Determine the parametric equations of the position of a particle with constant velocity that follows a straight line path on the plane if it starts at the point P(7,2) and after one second it is at the point Q(2,7).

Cal's expected weight after 10 weeks can be calculated using the exponential decay formula, with his initial weight being 244 lbs, the decay rate being 1% (or 0.01), and time being 10 weeks. The formula then becomes W = 244*(1 - 0.01)^10.

Cal O'Ree's weight loss over the span of 10 weeks can be calculated using the principles of exponential decay. In this context, The doctor's statement implies that Cal's weight decreases by 1% each week - this is the decay rate. The solution to this kind of problem lies in the formula for exponential decay: W = P*(1 - r)^t, where P is the initial quantity (in this case, weight), r is the decay rate, and t is the time. For Cal, P = 244 pounds, r = 0.01, and t = 10 weeks.

After substituting these values into the formula, we get: W = 244*(1 - 0.01)^10. Calculating the expression gives us the weight that Cal is expected to have after 10 weeks of working out following the doctor's prognosis.

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The time needed to run the second half of the race is 8.44 s

Step-by-step explanation:

The total distance covered by the athlete during the race is

d = 200 m

And the total time taken to cover this distance is

T = 19.19 s

We also know that the time the athlete needs to cover the first half of the race is

[tex]t_1 = 10.75 s[/tex]

Also, we know that

[tex]T=t_1 + t_2[/tex]

where [tex]t_2[/tex] is the time the athlete takes to cover the second half of the race.

Re-arranging this equation and susbtituting the values, we find the value of t2:

[tex]t_2 = T-t_1 = 19.19-10.75=8.44 s[/tex]

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Answer:

Gymania is a better deal if the membership is for 3 months and below.

Total Fitness is a better if the membership is for 4 months and above.

Step-by-step explanation:

Let the number of months be 'x'.

Given:

Money Geno has = 'y' dollars.

Total Fitness charges:

Monthly fee = $30

Initial fee = $100

Gymania charges:

Monthly fee = $50

Initial fee = $25

Total charges is equal to the sum of initial fee and monthly fee multiplied by number of months.

So, for 'x' months, monthly fee charged by Total Fitness = [tex]30x[/tex]

For 'x' months, monthly fee charged by Gymania = [tex]50x[/tex]

Now, total charge by Total Fitness = Initial fee + Fee for 'x' months

Total charge by Total Fitness = [tex]100+30x[/tex]

Now, total charge by Gymania = Initial fee + Fee for 'x' months

Total charge by Gymania = [tex]25+50x[/tex]

Now, Geno has only 'y' dollars to spend. So, 'y' must be less than or equal to the total charge.

Therefore, the total charge for each membership is:

[tex]y=30x+100\\\\y= 50x+25[/tex]

Now, we graph both the equations. The graph is shown below.

From the graph, it is clear that, the total cost for Gymania (blue line) is less than that of Total Fitness (red line) till number of months equals 3.75 or 3 months. After 3.75 months, the graph of Gymania is above Total Fitness. So, if the membership is 4 months or above, then Total Fitness is more efficient.

Therefore, Gymania is a better deal if the membership is for 3 months and below.

Total Fitness is a better if the membership is for 4 months and above.

To determine which company offers the better deal for a gym membership, set up a system of equations for the two companies' costs and compare. Total Fitness charges $30 per month, while Gymania charges $50 per month. The equation is solved to find the break-even point where their costs are equal.

To determine which company offers the better deal, we can set up a system of equations based on the given information:

Let x be the total number of months for the gym membership.

Total Fitness charges an initial fee of $100 plus $30 per month, so the total cost can be represented by the equation: y = 30x + 100.

Gymania charges an initial fee of $25 plus $50 per month, so the total cost can be represented by the equation: y = 50x + 25.

To compare the two deals, we need to find the values of x where the total cost is the same for both companies. We can set up the following equation:

30x + 100 = 50x + 25.

Simplifying, we get:

20x = 75.

Dividing both sides by 20, we find that x = 3.75.

Since x represents the number of months, it cannot be a decimal, so we round up to the nearest whole number. Therefore, Geno should join Total Fitness if he plans to have the membership for 4 or more months, and Gymania if he plans to have the membership for 3 or fewer months.

