En el triángulo rectángulo ABC, recto en “B”, SI Tg C = 3/5, hallar el valor “cosec A”

Answer:

(a) The value of E (X) is 4/7.

    The value of V (X) is 3/98.

(b) The value of P (X ≤ 0.5) is 0.3438.

Step-by-step explanation:

The random variable X is defined as the proportion of surface area in a randomly selected quadrant that is covered by a certain plant.

The random variable X follows a standard beta distribution with parameters α = 4 and β = 3.

The probability density function of X is as follows:

[tex]f(x) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)} ; \hspace{.3in}0 \le x \le 1;\ \alpha, \beta > 0[/tex]

Here, B (α, β) is:

[tex]B(\alpha,\beta)=\frac{(\alpha-1)!\cdot\ (\beta-1)!}{((\alpha+\beta)-1)!}[/tex]

            [tex]=\frac{(4-1)!\cdot\ (3-1)!}{((4+3)-1)!}\\\\=\frac{6\times 2}{720}\\\\=\frac{1}{60}[/tex]

So, the pdf of X is:

[tex]f(x) = \frac{x^{4-1}(1-x)^{3-1}}{1/60}=60\cdot\ [x^{3}(1-x)^{2}];\ 0\leq x\leq 1[/tex]

(a)

Compute the value of E (X) as follows:

[tex]E (X)=\frac{\alpha }{\alpha +\beta }[/tex]

         [tex]=\frac{4}{4+3}\\\\=\frac{4}{7}[/tex]

The value of E (X) is 4/7.

Compute the value of V (X) as follows:

[tex]V (X)=\frac{\alpha\ \cdot\ \beta}{(\alpha+\beta)^{2}\ \cdot\ (\alpha+\beta+1)}[/tex]

         [tex]=\frac{4\cdot\ 3}{(4+3)^{2}\cdot\ (4+3+1)}\\\\=\frac{12}{49\times 8}\\\\=\frac{3}{98}[/tex]

The value of V (X) is 3/98.

(b)

Compute the value of P (X ≤ 0.5) as follows:

[tex]P(X\leq 0.50) = \int\limits^{0.50}_{0}{60\cdot\ [x^{3}(1-x)^{2}]} \, dx[/tex]

                    [tex]=60\int\limits^{0.50}_{0}{[x^{3}(1+x^{2}-2x)]} \, dx \\\\=60\int\limits^{0.50}_{0}{[x^{3}+x^{5}-2x^{4}]} \, dx \\\\=60\times [\dfrac{x^4}{4}+\dfrac{x^6}{6}-\dfrac{2x^5}{5}]\limits^{0.50}_{0}\\\\=60\times [\dfrac{x^4\left(10x^2-24x+15\right)}{60}]\limits^{0.50}_{0}\\\\=[x^4\left(10x^2-24x+15\right)]\limits^{0.50}_{0}\\\\=0.34375\\\\\approx 0.3438[/tex]

Thus, the value of P (X ≤ 0.5) is 0.3438.

Answer:

probability of the event E = 1/5

Step-by-step explanation:

We are given;

Sample space, S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10},

Number of terms in sample S is;

n(S) = 10

We are given the event; E = {1, 2}

Thus, number of terms in event E is;

n(E) = 2

Now, Probability = favorable outcomes/total outcomes

Thus, the probability of the event E is;

P(E) = n(E)/n(S)

P(E) = 2/10

P(E) = 1/5

Answer:

[tex]-\frac{161}{9}=\\or\\-16\frac{8}{9}[/tex]

Step-by-step explanation:

[tex]-6\frac{4}{9}-3\frac{2}{9}-8\frac{2}{9}=\\\\-\frac{58}{9}-\frac{29}{9}-\frac{74}{9}=\\\\-\frac{161}{9}=\\\\-16\frac{8}{9}[/tex]

Answer:

The full name of the place is the "Danat Jebel Dhanna".  The Jebel Dhanna is currently located in the Abu Dhabi.  It is said that it is one of the most best beach in the UAE, they also say that it is the biggest resort, of course, with a bunch of hotels.

hope this helps ;)

best regards,

`FL°°F~` (floof)

Answer:

there will be 9 id no. which it contains

Answer:

X is 3 units.

Step-by-step explanation:

Volume of prism is cross sectional area multiplied by length. So 1/2 ×2× x ×2 into 3x, which is equal to 6x^2. So, 6x^2=54. Therefore, x=3.

