SCALCET8 3.11.501.XP. Find the numerical value of each expression. (Round your answers to five decimal places.) (a) sinh(ln(4)) (b) sinh(4)

Given that exp(2x) is a solution, we assume another solution of the form

y(x) = v(x) exp(2x) = v exp(2x)

with derivatives

y' = v' exp(2x) + 2v exp(2x)

y'' = v'' exp(2x) + 4v' exp(2x) + 4v exp(2x)

Substitute these into the equation:

(2x + 5) (v'' exp(2x) + 4v' exp(2x) + 4v exp(2x)) - 4 (x + 3) (v' exp(2x) + 2v exp(2x)) + 4v exp(2x) = 0

Each term contains a factor of exp(2x) that can be divided out:

(2x + 5) (v'' + 4v' + 4v) - 4 (x + 3) (v' + 2v) + 4v = 0

Expanding and simplifying eliminates the v term:

(2x + 5) v'' + (4x + 8) v' = 0

Substitute w(x) = v'(x) to reduce the order of the equation, and you're left with a linear ODE:

(2x + 5) w' + (4x + 8) w = 0

w' + (4x + 8)/(2x + 5) w = 0

I'll use the integrating factor method. The IF is

µ(x) = exp( ∫ (4x + 8)/(2x + 5) dx ) = exp(2x - log|2x + 5|) = exp(2x)/(2x + 5)

Multiply through the ODE in w by µ :

µw' + µ (4x + 8)/(2x + 5) w = 0

The left side is the derivative of a product:

[µw]' = 0

Integrate both sides:

∫ [µw]' dx = ∫ 0 dx

µw = C

Replace w with v', then integrate to solve for v :

exp(2x)/(2x + 5) v' = C

v' = C (2x + 5) exp(-2x)

∫ v' dx = ∫ C (2x + 5) exp(-2x) dx

v = C₁ (x + 3) exp(-2x) + C₂

Replace v with y exp(-2x) and solve for y :

y exp(-2x) = C₁ (x + 3) exp(-2x) + C₂

y = C₁ (x + 3) + C₂ exp(2x)

It follows that the second fundamental solution is y = x + 3. (The exp(2x) here is already accounted for as the first solution.)

Answer:

Step-by-step explanation:

First you have to know two definitions. Well, you only have to know one for this problem, but you should probably learn the 2nd just to be thorough.

Definition 1: Complementary angles are two angles whose sum is 90 degrees.

Definition 2: Supplementary angles are two angles whose sum is 180 degrees.

For this problem, we'll work with the definition that says two complementary angles have a sum of 90 degrees.

Soooo, here are the facts from your problem: if one angle is 15 degree more than 2 times the other.find the measure of two angles.

Let's let the larger angle equal this: 15 + 2(x) (<--See how it is 15 degrees MORE than 2 times the other?)

Let's let the smaller angle equal: x

SO now our total equation is:

15 + 2(x) + x = 90

3x + 15 = 90 (combined like terms)

3x = 75 (subtracted 15 from both sides)

x = 25 (divided both sides by 3)

Now we know that one angle is 25. The other angle must add to 25 to make 90 degrees, so 90 - 25 = 65.

Therefore, your two angles are 25 and 65 degrees.

Does this check out? Let's see...

First: 25 + 65 = 90 Therefore, this checks out.

Second: The angle that is 65 degrees must be 15 degrees more than twice the other. So, let's take twice the other...... 25 * 2 = 50. And, let's add 15....50 + 15 = 65. Therefore YES, the 2nd angle is 15 more than 2 times the angle that was 25 degrees.

I hope this is helpful. :-)

Answer:

[tex]\frac{-1}{4} x^{2}[/tex]

[tex]\frac{-1}{4} g(x)[/tex]

Step-by-step explanation:

Answer:

a. The mean is 3, the variance is 2.25 and the standard deviation is 1.5.

b. 0.0401 = 4.01% probability that the number of people who own individual stocks is exactly six.

c. 0.1584 = 15.84% probability that the number of people who say they own individual stocks is at least two.

d. 0.3907 = 39.07% probability that the number of people who say they own individual stocks is at most two

e. Both cases include one common outcome, that is, 2 people owning stocks, so the events are not mutually exclusive.

Step-by-step explanation:

For each person, there are only two possible outcomes. Either they own stocks, or they do not. The probability of a person owning stocks is independent of any other person, which means that the binomial probability distribution is used to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

One in four people in the US owns individual stocks.

This means that [tex]p = \frac{1}{4} = 0.25[/tex]

You randomly select 12 people and ask them if they own individual stocks.

This means that [tex]n = 12[/tex]

a. Find the mean, variance, and standard deviation of the resulting probability distribution.

