what is the slope of a line that is perpendicular to the line y=-1/2x+5

The true statements for the given function f(x) are:

The value of g(1) is 3 and the y- intercept of g(x) is at the point (0, 1) .

The function g(x) = f( x - 3 )

g (1) = f (1 -3 )

       = f (-2 )

       = 3

g (-1) = f (-1 -3)

       = f (-4)

       = - 1

Substituting , x = 0 to find the y intercept of g(x)

g ( 0 ) = f ( 0 - 3)

          =f (-3)

          =1

The y intercept of g(x) is at the point (0, 1)

Thus, options 1 and 4 are the true statements for the given function.

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The probability that a package of 5 tasks is processed in less than 8 minutes is 0.963.

Let X denote the time required to process a package of five tasks. X is an exponentially distributed random variable with mean 2 minutes.

The probability of X being less than 8 minutes is given by:

P(X ≤ 8) = 1 - P(X > 8)

= \(1 - (1 - e^{(-8/2)}^{5}\)

= 0.963

Therefore, the probability that a package of 5 tasks is processed in less than 8 minutes is 0.963.

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Answer:

5

Step-by-step explanation:

Answer:

No

Step-by-step explanation:

Because after u substitute the 15 for the x u get 38>42 which is incorrect

Answer:

Step-by-step explanation:

1. B. lateral faces

2. D. surface area

3. B. lateral faces

4. A. lateral area

5. E. cylinder

Answer:

x < -20

Step-by-step explanation:

To solve x/5 + 6 < 2,

First subtract 6 from both sides of the equation.

This makes x/5 < -4.

Then multiply both sides of the equation by 5.

This makes x < -20.

Therefore, the solution to the equation is x < -20.

Answer:

(a) To prove that if n is an odd integer, then n^2 is an odd integer, we assume that n is an odd integer and prove that n^2 is also an odd integer.

Since n is an odd integer, we can express it as n = 2k + 1, where k is an integer.

Now, let's square both sides of the equation:

n^2 = (2k + 1)^2

Expanding the equation:

n^2 = 4k^2 + 4k + 1

We can rewrite the equation as:

n^2 = 2(2k^2 + 2k) + 1

Let's define a new integer m = 2k^2 + 2k. Since m is an integer, we can rewrite the equation as:

n^2 = 2m + 1

The equation shows that n^2 can be expressed in the form 2m + 1, where m is an integer. Therefore, n^2 is an odd integer.

(b) To prove that for any positive real numbers x and y, xy^2 is positive, we assume that x and y are positive real numbers and prove that xy^2 is also positive.

Since x and y are positive real numbers, they are greater than zero: x > 0 and y > 0.

Multiplying x and y^2, we have:

xy^2 > 0 * y^2

xy^2 > 0

Therefore, xy^2 is positive.

(c) To prove that if x is a real number and x ≤ 3, then 12 - 7x + x^2 ≥ 0, we assume that x is a real number and x ≤ 3, and prove that 12 - 7x + x^2 is greater than or equal to zero.

We start with the quadratic expression 12 - 7x + x^2 and simplify it:

12 - 7x + x^2 = x^2 - 7x + 12

To determine the sign of the expression, we factor it:

x^2 - 7x + 12 = (x - 3)(x - 4)

Since x ≤ 3, both factors (x - 3) and (x - 4) are less than or equal to zero.

Multiplying two negative or non-positive numbers yields a non-negative or positive result:

(x - 3)(x - 4) ≥ 0

Therefore, 12 - 7x + x^2 ≥ 0 when x is a real number and x ≤ 3.

(d) To prove that the product of two odd integers is an odd integer, we assume that m and n are odd integers and prove that their product mn is also an odd integer.

Since m and n are odd integers, we can express them as m = 2k + 1 and n = 2j + 1, where k and j are integers.

Now, let's multiply m and n:

mn = (2k + 1)(2j + 1)

Expanding the equation:

mn = 4kj + 2k + 2j + 1

We can rewrite the equation as:

mn = 2(2kj + k + j) + 1

Let's define a new integer p = 2kj + k + j. Since p is an integer, we can rewrite the equation as:

mn = 2p + 1

The equation shows that mn can be expressed in the form 2p + 1, where p is an integer. Therefore, mn is an odd integer.

