Sam bought 2 granola bars and Haley Bought 5 granola bars each granola bar is the same price right in expression for the total price of the granola bars then simplify the expression to create an equivalent expression

Answer:

20

Step-by-step explanation:

The question states that the sample size is 16 and standard deviation of sampling distribution of sample mean also known as standard error is 5. This information can be written as

σxbar=standard error=5 ,n=sample size=16.

We have to find population standard deviation σ.

We know that

[tex]Standard error=\frac{population standard deviation}{\sqrt{n} }[/tex]

[tex]population standard deviation=\sqrt{n} *(standard erorr)[/tex]

[tex]\sqrt{n} =\sqrt{16} =4[/tex]

Population standard deviation=σ=4*5=20

The percent error For Diego measurement is 4.3 % decrease

Solution:

Given that, Diego measured the length of a pain to be 22 cm

The actual length of the pen is 23 cm

To find: percent error

Percent error is the difference between a measured and actual value, divided by the actual value, multiplied by 100%

The formula for percent error is given as:

[tex]\text{Percent error } = \frac{\text{Measured value - actual value}}{\text{Actual value}} \times 100[/tex]

Here given that,

Measured value = 22 cm

Actual value = 23 cm

Substituting the values in formula,

[tex]Percent\ Error = \frac{22-23}{23} \times 100\\\\Percent\ Error = \frac{-1}{23} \times 100\\\\Percent\ Error = -0.043 \times 100\\\\Percent\ Error = -4.3[/tex]

Here, negative sign denotes percent decrease

Thus percent error For Diego measurement is 4.3 % decrease

Answer:

a)[tex]P(X>300)=P(\frac{X-\mu}{\sigma}>\frac{300-\mu}{\sigma})=P(Z>\frac{300-300}{25})=P(z>0)= 0.5[/tex]

[tex]P(X>335)=P(\frac{X-\mu}{\sigma}>\frac{335-\mu}{\sigma})=P(Z>\frac{335-300}{25})=P(z>1.4)=0.0808[/tex]

b)[tex]P(\bar X>300)=P(\frac{\bar X-\mu}{\sigma_{\bar x}}>\frac{300-\mu}{\sigma_{\bar x}})=P(Z>\frac{300-300}{17.5})=P(z>0)= 0.5[/tex]

[tex]P(\bar X>335)=P(\frac{\bar X-\mu}{\sigma_{\bar x}}>\frac{335-\mu}{\sigma_{\bar x}})=P(Z>\frac{335-300}{17.5})=P(z>2)=0.0228[/tex]

Step-by-step explanation:

Assuming the following questions:

a) Choose one twelfth-grader at random. What is the probability that his or her score is higher than 300? Higher than 335?

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

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

[tex]X \sim N(300,35)[/tex]  

Where [tex]\mu=300[/tex] and [tex]\sigma=35[/tex]

We are interested on this probability

[tex]P(X>300)[/tex]

And the best way to solve this problem is using the normal standard distribution and the z score given by:

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

If we apply this formula to our probability we got this:

[tex]P(X>300)=P(\frac{X-\mu}{\sigma}>\frac{300-\mu}{\sigma})=P(Z>\frac{300-300}{25})=P(z>0)= 0.5[/tex]

We find the probabilities with the normal standard table or with excel.

And for the other case:

[tex]P(X>335)=P(\frac{X-\mu}{\sigma}>\frac{335-\mu}{\sigma})=P(Z>\frac{335-300}{25})=P(z>1.4)=0.0808[/tex]

b) Now choose an SRS of four twelfth-graders. What is the probability that his or her mean score is higher than 300? Higher than 335?