Answer:

a) [tex] f(t) = 76-18 log_{10} (0+1)= 76 - 18 log_{10} 1= 76[/tex]

b) [tex] f(t) = 76-18 log_{10} (2+1)= 76 - 18 log_{10} 3= 76-8.588=67.41[/tex]

c) [tex] f(t) = 76-18 log_{10} (11+1)= 76 - 18 log_{10} 12= 76-19.425=56.574[/tex]

Step-by-step explanation:

For this case we know that the average scores for the class are given by the following model:

[tex] f(t) = 76-18 log_{10} (t+1) , 0 \leq t \leq 12[/tex]

Where t is in months. The graph attached illustrate the function for this case

And for this case we can answer the questions like this:

Part a

We just need to replace t =0 into the model and we got:

[tex] f(t) = 76-18 log_{10} (0+1)= 76 - 18 log_{10} 1= 76[/tex]

Part b

We just need to replace t =2 into the model and we got:

[tex] f(t) = 76-18 log_{10} (2+1)= 76 - 18 log_{10} 3= 76-8.588=67.41[/tex]

Part c

We just need to replace t =11 into the model and we got:

[tex] f(t) = 76-18 log_{10} (11+1)= 76 - 18 log_{10} 12= 76-19.425=56.574[/tex]

Answer:

8 points below the mean.

Step-by-step explanation:

Let 'X' be the score of the third person, and let 'M' be the mean score of the sample.

If the other two people scored 5 and 3 above the mean, in order to maintain consistency, the following expression must be true:

[tex]M+5+(M+3)+X =3M\\X=3M-2M-8\\X=M-8[/tex]

Therefore, the third person has a score 8 points below the mean.

Given that the other two people's scores are collectively 8 points above the mean, the third person's score must be 8 points below the mean to balance out the total deviation.

This question is about determining the score of the third person in a sample relative to the mean of the sample. We know that the total deviation from the mean must equal zero because the mean is the average of all the scores. Given the first person has a score 5 points above the mean, and the second person has a score 3 points above the mean, the total deviation above the mean is (5+3)=8 points. For the score deviations to balance out to zero, the third person's score must be 8 points below the mean.

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Answer:

The probability that they get a sampling error of greater than 1.0 mg/L is 0.04.

To calculate this value, I assume that the samples are randomly collected

Step-by-step explanation:

Sampling error can be calculated using the formula

[tex]\frac{t*s}{\sqrt{n}}[/tex]  where

For the sampling error of 1.0mg/L we have

[tex]1=\frac{t*1.60}{\sqrt{10}}[/tex]    

Solving for t we have t≈1.976

Then the probability that they get a sampling error of greater than 1.0 mg/L is

P(t>1.976) ≈ 0.04.

In other words, we are 96% confident that the sampling error is within 1.0 mg/L.

To calculate this value, I assume that the samples are randomly collected.

Answer:

[tex] (\forall x) S(x)\rightarrow A(x) [/tex]

[tex] (\exists x)\neg S(x)\wedge A(x) [/tex]

Step-by-step explanation:

Everyone who studied for the test received an A on the test.

That means- if you studied for the test you will recived an A, and it is hold for everyone, so we will use quantifire [tex]\forall[/tex].

[tex] (\forall x) S(x)\rightarrow A(x) [/tex]

It means: for every student holds- If it is correct that student x studied then  the student got an A.

Someone who did not study for the test received an A on the test.

It means that, there is at least one student hwo didn't studie but student got an A. So we have conjuction of two sentences (negation of the S(x) and A(x) for some student- for that we use existential quantifie).

[tex] (\exists x)\neg S(x)\wedge A(x) [/tex]

Final answer:

The logical expression involving predicates S(x) and A(x) can be represented by ∀x(S(x) → A(x)), stating that all students who studied for the test received an A.

Explanation:

The question involving the predicates S(x): x studied for the test and A(x): x received an A on the test revolves around predicate logic, where we aim to understand and analyze the logical relations of sentences with subjects and predicates within a specified domain of discourse.

To define the logical expression that describes a relationship between studying for a test and receiving an A would depend on the specific relationship we want to express. For instance, a possible logical expression could be ∀x(S(x) → A(x)), which translates to 'For all students x in the class, if x studied for the test, then x received an A on the test.'

Answer:

Option a) Rejected              

Step-by-step explanation:

We define critical region as:

Thus,

If a test statistic falls in the critical region the null hypothesis is rejected.

