Graph the system of equations on graph paper. {8x+6y=48 {2x−3y=−6 Which statements are true about the solution to the system of equations? Select each correct answer.

The x-coordinate of the solution is −3 .

The x-coordinate of the solution is 3.

The ordered pair that is the solution to the system lies in Quadrant I .

The y-coordinate of the solution is 4.

The ordered pair that is the solution to the system lies in Quadrant II .

The y-coordinate of the solution is 3.

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:

[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:

a) The physicians since we want to measure about behaviors in the natural clinical settings from successful physicians. And the obervational unit would be the patients from these physicians.

b) They have in total 10 physicians who have been rated highly by patients but they use random sampling and they select just 5 physician-patient encounters from the 10 physicians

c) For this case based on the survey, we only have 10 doctors who have been highly qualified by the patients, so our goal is to analyze the information on these 10, and the sample size is 5. and that represent 50% of the original objective, so in this case we can say that this sample size would be sufficient to extrapolate the sample size information from 5 to the total number of doctors 10 of interest.

Step-by-step explanation:

Assuming this complete question: "Suppose you are interested in the behaviors of physicians that have high ratings of patient satisfaction. The research goal is to identify the behaviors in the natural clinical settings of these successful physicians so that these behaviors can be built into the curricula of medical preparation programs. The main data were collected by the video recording of five randomly selected physician-patient encounters from 10 physicians who have been rated highly by patients in a reliable satisfaction survey. In analyzing the recordings, what would you define as the unit for analysis? Why? How many data units (in rough estimates) are you likely to get based on this decision? Does the estimated number of data units seem adequate? Why or why not? "

In analyzing the recordings, what would you define as the unit for analysis? Why?

The physicians since we want to measure about behaviors in the natural clinical settings from successful physicians. And the obervational unit would be the patients from these physicians.

How many data units (in rough estimates) are you likely to get based on this decision?

They have in total 10 physicians who have been rated highly by patients but they use random sampling and they select just 5 physician-patient encounters from the 10 physicians

Does the estimated number of data units seem adequate? Why or why not?

For this case based on the survey, we only have 10 doctors who have been highly qualified by the patients, so our goal is to analyze the information on these 10, and the sample size is 5. and that represent 50% of the original objective, so in this case we can say that this sample size would be sufficient to extrapolate the sample size information from 5 to the total number of doctors 10 of interest.

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.

More about the data handling link is given below.

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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:

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:

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:

x = 6.1 and y = 12.6

Step-by-step explanation:

sin 26 = x/14

x = 14sin26

x = 6.14

x= 6.1

cos 26 = y/14

y = 14cos26

y = 12.58

y = 12.6

Answer:the height of the ramp is 6.1ft

the length of the base of the ramp is 12.6 ft

Step-by-step explanation:

Triangle ABC is a right angle triangle.

The length of the ramp represents the hypotenuse of the right angle triangle.

With 26 degrees being the reference angle,

The opposite side of the right angle triangle = x

The adjacent side of the right angle triangle = y

To determine x, we would apply the Sine trigonometric ratio.

Sin θ = opposite side/hypotenuse

Sin 26 = x/14

x = 14Sin26 = 14 × 0.4384 =

x = 6.1

To determine y, we would apply the Cosine trigonometric ratio.

Cos θ = adjacent side /hypotenuse

Cos 26 = y/14

x = 14Cos26 = 14 × 0.8988 =

x = 12.6

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

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:

sin C = 7/27, cos C = 24/27 and tan C = 7/24

Step-by-step explanation:

sin C = opposite/hypotenuse

         = 7/27

cos C = adjacent/hypotenuse

          = 24/27

tan C = opposite/ adjacent

         = 7/24

Answer:

sin c - 25/7

cos c - 7/24

tan c - 25/24

Step-by-step explanation:

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:

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:

Step-by-step explanation:

Triangle RST is a right angle triangle.

From the given right angle triangle

RT represents the hypotenuse of the right angle triangle.

With 26 degrees as the reference angle,

ST represents the adjacent side of the right angle triangle.

RS represents the opposite side of the right angle triangle.

