a. Find all regular points of the set {x E R3|x1 + x2 + x3 = 3, x1 + x2 + xy = 3}.
b. Find all irregular points of the set {x E R4 | 2(x1 + x2 + x3)3 +3(x1+x2 + x3)2 = 1, xỉ + x2+xz+x4 = 1}.

The margin of error with 99% confidence is 62.15 (rounded to 2 decimal places).

a. To calculate the margin of error with 99% confidence, we first need to find the sample mean and the sample standard deviation.

The sample mean is: 162.5

where z* is the z-score for 99% confidence level and n is the sample size.

From the z-score table, we find that the z-score for 99% confidence level is 2.576.

Thus, the margin of error is:

ME = 62.15

Therefore, the margin of error with 99% confidence is 62.15 (rounded to 2 decimal places).

b. To construct the 99% confidence interval for the average number of customers served on weekdays,

CI is the confidence interval, x is the sample mean, z is the z-score for 99% confidence level, s is the sample standard deviation, and n is the sample size.

Substituting the values, we get:

CI = = (89.29, 235.71)

Therefore, the 99% confidence interval for the average number of customers served on weekdays is (89.29, 235.71).

The margin of error reported in part a can be reduced by either increasing the sample size or reducing the variability in the data.

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The first step in the construction is: A) place the compass point at A. For the construction of the line segment containing point A and and perpendicular to segment BC using straight edge and compass we have to follow the steps as:

1) Place the point of the compass on the given point and draw a arc on the line on either side of the given point.

2) Then increase the width of the compass and place the point on the compass on the new point where the above arc intersect the line segment. and make arc from both the points.

3) Join the point of intersection of the new arc to the original point A and hence obtain the perpendicular line.

Hence, the first step in the construction is:

A) place the compass point at A.

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Full Question ;

Suppose we wish to construct a line segment containing point A and perpendicular to segment BC. To do so with the fewest compass measurements, we should first A) place the compass point at A. B) place the compass point at B. C) place the straightedge on segment BC. D) place the straightedge on points A and B.

An advantage of stem-and-leaf plots compared to most frequency distributions is that provide more information about the distribution of the data.

Stem-and-leaf plots offer several advantages over most frequency distributions.

One advantage is that stem-and-leaf plots provide a more detailed representation of the data than frequency distributions.

They allow you to see the individual data values and their magnitudes, which can provide more information about the distribution, such as the spread, central tendency, and outliers.

Additionally, stem-and-leaf plots can be easier to read and interpret than frequency distributions, especially for small data sets.

They can reveal patterns and trends in the data that might not be apparent in a frequency distribution.

Finally, stem-and-leaf plots can be used to compare different data sets or to identify similarities or differences within a single data set.

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The volume of the balloon when the atmospheric pressure is 1.20 atm = 144.08 ml

We know that the Boyle's Law states that the pressure is inversely proportioal to the volume.

From Boyle's Law also states: PV = k

where P is pressure,

V is volume

and k = the proportionality constant

Using Boyle's law we get an equation,

P₁V₁ = P₂V₂

Let P₁ = 0.988 atm,

V₁ = 175 ml

P₂ = 1.20 atm

V₂ = ?

Substitute these values in above equation,

0.988 × 175 = 1.20 × V₂

V₂ = 172.9 / 1.20

V₂ = 144.08 ml

This is the required volume of balloon.

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The complete question is:

A weather balloon has a volume of 175 L when filled with hydrogen at a pressure of 0.988 atm. Calculate the volume of the balloon when the atmospheric pressure is 1.20 atm. Assume that temperature is constant.

Answer:

x ≈ 62,15°

Step-by-step explanation:

Given:

A right triangle

FG = 8,4

FH = 9,5

Find: x - ?

Use trigonometry:

[tex] \sin(x°) = \frac{fg}{fh} [/tex]

[tex] \sin(x°) = \frac{8.4}{9.5} ≈0.8842[/tex]

[tex]x≈62.15°[/tex]

a, b, c, and d are distinct real numbers, the terms involving products of three distinct numbers (abc, abd, acd, bcd) are all non-zero. The given equation cannot be factored into linear factors and is irreducible over the real numbers.

