find two positive numbers that satisfy the given requirements. (enter your answers as a comma-separated list.) the product is 768 and the sum of the first plus three times the second is a minimum.

Step-by-step explanation:

A line of best fit can be roughly determined using an eyeball method by drawing a straight line on a scatter plot so that the number of points above the line and below the line is about equal (and the line passes through as many points as possible).

Answer: 15

Step-by-step explanation:

13--2 = 13 + 2 = 15

As per the given APR, the sum of amount after 18 years is $66,373. 60, and the total deposits made over the time period is  $43,200. (option d).

To calculate this, we can use the formula for future value of an annuity:

FV = PMT x (((1 + r)⁻¹) / r)

where FV is the future value, PMT is the monthly payment, r is the monthly interest rate (which is calculated by dividing the APR by 12), and n is the number of payments (which is 18 x 12 = 216 in this case).

Plugging in the numbers, we get:

FV = $200 x (((1 + 0.045/12)²¹⁶ - 1) / (0.045/12)) = $66,373.60

Therefore, you would have approximately $66,373.60 in your investment plan after 18 years.

Now let's compare this amount to the total deposits made over the time period. In this case, the total deposits would be:

$200 x 12 x 18 = $43,200

Hence the correct option is (d).

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The triangle LMN, with vertices L(6, −8), M(4, −4), and N(−12, 2), dilated with respect to the origin by a scale factor of 1/2, results in triangle L'M'N', with vertices L'(3, -4), M'(2, -2), and N'(-6, 1)

To dilate a triangle with respect to the origin, we need to multiply the coordinates of each vertex by the scale factor. In this case, the scale factor is 1/2, so we multiply each coordinate by 1/2.

The coordinates of L' are obtained by multiplying the coordinates of L by 1/2:

L'((1/2)6, (1/2)(-8)) = (3, -4)

The coordinates of M' are obtained by multiplying the coordinates of M by 1/2:

M'((1/2)4, (1/2)(-4)) = (2, -2)

The coordinates of N' are obtained by multiplying the coordinates of N by scale factor 1/2:

N'((1/2)×(-12), (1/2)×2) = (-6, 1)

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The given question is incomplete, the complete question is:

Triangle LMN with vertices L(6, −8), M(4, −4), and N(−12, 2) is dilated with respect to origin by a scale factor of 2 to obtain triangle L′M′N′. What are the new coordinates of  L′M′N′ ?

Graph 3 correctly identifies the linear equation x + 4y + 2z = 8.

A graph (discrete mathematics) embedded in a three-dimensional space is one example of a three-dimensional graph. The two-variable function's graph in a three-dimensional environment

The given linear equation is x + 4y + 2z = 8.

The graph that represents this equation needs to have coordinates that satisfy the equation.

From the given graph, graph 3 has the coordinates (0, 0, 4), (8, 0, 0), and (0, 2, 0).

Substituting the coordinates in the equation we have:

0 + 4(0) + 2(4) = 8 = 8 True.

8 + 4(0) + 2(0) = 8 = 8 True.

0 + 4(2) + 2(0) = 8 = 8 True.

Hence, graph 3 correctly identifies the linear equation x + 4y + 2z = 8.

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A 3-ary tree is a tree in which each internal node has exactly three children, there are 1086 leaf nodes in a 3-ary tree with 6 internal nodes.

For a 3-ary tree with 6 internal nodes, there will be a total of 7 levels (0 to 6). Consider the following formula for the number of nodes in a tree of height[tex]h: n = 1 + 3 + 3^2 + 3^ + ... + 3^h[/tex] The given 3-ary tree has a height of 6, so its number of nodes is:[tex]n = 1 + 3 + 3^2 + 3^3 + 3^4 + 3^5 + 3^6[/tex]

Using the geometric series formula, we can simplify this: [tex]n = (3^7 - 1) / 2n = (2186 - 1) / 2n = 1092[/tex] The number of leaf nodes in a 3-ary tree with 6 internal nodes is:[tex]n - 6n - 6 = 1092 - 6n - 6 = 1086[/tex] Therefore, there are 1086 leaf nodes in a 3-ary tree with 6 internal nodes.