For this case since the distribution for X is normal then the distribution for the sample mean is also normal and given by:

[tex] \bar X = \sim N(\mu = 300, \sigma_{\bar x} = \frac{35}{\sqrt{4}}=17.5)[/tex]

The new z score is given by:

[tex]z=\frac{\bar X -\mu}{\sigma_{\bar x}}[/tex]

And using the formula we got:

[tex]P(\bar X>300)=P(\frac{\bar X-\mu}{\sigma_{\bar x}}>\frac{300-\mu}{\sigma_{\bar x}})=P(Z>\frac{300-300}{17.5})=P(z>0)= 0.5[/tex]

We find the probabilities with the normal standard table or with excel.

And for the other case:

[tex]P(\bar X>335)=P(\frac{\bar X-\mu}{\sigma_{\bar x}}>\frac{335-\mu}{\sigma_{\bar x}})=P(Z>\frac{335-300}{17.5})=P(z>2)=0.0228[/tex]

Answer: 21.9 pounds of his weight is made up of fat.

Step-by-step explanation:

The total weight of the individual is 129 pounds. The individual has a body fat percentage of 17.7%.

Therefore, the number of pounds of his body that is made up of fat would be

17.7/100 × 129 = 0.177 × 129 = 21.93 pounds.

Approximating to the nearest tenth, it becomes 21.9 pounds.

Answer:

The answer to your question is the second option

Step-by-step explanation:

Process

Simplify using exponents laws, first inside the parentheses and then outside the parentheses.

                             [tex][\frac{a^{-2}b^{2}}{a^{2}b^{-1}} ]^{-3}[/tex]

a) Simplify a

                        a⁻² a⁻² = a⁻⁴

b) Simplify b

                        b² b¹ = b³

c) Write the result

                        [tex][\frac{b^{3}}{a^{4}}]^{-3}[/tex]

d)                      [tex][\frac{a^{4}}{b^{3}}]^{3}[/tex]

e) Simplify

                       [tex]\frac{(a^{4})^{3}}{(b^{3})3}[/tex]

f) Result

                      [tex]\frac{a^{12}}{b^{9}}[/tex]

The percent decrease of the number of students home sick is 24.14%.

Solution:

The number of students home this last week was 145

This week there were only 110 students home sick

To find: Percent decrease

The percent decrease is given by formula:

[tex]\text{Percent Decrease } = \frac{\text{Final value-initial value}}{\text{Initial value}} \times 100[/tex]

Here given that,

Initial value = last week = 145

Final value = this week = 110

Substituting the values in formula, we get,

[tex]\text{Percent Decrease } = \frac{110-145}{145} \times 100\\\\\text{Percent Decrease } = \frac{-35}{145} \times 100\\\\\text{Percent Decrease } = -24.14[/tex]

Here negative sign denotes decrease in percent

Thus the percent decrease of the number of students home sick is 24.14%.

Answer:

B. [tex]-6b-4[/tex]

Step-by-step explanation:

Given expression is [tex]-6(b+2)+8[/tex]

Simplifying the expression now

Lets use distributive property of multiplication to solve it.

As given, [tex]-6(b+2)+8[/tex]

Distributing -6 with b and 2.

= [tex](-6b-12)+8[/tex]

Opening the parenthesis

= [tex]-6b-12+8[/tex]

= [tex]-6b-4[/tex]

∴ As from the given options, [tex]-6b-4[/tex] is the correct choice.

Answer:

B

Step-by-step explanation:

Its right got it from khan

Answer:

in the year 2023

Step-by-step explanation:

Initial population of China = 1.355  billion

Initial population of India = 1.255  billion

Annual population growth rate of China = 0.44% = 0.0044

Annual population growth rate of India = 1.25% = 0.0125

Now,

Final population = P₀ [tex]\times e^{\text{rate}\times t}[/tex]

Here,

P₀ = initial population

t = time

Thus,

Population of India > Population of China

1.255[tex]\times e^{\text{0.0125}\times t}[/tex]  > 1.355[tex]\times e^{\text{0.0044}\times t}[/tex]  

or

[tex]e^{0.0081t}[/tex] > 1.07968

taking natural log both sides

0.0081t > ln (1.07968 )

or

0.0081t > 0.0766

or

t > 9.465

Hence,

9.465 year after 2014

i.e

in 2014 + 9.465 = 2023.46

in the year 2023

Using the formula for exponential growth and the provided population data for China and India, we can approximate that India's population will exceed China's in roughly 20 years.