(e) To prove that if r and s are rational numbers, then the product of r and s is a rational number, we assume that r and s are rational numbers and prove that their product rs is also a rational number.

Since r and s are rational numbers, we can express them as r = a/b and s = c/d, where a, b, c, and d are integers and b ≠ 0, d ≠ 0.

Now, let's multiply r and s:

rs = (a/b)(c/d)

Multiplying the numerators and denominators:

rs = (ac)/(bd)

Since ac and bd are both integers and bd ≠ 0, rs can be expressed as a fraction with integers in the numerator and denominator. Therefore, rs is a rational number.

a) The n² is an odd integer. b) We have proved that for any positive real numbers x and y, their product xy is also positive.

(a) If n is an odd integer, then n² is an odd integer.

To prove this statement, we will assume that n is an odd integer and show that n² is also an odd integer.

Assumption: n is an odd integer, so n = 2k + 1, where k is an integer.

Proof:

n² = (2k + 1)² [Substituting the value of n from the assumption]

= 4k² + 4k + 1 [Expanding the square]

Now, let's express 4k² + 4k as 2m, where m is an integer:

4k² + 4k = 2(2k² + 2k) = 2m

Substituting this back into the expression for n²:

n² = 2m + 1

We have expressed n² in the form 2m + 1, where m = 2k² + 2k. Since m is an integer, n² can be expressed as 2 times an integer plus 1. Therefore, n² is an odd integer.

Hence, we have proved that if n is an odd integer, then n² is an odd integer.

(b) For any positive real numbers x and y, xy > 0.

To prove this statement, we will assume x and y are positive real numbers and show that their product is also positive.

Assumption: x and y are positive real numbers.

Proof:

Since x and y are positive real numbers, we know that both x and y are greater than zero: x > 0 and y > 0.

Multiplying two positive numbers results in a positive number. Therefore, we have:

x * y > 0

Hence, we have proved that for any positive real numbers x and y, their product xy is also positive.

(c) If x is a real number and x ≤ 3, then 12 - 7x + x^2 ≥ 0.

To prove this statement, we will assume that x is a real number and x ≤ 3, and show that the expression 12 - 7x + x^2 is greater than or equal to zero.

Assumption: x is a real number and x ≤ 3.

Proof:

We can rewrite the expression 12 - 7x + x^2 as (x - 3)(x - 4).

We know that x ≤ 3, so (x - 3) ≤ 0. Similarly, (x - 4) ≤ 0.

Multiplying two non-positive numbers or two non-negative numbers results in a non-negative number. Therefore, we have:

(x - 3)(x - 4) ≥ 0

Hence, we have proved that if x is a real number and x ≤ 3, then 12 - 7x + x^2 ≥ 0.

(d) The product of two odd integers is an odd integer.

To prove this statement, we will assume that m and n are odd integers and show that their product is also an odd integer.

Assumption: m and n are odd integers.

Proof:

Since m and n are odd integers, we can express them as m = 2k + 1 and n = 2l + 1, where k and l are integers.

The product of m and n is:

m * n = (2k + 1)(2l + 1)

= 4kl + 2k + 2l + 1

= 2(2kl + k + l) + 1

Let p = 2kl + k + l. Since k, l, and p are integers, we can rewrite the expression as:

m * n = 2p + 1

We have expressed the product m * n as 2p + 1, where p is an integer. Therefore, the product of two odd integers is an odd integer.

Hence, we have proved that the product of two odd integers is an odd integer.

(e) If r and s are rational numbers, then the product of r and s is a rational number.

To prove this statement, we will assume that r and s are rational numbers and show that their product is also a rational number.

Assumption: r and s are rational numbers.

Proof:

Since r and s are rational numbers, we can express them as fractions: r = a/b and s = c/d, where a, b, c, and d are integers and b, d ≠ 0.

The product of r and s is:

r * s = (a/b)(c/d)

= ac / bd

The product ac and bd is the product of two integers, which is also an integer. Furthermore, since b and d are nonzero integers, their product bd is also nonzero.