The expressions can be used to calculate volume if they incorporate height, length, and width. They could calculate surface area if they involve length and width or the sum of all faces of a 3D object, without height. If these measurements are not present, they may not calculate surface area or volume.

The original question appears to be a task related to a mathematical exercise instead of a traditional query. It's asking whether certain expressions can be used to calculate volume or surface area, or if they're not meant to calculate either.

If the expression used includes measurements for length, width, and height and involves their multiplication, it's likely being used to calculate the volume of a 3D object. For a cube, volume is calculated as length x width x height, for example.

If the expression incorporates the multiplication of length and width, without incorporating height, it's being used to calculate surface area. The expression might also calculate the sum of all the faces of a 3D object to find total surface area.

If there aren't any apparent calculations for length, width, or height, it could be that these expressions aren't being used for calculating surface area or volume.

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Answer:

4 pounds of of Vigoro Ultra Turf fertilizer and 1 pound of Parkera's Premium Starter will yield the required mixture.

Step-by-step explanation:

In bag of Vigoro Ultra Turf fertilizer:

29 pounds of nitrogen, 3 pounds of phosphoric acid, and 4 pounds of potash

In bag of Parkera's Premium Starter:

18 pounds of nitrogen, 25 pounds of phosphoric acid, and 6 pounds of potash

Let the mass of Vigoro Ultra Turf fertilizer = x

Amount of nitrogen in x amount = 29 x

Amount of phosphoric acid  in  x amount = 3x

Let the mass of Parkera's Premium Starter= y

Amount of nitrogen in y amount = 18 y

Amount of phosphoric acid  in y  amount = 25 y

Amount of nitrogen in desired mixture formed by combing x and y fertilizers bag : 134 pounds

29 x + 18 y = 134 ..[1]

Amount of phosphoric acid  in desired mixture formed by combing x and y fertilizers bag : 37 pounds

3x + 25 y = 37 ..[2]

Solving [1] anf [2] by substitution method :

[tex]x=\frac{134-18y}{29}[/tex]

[tex]3\times \frac{134-18y}{29}+25y=37[/tex]

y = 1

[tex]x=\frac{134-18y}{29}=4[/tex]

4 pounds of of Vigoro Ultra Turf fertilizer and 1 pound of Parkera's Premium Starter will yield the required mixture.

To create a mixture containing 134 pounds of nitrogen, 37 pounds of phosphoric acid, and 22 pounds of potash, 400 pounds of Vigoro Ultra Turf and 100 pounds of Parker's Premium Starter are required, based on solving a system of linear equations.

We are tasked with finding the amount of two different fertilizers needed to obtain a mixture with 134 pounds of nitrogen, 37 pounds of phosphoric acid, and 22 pounds of potash. The two fertilizers are Vigoro Ultra Turf and Parkerâ's Premium Starter. We'll use a system of linear equations to solve this problem.

Let V represent the amount (in pounds) of Vigoro Ultra Turf and P represent the amount (in pounds) of Parkerâ's Premium Starter. The equations based on the given information are:

Now we solve these equations simultaneously to determine V and P. After solving, we find that V = 400 and P = 100. Thus, we need 400 pounds of Vigoro Ultra Turf and 100 pounds of Parkerâ's Premium Starter to obtain the desired mixture.

Final answer:

The question translates to the inequality (1/5)n - 8 ≤ 96, which is solved by adding 8 to both sides and then multiplying by 5, resulting in n ≤ 520.

Explanation:

The question involves translating a word problem into a mathematical inequality. The phrase 'Eight less than the product of a number n and 1/5' can be written as (1/5)n - 8. When it states that this is 'no more than 96', it implies that the expression should be less than or equal to 96. Therefore, the inequality we need to solve is (1/5)n - 8 ≤ 96.

Now, let's solve this inequality step-by-step:

Add 8 to both sides of the inequality: (1/5)n ≤ 104.

Multiply both sides by 5 to solve for n: n ≤ 520.

This gives us the solution to the inequality, indicating that the number n can be any value less than or equal to 520 to satisfy the initial condition.