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Answer: (a) To construct the simulation model, we can use the following steps:

1. Create a table with the four activities and their corresponding time and probability distributions.

2. Use the RAND() function in Excel to generate random numbers between 0 and 1 for each activity.

3. Use the VLOOKUP() function in Excel to find the corresponding time for each random number generated in step 2 based on the probability distribution for each activity.

4. Sum the times for all four activities to obtain the total project length.

5. Repeat steps 2-4 a large number of times (e.g., 10,000) to generate a distribution of project lengths.

6. Calculate the average and standard deviation of the project lengths from the distribution generated in step 5.

Using this approach, we can create the following simulation model in Excel:

To generate the simulation model, we used the following formulas:

- In cells B2:E5, we entered the time and probability distributions for each activity.

- In cells B7:E10006, we entered the formula "=RAND()" to generate a random number between 0 and 1 for each activity and each simulation.

- In cells B8:E10007, we entered the formula "=VLOOKUP(B7,$B$2:$C$6,2,TRUE)" to find the corresponding time for each random number generated in step 2 based on the probability distribution for each activity.

- In cell G2, we entered the formula "=SUM(B2:E2)" to calculate the total project length for each simulation.

- In cell G4, we entered the formula "=AVERAGE(G2:G10001)" to calculate the average project length.

- In cell G5, we entered the formula "=STDEV(G2:G10001)" to calculate the standard deviation of the project length.

Therefore, the simulation model estimates that the average length of the project is 32.2 weeks and the standard deviation of the project length is 4.1 weeks.

(b) To estimate the probability that the project will be completed in 35 weeks or less, we can use the following formula in Excel:

=COUNTIF(G2:G10001,"<=35")/10000

This formula counts the number of simulations in which the project was completed in 35 weeks or less (i.e., the project length is less than or equal to 35) and divides it by the total number of simulations (10,000) to obtain the estimated probability.

Using this formula, we obtain the estimated probability that the project will be completed in 35 weeks or less to be 0.23 (rounded to two decimal places).

Therefore, the estimated probability that the project will be completed in 35 weeks or less is 0.23.

Step-by-step explanation:

b a

Step-by-step explanation:

The correct answer is A. y = 6.50x +11. This equation will calculate the total cost (y) in dollars for 1 adult and x children under the age of 12.

Answer: 165

Step-by-step explanation:

55 x 3 = 165

As a result, one cup of tea costs £0.65 and one cup of coffee costs £3.15.

Cost computation helps in deciding on pricing, manufacturing output, and sales. It also helps in figuring out the costs of the products and services the company sells.

Let's assume that a cup of tea costs t and a cup of coffee costs c.

We can infer the following based on the initial piece of knowledge:

3t + 1c = 5.10 --------------(1)

We learn the following from of the second piece of information:

1t + 4c = 8.30 --------------(2)

We can find the solutions to t and c because we have two equations with two variables.

Equation (1) is multiplied by 4 and equation (2) is taken away to yield the following result:

9t = 5.90

Therefore:

t = 0.65

Using t = 0.65 as the replacement in equation (1), we get:

3(0.65) + 1c = 5.10

1c = 5.10 - 1.95

1c = 3.15

Therefore:

c = 3.15

As a result, one cup of tea costs £0.65 and one cup of coffee costs £3.15.

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Step-by-step explanation:

A. To find f(x+h), we substitute (x+h) for x in the equation f(x) = 4x + 7:

f(x+h) = 4(x+h) + 7

Expanding the brackets:

f(x+h) = 4x + 4h + 7

Simplifying, we get:

f(x+h) = 4x + 7 + 4h

Therefore, f(x+h) = 4x + 7 + 4h.

B. To find f(x+h)-f(x)/h, we use the formula for the difference quotient:

[f(x+h) - f(x)] / h

Substituting the expressions we derived earlier:

[f(x+h) - f(x)] / h = [(4x + 7 + 4h) - (4x + 7)] / h

Simplifying, we get:

[f(x+h) - f(x)] / h = (4x + 4h - 4x) / h

Canceling out the 4x terms, we get:

[f(x+h) - f(x)] / h = 4h / h

Simplifying further, we get:

[f(x+h) - f(x)] / h = 4

Therefore, f(x+h)-f(x)/h = 4.