Therefore, ac / bd is a fraction where the numerator and denominator are both integers. Hence, the product of r and s is a rational number.

Hence, we have proved that if r and s are rational numbers, then the product of r and s is a rational number.

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Answer:

60

Step-by-step explanation:

Answer:

10% discounted

Step-by-step explanation:

500× X = 450

while X is the sale price

so we solve for X

(500X)÷500 = 450 ÷ 500

X = 0.9

then we turn that into percentage

0.9 × 100 = 90%

finding the discount percentage

100% - 90% = 10%

If the length of rectangle is 5.25, then the area of rectangle will be less than 25 square meters.

What is area of rectangle?

Area of rectangle is the region covered by the rectangle in a two-dimensional plane. A rectangle is a type of quadrilateral, a 2d shape that has four sides and four vertices.

We know the perimeter of rectangle is 20 meters. So according to formula of Perimeter of rectangle = 2 (L + B) we can conclude that sum of length and width will be 10 meters.

We are given length as 1, 3, 5, 7, and 9.

So width of the rectangle will be 9, 7, 5, 3 and 1.

Area of Rectangle = L x B

So area of rectangle will be 9, 21, 25, 21, and 9.

If the length of rectangle is 5.25, then the area of rectangle will be less than 25 square meters.

Values are changing in a linear way and not in an exponential way.

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Answer:

vertex = (- 1, - 4 )

Step-by-step explanation:

the equation of a parabola in vertex form is

f(x) = a(x - h)² + k

where (h, k ) are the coordinates of the vertex and a is a multiplier

f(x) = 3(x + 1)² - 4 ← is in vertex form

with vertex = (- 1, - 4 )

Answer:

(−1,−4)

Step-by-step explanation:

f(x)=3(x+1)2−4

Answer:

x = -1

Step-by-step explanation:

LOCO LOOOCOOOO LOCO LOOOOCOOO

The total sum of money is 77 dollars .

Let the total sum of money be x .

⇒ Aziz's share = 4x / 11

   Ahmad's share = 7x / 11

Given : Aziz received 21 dollars less than Ahmad

⇒ 4x / 11 = ( 7x / 11 ) - 21

Solving the above linear equation in one variable ,

( 7x - 4x ) / 11 = 21                      ( taking like terms together )

3x = 21 * 11

x = 7 * 11

x = 77

Aziz's share = ( 4 / 11 ) * 77 = 28

Ahmad's share = ( 7 / 11 ) * 77 = 49

Total Sum of money = 28 + 49 = 77 dollars .

Hence , the total sum of money = 77 dollars .

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Hello !

\(\begin{cases} 5x+3y - 35&=0 \\ 3x - 4y &= - 8 \end{cases}\)

\(\Leftrightarrow\begin{cases} 5x+3y &=35 \\ 3x - 4y &= - 8 \end{cases}\)

\(\Leftrightarrow AX = B \)

With

\(A=\left[\begin{array}{ccc}5&3\\3& - 4\end{array}\right] \)

\(X=\left[\begin{array}{ccc}x\\y\\\end{array}\right] \)

\(B=\left[\begin{array}{ccc}35\\ - 8\\\end{array}\right] \)

The solution is given by \(X=A^{-1}B\).

\(X= {\left[\begin{array}{ccc}5&3\\3& - 4\end{array}\right] }^{ - 1} \left[\begin{array}{ccc}35\\ - 8\\\end{array}\right] \)

\(X=\left[\begin{array}{ccc}4\\ 5\\\end{array}\right] \)

The point of intersection between the lines is (4;5).

Have a nice day

Answer:

Point of intersection (4,5)

Step-by-step explanation:

5x + 3y - 35 = 0

3x - 4y = -8

 ⇒ 5x + 3y = 35

     3x - 4y = -8

  Matrix A will be formed by the coefficient of x and y. Matrix B will be formed by the constants.

\(\sf A = \left[\begin{array}{cc}5&3\\3&-4\end{array}\right]\)

\(\sf B = \left[\begin{array}{c}35&-8\end{array}\right]\)

AX = B

 \(\sf X =A^{-1}B\)

\(Now ,\ we \ have \ to \ find \ A^{-1}\),

Find the workout in the document attached.