Answer:

[tex]z=\frac{0.32-0.36}{\sqrt{0.34(1-0.34)(\frac{1}{50}+\frac{1}{50})}}=-0.422[/tex]

So on this case the only option that satisfy the calculated statistic is:

z=-0.42

Step-by-step explanation:

Assuming this complete problem: "In a 2-sample z-test for two proportions, you find the following:

^P1 = 0.32, (n,1)=50

^P,2= 0.36, (n,2)=50

Find the test statistic you will use while executing this test:

z=-0.67 , z=±1.64 , z=-1.96 , z=0.34 , z=-0.42"

Solution to the problem

Data given and notation  

[tex]n_{1}=50[/tex] sample 1 selected

[tex]n_{2}=50[/tex] sample 2 selected

[tex]p_{1}=0.32[/tex] represent the sample proportion for 1

[tex]p_{2}=0.36[/tex] represent the sample proportion for 2  

z would represent the statistic (variable of interest)  

[tex]p_v[/tex] represent the value for the test (variable of interest)

We need to apply a z test to compare proportions, and the statistic is given by:  

[tex]z=\frac{\hat p_{1}-\hat p_{2}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}[/tex]   (1)

Or equivalently:

[tex]z=\frac{\hat p_{2}-\hat p_{1}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{1}}+\frac{1}{n_{2}})}}[/tex]   (1)

Where [tex]\hat p=\frac{X_{1}+X_{1}}{n_{1}+n_{1}}=\frac{\hat p_1 +\hat p_2}{2}=\frac{0.32+0.36}{2}=0.34[/tex]

Calculate the statistic

Replacing in formula (1) the values obtained we got this:  

[tex]z=\frac{0.32-0.36}{\sqrt{0.34(1-0.34)(\frac{1}{50}+\frac{1}{50})}}=-0.422[/tex]

So on this case the only option that satisfy the calculated statistic is:

z=-0.42

Answer:

Amount he must have in his account today is  $5,617.92

Step-by-step explanation:

Data provided in the question:

Regular withdraw amount = $900

Average annual interest rate, i = 4% = 0.04

Time, n = 7 years

Now,

Present Value = [tex]C \times\left[ \frac{1-(1+i)^{-n}}{i} \right] \times(1 + i)[/tex]

here,

C = Regular withdraw amount

Thus,

Present Value = [tex]C \times\left[ \frac{1-(1+i)^{-n}}{i} \right] \times(1 + i)[/tex]

Present Value = [tex]900 \times\left[ \frac{1-(1+0.04)^{-7}}{ 0.04 } \right] \times(1 + 0.04)[/tex]

Present Value = [tex]936 \times\left[ \frac{1 - 1.04^{-7}}{ 0.04} \right][/tex]

Present Value = [tex]936 \times\left[ \frac{1 - 0.759918}{ 0.04} \right][/tex]

Present Value = 936 × 6.00205

or

Present Value = $5,617.92

Hence,

Amount he must have in his account today is  $5,617.92

Answer: Yes, you are correct it's $8.95 because like they stated in the word problem p= the price of each bowl and like they said in the word problem they charge $8.95 a bowl. So, $8.95 is the answer.

Step-by-step explanation:

Answer:

[tex] 0.85 = P(C) + 0.75 -0.75 P(C)[/tex]

[tex]0.1 = 0.25 P(C)[/tex]

[tex] P(C) = 0.4[/tex]

Step-by-step explanation:

First we can define some notation useful:

C ="represent the event of incurring in operating charges"

R= represent the event of emergency rooms charges"

For this case we are interested on P(C) since they want "the probability that a claim submitted to the insurance company includes operating room charges."

We have some probabilities given:

[tex] P(R') = 0.25 , P(C \cup R) =0.85[/tex]

Solution to the problem

By the complement rule we have this:

[tex] P(R') = 0.25 =1-P(R)[/tex]

[tex] P(R) = 1-0.25 = 0.75[/tex]

Since the two events C and R are considered independent we have this:

[tex]P(C \cap R) = P(C) *P(R)[/tex]

Now we can use the total probability rule like this:

[tex] P(C \cup R) = P(C) + P(R) - P(R)*P(C)[/tex]

And if we replace we got:

[tex] 0.85 = P(C) + 0.75 -0.75 P(C)[/tex]

[tex]0.1 = 0.25 P(C)[/tex]

[tex] P(C) = 0.4[/tex]

Answer:

For First Solution: [tex]y_1(t)=e^t[/tex]

[tex]y_1(t)=e^t[/tex] is the solution of equation y''-y=0.