Answer:

Step-by-step explanation:

Given that;

the following procedure for accomplishing our task are:

1. Flip the coin.

2. Flip the coin again.

From here will know that the coin is first flipped twice

3. If both flips land on heads or both land on tails, it implies that we return to step 1 to start again. this makes the flip to be insignificant since both flips land on heads or both land on tails

But if the outcomes of the two flip are different i.e they did not land on both heads or both did not land on tails , then we will consider such an outcome.

Let the probability of head = p

so P(head) = p

the probability of tail be = (1 - p)

This kind of probability follows a conditional distribution and the probability  of getting heads is :

[tex]P( \{Tails, Heads\})|\{Tails, Heads,( Heads ,Tails)\})[/tex]

[tex]= \dfrac{P( \{Tails, Heads\}) \cap \{Tails, Heads,( Heads ,Tails)\})}{ {P( \{Tails, Heads,( Heads ,Tails)\}}}[/tex]

[tex]= \dfrac{P( \{Tails, Heads\}) }{ {P( \{Tails, Heads,( Heads ,Tails)\}}}[/tex]

[tex]= \dfrac{P( \{Tails, Heads\}) } { {P( Tails, Heads) +P( Heads ,Tails)}}[/tex]

[tex]=\dfrac{(1-p)*p}{(1-p)*p+p*(1-p)}[/tex]

[tex]=\dfrac{(1-p)*p}{2(1-p)*p}[/tex]

[tex]=\dfrac{1}{2}[/tex]

Thus; the probability of getting heads is [tex]\dfrac{1}{2}[/tex] which typically implies that the coin is fair

(b) Could we use a simpler procedure that continues to flip the coin until the last two flips are different and then lets the result be the outcome of the final flip?

For a fair coin (0<p<1) , it's certain that both heads and tails at the end of the flip.

The procedure that is talked about in (b) illustrates that the procedure gives head if and only if the first flip comes out tail with probability 1 - p.

Likewise , the procedure gives tail if and and only if the first flip comes out head with probability of  p.

In essence, NO, procedure (b) does not give a fair coin flip outcome.

Answer:

y = 2x+4

Step-by-step explanation:

First we need to find the slope using two points

(-2,0) and (0,4)

m = (y2-y1)/(x2-x1)

m = (4-0)/(0--2)

   = 4/+2

   = 2

we have the y intercept  which is 4

Using the slope intercept form of the line

y = mx+b where m is the slope and b is the y intercept

y = 2x+4

Answer:

Length: 18

Width: 9

Step-by-step explanation:

Denote the width as x, hence the length is x+9. As a result, you can create the equation x(x+9) = 162. Solving, you find x = 9.

Answer:

[tex]solution \\ 3(5x) + 8(2x) \\ = 3 \times 5x + 8 \times 2x \\ = 15x + 16x \\ = 31x[/tex]

hope this helps...

Good luck on your assignment...

The expression  [tex]3(5x) + 8(2x)[/tex] in simplest form is 31x.

To simplify the expression [tex]3(5x) + 8(2x)[/tex], we can apply the distributive property:

[tex]3(5x) + 8(2x)[/tex]

[tex]= 15x + 16x[/tex]

Combining like terms, we have:

[tex]15x + 16x = 31x[/tex]

Therefore, the expression [tex]3(5x) + 8(2x)[/tex] simplifies to [tex]31x.[/tex]

To learn more on Expressions click:

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Answer:

B

Step-by-step explanation:

Answer:

Approximately 34% of people sleep between 6 and 7.5 hours per night

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 6

Standard deviation = 1.5

The normal distribution is symmetric, which means that 50% of the measures are below the mean and 50% are above.

What percentage of people sleep between 6 and 7.5 hours per night

6 is the mean.

7.5 is 1 one standard deviation above the mean.

By the Empirical Rule, of the 50% of the measures that are above the mean, 68% are within 1 standard deviation of the mean(between 6 and 7.5).