The measure of angle a is 15 degrees and this can be determined by using the properties of the isosceles triangle.

In geometry, interior angles are formed in two ways. One is inside a polygon, and the other is when parallel lines cut by a transversal. Angles are categorized into different types based on their measurements.

Given:

The following steps can be used in order to determine the measure of angle a:

Step 1 - According to the given data, it can be concluded that triangle PQR and triangle PSR are isosceles triangles.

Step 2 - Apply the sum of interior angle property on triangle PQR.

\(\angle\text{Q}+\angle\text{P}+\angle\text{R}=180\)

\(\angle\text{Q}+2\angle\text{R}=180\)

\(2\angle\text{R}=180-60\)

\(\angle\text{R}=60^\circ\)

Step 3 - Now, apply the sum of interior angle property on triangle PSR.

\(\angle\text{P}+\angle\text{S}+\angle\text{R}=180\)

\(\angle\text{S}+2\angle\text{R}=180\)

\(2\angle\text{R}=180-90\)

\(\angle\text{R}=45^\circ\)

Step 4 - Now, the measure of angle a is calculated as:

\(\angle\text{a}=60-45\)

\(\angle\text{a}=15\)

The measure of angle a is 15 degrees.

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Solution: B. 1/2

Analysis

We are transforming triangle ABC to DFE. If we see the graph, we can see the measures of each side of the triangle, decrease by half. According to that, it would be 1/2 of the original measure.

Answer:

Step-by-step explanation:

r

Answer:

The slope intercept form of (0,3) and (4,2) is y= -1/4x+3

Step-by-step explanation:

The distance from our y values from 3 to 2 is -1.

The distance from our x values is 4.

We can put these values as our slope and we know that the Y values are always rise and the x values are always run

Therefore we get y= -1/4x+3 (side note: The 3 is our y intercept, this means that the x value of the three is always 0. For example, in (0,6) the y intercept would be 6.

Hope this helps! :)

Answer:

The answer and work is both located in the screenshot provided!

Step-by-step explanation:

Hoped this helped!

Based on the given information, we know that a new hair cream was tested on a random sample of people who were either bald or currently losing their hair.

The results of the test showed that a certain percentage of people either started growing back hair or stopped losing their hair, but there was a margin of error of which means that the results could vary by that amount. Unfortunately, we do not have the specific percentage of people who saw an improvement in their hair growth, but we do know that the margin of error is . This means that if the test were to be repeated with another sample of people, the results could vary by that amount in either direction. Therefore, we cannot give an exact number of people who are expected to see an improvement in their baldness if the new hair cream is administered to people. However, we can estimate that between [dropdown1] and [dropdown2] people are predicted to see an improvement in their hair growth based on the results of the test and the margin of error.

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Answer:

drop down 1: 2226

drop down 2: 3241

Step-by-step explanation:

Step-by-step explanation:

(a) To find the lower bound for the length of the rectangle, we need to subtract half of the smallest unit of measurement from the given measurement.

The smallest unit of measurement given is 1 mm, so half of that is 0.5 mm.

Therefore, the lower bound for the length of the rectangle is:

64 mm - 0.5 mm = 63.5 mm

(b) The perimeter of a rectangle is calculated by adding up the length of all four sides.

The lower bounds for the length and width of the rectangle are 63.5 mm and 36.5 mm, respectively.

So, the lower bound for the perimeter is:

2(63.5 mm + 36.5 mm) = 2(100 mm) = 200 mm

Answer and explanation below. Credits to Chris, Mechanical engineering college, Master's degree

The differential equation that models the concentration of the drug in the bloodstream as a function of time is dc/dt = -k c(t).

where c(t) is the concentration of the drug in the bloodstream at time t and k is the constant of proportionality.

The side condition is:

c(0) = c0

which states that the initial concentration of the drug in the bloodstream is c0 g/mL.