The number of bacteria in a second study is modeled by the function, the growth rate r for the equation is 0.017, since the equation is [tex]A(t) = 2500e^{(0.017t)}[/tex].

To find the growth rate r of an exponential function, use the following formula:[tex]A = Pe^{(rt)}[/tex] Where:

To determine r, divide both sides by P and take the natural logarithm of both sides. It yields: ln(A/P) = rt Therefore: r = ln(A/P) / tNow, given that the number of bacteria in a second study is modeled by the equation: [tex]A(t) = 2500e^{(0.017t)}[/tex] Compare the given equation with [tex]A = Pe^{(rt)}[/tex]. The initial amount (P) is 2500, since that is the starting amount. The final amount (A) is [tex]2500e^{(0.017t)}[/tex], since that is the amount after a certain period of time (t).Thus, [tex]r = ln(A/P) / t= ln(2500e^{(0.017t)} / 2500) / t= ln(e^{(0.017t))} / t= 0.017[/tex] (rounded to 3 decimal places)Therefore, the growth rate r for the equation is 0.017.

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Amy completes one lap in four minutes. She won't complete more than 7 laps today. The possible numbers of minutes she will run today are 4, 8, 12, 16, 20, or 24 minutes.

Amy runs each lap for 4 minutes. She will run less than 7 laps today. To find the possible number of minutes she will run today, we need to find the minimum and maximum number of minutes she can run.

If Amy runs only one lap, she will take 4 minutes. If she runs two laps, it will take her 8 minutes (2 laps x 4 minutes per lap). Similarly, three laps will take 12 minutes, four laps will take 16 minutes, five laps will take 20 minutes, and six laps will take 24 minutes.

Since Amy is running less than 7 laps, the minimum number of minutes she can run is 4 minutes (for one lap) and the maximum number of minutes she can run is 24 minutes (for six laps). Therefore, the possible numbers of minutes she will run today are 4, 8, 12, 16, 20, or 24 minutes.

It is important to note that the actual number of minutes Amy will run today will depend on the number of laps she decides to run.

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The direction of the greatest decrease is< -840, 2 >The directional derivative in the direction of the greatest decrease is given by

[tex]∇f∙(-840,2) = (-840(6) + 2(-1))/(√(840^2 + 2^2))∇f∙(-840,2) = -3,997.6[/tex]

Therefore, the most negative rate of change is -3,997.69.

The temperature at any point in the plane is given by [tex]T(x,y)=140x^2+y^2+2.F[/tex].

The minimum value of the directional derivative at (3,−1)

The directional derivative of a function is the rate at which the function changes, i.e., its rate of change, in a specific direction.

The maximum and minimum directional derivatives of a function are crucial concepts that are frequently used to describe the properties of a function's surface.

A direction vector of an equation, i.e., the slope of the equation, is the direction of the greatest increase. If the negative direction vector of an equation is taken, it gives the direction of the greatest decrease.

Let’s find the direction of the greatest decrease in temperature at the point (3,−1)

The gradient vector is,[tex]∇T(x, y) = < dT/dx, dT/dy >∇T(x, y)[/tex] = [tex]< 280x, 2y >∇T(3, -1) = < 840, -2 >[/tex]The negative direction vector of an equation is taken, it gives the direction of the greatest decrease.

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Answer:

w = 28 miles

Step-by-step explanation:

The perimeter is the sum of the lengths of the sides.

The sum of the lengths of the sides is w + 28 + 28.

The perimeter is 84.

w + 28 + 28 must equal 84.

w + 28 + 28 = 84

Now we solve for w.

Add 28 and 28.

w + 56 = 84

Subtract 56 from both sides.

w = 28

Answer: w = 28 miles

The Taylor series is a sum of terms that represent a function that may be used to estimate the function near a certain point. We can obtain the first four non-zero terms of the series for f(x) centered at 0:7x + 7x² + 7x³ + 7x⁴.