0.5*0.68 = 0.34

Approximately 34% of people sleep between 6 and 7.5 hours per night

Answer:

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

And using the probability mass function we can find the individual probabilities:

[tex]P(X=0)=(8C0)(0.18)^0 (1-0.18)^{8-0}=0.2044[/tex]

[tex]P(X=1)=(8C1)(0.18)^1 (1-0.18)^{0-1}=0.3590[/tex]

And replacing we got:

[tex] P(X\geq 2)=1 -[0.2044 +0.3590]= 0.4366[/tex]

Then the probability that at least 2 disapprove of daily pot smoking is 0.4366

Step-by-step explanation:

Let X the random variable of interest "number of seniors who disapprove of daily smoking ", on this case we now that:

[tex]X \sim Binom(n=8, p=0.18)[/tex]

The probability mass function for the Binomial distribution is given as:

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

Where (nCx) means combinatory and it's given by this formula:

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

And we want to find this probability:

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

And using the probability mass function we can find the individual probabilities:

[tex]P(X=0)=(8C0)(0.18)^0 (1-0.18)^{8-0}=0.2044[/tex]

[tex]P(X=1)=(8C1)(0.18)^1 (1-0.18)^{0-1}=0.3590[/tex]

And replacing we got:

[tex] P(X\geq 2)=1 -[0.2044 +0.3590]= 0.4366[/tex]

Then the probability that at least 2 disapprove of daily pot smoking is 0.4366

Answer:

  7

Step-by-step explanation:

The number of cells in a tile is 4. If colored alternately, there are 3 of one color and 1 of the alternate color. To balance the coloring, an even number of tiles is needed. Hence the board dimensions must be multiples of 4.

In the given range, there are 7 such boards:

  4×4, 8×8, 12×12, 16×16, 20×20, 24×24, and 28×28

Answer:

  The line of g(x) is steeper and has a lower y-intercept.

Step-by-step explanation:

Doubling the function expands it vertically by a factor of 2. Everything is twice as far from the x-axis as it was. The line becomes steeper, and the y-intercept moves twice as far away. It was at -1, now is lower, at -2.

The line of g(x) is steeper and has a lower y-intercept.

Answer:

the length of his shadow on the building is decreasing at the rate of 0.525 m/s

Step-by-step explanation:

From the diagram attached below;

the man is standing at point D with his head at point E

During that time, his shadow on the wall is y = BC

ΔABC and Δ ADE are similar in nature; thus their corresponding sides have equal ratios; i.e

[tex]\dfrac{AD}{AB} = \dfrac{DE}{BC}[/tex]

[tex]\dfrac{8}{12} = \dfrac{2}{y}[/tex]

8y = 24

y = 24/8

y = 3 meters

Let take an integral look  at the distance of the man from the building as x, therefore the distance from the spotlight to the man is  12 - x

∴

[tex]\dfrac{12-x}{12}=\dfrac{2}{y}[/tex]

[tex]1- \dfrac{1}{12}x = 2* \dfrac{1}{y}[/tex]

To find the derivatives of both sides ;we have:

[tex]- \dfrac{1}{12}dx = 2* \dfrac{1}{y^2}dy[/tex]

[tex]- \dfrac{1}{12} \dfrac{dx}{dt} = 2* \dfrac{1}{y^2} \dfrac{dy}{dt}[/tex]

During that time ;

[tex]\dfrac{dx}{dt }= 1.4 \ m/s[/tex]   and y = 3

So; replacing the value into above ; we have:

[tex]-\dfrac{1}{12}(1.4) = - \dfrac{2}{9} \dfrac{dy}{dt}[/tex]

[tex]\dfrac{dy}{dt} = \dfrac{\dfrac{ 1.4} {12 } }{ \dfrac{2}{9}}[/tex]

[tex]\dfrac{dy}{dt} = {\dfrac{ 1.4} {12 } }*{ \dfrac{9}{2}}[/tex]

[tex]\dfrac{dy}{dt} =0.525 \ m/s[/tex]

Thus; the length of his shadow on the building is decreasing at the rate of 0.525 m/s

Step-by-step explanation:

4m2 - 24m + 8m - 32

4m2 - 16m - 32

Answer:

The 88% confidence interval for the proportion of defectives today is (0.053, 0.123)

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 zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].

For this problem, we have that:

[tex]n = 160, \pi = \frac{14}{160} = 0.088[/tex]

88% confidence level

So [tex]\alpha = 0.12[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.12}{2} = 0.94[/tex], so [tex]Z = 1.555[/tex].