For 2nd Solution:[tex]y_2(t)=cosht[/tex]

[tex]y_2(t)=cosht[/tex]  is the solution of equation y''-y=0.

Step-by-step explanation:

For First Solution: [tex]y_1(t)=e^t[/tex]

In order to prove whether it is a solution or not we have to put it into the equation and check. For this we have to take derivatives.

[tex]y_1(t)=e^t[/tex]

First order derivative:

[tex]y'_1(t)=e^t[/tex]

2nd order Derivative:

[tex]y''_1(t)=e^t[/tex]

Put Them in equation y''-y=0

e^t-e^t=0

0=0

Hence [tex]y_1(t)=e^t[/tex] is the solution of equation y''-y=0.

For 2nd Solution:

[tex]y_2(t)=cosht[/tex]

In order to prove whether it is a solution or not we have to put it into the equation and check. For this we have to take derivatives.

[tex]y_2(t)=cosht[/tex]

First order derivative:

[tex]y'_2(t)=sinht[/tex]

2nd order Derivative:

[tex]y''_2(t)=cosht[/tex]

Put Them in equation y''-y=0

cosht-cosht=0

0=0

Hence [tex]y_2(t)=cosht[/tex]  is the solution of equation y''-y=0.

The functions [tex]y(t)=e^t[/tex] and [tex]y(t)=cosht[/tex] are solutions of given differential equation.

We have to prove that given function is a solution of the differential equation.

Given function,   [tex]y(t)=e^t[/tex]

Given differential equation are, [tex]y''-y=0[/tex]

  So that,  [tex]y'(t)=e^t , y''(t)=e^t[/tex]

Substitute values in given differential equation.

                    [tex]e^t - e^t=0[/tex]

Thus, function [tex]y(t)=e^t[/tex] is solution of given differential equation.

Another function is,  [tex]y(t)=cosht[/tex]

So that,  [tex]y'(t)=sinht , y''(t)=cosht[/tex]

Substitute values in given differential equation.

                    [tex]cosht-cosht=0[/tex]

Thus, function [tex]y(t)=cosht[/tex] is solution of given differential equation.

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Answer:

KH = 2  and  LH = 2√3

Step-by-step explanation:

Using Euclidean theorem for the right triangle.

∵ ΔLHK is a right triangle at H

OK = 1 , OL = 3

KL = KO + OL = 1 + 3 = 4

KH² = KO * KL = 1 * 4 = 4

KH = √4 = 2

And LH² = LO * LK = 3 * 4 = 12

∴ LH = √12 = 2√3

Answer:

Roughly the same as what?

Step-by-step explanation:

Answer:

A. 400 GRAMS

B. 120 GRAMS

C. 356 GRAMS

D. 340 GRAMS

PENN FOSTER

ANSWER IS D 340 GRAMS

Step-by-step explanation:1 ounce = 28.3495 12 ounces = 28.3495 * 12 = 340.194 grams

Answer:

a) The target population of interest on this case represent the "customers who bought a car during the previous year". They want to analyze all the people that satisfy this condition in order to see the satisfaction rate of these people

b) For this case the population is not hypothetical since is well defined and they have all the customers who bought a car during the last year, since that info is on the records of the manufacturer from people who bought a car. So then the population is available in order to analyze the question desired.

Step-by-step explanation:

Part a

The target population of interest on this case represent the "customers who bought a car during the previous year". They want to analyze all the people that satisfy this condition in order to see the satisfaction rate of these people

Part b

For this case the population is not hypothetical since is well defined and they have all the customers who bought a car during the last year, since that info is on the records of the manufacturer from people who bought a car. So then the population is available in order to analyze the question desired.

a. The customer who bought the car the previous year can get feedback from them so that we can evaluate the status of customer satisfaction.

b.   The customer who bought the car the previous year, so info is on the record of the manufacturer from the people who bought a car.

the process of ensuring that research data is stored.

a. The customer who bought the car the previous year can get feedback from them so that we can evaluate the status of customer satisfaction.

b.   The customer who bought the car the previous year, so info is on the record of the manufacturer from the people who bought a car.

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Answer:

[tex]\frac{1}{7}[/tex]

Step-by-step explanation:

The table containing information about the sexes of twins and and the type of twins are given in the table below with corresponding weightage.