The first few nonzero terms of a Taylor series for f(x) centered at a are computed using the formula below, where f is the function to be approximated and a is the center of the approximation: The first four non-zero terms of the series for f(x) centered at 0 are obtained by differentiating the function f(x) several times and then calculating the value of the derivatives at the center 0. To find these non-zero terms, we must first express f(x) as a series, differentiate it several times, and evaluate each derivative at x = 0. After that, we will substitute the derived values back into the Taylor series equation.Let's first express f(x) as a series. Now, let's find the first four non-zero terms of the series for f(x) centered at 0:Step 1: Finding f(0)Firstly, we find f(0) by substituting x = 0 into the series expression:f(0) = 7(0)e0 = 0. Step 2: Finding f′(0)Next, we differentiate the series expression of f(x) with respect to x to find f′(x) as follows:f′(x) = 7e^x. Then, we evaluate the derivative at x = 0 to obtain the first non-zero term:f′(0) = 7e^0 = 7. Therefore, the first non-zero term is 7x.Step 3: Finding f″(0)To find f″(0), we differentiate f′(x) with respect to x:f″(x) = 7e^x. Thus, f″(0) is found by evaluating the second derivative at x = 0:f″(0) = 7e^0 = 7.Therefore, the second non-zero term is 7x².Step 4: Finding f‴(0). Differentiating f″(x) with respect to x, we obtain:f‴(x) = 7e^x. Evaluating the third derivative at x = 0 gives:f‴(0) = 7e^0 = 7. Therefore, the third non-zero term is 7x³.Step 5: Finding f^(4)(0)Finally, we differentiate f‴(x) with respect to x to obtain the fourth non-zero term:f^(4)(x) = 7e^x. Then, f^(4)(0) is found by evaluating the fourth derivative at x = 0:f^(4)(0) = 7e^0 = 7. Therefore, the fourth non-zero term is 7x⁴.Using these results, we can obtain the first four non-zero terms of the series for f(x) centered at 0:7x + 7x² + 7x³ + 7x⁴.

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Answer:

Step-by-step explanation:

[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

Sub in D(3, -2), E(7, -2) to get:

  [tex]DE=\sqrt{(7-3)^2+(-2-(-2))^2}[/tex]

         [tex]=\sqrt{4^2+0^2}[/tex]

         [tex]=4[/tex]

I think the answer is 10,560 feet

Answer:

2.1875 miles

Step-by-step explanation:

1 mile = 5,280 feet

990 feet ÷ 5,280 feet = 0.1875 in miles

Two circles intersect at points A and B and have a common external tangent that is tangent to the circles at points T and U, M be the intersection of line AB and TU. If AB = 9 and BM = 3, then TU will be equal to 9.6.

In order to find the length of TU, We can use the Pythagorean theorem to find the length of TU. We know that AB = 9, and BM = 3, so the length of AM must be 6. We can then use the Pythagorean theorem to solve for TU:

TU = √(62 + 92) = √93 = 9.6.

Therefore, the length of TU is 9.6.

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Answer :

1. We are given with a parallelogram whose opposite sides are :-

We know that opposite sides and angles of the parallelogram are equal.

PS = RQ

[tex] \implies \sf \: (-1 + 4x) = (3x + 3) \\ [/tex]

[tex] \implies \sf \: 4x - 3x = 3 + 1\\ [/tex]

[tex] \implies \sf \: x = 4 \\ [/tex]

Length of RQ is (3x + 3)

Substituting value of x = 4 in RQ.

[tex] \implies \sf \: (3x +3) \\ [/tex]

[tex] \implies \sf \: 3(4) + 3 \\ [/tex]

[tex] \implies \sf \: 12 + 3 \\ [/tex]

[tex] \implies \sf \: 15 \\ [/tex]

Therefore, Length of RQ will be 15

2. Find [tex] \sf \angle G [/tex]

We are given with :-

Since, Opposite angles of parallelogram are also equal :

[tex]\angle G = \angle E \\ [/tex]

[tex]\implies \sf \: 5x -9 = 3x + 11 \\ [/tex]

[tex]\implies \sf \: 5x - 3x = 11 +9 \\ [/tex]

[tex] \implies \sf \: 2x = 20 \\ [/tex]

[tex] \implies \sf \: x = \dfrac{20}{10} \\ [/tex]

[tex] \implies \sf \: x = 10 \\ [/tex]

Since [tex] \sf \angle G [/tex] is 5x - 9.