The given equation can be simplified using the distributive property of multiplication and combining like terms:

(x - b)(x - c)(x - d) + (x - a)(x - c)(x - d) + (x - a)(x - b)(x - d) + (x - a)(x - b)(x - c)

Expanding each of the terms gives:

(x^3 - (b+c+d)x^2 + (bc+cd+bd)x - bcd) + (x^3 - (a+c+d)x^2 + (ac+cd+ad)x - acd) + (x^3 - (a+b+d)x^2 + (ab+bd+ad)x - abd) + (x^3 - (a+b+c)x^2 + (ab+ac+bc)x - abc)

Combining like terms gives:

4x^3 - 2(a+b+c+d)x^2 + 3(ab+ac+ad+bc+bd+cd)x - 6abc - 6abd - 6acd - 6bcd

Since a, b, c, and d are distinct real numbers, the terms involving products of three distinct numbers (abc, abd, acd, bcd) are all non-zero. The given equation cannot be factored into linear factors and is irreducible over the real numbers.

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If a, b and c are distinct real numbers, prove that the equation

(x−a)(x−b)+(x−b)(x−c)+(x−c)(x−a)=0

has real and distinct roots.

The equations have been solved below

The solutions to the equations is as follows;

1) w + 7/8 = 5

w = 5 - 7/8

w = 40 - 7/8

w = 33/8 or 4 1/8

2) 5h - 2/3 = 6

5h = 6 + 2/3

5h = 18 + 2/3

5h = 20/3

h = 20/3 * 1/5

h = 4/3 = 1 1/3

3) v/9 + 2 = 7

v/9 = 7 - 2

v/9 = 5

v = 9 * 5

v = 45

4) 4r/3 - 1 = 8/3

4r = 8/3 + 1

4r = 8 + 3/3

4r = 11/3

r = 11/3 * 1/4

r = 11/12

5) 5y = 13/4

y = 13/4 * 1/5

y = 13/20

6) 3f/2 + 1/2 = 7/2

3f/2 = 7/2 - 1/2

3f/2 = 6/2

3f/2 = 3

f = 3 * 2/3

f = 2

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Answer:

Each of the four arrangements contains 4 gray tiles.

Step-by-step explanation:

In each of the four arrangements, the black and gray square tiles are arranged in a 2 x 2 grid. Since one of the tiles is black, the remaining three tiles are gray. Therefore, each of the four arrangements contains three gray tiles.

The surface area of a rectangular prism of dimensions 14 cm, 4.5 cm and 32 cm is given as follows:

1310 cm².

The surface area of a rectangular prism of height h, width w and length l is given by:

S = 2(hw + lw + lh).

This means that the area of each rectangular face of the prism is calculated, and then the surface area is given by the sum of all these areas.

The dimensions for this problem are given as follows:

14 cm, 4.5 cm and 32 cm.

Hence the surface area of the prism is given as follows:

S = 2 x (14 x 4.5 + 14 x 32 + 4.5 x 32)

S = 1310 cm².

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Meg can get 1 and 3/4 servings of 1/2 jug from her 7/8 jug of orange juice.

Meg has a 7/8 jug of orange juice, and she wants to know how many 1/2 jug servings she can get from it. To solve this problem, we need to divide the total amount of orange juice by the amount of orange juice in each serving.

First, we need to convert the 7/8 jug to an equivalent fraction with a denominator of 2. To do this, we can multiply both the numerator and denominator of 7/8 by 2, which gives us 14/16.

Next, we can divide 14/16 by 1/2 to find out how many 1/2 jug servings Meg can get from the jug. To divide fractions, we invert the second fraction and multiply. So we have:

14/16 ÷ 1/2 = 14/16 x 2/1 = 28/16

Now, we need to simplify this fraction by dividing the numerator and denominator by their greatest common factor, which is 4. So we have:

28/16 = (28 ÷ 4) / (16 ÷ 4) = 7/4

Therefore, Meg can get 7/4 or 1 and 3/4 servings of 1/2 jug from her 7/8 jug of orange juice.

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The appropriate confidence interval for determining the percent of American adults who believe in the existence of angels would be a confidence interval for a population proportion. This is because we are interested in the proportion or percentage of American adults who hold a particular belief.

A confidence interval is a range of values that we can be reasonably sure contains the true population parameter. In this case, we want to estimate the proportion of American adults who believe in angels and we can use statistical methods to estimate this parameter.

A confidence interval for a population proportion is typically calculated using the sample proportion and the sample size. The margin of error is also taken into consideration when calculating the interval. This type of interval would allow us to estimate the proportion of American adults.

It is important to note that the confidence interval only gives us an estimate of the population parameter and not an exact value. The confidence level indicates how confident we can be that the true population parameter falls within the interval.