In order to determine when India's population will exceed that of China, we can use the formula for exponential growth, which is P = P_0 * ert , where P is the future population, P_0 is the initial population, r is the growth rate, and t is time. As per the data provided, for China, P = 1.355 billion, r = 0.44%, and for India, P = 1.255 billion, r = 1.25%. We have to find time t when population of India will exceed that of China. This requires solving the equation: 1.355 * e0.0044t = 1.255 * e0.0125t.

This equation can be solved using algebra and logarithms. However, by making some approximations and using a spreadsheet or calculator, we can find that the population of India will exceed that of China after approximately 20 years, based on the growth rates provided.

https://brainly.com/question/18415071

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Answer:

Option C) population

Step-by-step explanation:

We are given the following situation in the question:

A researcher is curious about the average IQ of registered voters in the state of FL.

Sample:

It is a part of a population. It is always smaller than the population.

Statistic:

Any numerical value or any other measure describing a sample of a population is known as statistic.

Population:

It is the universal data set. Every observation belongs to this group which is of interest. Sampling is done within population to obtain small groups of sample.

Parameter:

Any numerical value or any other measure describing the population is known as a parameter.

In this situation, the entire group of registered voters in FL is an example of a population because it contains all the individual for variable of interest.

Variable of interest:

Average IQ of registered voters in the state of FL.

Individual of interest:

Registered voters in the state of FL.

Answer:

22.8 MPG

Step-by-step explanation:

356 divided by 15.6 = 22.8

Answer:

See explanation.

Step-by-step explanation:

Let us first analyze some principle theory. By definition we know that the velocity ( [tex]v[/tex] ) is a function of a distance  ( [tex]d[/tex] ) covered in some time  ( [tex]t[/tex] ), whilst acceleration ( [tex]a[/tex] ) is the velocity achieved in some time. These can also been expressed as:

[tex]v = \frac{d}{t}\\[/tex] and [tex]a=\frac{v}{t}[/tex]

We also know that both velocity and acceleration are vectors (therefore they are characterized by both a magnitude and a direction). Finally we know that given a position vector we can find the velocity and the acceleration, by differentiating the vector with respect to time, once and twice, respectively.

Let us now solve our problem. Here we are givine the Position vector of a particle P (in two dimensional space of [tex]i-j[/tex] ) as:

[tex]r=(3t^3-t+3)i+(2t^2+2t-1)j[/tex]         Eqn.(1)

Let us solve.

Part (a) Velocity: we need to differentiate Eqn.(1) with respect to time as:

[tex]v(t)=\frac{dr}{dt}\\\\ v(t)=[(3)3t^2-1]i+[(2)2t+2]j\\\\v(t)=(9t^2-1)i+(4t+2)j[/tex]            Eqn.(2)

Part (b) Acceleration: we need to differentiate Eqn.(2) with respect to time as:

[tex]a(t)=\frac{dv}{dt}\\ \\a(t)=[(2)9t]i+4j\\\\a(t)=(18t)i+4j[/tex]

Thus the expressions for the velocity and the acceleration of particle P in terms of t are

[tex]v(t)=(9t^2-1)i+(4t+2)j[/tex]   and   [tex]a(t)=(18t)i+4j[/tex]

Answer: 240

Step-by-step explanation:

Answer:

Non Arguement passage.