The mean of the binomial distribution is:

[tex]E(X) = np[/tex]

So

[tex]E(X) = 12(0.25) = 3[/tex]

The variance is:

[tex]V(X) = np(1-p)[/tex]

So

[tex]V(X) = 12(0.25)(0.75) = 2.25[/tex]

Standard deviation is the square root of the variance, so:

[tex]\sqrt{V(X)} = \sqrt{2.25} = 1.5[/tex]

The mean is 3, the variance is 2.25 and the standard deviation is 1.5.

b. Find the probability that the number of people who own individual stocks is exactly six.

This is P(X = 6). So

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 6) = C_{12,6}.(0.25)^{6}.(0.75)^{6} = 0.0401[/tex]

0.0401 = 4.01% probability that the number of people who own individual stocks is exactly six.

c. Find probability that the number of people who say they own individual stocks is at least two.

This is:

[tex]P(X \geq 2) = 1 - P(X < 2)[/tex]

In which

[tex]P(X < 2) = P(X = 0) + P(X = 1)[/tex]

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 0) = C_{12,0}.(0.25)^{0}.(0.75)^{12} = 0.0317[/tex]

[tex]P(X = 1) = C_{12,1}.(0.25)^{1}.(0.75)^{11} = 0.1267[/tex]

[tex]P(X < 2) = P(X = 0) + P(X = 1) = 0.0317 + 0.1267 = 0.1584[/tex]

0.1584 = 15.84% probability that the number of people who say they own individual stocks is at least two.

d. Find the probability that the number of people who say they own individual stocks is at most two.

This is:

[tex]P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)[/tex]

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 0) = C_{12,0}.(0.25)^{0}.(0.75)^{12} = 0.0317[/tex]

[tex]P(X = 1) = C_{12,1}.(0.25)^{1}.(0.75)^{11} = 0.1267[/tex]

[tex]P(X = 2) = C_{12,2}.(0.25)^{2}.(0.75)^{10} = 0.2323[/tex]

[tex]P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.0317 + 0.1267 + 0.2323 = 0.3907[/tex]

0.3907 = 39.07% probability that the number of people who say they own individual stocks is at most two.

e. Are the events in part c. and in part d. mutually exclusive

Both cases include one common outcome, that is, 2 people owning stocks, so the events are not mutually exclusive.

Answer:

[tex]x=6\\y=4[/tex]

Step-by-step explanation:

Elimination method:

[tex]x+5y=26[/tex]

[tex]-x+7y=22[/tex]

Add these equations to eliminate x:

[tex]12y=48[/tex]

Then solve [tex]12y=48[/tex] for y:

[tex]12y=48[/tex]

[tex]y=48/12[/tex]

[tex]y=4[/tex]

Write down an original equation:

[tex]x+5y=26[/tex]

Substitute 4 for y in [tex]x+5y=26[/tex]:

[tex]x+5(4)=26[/tex]

[tex]x+20=26[/tex]

[tex]x=26-20[/tex]

[tex]x=6[/tex]

{ [tex]x=6[/tex] and [tex]y=4[/tex] }    ⇒ [tex](6,4)[/tex]

hope this helps...

Answer:

x = 6, y = 4

Step-by-step explanation:

x + 5y = 26

- x + 7y = 22

_________

0 + 12y = 48

12y = 48

y = 48 / 12

y = 4

Substitute y = 4 in eq. x + 5y = 26,

x + 5 ( 4 ) = 26

x + 20 = 26

x = 26 - 20

x = 6

Hello!

[tex]\large\boxed{(x + 2)(x + 3)}[/tex]

x² + 5x + 6

Find two numbers that add up to 5 and multiply to 6. We get:

2, 3

Therefore:

(x + 2)(x + 3)

Answer:

Step-by-step explanation:

                    Statements                                        Reasons

1). ΔABC with side lengths a, b, c, and h      1). Given

2). Area = [tex]\frac{1}{2}bh[/tex]                                                 2). Triangle area formula

3). [tex]\text{sin}C=\frac{h}{a}[/tex]                                                    3). Definition of sine

4). asin(C) = h                                                4). Multiplication property of

                                                                          equality.

5). Area = [tex]\frac{1}{2}ba\text{sin}C[/tex]                                         5). Substitution property

6). Area = [tex]\frac{1}{2}ab\text{sin}C[/tex]                                         6). Commutative property of

                                                                           multiplication.

Hence, proved.

Answer:

0.0409 = 4.09% probability that the mean height of the 35 NBA players will be more than 80 inches.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

The average height of a current NBA player is 79 inches with a standard deviation of 3.4 inches.

This means that [tex]\mu = 79, \sigma = 3.4[/tex]

A random sample of 35 current NBA players is taken.

This means that [tex]n = 35, s = \frac{3.4}{\sqrt{35}}[/tex]

What is the probability that the mean height of the 35 NBA players will be more than 80 inches?