In conclusion, to determine the percent of American adults who believe in the existence of angels, an appropriate confidence interval would be a confidence interval for a population proportion. This would provide us with an estimate of the proportion with a certain level of confidence.

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When using the "rule of thirds" when examining an extremity, the bone is divided into thirds. Therefore, the correct option is option C.

First aid is the initial and urgent help provided to anyone who has a little or major disease or injury,[1] with the goal of preserving life, preventing the condition from getting worse, or promoting recovery until medical help arrives. First aid is typically administered by a person with only little medical training. The idea of first aid is expanded to include mental health in mental health first aid. When using the "rule of thirds" when examining an extremity, the bone is divided into thirds.

Therefore, the correct option is option C.

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The probability of the electron tunneling through the barrier of width 0.80 nm is 0.024, or 2.4%.

The probability of the electron tunneling through the barrier of width 0.40 nm is 0.155, or 15.5%.

The probability of an electron tunneling through a barrier can be calculated using the transmission coefficient:

[tex]T = e^(-2κL)[/tex]

where c, L is the width of the barrier, and e is the base of the natural logarithm.

The wave vector can be calculated using the following formula:

κ = sqrt(2m(E - V))/ħ

where m is the mass of the electron, E is the energy of the incident electron, V is the height of the barrier, and ħ is the reduced Planck constant.

Substituting the given values:

m = 9.10938356 × 10^-31 kg (mass of electron)

E = 6.0 eV (energy of incident electron)

V = 11.0 eV (height of the barrier)

[tex]ħ = 1.054571817 × 10^-34 J s (reduced Planck constant)[/tex]

a) For a barrier width of 0.80 nm:

[tex]L = 0.80 × 10^-9 m[/tex]

[tex]κ = sqrt(2 × 9.10938356 × 10^-31 kg × (6.0 eV - 11.0 eV))/1.054571817 × 10^-34 J s[/tex]

= 2.317 × 10^10 m^-1

[tex]T = e^(-2κL) = e^(-2 × 2.317 × 10^10 m^-1 × 0.80 × 10^-9 m)[/tex]

[tex]= e^(-3.731)[/tex]

= 0.024

Therefore, the probability of the electron tunneling through the barrier is 0.024, or 2.4%.

b) For a barrier width of 0.40 nm:

L = 0.40 × 10^-9 m

[tex]κ = sqrt(2 × 9.10938356 × 10^-31 kg × (6.0 eV - 11.0 eV))/1.054571817 × 10^-34 J s[/tex]

[tex]= 2.317 × 10^10 m^-1[/tex]

[tex]T = e^(-2κL) = e^(-2 × 2.317 × 10^10 m^-1 × 0.40 × 10^-9 m)[/tex]

[tex]= e^(-1.866)[/tex]

= 0.155

Therefore, the probability of the electron tunneling through the barrier is 0.155, or 15.5%.

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The area of the region that lies inside the circle r=3sinΘ and outside the cardioid r=1+sinΘ is 9π/4 - √3/2.

To find the area, we first need to determine the values of θ at which the two curves intersect. Setting r=3sinΘ equal to r=1+sinΘ, we get sinΘ = 1/2, which gives Θ = π/6 and Θ = 5π/6.

Next, we can use the area formula for polar coordinates: A=1/2∫βα(f(θ))2dθ. Since the cardioid is inside the circle for Θ between π/6 and 5π/6, we need to find the area of the circle minus the area of the cardioid. Thus, we have:

A = 1/2 [(∫0^(π/6) (3sinΘ)^2 dΘ) + (∫5π/6^π (3sinΘ)^2 dΘ) - (∫π/6^(5π/6) (1+sinΘ)^2 dΘ)]

Simplifying and evaluating the integrals, we get: A = 9π/4 - √3/2

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To find the probability that a plant chosen at random will be between and cm tall, we need to use the normal distribution formula. We know that the mean height of the plant is cm and the standard deviation is cm.

First, we need to standardize the values of and by subtracting the mean and dividing by the standard deviation:

Z1 = ( - ) / = ( - ) /
Z2 = ( - ) / = ( - ) /

Next, we use a standard normal distribution table or calculator to find the area under the curve between these two Z-values. This area represents the probability that a plant chosen at random will have a height between and cm.

Alternatively, we can use the normal distribution function on a calculator or software to find the probability directly. The formula for the normal distribution function is:

P( < X < ) = 1/2[erf(( - )/sqrt(2)) - erf(( - )/sqrt(2))]

where erf is the error function.