Step-by-step explanation:

The passage given is a non arguement passage , the passage is more of a report especially the introductory part where the author said ''Since the 1950s a malady called whirling disease has invaded U.S. fishing streams, frequently attacking rainbow trout.'' this highlighted phrase is a report gathered or investigated by the author which was gotten as a result of his own personal findings or from history. For an argument passage, the introductory part will have portrayed what the author implied, there will be an indication of the authors stance or favoured opinion which of course will be backed by evidence from his or her findings. as such, there is nothing of such which may serve as a precursor to indicate or informed us if the passage is that of an arguement. Again, the passage is a report and not an argument. as nothing can be inferred from the paragraph to point to us if it is an argument passage.

However, there is a conclusion in the passage and conclusions has arrived by the author must have been from a detailed findings and research, if possible an experimental study before a conclusion can be reached as the last line of the paragraph says ''A parasite deforms young fish, which often chase their tails before dying, hence the name.'' The conclusion is that parasite are known to cause deformation in young fish.

An inequality that shows the distance Johnathan could of ran any day this week is:

[tex]x\leq 3.5[/tex]

Solution:

Let "x" be the distance Johnathan can run any day of this week

Given that,

Johnathan ran 5 days this week. The most he ran in one day was 3.5 miles

Therefore,

Number of days ran = 5

The most he ran in 1 day = 3.5 miles

Thus, the maximum distance he ran in a week is given as:

[tex]distance = 5 \times 3.5 = 17.5[/tex]

The maximum distance he ran in a week is 17.5 miles

If we let x be the distance he can run any day of this week then, we get a inequality as:

[tex]x\leq 3.5[/tex]

If we let y be the total distance he can travel in a week then, we may express it as,

[tex]y\leq 17.5[/tex]

Answer:

Mass of the cat  = 16800kg

Step-by-step explanation:

The volume of the box = its length * its breath * its height- 4m × 3m × 2m = 24[tex]m^{3}[/tex]

The standard density of water is 1000 Kg/[tex]m^{3}[/tex]

By Archimedes principle, the mass of a body  floating body is equivalent to the mass of the volume liquid displaced

in this case we have

Mass of water displaced = Density of the water × Volume of the water  displaced =   1000 Kg/[tex]m^{3}[/tex] × 24[tex]m^{3}[/tex] = 24000kg

The mass of the box = Box density × Box volume = 24[tex]m^{3}[/tex] × 300kg/[tex]m^{3}[/tex] = 7200 kg Hence the mass of water displaced = mass of the foating box + mass of th cat

24000kg =  mass of cat +7200kg

mass of cat = 24000kg - 7200kg = 16800kg

Answer:

(-9,-9)

Step-by-step explanation:

Answer:

(-9,-9)

Step-by-step explanation:

Answer:

The answer to your question is

                       Number of stickers = number of days + 3

Step-by-step explanation:

- To find the equation of the line that represents the situation, first, find the slope.

Slope = m = [tex]\frac{y2 - y1}{x2 - x1}[/tex]

             m = [tex]\frac{4 - 3}{1 - 0} = \frac{1}{1} = 1[/tex]

- Find the equation of the line

                      y - y1 = m(x - x1)

                      y - 4 = 1(x - 1)

                      y - 4 = x - 1

                           y = x - 1 + 4

                           y = x + 3

y = number of stickers

x = days

                     Number of stickers = number of days + 3

Answer:

is x = y + 3

Step-by-step explanation:

See the attached picture.

Answer could change slightly depending on how all the steps are rounded.

Answer:

The constant of proportionality is 2.5 dollar per cupcake and then the required equation would be [tex]y=2.5x[/tex].

Step-by-step explanation:

Given:

Let the number of cupcakes be represent by [tex]'x'[/tex]

Also let the total cost be represented by [tex]'y'.[/tex]

We know that two proportional quantities are in for;

[tex]y=kx[/tex]

where, k⇒ represents constant of proportionality.

Now we know that;

1 dozen = 12

Half dozen = 6

Now Given:

half dozen cupcakes cost $15.