This is 1 subtracted by the p-value of Z when X = 80. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{80 - 79}{\frac{3.4}{\sqrt{35}}}[/tex]

[tex]Z = 1.74[/tex]

[tex]Z = 1.74[/tex] has a p-value of 0.9591

1 - 0.9591 = 0.0409

0.0409 = 4.09% probability that the mean height of the 35 NBA players will be more than 80 inches.

9514 1404 393

Answer:

Step-by-step explanation:

I find this approach the most straightforward of the various ways that trinomial factoring is explained or diagramed.

You want two factors of "ac" that have a total of "b". Here, that means you want factors of 2·50 = 100 that have a total of 25. It is helpful to know your times tables.

  100 = 1·100 = 2·50 = 4·25 = 5·20 = 10·10

The sums of these factor pairs are 101, 52, 29, 25, and 20. We want the pair with a sum of 25, so that's 5 and 20.

The trinomial can be rewritten using these factors as ...

  2x^2 +5x +20x +50

Then it can be factored by grouping consecutive pairs:

  (2x^2 +5x) +(20x +50) = x(2x +5) +10(2x +5) = (x +10)(2x +5)

_____

Additional comment

It doesn't matter which of the factors of the pair you write first. If our rewrite were ...

  2x^2 +20x +5x +50

Then the grouping and factoring would be (2x^2 +20x) +(5x +50)

  = 2x(x +10) +5(x +10) = (2x +5)(x +10) . . . . . same factoring

Answer:

y=2x-4

Step-by-step explanation:

If you are asking for point slope form, that would be it

Answer:

108801

Step-by-step explanation:

Palindrome as defined in the given question as a number which reads the same forward and backward. Examples are: 1001, 20202, 1331 etc.

Thus, to determine the smallest 6-digit palindrome divisible by 99 without a remainder, the digits should be in the form of abccba.

Therefore, the smallest 6-digit palindrome that can be divided by 99 is 108801.

So that,

108801 ÷ 99 = 1099

Answer:

p and q are two numbers.whrite down an expression of

Answer:

Who was the first president of United States?

9514 1404 393

Answer:

  sin(D) = cos(E) = (√3)/2

  cos(D) = sin(E) = 1/2

Step-by-step explanation:

The mnemonic SOH CAH TOA is intended to remind you of the relationships between trig functions and right triangle sides.

  Sin = Opposite/Hypotenuse

  Cos = Adjacent/Hypotenuse

For this diagram, this means ...

  sin(D) = cos(E) = (13√3)/26 = (√3)/2

  cos(D) = sin(E) = 13/26 = 1/2

Answer/Step-by-step explanation:

Given:

f(x) = 2x + 2

Domain = {-5, -1, 2, 3}

To write the range of f using set notation, substitute each domain value into f(x) = 2x + 2 to get each corresponding range value that will make up the set.

Thus:

✔️f(-5) = 2(-5) + 2

= -10 + 2

f(-5) = -8

✔️f(-1) = 2(-1) + 2

= -2 + 2

f(-1) = 0

✔️f(2) = 2(2) + 2

= 4 + 2

f(-1) = 6

✔️f(3) = 2(3) + 2

= 6 + 2

f(3) = 8

Range of f using set notation = {-8, 0, 6, 8}

✔️Graph f by plotting the domain values on the x-axis against the corresponding range values on the y-axis as shown in the attachment below:

*See attachment for the graph of f

Answer: [tex]t\in [\dfrac{1}{4},2][/tex]

Step-by-step explanation:

Given

Inequality is [tex]4t^2\leq9t-2[/tex]

Taking variables one side

[tex]\Rightarrow 4t^2-9t+2\leq0\\\Rightarrow 4t^2-8t-t+2\leq0\\\Rightarrow 4t(t-2)-1(t-2)\leq0\\\Rightarrow (4t-1)(t-2)\leq0[/tex]

Using wavy curve method

[tex]t\in [\dfrac{1}{4},2][/tex]

Answer:

Terms:

Like Terms:

Coefficients:

Constants:

You simplify the expression by combining like terms:

-5x + 4 - x - 1 = -6x + 5

(7b - 4) + (-2b + a + 1) = 7b - 4 - 2b + a + 1 = 5b + a - 3

Answer:

The postage per ounce would be of $2.02.

Step-by-step explanation:

Exponential model:

The postage, in t years after 1963, follows the following format:

[tex]P(t) = P(0)(1+r)^t[/tex]

In which P(0) is the initial value and r is the growth rate, as a decimal.

In 1963, postage was 5 cents per ounce.

This means that [tex]P(0) = 5[/tex]

So

[tex]P(t) = P(0)(1+r)^t[/tex]

[tex]P(t) = 5(1+r)^t[/tex]

In 1981, postage was 18 cents per ounce.