The lower limit of this interval is:

[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.088 - 1.555\sqrt{\frac{0.088*0.912}{160}} = 0.053[/tex]

The upper limit of this interval is:

[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.088 + 1.555\sqrt{\frac{0.088*0.912}{160}} = 0.123[/tex]

The 88% confidence interval for the proportion of defectives today is (0.053, 0.123)

Answer:

Step-by-step explanation:

The question is incomplete. The complete question is:

It is known that the number of hours a student sleeps per night has a normal distribution. The sleeping time in hours of a random sample of 8 students is given below. 7.4, 6.2, 8.5, 6.3, 5.4, 5.5, 6.3, 8.3 Compute a 98% confidence interval for the true mean time a student sleeps per night and fill in the blanks appropriately. We have 98% confidence that the true mean time a student sleeps per night is between _____ and ____ hours. (Keep 3 decimal places)

Solution:

Mean = (7.4 + 6.2 + 8.5 + 6.3 + 5.4 + 5.5 + 6.3 + 8.3)/8 = 6.7375

Standard deviation = √(summation(x - mean)²/n

Summation(x - mean)² = (7.4 - 6.7375)^2 + (6.2 - 6.7375)^2 + (8.5 - 6.7375)^2 + (6.3 - 6.7375)^2 + (5.4 - 6.7375)^2 + (5.5 - 6.7375)^2 + (6.3 - 6.7375)^2 + (8.3 - 6.7375)^2 = 9.97875

Standard deviation = √(9.97875/8

s = 1.12

Confidence interval is written in the form,

(Sample mean - margin of error, sample mean + margin of error)

The sample mean, x is the point estimate for the population mean.

Margin of error = z × s/√n

Where

sample standard deviation

number of samples

From the information given, the population standard deviation is unknown and the sample size is small, hence, we would use the t distribution to find the z score

In order to use the t distribution, we would determine the degree of freedom, df for the sample.

df = n - 1 = 8 - 1 = 7

Since confidence level = 98% = 0.98, α = 1 - CL = 1 - 0.98 = 0.02

α/2 = 0.02/2 = 0.01

the area to the right of z0.01 is 0.01 and the area to the left of z0.01 is 1 - 0.01 = 0.99

Looking at the t distribution table,

z = 2.998

Margin of error = 2.998 × 1.12/√8

= 1.19

the lower limit of this confidence interval is

6.738 - 1.19 = 5.548

the upper limit of this confidence interval is

6.738 + 1.19 = 7.928

We have 98 % confidence that the true mean time a student sleeps per night is between 5.548 hours and 7.928 hours.

Answer:

Luther is wrong

Wade is right

Step-by-step explanation:

Luther's case 2^3 = 2x2x2 = 8

Wade's case 3^3 = 3 x 3 = 9

Answer:

Luther is incorrect, while Wade is correct. (2)(2)(2)=8, not 9. (3)(3)= 9.

Step-by-step explanation:

I put that as my answer and it was counted as right.

Answer:

m = 3/2, y intercept = 3

Step-by-step explanation:

The y intercept is where it crosses the y axis.  It crosses at 3

The slope is  found by using two points on the line

(-2,0) and (0,3)

m= (y2-y1)/(x2-x1)

   = (3-0)/(0- -2)

  = 3 / +2

  =3/2

Answer:

[tex]P(11.5<X<15.5)=P(\frac{11.5-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{13.5-\mu}{\sigma})=P(\frac{11.5-13.5}{2}<Z<\frac{15.5-13.5}{2})=P(-1<z<1)[/tex]

And we can find the probability with this difference

[tex]P(-1<z<1)=P(z<1)-P(z<-1)[/tex]

And we can use the normal standard distribution or excel and we got:

[tex]P(-1<z<1)=P(z<1)-P(z<-1)=0.841-0.159=0.682[/tex]

So then we expect a proportion of 0.682 between 11.5 and 13.5

Step-by-step explanation:

Let X the random variable that represent the price earning ratios of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(13.5,2)[/tex]  

Where [tex]\mu=13.5[/tex] and [tex]\sigma=2[/tex]

We want to find the following probability

[tex]P(11.5<X<15.5)[/tex]

And we can use the z score formula given by:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

Using this formula we got:

[tex]P(11.5<X<15.5)=P(\frac{11.5-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{13.5-\mu}{\sigma})=P(\frac{11.5-13.5}{2}<Z<\frac{15.5-13.5}{2})=P(-1<z<1)[/tex]

And we can find the probability with this difference

[tex]P(-1<z<1)=P(z<1)-P(z<-1)[/tex]

And we can use the normal standard distribution or excel and we got:

[tex]P(-1<z<1)=P(z<1)-P(z<-1)=0.841-0.159=0.682[/tex]

So then we expect a proportion of 0.682 between 11.5 and 13.5

Answer:

x = 6√2

Step-by-step explanation:

It is a 45°45°90° triangle so you can use the ratio.

x : x√2

x = 6√2

Answer:

The amount of money separating the lowest 80% of the amount invested from the highest 20% in a sampling distribution of 10 of the family's real estate holdings is $238,281.57.