So Let us substitute [tex]x=6[/tex] and  [tex]y=15[/tex] in above equation we get;

[tex]15 =k \times 6[/tex]

Dividing both side by 6 we get;

[tex]\frac{15}{6}=\frac{k6}{6}\\\\k= 2.5 \ \$/cupcake[/tex]

Hence the constant of proportionality is 2.5 dollar per cupcake and then the required equation would be [tex]y=2.5x[/tex].

Final answer:

The constant of proportionality relating the number of cupcakes to the cost is $2.50 per cupcake. The equation representing the relationship is C = $2.50 × n.

Explanation:

To find the constant of proportionality for the number of cupcakes and total cost, we use the given information: half a dozen cupcakes (which is 6 cupcakes) cost $15. Therefore, we can divide 15 by 6 to find the cost per cupcake.
C = k × n

Where C is the total cost, n is the number of cupcakes, and k is the constant of proportionality (cost per one cupcake). First, find the constant:
k = C/n = $15/6 cupcakes = $2.50 per cupcake

The equation that represents the relationship between the number of cupcakes (n) and the total cost (C) is:
C = $2.50 × n

When analyzing data on homes in a neighborhood, focus on variables like value, size, year built, and basement presence. The sample data set of lawn areas is quantitative and continuous.

Data Analysis Process:

When collecting data on homes in a neighborhood, you can analyze it by looking at variables like value, size, year built, and basement presence. To make sense of the data, you can calculate statistics like the median home value and the variation in values to get a clearer picture of the housing market.

Data Types:

The areas of lawns in square feet sampled from five houses represent quantitative data, specifically continuous data, as they can take on any value within a range without restrictions.

Answer:36 burgers were sold.

Step-by-step explanation:

Let x represent the total number of burgers, with or without cheese that was sold.

The total number of burgers with cheese that the fast food sold is 30.

if the ratio burger sold with cheese compared to without cheese was

5 : 1 , the total ratio would be the sum of both proportions. It becomes

5 + 1 = 6

Therefore

5/6 × x = 30

5x/6 = 30

Cross multiplying by 6, it becomes

5x = 30 × 6 = 180

x = 180/5 = 36

Therefore, the number of burgers without cheese sold would be

36 - 30 = 6

Answer:

−438°, -78°, 642°

Step-by-step explanation:

Given angle:

282°

To find the co-terminal angles of the given angle.

Solution:

Co-terminal angles are all those angles having same initial sides as well as terminal sides.

To find the positive co-terminal of an angle between 360°-720° we will add the angle to 360°

So, we have: [tex]282\°+360\°=642\°[/tex]

To find the negative co-terminal of an angle between 0° to -360° we add it to -360°

So, we have:  [tex]282\°-360\°=-78\°[/tex]

To find the negative co-terminal of an angle between -360° to -720° we add it to -720°

So, we have:   [tex]282\°-720\°=-438\°[/tex]

Thus, the co-terminal angles for  282° are:

−438°, -78°, 642°

The correct coterminal angles with 282° are -78° option(2) and 642° option(5). The angles -438°, 78°, and 572° are not coterminal with 282°.

In mathematics, coterminal angles are angles that share the same initial and terminal sides. To find coterminal angles for a given angle, we add or subtract multiples of 360° (a full rotation). For the angle 282°:

1. Subtracting 360°:

⇒ 282° - 360° = -78°

2. Adding 360°:

⇒ 282° + 360° = 642°

Therefore, the angle measures -78° and 642° are coterminal with 282°. The other values, -438°, 78°, and 572°, are not coterminal with 282° as they do not share the same position after full rotations.

In summary, the correct coterminal angles with 282° are -78° and 642°.

Complete question:

Which angles are coterminal with an angle drawn in standard position measuring 282°?

Select all correct angle measures.