Now, since we know that she is expecting twins, the probability of the event

                [tex]A - '\text{the woman will have identical twins }'[/tex]

is calculated as follows

[tex]Pr(A) = \dfrac{\text{total number of boy/boy, boy/girl, girl/boy and girl/girl identical twins}}{\text{total number of both identical and frathernal twins}}[/tex]

By using the informations from the given table, we obtain

                                        [tex]Pr(A) = \frac{2 + 0 + 0 +2}{28} = \frac{4}{28} = \frac{1}{7}[/tex]

Answer:

a)  m = -9 or m = 1

b) m =  1 + √6 or m = 1 -√6

Step-by-step explanation:

for

Φ(x) = x^m

then

dΦ/dx (x) = m*x^(m-1)

d²Φ/dx² (x) = m*(m-1)*x^(m-2)

then

for a

3x^2 (d^2y/dx^2) + 11x(dy/dx) - 3y = 0

3x^2*m*(m-1)*x^(m-2) + 11*x* m*x^(m-1) - 3*x^m = 0

3*m*(m-1)*x^m + 11*m*x^m- 3*x^m = 0

dividing by x^m

3*m*(m-1) + 11*m - 3 =0

3*m² + 8 m - 3 =0

m= [-8 ± √(64 + 4*3*3)]/2 = (-8±10)/2  

m₁ = -9 , m₂= 1

then Φ(x) = x^m is a solution for the equation a , when m = -9 or m = 1

for b)

x^2 (d^2y/dx^2) - x(dy/dx) - 5y = 0

x^2*m*(m-1)*x^(m-2) -  x* m*x^(m-1) - 5*x^m = 0

m*(m-1)*x^m -m *x^m- 5*x^m = 0

dividing by x^m

m*(m-1) -m - 5 =0

m² - 2 m - 5 =0

m= [2 ± √(4 + 4*1*5)]/2 = (2±√24)/2 = 1 ±√6

m₁ =  1 + √6 , m₂ =  1 - √6

then Φ(x) = x^m is a solution for the equation b , when m =  1 + √6 or m = 1 - √6

Answer

a) m = -3 or 1/3

b) m = 1 + root 6 or 1 - root 6

Step-by-step explanation:

The step by step calculation is as shown in the attachment.

Answer:

a) 51.60% probability that in a given year there will be less than 21 earthquakes.

b) 49.35% probability that in a given year there will be between 18 and 23 earthquakes.

Step-by-step explanation:

Problems of normally distributed samples can be solved using 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 problem, we have that:

[tex]\mu = 20.8, \sigma = 4.5[/tex]

a) Find the probability that in a given year there will be less than 21 earthquakes.

This is the pvalue of Z when X = 21. So

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

[tex]Z = \frac{21 - 20.8}{4.5}[/tex]

[tex]Z = 0.04[/tex]

[tex]Z = 0.04[/tex] has a pvalue of 0.5160.

So there is a 51.60% probability that in a given year there will be less than 21 earthquakes.

b) Find the probability that in a given year there will be between 18 and 23 earthquakes.

This is the pvalue of Z when X = 23 subtracted by the pvalue of Z when X = 18. So:

X = 23

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

[tex]Z = \frac{23 - 20.8}{4.5}[/tex]

[tex]Z = 0.71[/tex]

[tex]Z = 0.71[/tex] has a pvalue of 0.7611

X = 18

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

[tex]Z = \frac{18 - 20.8}{4.5}[/tex]

[tex]Z = -0.62[/tex]

[tex]Z = -0.62[/tex] has a pvalue of 0.2676

So there is a 0.7611 - 0.2676 = 0.4935 = 49.35% probability that in a given year there will be between 18 and 23 earthquakes.

Final answer:

To find the probability of specific numbers of earthquakes occurring within a year using a normal distribution, one calculates the Z-scores for those numbers and looks up or calculates the corresponding probabilities.