Step-by-step explanation:

Let the random variable X represent the amount of money that the family has invested in different real estate properties.

The random variable X follows a Normal distribution with parameters μ = $225,000 and σ = $50,000.

It is provided that the family has invested in n = 10 different real estate properties.

Then the mean and standard deviation of amount of money that the family has invested in these 10 different real estate properties is:

[tex]\mu_{\bar x}=\mu=\$225,000\\\\\sigma_{\bar x}=\frac{\sigma}{\sqrt{n}}=\frac{50000}{\sqrt{10}}=15811.39[/tex]

Now the lowest 80% of the amount invested can be represented as follows:

[tex]P(\bar X<\bar x)=0.80\\\\\Rightarrow P(Z<z)=0.80[/tex]

The value of z is 0.84.

*Use a z-table.

Compute the value of the mean amount invested as follows:

[tex]\bar x=\mu_{\bar x}+z\cdot \sigma_{\bar x}[/tex]

   [tex]=225000+(0.84\times 15811.39)\\\\=225000+13281.5676\\\\=238281.5676\\\\\approx 238281.57[/tex]

Thus, the amount of money separating the lowest 80% of the amount invested from the highest 20% in a sampling distribution of 10 of the family's real estate holdings is $238,281.57.

Answer:

Its D The Last Graph

Step-by-step explanation:

it just is my guy

Answer:145

Step-by-step explanation: $19=20 76=80 53=50 23=20 12=10 total = 180 325-180 =145

Answer:

a) P(x > 43) = 0.9599

b) P(x < 42) = 0.0228

c) P(x > 57.5) = 0.03

d) P(x = 42) = 0.

e) P(x<40 or x>55) = 0.1118

f) 43.42

g) Between 46.64 and 53.36.

h) Above 45.852.

Step-by-step explanation:

When the distribution is normal, we use the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore 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 pvalue, we get the probability that the value of the measure is greater than X.

In this question, we have that:

[tex]\mu = 50, \sigma = 4[/tex]

a) x>43

This is 1 subtracted by the pvalue of Z when X = 43. So

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

[tex]Z = \frac{43 - 50}{4}[/tex]

[tex]Z = -1.75[/tex]

[tex]Z = -1.75[/tex] has a pvalue of 0.0401

1 - 0.0401 = 0.9599

P(x > 43) = 0.9599

b) x<42

This is the pvalue of Z when X = 42.

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

[tex]Z = \frac{42 - 50}{4}[/tex]

[tex]Z = -2[/tex]

[tex]Z = -2[/tex] has a pvalue of 0.0228

P(x < 42) = 0.0228

c) x>57.5

This is 1 subtracted by the pvalue of Z when X = 57.5. So

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

[tex]Z = \frac{57.5 - 50}{4}[/tex]

[tex]Z = 1.88[/tex]

[tex]Z = 1.88[/tex] has a pvalue of 0.97

1 - 0.97 = 0.03

P(x > 57.5) = 0.03

d) P(x = 42)

In the normal distribution, the probability of an exact value is 0. So

P(x = 42) = 0.

e) x<40 or x>55

x < 40 is the pvalue of Z when X = 40. So

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

[tex]Z = \frac{40 - 50}{4}[/tex]

[tex]Z = -2.5[/tex]

[tex]Z = -2.5[/tex] has a pvalue of 0.0062

x > 55 is 1 subtracted by the pvalue of Z when X = 55. So

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

[tex]Z = \frac{55 - 50}{4}[/tex]

[tex]Z = 1.25[/tex]

[tex]Z = 1.25[/tex] has a pvalue of 0.8944

1 - 0.8944 = 0.1056

0.0062 + 0.1056 = 0.1118

P(x<40 or x>55) = 0.1118

f) 5% of the values are less than what X value?

X is the 5th percentile, which is X when Z has a pvalue of 0.05, so X when Z = -1.645.

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

[tex]-1.645 = \frac{X - 50}{4}[/tex]

[tex]X - 50 = -1.645*4[/tex]

[tex]X = 43.42[/tex]

43.42 is the answer.

g) 60% of the values are between what two X values (symmetrically distributed around the mean)?

Between the 50 - (60/2) = 20th percentile and the 50 + (60/2) = 80th percentile.

20th percentile:

X when Z has a pvalue of 0.2. So X when Z = -0.84.

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

[tex]-0.84 = \frac{X - 50}{4}[/tex]

[tex]X - 50 = -0.84*4[/tex]

[tex]X = 46.64[/tex]

80th percentile:

X when Z has a pvalue of 0.8. So X when Z = 0.84.