Answer:

k = -12/5

A = 125/12

B = -325/12

a = 5

Step-by-step explanation:

y = e^5t

Dy/dt = 5e^5t

d2y/dt2 = 25e^5t

Inputting the values of dy/dt and d2y/dt2 into the equation above, we have:

25e^5t - 13e^5t + 5k(e^5t) = 0

12e^5t + 5k(e^5t) = 0

e^5t(12 + 5k) = 0

12 + 5k = 0

k = -12/5

The equation becomes,

d2y/dt2 - 13dy/dt -12/5y = o

So rearranging the equation,

-5/12d2y/dt2 + 65/12dy/dt + y = 0

y = 5/12(25e^5t) - 65/12(5e^5t)

y = 125/12e^5t - 325/12e^5t

Therefore,

k = -12/5

A = 125/12

B = -325/12

a = 5

The value of the constant k is 40. The general form of the solution is y = A[tex]e^{5t}[/tex] + B[tex]e^{8t}[/tex]

Let's start by recognizing that if y = [tex]e^{5t}[/tex] is a solution to the differential equation d²y/dt² - 13 dy/dt + ky = 0, we need to find the value of the constant k and the general form  y = A[tex]e^{5t}[/tex] + B[tex]e^{at}[/tex] of the solution. To do this, we need to determine k and a.

1. First, calculate the first and second derivatives of y = [tex]e^{5t}[/tex]

2. Substitute these into the differential equation:

d²y/dt² - 13 dy/dt + ky = 0

3. Substituting, we get:

25[tex]e^{5t}[/tex] - 13(5[tex]e^{5t}[/tex]) + k[tex]e^{5t}[/tex] = 0

25[tex]e^{5t}[/tex] - 65[tex]e^{5t}[/tex] + k[tex]e^{5t}[/tex]= 0

(25 - 65 + k)[tex]e^{5t}[/tex] = 0

(-40 + k)[tex]e^{5t}[/tex] = 0

4. For this to hold true, the following must be true:

k - 40 = 0

Thus:

k = 40

y = A[tex]e^{5t}[/tex] + B[tex]e^{at}[/tex]

1. To find a, substitute y = [tex]e^{at}[/tex] into the equation:

d²([tex]e^{at}[/tex])/dt² - 13 d([tex]e^{at}[/tex])/dt + 40[tex]e^{at}[/tex] = 0

We get:

a²[tex]e^{at}[/tex]- 13a[tex]e^{at}[/tex] + 40[tex]e^{at}[/tex]= 0

(a² - 13a + 40)[tex]e^{at}[/tex] = 0

2. The characteristic equation is:

a² - 13a + 40 = 0

3. Solve for a using the quadratic formula:

a = [13 ± √(13² - 4⋅40)] / 2

a = [13 ± √(169 - 160)] / 2

a = [13 ± √9] / 2

a = [13 ± 3] / 2

4. The roots are:

Since y = [tex]e^{5t}[/tex] is a solution already, the other root a = 8 is the additional solution. Thus, the general solution to the differential equation is:

y = A[tex]e^{5t}[/tex] + B[tex]e^{8t}[/tex].

Answer:

3.93

Step-by-step explanation:

Let x is the number of hours of television watched per week by six students.

X 8 10 5 14 3 6

Standard deviation for sample data is

[tex]Standard deviation=S=\sqrt\frac{{sum(x-xbar)^2} }{n-1}[/tex]

[tex]xbar=\frac{sum(x)}{n}[/tex]

[tex]xbar=\frac{8+10+5+14+3+6}{6}[/tex]

[tex]xbar=\frac{46}{6}[/tex]

xbar=7.67

[tex]sum(x-xbar)^2=(8-7.67)^2+(10-7.67)^2+(5-7.67)^2+(14-7.67)^2+(3-7.67)^2+(6-7.67)^2[/tex][tex]sum(x-xbar)^2=0.11+5.44+7.11+40.11+21.78+2.78=77.33[/tex]

[tex]Standard deviation=S=\sqrt\frac{{(77.33)} }{5}[/tex]

[tex]S=\sqrt15.466[/tex]

S=3.93

Answer: The resultant would be the sum and the difference between the vectors.