1) To determine RS, we would apply trigonometric ratio

Sin θ = opposite side/hypotenuse Therefore,

Sin 26 = RS/9.1

RS = 9.1Sin26 = 9.1 × 0.4384

RS = 4.0

2) To determine ST, we would apply trigonometric ratio

Cos θ = adjacent side/hypotenuse Therefore,

Cos 26 = ST/9.1

ST = 9.1Cos26 = 9.1 × 0.8988

ST = 8.1

3) The sum of the angles in a triangle is 180 degrees. Therefore,

∠R + 26 + 90 = 180

∠R = 180 - (26 + 90)

∠R = 64 degrees

The options are properly listed below:

A. a proven scientific fact

B. an instrument that is used to examine environmental conditions

C. a testable proposition that explains an observed phenomenon or answers a question

D. the design of an experiment that can be used in scientific inquiry

E. a prediction about something that has not yet been observed

ANSWER:

C. a testable proposition that explains an observed phenomenon or answers a question

EXPLANATION:

HYPOTHESIS is a proposed explanation for an observable fact, but that cannot satisfactorily be explained with the available scientific theories. It is based on the information acquired from a primary source or data collected during a scientific activity.

Also, for a hypothesis to be a scientific hypothesis, the scientific method requires that it is testable.

It is also a trial solution to a question.

Hypothesis can be provisionally accepted as a starting point for further research.

Answer:a) μ = 5 and σ = 16

b) z-score are -0.3125, 0.0625, -0.0625, -0.125, 0.4375

c) New population of N=5 scores are 93.75, 101.25, 98.75, 97.5, 108.75

Step-by-step explanation:

The detailed explanation can be found in the attached pictures

The new population of N = 5 scores with a mean of µ = 100 and a standard deviation of σ = 20 are 125, 105, 95, 90 and 135

(a) Compute µ and σ for the population.

The dataset is given as:

0, 6, 4, 3, and 12.

The mean is calculated as:

[tex]\mu = \frac{\sum x}n[/tex]

So, we have:

[tex]\mu = \frac{0 + 6 + 4 + 3 + 12}5[/tex]

[tex]\mu = 5[/tex]

The standard deviation is calculated as:

[tex]\sigma = \sqrt{\frac{\sum(x - \bar x)^2}n}[/tex]

This gives

[tex]\sigma = \sqrt{\frac{(0 - 5)^2 + (6- 5)^2 + (4- 5)^2 + (3- 5)^2 + (12- 5)^2}5[/tex]

[tex]\sigma = \sqrt{\frac{80}5[/tex]

[tex]\sigma = \sqrt{16[/tex]

[tex]\sigma = 4[/tex]

Hence, the values of μ and σ are μ = 5 and σ = 4

(b) The z-scores

This is calculated as:

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

When x = 0, 6, 4, 3, and 12.

We have:

[tex]z = \frac{0 - 5}{4} = 1.25[/tex]

[tex]z = \frac{6 - 5}{4} = 0.25[/tex]

[tex]z = \frac{4 - 5}{4} = -0.25[/tex]

[tex]z = \frac{3 - 5}{4} = -0.5[/tex]

[tex]z = \frac{12 - 5}{4} = 1.75[/tex]

Hence, the z-scores are 1.25, 0.25, -0.25, -0.5 and 1.75

(c) Transform the new population

We have:

N = 5, µ = 100 and σ = 20.

In (b), we have:

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

Make x the subject

[tex]x = \mu + z\sigma[/tex]

This gives

[tex]x_i = \mu + z_i\sigma[/tex]

So, we have:

[tex]x_1 = 100 + 1.25* 20 = 125[/tex]

[tex]x_2 = 100 + 0.25* 20 = 105[/tex]

[tex]x_3 = 100 - 0.25* 20 = 95[/tex]

[tex]x_4 = 100 - 0.5* 20 = 90[/tex]

[tex]x_5 = 100 + 1.75* 20 = 135[/tex]

Hence, the new population of N = 5 scores with a mean of µ = 100 and a standard deviation of σ = 20 are 125, 105, 95, 90 and 135

Read more about z-scores at:

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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]

To find the area of a parallelogram, you can use the cross product of two vectors created from the vertices. The magnitude of the resulting vector represents the area of the parallelogram. Ensure to put an absolute value on the final result.