Supporting answer: The differential equation dc/dt = -k c(t) is a first-order homogeneous linear ordinary differential equation, which means it can be solved using separation of variables:

dc/c(t) = -k dt

Integrating both sides gives:

ln|c(t)| = -k t + C

where C is the constant of integration. Exponentiating both sides of the equation yields:

c(t) = e^(C-k t)

To find the value of C, we use the initial condition c(0) = c0:

c(0) = e^C

C = ln(c0)

Therefore, the solution to the differential equation with the side condition is:

c(t) = c0 e^(-k t)

This is an exponential function that decays over time with a decay constant of k, which represents the rate of elimination of the drug from the bloodstream. The larger the value of k, the faster the drug is eliminated and the shorter its half-life. The concentration of the drug in the bloodstream at any time t is proportional to its initial concentration c0, but inversely proportional to the exponential decay factor e^(k t).

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The solution to the given differential equation 2xy(dy/dx) = y², with the initial condition y(2) = 3, is y = (27 * e⁽ˣ⁻²⁾\()^{1/4}\).

To solve the given differential equation

2xy(dy/dx) = y²

We will use separation of variables and integrate to find the solution.

Start with the given equation

2xy(dy/dx) = y²

Divide both sides by y²:

(2x/y) dy = dx

Integrate both sides:

∫(2x/y) dy = ∫dx

Integrating the left side requires a substitution. Let u = y², then du = 2y dy:

∫(2x/u) du = ∫dx

2∫(x/u) du = ∫dx

2 ln|u| = x + C

Replacing u with y²:

2 ln|y²| = x + C

Using the properties of logarithms:

ln|y⁴| = x + C

Exponentiating both sides:

|y⁴| = \(e^{x + C}\)

Since the absolute value is taken, we can remove it and incorporate the constant of integration

y⁴ = \(e^{x + C}\)

Simplifying, let A = \(e^C:\)

y^4 = A * eˣ

Taking the fourth root of both sides:

y = (A * eˣ\()^{1/4}\)

Now we can incorporate the initial condition y(2) = 3

3 = (A * e²\()^{1/4}\)

Cubing both sides:

27 = A * e²

Solving for A:

A = 27 / e²

Finally, substituting A back into the solution

y = ((27 / e²) * eˣ\()^{1/4}\)

Simplifying further

y = (27 *  e⁽ˣ⁻²⁾\()^{1/4}\)

Therefore, the solution to the given differential equation with the initial condition y(2) = 3 is

y = (27 * e⁽ˣ⁻²⁾\()^{1/4}\)

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Step-by-step explanation:

7x - 5 = 8

7x = 8 + 5

7x = 13 (divide both sides by 7 to get x)

7x/7 = 13/7

x = 1.85714285714

a) The relevant population is the heavy soft-drink users in the given case, and the appropriate sampling design that should be used is  stratified random sampling. The list of all heavy soft-drink users is the sampling frame.

b) Cluster sampling refers to a sampling design where population is divided into naturally occurring groups and a random sample of clusters is chosen.

The advantages are efficient, easy to perform, and used when the population is widely dispersed. The disadvantages are sampling errors, have lower level of precision, and have the standard error of the estimate.

a) The relevant population for the public relations research department to investigate the initial reactions of heavy soft-drink users to a new all-natural soft drink is heavy soft-drink users. The appropriate sampling design that can be used to investigate the issues is stratified random sampling.

Stratified random sampling is a technique of sampling in which the entire population is divided into subgroups (or strata) based on a particular characteristic that the population shares. Then, simple random sampling is done from each stratum. Stratified random sampling is appropriate because it ensures that every member of the population has an equal chance of being selected.

Moreover, it ensures that every subgroup of the population is adequately represented, and reliable estimates can be made concerning the entire population. The list of all heavy soft-drink users can be the sampling frame.

b) Cluster sampling is a type of sampling design in which the population is divided into naturally occurring groups or clusters, and a random sample of clusters is chosen. The elements within each chosen cluster are then sampled.

The advantages of cluster sampling are:

The disadvantages of cluster sampling are:

A situation where cluster sampling would be appropriate is in investigating the effects of a new medication on various groups of people. In this case, the population can be divided into different clinics, and a random sample of clinics can be selected. Then, all patients who meet the inclusion criteria from the selected clinics can be recruited for the study. This way, the study will be less expensive, and it will ensure that the sample is representative of the entire population.