Step by step explanation: 1. The possible resultant is between the sum of the 2 vectors and the difference between the two vectors.

2. The greatest magnitude is when the vectors lie in the same direction and the sum would be the scalar sum of the two vectors. The angle between the two would be zero degree.

Answer:they would pay a total amount of $9

Step-by-step explanation:

Scott and Tom rent a boat at Stow Lake. They start at 10:15 and end at 11:45. The number of hours for which they rented the boat would be

11:45 - 10:15 = 1 hour 30 minutes = 1.5 hours.

Converting to minutes, It becomes

60 + 30 = 90 minutes.

The boat rental costs $1.50 for every 15 minutes. Therefore, the total amount of money that they would pay is

90/15 × 1.5 = $9

Answer:

(a) -7 , - 9 , - 11

(b) Arithmetic sequence

(c) There is a common difference of -2

(d) -53

Step-by-step explanation:

(a) To find the next three terms , we must firs check if it is arithmetic sequence or a geometric sequence . For it to be an arithmetic sequence , there must be a common difference :

check :

-3 - (-1) = -5 - (-3) = -7 - (-5)  = -2

This means that there is a common difference of -2 , which means it is an arithmetic sequence.

The next 3 terms we are to find are: 5th term , 6th term and 7th term.

[tex]t_{5}[/tex] = a + 4d

[tex]t_{5}[/tex] = - 1 + 4 ( -2 )

[tex]t_{5}[/tex] = -1 - 8

[tex]t_{5}[/tex] = - 9

6th term = a +5d

[tex]t_{6}[/tex] = -1 + 5(-2)

[tex]t_{6}[/tex] = -1 - 10

[tex]t_{6}[/tex] = - 11

[tex]t_{7}[/tex] = a + 6d

[tex]t_{7}[/tex] = -1 + 6 (-2)

[tex]t_{7}[/tex] = -1 - 12

[tex]t_{7}[/tex] = -13

Therefore : the next 3 terms are : -9 , -11 , - 13

(b) it is an arithmetic sequence because there is a common difference which is -2

(c) Because of the existence of common difference

(d) [tex]t_{27}[/tex] = a + 26d

[tex]t_{27}[/tex] = -1 + 26 ( -2 )

[tex]t_{27}[/tex] = -1 - 52

[tex]t_{27}[/tex] = - 53

Answer:

Therefore the equation of the required line is y = [tex]\frac{-1}{2}[/tex]x + 2 or 2y + x = 4.

Step-by-step explanation:

i) when x = -2 then y = 3 so the line from x = -2 to x = 2 has the point (-2, 3)

ii)when x = 2 then y = 1 so the line from x = -2 to x = 2 has the point (2, 1)

iii) if two points in a line are given then slope of equation passing through the lines is given by

slope m = [tex]\frac{y_{2} - y_{1} }{x_{2} - x_{1} }[/tex] = [tex]\frac{1 - 3}{2 - (-2)}[/tex] = [tex]\frac{-2}{4}[/tex]  =  [tex]\frac{-1}{2}[/tex]

So from the general equation of a line y = mx + c

we get y = [tex]\frac{-1}{2}[/tex]x + c and substituting for x and y with (-2, 3) respectively we get

3 = 1 + c. Therefore c = 2.

Therefore the equation of the required line is y = [tex]\frac{-1}{2}[/tex]x + 2 or 2y + x = 4.