The subject of this question is geometry, specifically finding the area of a parallelogram given the vertices. You can calculate the area of a parallelogram using the cross product of two vectors. First, create two vectors from the given vertices, for example, PQ = Q - P and PR = R - P. For your question, let's take P as s1, 0, 2d, Q as s3, 3, 3d, and R as s7, 5, 8d. So, PQ = s2, 3, 1d and PR = s6, 5, 6d. The cross product of these two vectors would give a vector perpendicular to both, whose magnitude represents the area of the parallelogram spanned by the vectors PQ and PR. The magnitude (or length) of a vector ABC = sA, B, Cd is calculated as √(A² + B² + C²).

Using these formulas and calculations, you should be able to find the area of your parallelogram. Remember to put absolute value on the final answer as area cannot be negative.

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Answer:

Step-by-step explanation:

For testing of significance of correlation coefficient denoted by r, we create hypotheses in three ways

They are one tailed, two tailed.  One tailed can be stated as right tailed and also left tailed.

The null hypothesis would normally be as r=0

Verbally we can say this there is no association between the dependent and independent variable (linear)

Against this alternate hypothesis is created as

either r not equal to 0

or r>0 or r<0

If r not equal to 0, we say two tailed hypothesis test

If r>0 is alternate hypothesis, it is right tailed test

If r<0 is alternate hypothesis, then it is left tailed test.

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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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 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.

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:

[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

Step-by-step explanation:

If x is the volume of 39% solution, and y is the volume of 70% solution, then:

x + y = 40

0.39x + 0.70y = 0.55(40)

Solve the system of equations.

0.39x + 0.70(40 − x) = 0.55(40)

0.39x + 28 − 0.70x = 22

6 = 0.31x

x = 19.4

y = 20.6

The chemist needs 19.4 liters of 39% solution and 20.6 liters of 70% solution.

Answer:

0.6547 or 65.47%

Step-by-step explanation:

One minute equals 1/60 of an hour, the mean number of occurrences in that interval is:

[tex]\lambda =\frac{400}{60}=6.6667[/tex]

The poisson distribution is described by the following equation:

[tex]P(x) =\frac{\lambda^{x}*e^{-\lambda}}{x!}[/tex]

The probability that more than 5 vehicles will arrive is:

[tex]P(x>5)= 1-(P(0)+P(1)+P(2)+P(3)+P(4)+P(5))\\P(x>5) = 1-(\frac{6.667^{0}*e^{-6.667}}{1}+\frac{6.667^{1}*e^{-6.667}}{1}+\frac{6.667^{2}*e^{-6.667}}{2}+\frac{6.667^{3}*e^{-6.667}}{3*2}+\frac{6.667^{4}*e^{-6.667}}{4*3*2}+\frac{6.667^{5}*e^{-6.667}}{5*4*3*2})\\P(x>5)=1-(0.00127+0.00848+0.02827+ 0.06283+0.10473+0.13965)\\P(x>5)=0.6547[/tex]

The probability that more than five vehicles will arrive in a one-minute interval is 0.6547 or 65.47%.

The probability that more than five vehicles will arrive in a one-minute interval is approximately 0.6582.

Step 1

Given that vehicles arrive at an intersection at a rate of 400 vehicles per hour, and this follows a Poisson distribution, we want to find the probability that more than five vehicles will arrive in a one-minute interval.

First, convert the arrival rate to a one-minute interval. Since there are 60 minutes in an hour, the arrival rate per minute is:

[tex]\[ \lambda = \frac{400 \, \text{veh/h}}{60} = \frac{400}{60} \approx 6.67 \, \text{veh/min} \][/tex]

The Poisson distribution formula for the probability of observing k events in an interval is:

[tex]\[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \][/tex]

We need the probability that more than five vehicles arrive in one minute:

[tex]\[ P(X > 5) = 1 - P(X \leq 5) \][/tex]

Step 2

First, calculate [tex]\( P(X \leq 5) \)[/tex] by summing the probabilities for k = 0 to k = 5 :

[tex]\[ P(X \leq 5) = \sum_{k=0}^{5} \frac{e^{-6.67} 6.67^k}{k!} \][/tex]

Let's calculate these probabilities:

- For k = 0 :

 [tex]\[ P(X = 0) = \frac{e^{-6.67} 6.67^0}{0!} = e^{-6.67} \][/tex]

 - For k = 1 :

 [tex]\[ P(X = 1) = \frac{e^{-6.67} 6.67^1}{1!} = e^{-6.67} \times 6.67 \][/tex]