Using either method, we can find that the probability that a plant chosen at random will be between and cm tall is approximately %.

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The value of angle FCE is 100⁰.

The value of arc DE is 125⁰.

The value of angle DCA is 125⁰.

The value of arc FAE is 180⁰.

The measure of angle FCE is calculated by applying the following formula.

Based on the angle of intersecting chord theorem, we will have the following equation.

m∠ECB = 25⁰ (intersecting chord theorem)

m∠FCE = 180 - (55 + 25) (sum of angles on a straight line)

m∠FCE = 100⁰

Angle DCE = FCE + FCD

FCD = 55 (vertical opposite angles)

Angle DCE = 100 + 55 = 155⁰

Arc DE = 155⁰ (intersecting chord theorem)

Angle DCA = 360 - (155 + 25 + 55) = 125⁰

Arc FAE = 180 (semi circle)

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TThe first term converges to zero by the p-series test, while the second and third terms diverge. Therefore, the original series diverges.

To determine the convergence/divergence of the given series, we can apply various convergence/divergence tests. For instance, the series sin(n) is oscillatory and therefore does not converge. The series 1/n! converges by the ratio test or the root test, as both approaches lead to the limit zero. The series 1/n(n+2) is telescoping and can be written as a difference of two terms, which makes it convergent. The series n^2/(n^3+1) can be bounded by a p-series with p=2, so it also converges. The series In(n) diverges by the integral test, as the function ln(x) increases without bound as x approaches infinity.

The series with terms given by the expression 20907 + n^3/n^(1/3) + n^5/n^2 can be simplified by dividing each term by n^(5/3), leading to the series 20907/n^(5/3) + n^(4/3) + n^(10/3).

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The correct option will be option C: Multiply/divide both sides by the same non-zero constant.

To solve the linear equation of one variable;

Step-1: we have to balance each side by simplifying the equation

Step-2: add/subtract constant term on both sides of the equation to separate variable and constant term on both sides

Step-3: divide the coefficient of the variables on both sides.

So according to the question,

the given equations are:

AAA: 3(x+2)=18

BBB: x+2=6

We have to find a way from equation AAA to Equation BBB

from the above equation, it is clear BBB is factor AAA.

to get BBB from equation AAA, we have to just divide 3 on both sides of the equation AAA.

Therefore option C, will be correct as 3 is a non-zero constant. we have to divide both sides by this same non-zero constant.

Therefore The correct option will be option C:

Multiply/divide both sides by the same non-zero constant.

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Lawnco should produce 300 bags of grade a, 400 bags of grade b, and 200 bags of grade c fertilizer in order to meet the given nutrient requirements.

Let x, y, and z denote the number of 100-lb bags of grade a, b, and c fertilizer respectively.
Then, we can create the following system of equations based on the given information:
16x + 20y + 24z = 26200 (total nitrogen)
6x + 4y + 3z = 4700 (total phosphate)
7x + 4y + 6z = 6600 (total potassium)
Solving this system of equations, we get:
x = 300 (number of bags of grade a)
y = 400 (number of bags of grade b)
z = 200 (number of bags of grade c)
To find out how many 100-lb bags of each of the three grades of fertilizers Lawnco should produce, we need to set up a system of linear equations using the given information and solve for a, b, and c.
Equation 1 (nitrogen): 16a + 20b + 24c = 26,200
Equation 2 (phosphate): 6a + 4b + 3c = 4,700
Equation 3 (potassium): 7a + 4b + 6c = 6,600
Solving this system of linear equations will give you the number of bags of grade A, B, and C fertilizers Lawnco should produce to use all available nutrients.

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The limit of [tex](2x^2-y^2)^{11}/(x^2+2y^2)[/tex] as (x,y) approaches (0,0) is path-dependent and does not exist. The paths y=mx and x=my are used to demonstrate this. The expression approaches a value that depends on the constant m and the chosen path.

To show that the limit does not exist, we need to find two paths to the origin along which the limit has different values. Consider the path y = mx, where m is a constant. As (x,y) approaches (0,0) along this path, we have:

[tex](2x^2 - y^2)^{(11)} / (x^2 + 2y^2)[/tex]

[tex]= (2x^2 - (mx)^2)^{(11)} / (x^2 + 2(mx)^2)[/tex]

[tex]= (2 - m^2)^{11} / (1 + 2m^2)[/tex]

As x approaches 0, this expression approaches [tex](2 - m^2)^{11} / (2m^2)[/tex], which depends on the value of m. Thus, the limit depends on the path chosen, and so the limit does not exist.