Answer:

A. -25/27

Step-by-step explanation:

Given:

The equation is given as:

[tex]x^2+y^2=25[/tex]

To find: [tex]\frac{d^2 y}{dx^2}[/tex] at (4, 3)

Differentiating the above equation with respect to 'x', we get:

[tex]\frac{d}{dx}(x^2+y^2)=\frac{d}{dx}(25)\\\\2x+2yy'=0\\\\x+yy'=0\\\\yy'=-x\\\\y'=\frac{-x}{y}------- (1)[/tex]

Value of [tex]y'[/tex] at (4,3) is given as:

[tex]y'_{(4,3)}=-\dfrac{4}{3}-------- (2)[/tex]

Now, differentiating equation (1) with respect to 'x' again, we get:

[tex]y''=\frac{d}{dx}(\frac{-x}{y})\\\\y''=\frac{y(-1)-(-x)y'}{y^2}\\\\y''=\frac{-y+xy'}{y^2}[/tex]

Now, value of [tex]y''[/tex] at (4,3) is given as by plugging 4 for 'x', 3 for 'y' and [tex]\frac{-4}{3}[/tex] for [tex]y'[/tex]

[tex]y''_{(4,3)}=\frac{-3+(4)(-\frac{4}{3})}{3^2}\\\\y''_{(4,3)}=\frac{-3-\frac{16}{3}}{9}\\\\y''_{(4,3)}=\frac{-9-16}{3}\div 9\\\\y''_{(4,3)}=\frac{-25}{3}\div 9\\\\y''_{(4,3)}=\frac{-25}{3}\times \frac{1}{9}\\\\y''_{(4,3)}=-\frac{25}{27}[/tex]

Therefore, the value of the second derivative at (4, 3) is option (A) which is equal to -25/27.

Answers:

1) [tex]x^{8} y^{8}[/tex]

2) [tex]y^{3} \sqrt{y}[/tex]

3) [tex]5x^{4} \sqrt{6}[/tex]

4) [tex]\sqrt{7}[/tex]

5) [tex]\frac{\sqrt{z}}{z}[/tex]

Step-by-step explanation:

1) [tex]\sqrt{x^{16} y^{36}}[/tex]

Rewriting the expression:

[tex](x^{16} y^{36})^{\frac{1}{2}}[/tex]

Multiplying the exponents:

[tex]x^{\frac{16}{2}} y^{\frac{36}{2}}[/tex]

Simplifying:

[tex]x^{8} y^{8}[/tex]

2) [tex]\sqrt{y^{7}}[/tex]

Rewriting the expression:

[tex]\sqrt{y^{6} y}=(y^{6} y)^{\frac{1}{2}}[/tex]

Multiplying the exponents:

[tex]y^{\frac{6}{2}} y^{\frac{1}{2}}[/tex]

Simplifying:

[tex]y^{3} y^{\frac{1}{2}}=y^{3} \sqrt{y}[/tex]

3) [tex]\sqrt{150 x^{8}}[/tex]

Rewriting the expression:

[tex]\sqrt{(6)(25) x^{8}}[/tex]

Since [tex]\sqrt{25}=5[/tex]:

[tex]5x^{4}\sqrt{6}[/tex]

4) [tex]\frac{7}{\sqrt{7}}[/tex]

Multiplying numerator and denominator by [tex]\sqrt{7}[/tex]:

[tex]\frac{7}{\sqrt{7}} (\frac{\sqrt{7}}{\sqrt{7}})=\frac{7}{7\sqrt{7}}[/tex]

Simplifying:

[tex]\sqrt{7}[/tex]

5) [tex]\frac{5z}{\sqrt{25 z^{3}}}[/tex]

Rewriting the expression:

[tex]\frac{5z}{5z \sqrt{z}}[/tex]

Simplifying:

[tex]\frac{1}{\sqrt{z}}[/tex]

Since we do not want the square root in the denominator, we can multiply numerator and denominator by [tex]\sqrt{z}[/tex]:

[tex]\frac{1}{\sqrt{z}}(\frac{\sqrt{z}}{\sqrt{z}})[/tex]

Finally:

[tex]\frac{\sqrt{z}}{z}[/tex]