 - For k = 2 :

 [tex]\[ P(X = 2) = \frac{e^{-6.67} 6.67^2}{2!} = e^{-6.67} \times \frac{6.67^2}{2} \][/tex]

 - For k = 3:

 [tex]\[ P(X = 3) = \frac{e^{-6.67} 6.67^3}{3!} = e^{-6.67} \times \frac{6.67^3}{6} \][/tex]

 - For k = 4 :

 [tex]\[ P(X = 4) = \frac{e^{-6.67} 6.67^4}{4!} = e^{-6.67} \times \frac{6.67^4}{24} \][/tex]

 - For k = 5 :

 [tex]\[ P(X = 5) = \frac{e^{-6.67} 6.67^5}{5!} = e^{-6.67} \times \frac{6.67^5}{120} \][/tex]

Sum these probabilities to find [tex]\( P(X \leq 5) \)[/tex].

Step 3

Next, we calculate [tex]\( e^{-6.67} \)[/tex] and the terms:

[tex]\[e^{-6.67} \approx 0.00126\][/tex]

[tex]\[P(X = 0) \approx 0.00126\][/tex]

[tex]\[P(X = 1) \approx 0.00126 \times 6.67 = 0.0084\][/tex]

[tex]\[P(X = 2) \approx 0.00126 \times \frac{6.67^2}{2} = 0.0280\][/tex]

[tex]\[P(X = 3) \approx 0.00126 \times \frac{6.67^3}{6} = 0.0622\][/tex]

[tex]\[P(X = 4) \approx 0.00126 \times \frac{6.67^4}{24} = 0.1037\][/tex]

[tex]\[P(X = 5) \approx 0.00126 \times \frac{6.67^5}{120} = 0.1382\][/tex]

Sum these probabilities:

[tex]\[P(X \leq 5) \approx 0.00126 + 0.0084 + 0.0280 + 0.0622 + 0.1037 + 0.1382 = 0.34176\][/tex]

Finally, the probability that more than five vehicles will arrive in a one-minute interval is:

[tex]\[P(X > 5) = 1 - P(X \leq 5) = 1 - 0.34176 = 0.65824\][/tex]

The probability that more than five vehicles will arrive in a one-minute interval is approximately 0.6582.

Answer:

Yes, the claim can be concluded.

Step-by-step explanation:

We are given the following in the question:

Alpha, α = 0.05

The null hypothesis and alternate hypothesis can be designed in the following manner:

[tex]H_{0}: \mu_{500,000} = \mu_{100,000}\\H_A: \mu_{500,000} > \mu_{100,000}[/tex]

This is a one tailed(right) test.

[tex]z_{stat} = 16.2[/tex]

Now, we calculate the p - value from standard table.

P-value = 0.00001

Since the p value is less than the significance level, we fail to accept the null hypothesis and reject it.

We accept the alternate hypothesis.

Thus, we conclude that there is enough evidence to support the claim that the annual incomes of high school teachers in metropolitan areas having greater than 500,000 population are significantly greater than those paid in areas with fewer than 100,000 population.

Final answer:

To compute the covariance between Z1 and Z2, we need to calculate the covariance between X and X, X and Y, Y and X, and Y and Y individually. Using the given variances and calculations, we can find Cov(Z1, Z2) = 308 + 0.1(144 + 2Cov(X, Y) + 64).

Explanation:

To compute the covariance between Z1 and Z2, we need to calculate the covariance between X and X, X and Y, Y and X, and Y and Y first.

Now, we can calculate Cov(Z1, Z2) using the following formula:

Cov(Z1, Z2) = Cov(X + Y, X + 0.1X + 5) = Cov(Z1, Z1 + 0.1X + 5) = Cov(Z1, Z1) + Cov(Z1, 0.1X) + Cov(Z1, 5) = Var(Z1) + 0.1Cov(Z1, X) + 0 = Var(X + Y) + 0.1Cov(Z1, X) + 0 = 308 + 0.1(Cov(X, X) + Cov(X, Y) + Cov(Y, X) + Cov(Y, Y)) + 0 = 308 + 0.1(144 + Cov(X, Y) + Cov(Y, X) + 64) + 0 = 308 + 0.1(144 + 2Cov(X, Y) + 64)

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:

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