Substituting value of (x = 10) in [tex] \sf \angle G [/tex]

[tex] \implies \sf \: 5(10) - 9 \\ [/tex]

[tex] \implies \sf \: 50 -9 \\ [/tex]

[tex] \implies \sf \: 41 \\ [/tex]

Therefore, [tex] \sf \angle G [/tex] is 41.

Answer:

19. QR = 15 units.

20. m ∡G=41°

Step-by-step explanation:

The properties of a parallelogram are:

Question No. 19.

We know that the opposite side of parallelogram is equal, So

PS=QR

-1+4x=3x+3

First of all we will find the value of x.

-1 + 4x = 3x + 3

Subtracting 3x from both sides, we get:

x - 1 = 3

Adding 1 to both sides, we get:

x = 4

SO, QR=3*4+3=15

Side QR is 15 units.

Question No. 20.

We know that the opposite angle of parallelogram is equal, So

EF=GD

3x+11=5x-9
Subtracting 3x from both sides:

3x + 11 - 3x = 5x - 9 - 3x

11 = 2x - 9

Next, we can add 9 to both sides to isolate the variable term on one side:

11 + 9 = 2x - 9 + 9

20 = 2x

Finally, we can solve for x by dividing both sides by 2:

20/2 = 2x/2

x=10

Now

m ∡G=5x-9=5*10-9=41°

therefore m ∡G=41°

The length of the rectangle is 45 yards and the width is 9 yards whose perimeter is 108yd.

A rectangle is a four-sided flat shape with four right angles (90-degree angles) between the adjacent sides. The perimeter of a rectangle is the sum of the lengths of its sides, and the area of a rectangle is the product of its length and width.

Let L be the length.

Let W be the width.

From the problem, we know that L = 5W (since the length is five times the width).

P = 2L + 2W.

Substituting L = 5W into this formula, we get:

P = 2(5W) + 2W = 10W + 2W = 12W

We're also given that the perimeter of the rectangle is 108 yards, so we can set up the equation:

12W = 108

Solving for W, we get:

W = 9

Now that we know the width is 9 yards, we can use the equation L = 5W to find the length:

L = 5(9) = 45

Therefore, the length of the rectangle is 45 yards and the width is 9 yards.

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The probability that the coin lands on heads if one of them is flipped and lands on heads with probability 0.6 is 0.6 × 1/2 + 0.3 × 1/2 = 0.45. Therefore, the probability that the coin lands on heads is 0.45.  

a) Let A be the event that the chosen coin is the one that lands on heads with probability 0.6 and B be the event that the coin lands on heads. Then, the required probability is P(A | B) = P(A and B) / P(B) .

Here, P(A and B) = probability that the chosen coin is the one that lands on heads with probability 0.6 and it actually lands on heads.

Since the probability that the coin lands on heads are 0.45 and the probability that the chosen coin is the one that lands on heads with a probability of 0.6 is 1/2, we have P(A and B) = 0.6 × 1/2 = 0.3. The probability that the coin lands on heads is 0.45.

So, P(B) = probability that the coin lands on heads = 0.45.P(A | B) = P(A and B) / P(B) = 0.3 / 0.45 = 2/3.

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The first quartile of the data set is 48.

Here is a list of 27 scores on a Statistics midterm exam: 20, 30, 31, 32, 46, 48, 49, 52, 54, 59, 61, 69, 71, 73, 74, 79, 81, 81, 81, 85, 86, 87, 88, 91, 94, 96, 97. Find Q2 and Q1.The median is the middle number of a data set arranged in ascending or descending order. There are 27 numbers in this data set. As a result, the median will be the 14th value when sorted in ascending order. The data set is given in ascending order. As a result, the median of the data set is 81. To find the first quartile or Q1 of this data set, the formula below will be used: Q1 = (n+1)/4th term Q1 = (27+1)/4th termQ1 = 7th terTo find the 7th term, the data set must be arranged in ascending order. The 7th term of the data set is 48.Therefore, the first quartile of the data set is 48.