Similarly, we can consider the path x = my, where m is a constant, and obtain:

[tex](2x^2 - y^2)^{(11)} / (x^2 + 2y^2)[/tex]

[tex]= (2(my)^2 - y^2)^{(11)} / (m^2y^2 + 2y^2)[/tex]

[tex]= (2m^2 - 1)^{11} / (m^2 + 2)[/tex]

As y approaches 0, this expression approaches[tex](2m^2 - 1)^{11} / 2m^2[/tex], which again depends on the value of m. Therefore, the limit does not exist.

In summary, we showed that the limit of[tex](2x^2-y^2)^{11}/(x^2+2y^2)[/tex] as (x,y) approaches (0,0) does not exist, by considering two different paths to the origin and showing that the limit depends on the value of the parameter in each case.

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Someone in 1955 paid only 0.2% of today's price for a loaf of bread

To find what percentage of today's price someone in 1955 paid for a loaf of bread, we need to use the concept of inflation. Inflation is the increase in the general price level of goods and services in an economy over a period of time. In other words, the cost of goods and services increases over time due to inflation.

To calculate the inflation rate, we can use the following formula:

Inflation rate = (Current price - Base price) / Base price x 100%

Here, the base price is the price of bread in 1955, and the current price is the price of bread today.

Base price = 20 cents

Current price = $2.50

Using the formula, we get:

Inflation rate = ($2.50 - $0.20) / $0.20 x 100%

Inflation rate = $2.30 / $0.20 x 100%

Inflation rate = 1150%

This means that the price of bread has increased by 1150% since 1955 due to inflation. To find out what percentage of today's price someone in 1955 paid, we can divide the 1955 price by the inflation rate and multiply by 100%.

Percentage of today's price = (Base price / Inflation rate) x 100%

Percentage of today's price = (20 cents / 1150%) x 100%

Percentage of today's price = 0.002 x 100%

Percentage of today's price = 0.2%

Therefore, someone in 1955 paid only 0.2% of today's price for a loaf of bread.

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For a normal distribution of amount of foil on each roll of aluminum foil, the p-value or probability that the sample mean area is 249. 6 ft² or less for true claim is equals to the 0.2636 . So, option(e) is right one.

We have a manufacturer of a certain brand of aluminum and the amount of foil on each roll follows a Normal distribution. Mean of amount, μ = 250 ft²

Standard deviation, σ = 2 ft²

Sample size, n = 10

We have to determine the probability that the sample mean area is 249. 6 ft² or less if the manufacturer’s claim is true. Using the Z-Score formula for normal distribution, [tex]Z = \frac{ \bar X - \mu }{ \frac{\sigma}{ \sqrt{n}}} [/tex]where,

Now,[tex] Z = \frac{ 249.6 - 250 }{\frac{2}{\sqrt{10}} }[/tex]

= [tex] 0.2 \sqrt{10}[/tex]

= 0.632

Now, the probability that the sample mean area is 249. 6 ft² or less,

[tex]P ( \bar X ≤ 249.6 ) [/tex]

= [tex]P ( \frac{\bar X - \mu }{\frac{\sigma}{ \sqrt{n}}} ≤ \frac{ 249.6 - 250}{\frac{ 2}{\sqrt{10}}}) [/tex]

= P ( Z≤ 0.632 )

Using the distribution table, the probability value for Z ≤ 0.632 is equals to the 0.2636. Hence, required value is 2636.

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The best point estimate for the average number of credit hours per semester for all students at the local college would be 13.6, which is the average credit hours earned by the sample of 69 students.

A point estimate is a single value that is used to estimate an unknown parameter, in this case, the average credit hours per semester for all students at the local college. Since the sample size is large enough (69 students), the sample mean (13.6 credit hours) is a good point estimate of the population mean. However, it's important to note that there may be some variability in the estimates due to sampling error.


To find the best point estimate for the average number of credit hours per semester for all students at the local college, you can use the sample mean as an estimator.

Step 1: Identify the sample mean
In this case, the sample mean (also known as the average) is given as 13.6 credit hours per semester.

Step 2: Recognize the sample size
The sample size, denoted as "n", is 69 students.

Since the sample mean is the best point estimate for the population mean, the best estimate for the average number of credit hours per semester for all students at the local college is 13.6 credit hours.

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a. The tension in rope AB is T_AB = (FR/2) + (sqrt((FR/2)^2 + T_AC^2)).

b. The tension in rope AC is T_AC = (FR/2) + (sqrt((FR/2)^2 + T_AB^2)).