This means that [tex]P(1981 - 1963) = P(18) = 18[/tex]. We use this to find r. So

[tex]P(t) = 5(1+r)^t[/tex]

[tex]18 = 5(1+r)^{18}[/tex]

[tex](1+r)^{18} = \frac{18}{5}[/tex]

[tex]\sqrt[18]{(1+r)^{18}} = \sqrt[18]{3.6}[/tex]

[tex]1 + r = (3.6)^{\frac{1}{18}}[/tex]

[tex]1 + r = 1.0738[/tex]

So

[tex]P(t) = 5(1.0738)^t[/tex]

If the trend had continued through to 2015, what would the postage per ounce be?

2015 - 1963 = 52, so this is P(52).

[tex]P(52) = 5(1.0738)^{52} = 202[/tex]

202 cents, so $2.02.

Answer:

A. $7.00

Step-by-step explanation:

$82-$75=$7.00

Answer:

[tex]P(x>1)=0.9927[/tex]

Step-by-step explanation:

From the question we are told that:

Mean [tex]\=x =7[/tex]

Generally the Poisson equation for \=x is mathematically given by

[tex]P(x>1)=1-P(x \leq 1)[/tex]

Therefor

[tex]P(x>1)=1-(\frac{e^{-7}*7^0}{0!}+{\frac{e^{-7}*7^1}{1!})[/tex]

[tex]P(x>1)=1-(9.1*10^{-4}+6.3*10^{-3})[/tex]

[tex]P(x>1)=1-(7.3*10^{-3}[/tex]

[tex]P(x>1)=0.9927[/tex]

Answer:

Step-by-step explanation:

The difference of years:

The difference in profit over 6 years:

Average rate of change:

It has been 6 years,

The main difference in profit over 6 years between 1999 and 2005 is,

→ 206000 - 173000

→ 33000

Then the average rate of change is,

→ 33000/6

→ 5500

Hence, $ 5500 is the correct option.

Answer:

The length is of 59 cm.

Step-by-step explanation:

Perimeter of a rectangle:

The perimeter of a rectangle with width w and length l is given by:

[tex]P = 2(w + l)[/tex]

Width of 49 centimeters and a perimeter of 216 centimeters:

This means that [tex]w = 49, P = 216[/tex]

The length is cm.

We have to solve the equation for l. So

[tex]P = 2(w + l)[/tex]

[tex]216 = 2(49 + l)[/tex]

[tex]216 = 98 + 2l[/tex]

[tex]2l = 118[/tex]

[tex]l = \frac{118}{2}[/tex]

[tex]l = 59[/tex]

The length is of 59 cm.

Answer:

so the measure of both angle is 24°

Step-by-step explanation:

original angle = 48°

so since its bisected the both angles are equal and let the angles be x

so,

x + x = 48°

2x = 48°

x = 48°/2

so, x = 24°

Answer:

hello there here are your answers:

1) a- 12, 18, 24, 30, 36

2) b- 31

3) a-communitive property of addition

4) a- 6a

Step-by-step explanation:

1: go through all the numbers and add 6 like 12+6=16 etc.

2: the common difference is 4 so 27+4 =31

3: communitive property because you can change the number in any order and still get the same sum

4: 6a because only 24ab has a b in it

Answer:

A. c = -7

Step-by-step explanation:

Standard form of a quadratic equation is given as ax² + bx + c = 0, where,

a, b, and c are known values not equal to 0,

x is the variable.

Given a quadratic equation of -x² + 4x - 7, therefore,

a = -1

b = 4

c = -7

Answer:

0.007 = 0.7% probability that the proportion of Rolls Royce owners in a sample of 595 Americans would differ from the population proportion by more than 3%

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]

A statistician calculates that 8% of Americans own a Rolls Royce.

This means that [tex]p = 0.08[/tex]

Sample of 595:

This means that [tex]n = 595[/tex]

Mean and standard deviation:

[tex]\mu = p = 0.08[/tex]

[tex]s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.08*0.92}{595}} = 0.0111[/tex]

What is the probability that the proportion of Rolls Royce owners in a sample of 595 Americans would differ from the population proportion by more than 3%?

Proportion above 8% + 3% = 11% or below 8% - 3% = 5%. Since the normal distribution is symmetric, these probabilities are equal, and so we find one of them and multiply by 2.

Probability the proportion is less than 5%:

P-value of Z when X = 0.05. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{0.05 - 0.08}{0.0111}[/tex]

[tex]Z = -2.7[/tex]

[tex]Z = -2.7[/tex] has a p-value of 0.0035

2*0.0035 = 0.0070

0.007 = 0.7% probability that the proportion of Rolls Royce owners in a sample of 595 Americans would differ from the population proportion by more than 3%