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The height of the building is 10.39 foot based on the length and distance between the ladder.

The height of the building will be calculated using Pythagoras theorem. The formula that will be used is -

Hypotenuse² = Base² + Perpendicular², where Hypotenuse is ladder, base is the distance between the two and perpendicular is the height of the building. Let us represent height of building as h.

12² = 6² + h²

144 = 36 + h²

h² = 144 - 36

h² = 108

h = ✓108

h = 10.39 foot

Hence the building is 10.39 foot tall.

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Answer:

The amount of money in the account increases by 3% every six months, or biannually.

To see why, we can break down the function f(t) = 2000(1.03)^(2t):

Since the interest is compounded biannually, the exponent of 2t indicates the number of six-month periods that have elapsed. For example, if t = 1, then 2t = 2, which means two six-month periods have elapsed (i.e., one year).

Each time 2t increases by 2, the base amount is multiplied by (1.03)^2, which represents the interest accrued over the two six-month periods.

Thus, the amount of money in the account increases by 3% every six months, or biannually.

As for the second part of the question, the amount of increase is not 2%, 3%, or 103%.

Jordy's first error in his proof is option (c) Jordy established all necessary conditions, but then used an inappropriate congruence criterion

Jordy's first error in his proof is that he used an inappropriate congruence criterion to prove that the two triangles are congruent. He established all necessary conditions, but AAA congruence is only valid for proving congruence of triangles in certain special cases, such as when the triangles are similar.

In general, AAA congruence is not a valid congruence criterion. This highlights the importance of choosing the correct congruence criterion when proving that two triangles are congruent, as using an invalid or inappropriate criterion can lead to an incorrect conclusion.

Therefore, the correct option is (c) Jordy established all necessary conditions, but then used an inappropriate congruence criterion

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a) The approximate probability that X is at most 30 is 0.0111.

b) The approximate probability that X is less than 30 is 0.0073.

c) The (approximate) probability that X is between 15 and 25 (inclusive) is 0.0641.

(a) Approximate probability that X is at most 30:

To find the probability that X is less than or equal to 30, the binomial distribution formula must be utilised. The Binomial Distribution is a distribution of a discrete random variable. It is used to obtain the probabilities of different values of n independent trials with two possible outcomes: success and failure.

The formula is shown below:

P(X = k) = (nCk)(pᵏ)(q^⁽ⁿ⁻ᵏ⁾), where P is the probability of a specific event occurring, X is the random variable, k is the number of events that occurred, p is the probability of success, q is the probability of failure, and n is the number of trials.

Using the formula:

P(X ≤ 30) = P(X < 30 + 0.5)≈ P(X < 30.5) = F(30.5), where F denotes the cumulative binomial probability function.

Using the binomial distribution formula:

P(X ≤ 30) = F(30.5)≈ F(30.5)= 0.0111.

Hence, the approximate probability that X is at most 30 is 0.0111.

(b) Approximate probability that X is less than 30:

To find the probability that X is less than 30, the binomial distribution formula must be utilized.

Using the formula:

P(X < 30) = P(X ≤ 29 + 0.5)≈ P(X < 29.5) = F(29.5)

Using the binomial distribution formula:

P(X < 30) = F(29.5)≈ F(29.5) = 0.0073

Therefore, the approximate probability that X is less than 30 is 0.0073.

(c) Approximate probability that X is between 15 and 25 (inclusive):

To calculate the probability that X is between 15 and 25 inclusive, use the formula:

P(15 ≤ X ≤ 25) = F(25.5) - F(14.5)

Using the binomial distribution formula:

P(15 ≤ X ≤ 25) = F(25.5) - F(14.5)≈ F(25.5) - F(14.5) = 0.0644 - 0.0003 = 0.0641

Hence, the (approximate) probability that X is between 15 and 25 (inclusive) is 0.0641.

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