation:

bvxbvxbvxbvcvbbb cv

No. The F test has a requirement that samples be from the normally distributed populations, regardless of how large the samples are.

The F-test simply shows whether the variances that are in the numerator and the denominator are equal. The F-test can be applied on a large sampled population.

One main assumption of the F test is that the populations where the two samples are drawn are normally distributed.

Regarding the question, it's important to note that when using the F test with these​ data, it's not correct to reason that there is no need to check for "normality".

It should be noted that the F test has a requirement that samples are from the normally distributed populations, regardless of how large such samples are.

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Answer:

378 and 380

Step-by-step explanation:

The two even consecutive numbers that add up to 758 are going to be very close to half of 758. This is because two half of 758 are going to be the most similar addends of 758. This is important because the answers will be consecutive and therefore, must also be very similar. To solve, first, divide 758 by 2. This is 379, which is not an even number. So, to find the needed addends subtract and add 1 to 379. Both of these will be even and consecutive. These two numbers are 378 and 380. Then, to check you, can add them and see that they do sum 758.

Answer:

Step-by-step explanation:

Let the first number = x

Let the second number = x + 2

x + x + 2 = 758                      Collect like terms

2x + 2 = 758                         Subtract 2

2x = 758 - 2                          Combine

2x = 756                                Divide by 2

2x/2 = 756/2

x = 378

The first number is 378

The second number 380

If your teacher is really fussy, you can do it this way.

Let the first number = 2x

Let the second number = 2x + 2

The reason for this is to guarantee that both numbers were even to start with.

2x + 2x+2 = 758                    Combine like terms

4x + 2 = 758                          Subtract 2

4x = 756                                Divide by 4

x = 756/4

x = 189

Therefore 2x = 378

2x + 2 = 380                 Just as before.