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The dimensions of the rectangle are 10 feet (length) and 8 feet (width).

To find the dimensions of the rectangle with an area of 80 square feet and a width that is 2 feet less than the length,

follow these steps:

1. Let the length of the rectangle be L feet and the width be W feet.

2. According to the given information, W = L - 2.

3. The area of a rectangle is calculated by multiplying its length and width: Area = L × W.

4. Substitute the given area and the relationship between L and W into the equation: 80 = L × (L - 2).

5. Solve the quadratic equation: 80 = L² - 2L.

6. Rearrange the equation: L² - 2L - 80 = 0.

7. Factor the equation: (L - 10)(L + 8) = 0.

8. Solve for L: L = 10 or L = -8 (since the length cannot be negative, L = 10).

9. Substitute L back into the equation for W: W = 10 - 2 = 8.

So, the dimensions of the rectangle are 10 feet (length) and 8 feet (width).

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The function's formula for the ice sheet's thickness is S(t) = 0.2t + 1.02.

Mathematically, a linear function is sometimes referred to as an expression or the slope-intercept form of a straight line and it can be modeled (represented) by this mathematical expression;

S(t) = mt + b

Where:

After a time of 7 weeks and a rate of decrease of 0.2, the y-intercept or initial amount of ice can be calculated as follows;

S(t) = mt + b

2.42 = 0.2(7) + b

2.42 = 1.4 + b

y-intercept, b = 2.42 - 1.4

y-intercept, b = 1.02.

Therefore, the required linear function is given by S(t) = 0.2t + 1.02.

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Complete Question:

A lake near the Arctic Circle is covered by a thick sheet of ice during the cold winter months. When spring arrives, the warm air gradually melts the ice, causing its thickness to decrease at a rate of 0.2 meters per week. After 7 weeks, the sheet is only 2.42 meters thick.

Let S(t), denote the ice sheet's thickness S (measured in meters) as a function of time t (measured in weeks).

Write the function's formula.

S(t) = ?

The particular solution of the given differential equation with the initial condition x(0) = 4.

Any mathematical equation that connects a function and its derivatives to one or more independent variables is known as a differential equation. Many different physical phenomena, including as the behaviour of particles, fluids, and electrical circuits, are modelled using differential equations. They are used extensively in physics, engineering, and other disciplines. Differential equations' solutions frequently provide light on the behaviour of complicated systems and can be used to forecast how they will behave in the future.

Step 1: Write down the given differential equation and initial condition.
\(dx/dt = 4x\\x(0) = 4\)

Step 2: Rewrite the differential equation in a separable form.
\((1/x)dx = 4dt\)

Step 3: Integrate both sides of the equation.
\(\int\limits {x} \, dx (1/x)dx = \int\limits {x} \, dx 4dt\)

Step 4: Find the antiderivatives.
\(ln|x| = 4t + C\)

Step 5: Solve for x.
\(x = e^(4t + C)\\x = e^(4t) * e^C\)
Step 6: Apply the initial condition x(0) = 4.
\(4 = e^(4*0) * e^C\\4 = e^C\)

Step 7: Write the general solution, substituting the value of e^C.
\(x(t) = e^(4t) * 4\)

That's the particular solution of the given differential equation with the initial condition x(0) = 4.

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The margin of error for the point estimate using a 95% confidence level is 2.94, rounded to three decimal places.

To determine the margin of error for the point estimate using a 95% confidence level, we use the formula:Margin of Error = z* (standard error)Margin of error is always positive. The standard error formula is:Standard error = standard deviation / √nWhere n is the sample size and z is the z-score corresponding to the level of confidence.

We will use the 95% confidence level, which means z = 1.96. If the assumptions are not met, then we will enter 999 instead of any computed value. Let us assume we have a sample of 100 and a standard deviation of 15.Using the standard error formula, we can calculate the standard error: Standard error = 15/√100 = 1.5Now we can calculate the margin of error: Margin of Error = 1.96 * (1.5) = 2.94

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