To answer this question, we can use the fact that the net force acting on the boater must be zero (assuming they are not accelerating).

Let T_AB be the tension in rope AB and T_AC be the tension in rope AC.

a. In the x-direction: T_AB - T_AC = 0 (since the boater is not moving horizontally).
In the y-direction: T_AB + T_AC - FR = 0 (since the net force acting on the boater must be zero).

Using these two equations, we can solve for T_AB:

T_AB = (FR/2) + (sqrt((FR/2)^2 + T_AC^2))

b. Similarly, we can solve for T_AC:

T_AC = (FR/2) + (sqrt((FR/2)^2 + T_AB^2))

Note that we are given FR = 70 lb, the force exerted by the flowing water on the boater. We can substitute this value into the above equations to find the tensions in the ropes.

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The required central angle in the circle is as follows:

m∠2 = 30 degrees

The central angle of an arc is the central angle subtended by the arc.

Therefore, the measure of an arc is the measure of its central angle.

Hence, let's find the angle m∠2.

Therefore,

arc angle BD = 150 degrees

Therefore,

m∠4 = 150 degrees(central angle to the arc)

Let's find the value of m∠2.

Hence,

m∠4 = m∠3(vertically opposite angles)

m∠2 = 360 - 150 - 150 ÷ 2

m∠2 = 360 - 300 ÷ 2

m∠2 = 60 / 2

m∠2 = 30 degrees

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Answer:

It takes James and Renee 10 hours to catch up to Sarah.

Step-by-step explanation:

To solve this problem, we can use the formula:

time = distance / rate

Let's call the time it takes James and Renee to catch up to Sarah "t" and the distance they travel "d". We know that Sarah had a head start of 40 miles, so the distance they need to catch up to her is:

d = 40 miles

During the time "t", Sarah travels:

distance = rate x time = 46t

And James and Renee travel:

distance = rate x time = 50t

Since they both travel the same distance when they catch up, we can set these two distances equal to each other:

46t + 40 = 50t

Subtracting 46t from both sides, we get:

40 = 4t

Dividing both sides by 4, we get:

t = 10

So it takes James and Renee 10 hours to catch up to Sarah.

The function is :[tex]y(x) = 10 + (7/8) e^8x + (97/72) e^9x + 3 e^x[/tex]

To find y as a function of x, we need to solve the differential equation:

[tex]y′′′ − 17y′′ + 72y′ = 168e^x[/tex]

Step 1: Find the characteristic equation

[tex]r^3 - 17r^2 + 72r = 0[/tex]

Factor out r:

[tex]r(r^2 - 17r + 72) = 0[/tex]

Factor the quadratic:

r(r - 8)(r - 9) = 0

So the roots are:

r₁ = 0, r₂ = 8, r₃ = 9

Step 2: Find the general solution

The general solution will be of the form:

[tex]y(x) = C1 + C2e^8x + C3e^9x + y_p(x)[/tex]

where y_p(x) is a particular solution to the non-homogeneous equation.

Step 3: Find the particular solution

We can use the method of undetermined coefficients to find a particular solution. Since the right-hand side is an exponential function, we can guess that the particular solution is also an exponential function:

[tex]y_p(x) = A e^x[/tex]

[tex]y_p′(x) = A e^x[/tex]

[tex]y_p′′(x) = A e^x[/tex]

[tex]y_p′′′(x) = A e^x[/tex]

Substituting into the differential equation:

[tex]A e^x - 17A e^x + 72A e^x = 168 e^x[/tex]

Simplifying:

[tex]56A e^x = 168 e^x[/tex]

A = 3

So the particular solution is:

[tex]y_p(x) = 3 e^x[/tex]

Step 4: Find the constants using initial conditions

y(0) = C₁ + C₂ + C₃ + 3 = 16

y′(0) = 8C₂ + 9C₃ + 3 = 23

[tex]y′′(0) = 8^2 C2 + 9^2 C3 = 24[/tex]

Solving for the constants, we get:

C₁ = 10, C₂ = 7/8, C₃ = 97/72

Step 5: Write the final solution

Substituting the constants and the particular solution into the general solution, we get:

[tex]y(x) = 10 + (7/8) e^8x + (97/72) e^9x + 3 e^x[/tex]

So the function y(x) is:

[tex]y(x) = 10 + (7/8) e^8x + (97/72) e^9x + 3 e^x[/tex]

To learn more about the differential equation,

https://brainly.com/question/31583235

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