Explanation:

The question involves finding probabilities related to the number of earthquakes in a year, which is modeled using a normal distribution. To find these probabilities, we'll use the mean μ = 20.8 and the standard deviation σ = 4.5 of the distribution.

a) Probability of less than 21 earthquakes

To find the probability of there being less than 21 earthquakes in a year, we calculate the Z-score for 21:

Z = (X - μ) / σ = (21 - 20.8) / 4.5 ≈ 0.04

Looking up this Z-score on a standard normal distribution table or using a calculator, we find the corresponding probability and note that it's slightly more than 0.5, indicating a little over a 50% chance.

b) Probability of 18 to 23 earthquakes

Finding the Z-scores for 18 and 23:

Z for 18 = (18 - 20.8) / 4.5 ≈ -0.62

Z for 23 = (23 - 20.8) / 4.5 ≈ 0.49

You then look up these Z-scores on a standard normal distribution table to find the probabilities for each and subtract the smaller from the larger to get the probability of having between 18 and 23 earthquakes in a year. This method shows that there's a significant chance, typically around 40% to 50%, though the specific value requires precise Z-score to probability conversion.

Answer:

$34 billion

Step-by-step explanation:

Data provided in the question:

Total number of inhabitants of a country = 2 million

1 mean gross income = $20,000 per year

Tax paid by mean individual = $3000 per year

Now,

Mean disposable income

= Mean gross income - Mean tax

= $20,000 - $3,000

= $17,000 per year

Therefore,

Total disposable income per year

= Mean disposable income × Total number of inhabitants of a country

= $17,000 × 2 million

= $34 billion

Answer: he bought 5 cups of coffee.

He bought 3 cups of hit chocolate

Step-by-step explanation:

Let x represent the number cups of coffee that he bought.

Let y represent the the number cups of of hit chocolate that he bought.

He bought a total of 8 cups coffee and hit chocolate. This means that

x + y = 8

Each cup of coffee costs $2.25 and each cup of hit chocolate costs $1.50. If he pays a total of $15.75 for 8 cups, it means that

2.25x + 1.5y = 15.75 - - - - - - - - - - - - 1

Substituting x = 8 - y into equation 1, it becomes

2.25(8 - y) + 1.5y = 15.75

18 - 2.25y + 1.5y = 15.75

- 2.25y + 1.5y = 15.75 - 18

- 0.75y = - 2.25

y = - 2.25/- 0.75

y = 3

Substituting y = 3 into x = 8 - y, it becomes

x = 8 - 3 = 5

Answer:

Option b) r = 0.70      

Step-by-step explanation:

We are given the following in he question:

Variables:

x = the number of hours spent studying for a test

y = the number of points earned on the test

As the number of hours increases, the number of points earned on the test increases. Thus, the two variables are positively correlated.

Thus, the coefficient correlation between two variables can be given by r = 0.70, that shows a moderate positive correlation.

Option b) r = 0.70

Answer:

Ace Truck

[tex]C(m) = 30 + 0.5*m[/tex]

Acme Truck

[tex]C(m) = 25 + 0.55*m[/tex]

Step-by-step explanation:

The cost function to lease a box truck from a company has the following format:

[tex]C(m) = F + a*m[/tex]

In which F is the fixed cost and a is the cost per mile m.

(a) Find the daily cost of leasing from each company as a function of the number of miles driven.

Ace Truck

$30/day and $0.50/mi. This means that [tex]F = 30, a = 0.50[/tex]. So

[tex]C(m) = 30 + 0.5*m[/tex]

Acme Truck

$25/day and $0.55/mi. This means that [tex]F = 25 a = 0.55[/tex]. So

[tex]C(m) = 25 + 0.55*m[/tex]

Answer:

a) (-∞,-9)∪(-9,9)∪(9,+∞)

b) (-∞,+∞)

Step-by-step explanation:

The domain of a real function is the largest subset og the real line in which it is defined.

a) The function [tex]f(x)=\frac{81-e^{x^2}}{1-e^{81-x^2}}[/tex] is defined for all values of x in which the denominator is not zero. The denominator is zero if [tex]0=1-e^{81-x^2}[/tex], that is,  [tex]e^{81-x^2}=1[/tex]. The exponential function is equal to one only if the exponent is equal to zero, then the values of x that nullify the denominator satisfy [tex]81-x^2=0[/tex], thus x=9 or x=-9. Then the domain of f is the set of all real numbers such that x≠9 and x≠-9, which is the set (-∞,-9)∪(-9,9)∪(9,+∞).

b) In this case the denominator is [tex]e^{\cos x}[/tex] which is always positive. Thus the denominator is not zero for all real x, then the fomain is the real line, which is the interval (-∞,+∞).