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

[tex]0.84 = \frac{X - 50}{4}[/tex]

[tex]X - 50 = 0.84*4[/tex]

[tex]X = 53.36[/tex]

Between 46.64 and 53.36.

h) 85% of the values will be above what X value?

Above the 100 - 85 = 15th percentile, which is X when Z has a pvalue of 0.15. So X when Z = -1.037.

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

[tex]-1.037 = \frac{X - 50}{4}[/tex]

[tex]X - 50 = -1.037*4[/tex]

[tex]X = 45.852[/tex]

Above 45.852.

Answer: (a) Interval where f is increasing: (0.78,+∞);

Interval where f is decreasing: (0,0.78);

(b) Local minimum: (0.78, - 0.09)

(c) Inflection point: (0.56,-0.06)

Interval concave up: (0.56,+∞)

Interval concave down: (0,0.56)

Step-by-step explanation:

(a) To determine the interval where function f is increasing or decreasing, first derive the function:

f'(x) = [tex]\frac{d}{dx}[/tex][[tex]x^{4}ln(x)[/tex]]

Using the product rule of derivative, which is: [u(x).v(x)]' = u'(x)v(x) + u(x).v'(x),

you have:

f'(x) = [tex]4x^{3}ln(x) + x_{4}.\frac{1}{x}[/tex]

f'(x) = [tex]4x^{3}ln(x) + x^{3}[/tex]

f'(x) = [tex]x^{3}[4ln(x) + 1][/tex]

Now, find the critical points: f'(x) = 0

[tex]x^{3}[4ln(x) + 1][/tex] = 0

[tex]x^{3} = 0[/tex]

x = 0

and

[tex]4ln(x) + 1 = 0[/tex]

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

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

x = 0.78

To determine the interval where f(x) is positive (increasing) or negative (decreasing), evaluate the function at each interval:

interval                 x-value                      f'(x)                       result

0<x<0.78                 0.5                 f'(0.5) = -0.22            decreasing

x>0.78                       1                         f'(1) = 1                  increasing

With the table, it can be concluded that in the interval (0,0.78) the function is decreasing while in the interval (0.78, +∞), f is increasing.

Note: As it is a natural logarithm function, there are no negative x-values.

(b) A extremum point (maximum or minimum) is found where f is defined and f' changes signs. In this case:

Then, x=0.78 is a point of minimum and its y-value is:

f(x) = [tex]x^{4}ln(x)[/tex]

f(0.78) = [tex]0.78^{4}ln(0.78)[/tex]

f(0.78) = - 0.092

The point of minimum is (0.78, - 0.092)

(c) To determine the inflection point (IP), calculate the second derivative of the function and solve for x:

f"(x) = [tex]\frac{d^{2}}{dx^{2}}[/tex] [[tex]x^{3}[4ln(x) + 1][/tex]]

f"(x) = [tex]3x^{2}[4ln(x) + 1] + 4x^{2}[/tex]

f"(x) = [tex]x^{2}[12ln(x) + 7][/tex]

[tex]x^{2}[12ln(x) + 7][/tex] = 0

[tex]x^{2} = 0\\x = 0[/tex]

and

[tex]12ln(x) + 7 = 0\\ln(x) = \frac{-7}{12} \\x = e^{\frac{-7}{12} }\\x = 0.56[/tex]

Substituing x in the function:

f(x) = [tex]x^{4}ln(x)[/tex]

f(0.56) = [tex]0.56^{4} ln(0.56)[/tex]

f(0.56) = - 0.06

The inflection point will be: (0.56, - 0.06)

In a function, the concave is down when f"(x) < 0 and up when f"(x) > 0, adn knowing that the critical points for that derivative are 0 and 0.56:

f"(x) =  [tex]x^{2}[12ln(x) + 7][/tex]

f"(0.1) = [tex]0.1^{2}[12ln(0.1)+7][/tex]

f"(0.1) = - 0.21, i.e. Concave is DOWN.

f"(0.7) = [tex]0.7^{2}[12ln(0.7)+7][/tex]

f"(0.7) = + 1.33, i.e. Concave is UP.

Answer:

  5.03

Step-by-step explanation:

Answer:

5.03 = AC

Step-by-step explanation:

Since this is a right triangle, we can use trig functions

tan theta = opp/ adj

tan 40 = AC /6

6 tan 40 = AC

5.034597787 = AC

To the nearest hundredth

5.03 = AC