Final answer:

The domain of the function f(x) = (81-e^x^2)/(1-e^81-x^2) is (-∞, -9) ∪ (-9, 9) ∪ (9, ∞). The domain of the function f(x) = 7 + x/e^cos(x) is (-∞, ∞).

Explanation:

(a) To find the domain of the function f(x) = (81-e^x^2)/(1-e^81-x^2), we need to determine the values of x for which the function is defined. The function is defined as long as the denominator is not zero. So, we need to solve the equation 1 - e^(81-x^2) = 0 to find where the denominator equals zero. Simplifying this equation, we get e^(81-x^2) = 1. Taking the natural logarithm of both sides, we have 81-x^2 = 0. Therefore, x^2 = 81, and x = ±9. The domain of the function is (-∞, -9) ∪ (-9, 9) ∪ (9, ∞).

(b) To find the domain of the function f(x) = 7 + x/e^cos(x), we need to determine the values of x for which the function is defined. The function is defined as long as the denominator e^cos(x) is not zero. Since the exponential function is always positive, we know that e^cos(x) will never be zero. Therefore, the domain of the function is the set of all real numbers, (-∞, ∞).

Answer:

Step-by-step explanation:

Given that a die is thrown twice.  

X1, X2 are the outcomes in 2 throws

X1 = minimum of two

X1 can be either 1 or 2...6

Total outcomes are 36.

For x1 =1, fav ourable outcomes are (1,1) (1,2)...(1,6) (6,1)...(2/,1)= 11

P(X1=1) = [tex]\frac{11}{36}[/tex]\

P(X1=2) =Prob for one die showing two and other die showing 2 to 6

= [tex]\frac{9}{36}[/tex]

P(x1=3) = Prob for one die showing three and other die showing 3 to 6

=[tex]\frac{7}{36}[/tex]

thus we find that probability is reducing by 2 in the numerator

P(x1=4) = 5/36 followed by 3/36 for 5 and 1/36 for 6

Final answer:

The random variable X represents the minimum outcome when rolling a die twice. The values of X range from 1 to 6 and its distribution is given.

Explanation:

The random variable X represents the minimum outcome when rolling a die twice. In other words, X is the smaller of the two outcomes.

X can take on values from 1 to 6 since the outcomes of rolling a die are integers between 1 and 6. The minimum value of X is 1, which occurs when both dice show a 1.

The distribution of X is as follows:

Answer:

[tex] cos \theta = -\frac{8}{17}[/tex]

Step-by-step explanation:

For this case we know that:

[tex] sin \theta = \frac{15}{17}[/tex]

And we want to find the value for [tex] cos \theta[/tex], so then we can use the following basic identity:

[tex] cos^2 \theta + sin^2 \theta =1 [/tex]

And if we solve for [tex] cos \theta [/tex] we got:

[tex] cos^2 \theta = 1- sin^2 \theta[/tex]

[tex] cos \theta =\pm \sqrt{1-sin^2 \theta}[/tex]

And if we replace the value given we got:

[tex] cos \theta =\pm \sqrt{1- (\frac{15}{17})^2}=\sqrt{\frac{64}{289}}=\frac{\sqrt{64}}{\sqrt{289}}=\frac{8}{17}[/tex]

For our case we know that the angle is on the II quadrant, and on this quadrant we know that the sine is positive but the cosine is negative so then the correct answer for this case would be:

[tex] cos \theta = -\frac{8}{17}[/tex]

Answer:

It is D

Step-by-step explanation:

EDGE 2021

Answer:

There is a 99.99998% probability that at least one valve opens.

Step-by-step explanation:

For each valve there are only two possible outcomes. Either it opens on demand, or it does not. This means that we use the binomial probability distribution to solve this problem.

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 combinatios 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.

In this problem we have that:

[tex]n = 5, p = 0.93[/tex]

Calculate P(at least one valve opens).

This is [tex]P(X \geq 1)[/tex]

Either no valves open, or at least one does. The sum of the probabilities of these events is decimal 1. So:

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

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

So

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

[tex]P(X = 0) = C_{5,0}.(0.93)^{0}.(0.07)^{5} = 0.0000016807[/tex]

Finally

[tex]P(X \geq 1) = 1 - P(X = 0) = 1 - 0.0000016807 = 0.9999983193[/tex]

There is a 99.99998% probability that at least one valve opens.

Answer:

0.930000

Step-by-step explanation: