Which relation is a function?

Since A1, A2, ..., Am are mutually exclusive and exhaustive, answers to both parts of the question is;

a) We can use the same argument to show that P(A2) = P(A3) = ... = P(Am) = 1/m.
b) We have proved that if A = A1 ∪ A2 ∪ ... ∪ Ah and (a) holds, then P(A) = h/m.

(a) Since A1, A2, ..., Am are mutually exclusive and exhaustive, we have:

P(S) = P(A1) + P(A2) + ... + P(Am)

Since P(A1) = P(A2) = ... = P(Am), we can rewrite the above equation as:

P(S) = m * P(A1)

Since S is the sample space and its probability is 1, we have:

P(S) = 1

Therefore, we can solve for P(A1) as:

P(A1) = 1/m

Similarly, we can use the same argument to show that P(A2) = P(A3) = ... = P(Am) = 1/m.

(b) Since A1, A2, ..., Am are mutually exclusive and exhaustive, we have:

P(S) = P(A1) + P(A2) + ... + P(Am)

Using (a), we know that P(Ai) = 1/m for i = 1, 2, ..., m. Therefore, we can rewrite the above equation as:

1 = m * (1/m) + P(Ah+1) + ... + P(Am)

Simplifying this equation, we get:

P(Ah+1) + ... + P(Am) = (m - h) * (1/m)

Since A = A1 ∪ A2 ∪ ... ∪ Ah, we can write:

P(A) = P(A1) + P(A2) + ... + P(Ah) = h * (1/m)

Therefore, we have proved that if A = A1 ∪ A2 ∪ ... ∪ Ah and (a) holds, then P(A) = h/m.

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None of the options A, B, C, or D are the necessary step to verify InB + 2 = -2t as a solution to dB/dt = -2B.

what is differential equations?

Differential equations are mathematical equations that describe the relationship between an unknown function and its derivatives (or differentials).

To verify that InB + 2 = -2t is a solution to dB/dt = -2B, we can substitute InB + 2 for B in the differential equation and check if it satisfies the equation.

So, let's first differentiate InB + 2 with respect to t:

d/dt (InB + 2) = 1/B * dB/dt

Using the given differential equation, we can substitute dB/dt with -2B:

d/dt (InB + 2) = 1/B * (-2B)

Simplifying this expression, we get:

d/dt (InB + 2) = -2

Now, substituting InB + 2 for B in the original differential equation, we get:

dB/dt = -2(InB + 2)

We can differentiate this expression with respect to B to get:

d/dB (dB/dt) = d/dB (-2(InB + 2))

d²B/dt² = -2/B

Since we have already established that d/dt (InB + 2) = -2, we can differentiate this expression with respect to t to get:

d²B/dt² = d/dt (-2) = 0

Therefore, d²B/dt² = -2/B if and only if d/dt (InB + 2) = -2.

Now, let's check if the given solution satisfies this condition. Substituting InB + 2 = -2t in d/dt (InB + 2), we get:

d/dt (InB + 2) = d/dt (In(-2t) + 2) = -2/t

Since -2/t is not equal to -2, the given solution does not satisfy the differential equation dB/dt = -2B, and hence, we cannot verify it as a solution.

Therefore, none of the options A, B, C, or D are the necessary step to verify InB + 2 = -2t as a solution to dB/dt = -2B.

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The function g(t) = 3t⁵ + 40t⁴ + 150t³ + 120 is concave up on the interval (-∞, -2) and concave down on the interval (-2, ∞). There is an inflection point at t = -2.

1. Find the first derivative, g'(t) = 15t⁴ + 160t³ + 450t².
2. Find the second derivative, g''(t) = 60t³ + 480t² + 900t.
3. Factor out the common term, g''(t) = 60t(t² + 8t + 15).
4. Solve g''(t) = 0 to find critical points. In this case, t = 0 and t = -2.
5. Test the intervals to determine the concavity: For t < -2, g''(t) > 0, so it's concave up. For t > -2, g''(t) < 0, so it's concave down.
6. Since the concavity changes at t = -2, there is an inflection point at t = -2.

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The equation for the function, which is f(x) = -4x + 500 and is a linear function, is the answer to the given question based on the function.

A straight line on a graph is represented by a particular kind of mathematical function called a linear function. Two variables that are directly proportional to one another are modelled using linear functions. For instance, the distance-time relationship in a straight line motion is a linear function with speed as the slope.

Let's start by determining whether the function is exponential or linear. Given that Nick can write 120 words in 30 minutes, his word-per-minute rate is 120/30, or 4 words. In order to estimate how many words, he writes in x minutes, we can use this rate:

Write x words in x minutes and multiply by 4 = 4x

Since Nick wants to write 500 words per day, we can create an equation to roughly calculate how many words remain in his writing session after x minutes:

500 - 4x is the number of words remaining needed to meet the target.

Given that there is a constant pace of 4 words per minute between the number of words still needed and the amount of time left, this equation is linear. It can be expressed as a linear function with the formula f(x) = mx + b, where m denotes the slope (rate) and b the y-intercept (value at x=0).

Since Nick needs to write 500 words at the beginning of the writing session, the y-intercept is 500 and the slope is -4 (indicating that the rate of words still needed is falling at a rate of 4 words per minute):

f(x) = -4x + 500

As a result, the function's equation is f(x) = -4x + 500, indicating that it is a linear function.

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Yes, the mysteries equal 0.06 of the total books.

Marcella said that the mysteries equal 0.06 of the total books.

To check the mysteries equal 0.06 of the total books is correct or not.

We can follow these steps:

1. Identify the total number of books and the number of mysteries: Marcella read 100 books, and 60 of them were mysteries.

2. Calculate the fraction of mysteries: Divide the number of mysteries (60) by the total number of books (100) to find the fraction of mysteries.

3. Compare the fraction with Marcella's claim: If the calculated fraction equals 0.06, then she is correct.

Now let's perform the calculations:

60 mysteries ÷ 100 total books = 0.6

Since 0.6 ≠ 0.06, Marcella's claim that the mysteries equal 0.06 of the total books is incorrect. In reality, mysteries make up 0.6 or 60% of the total books she read.

A model to support this answer could be a pie chart, where the circle represents the 100 books, and the mysteries portion is shaded in. By dividing the circle into 10 equal sections, the mysteries would fill 6 of those sections, which represents 60% of the total books.

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Answer:

32.2

Step-by-step explanation:

Answer:

Step-by-step explanation:

You want the measures of angles A and B in right triangle ABC with hypotenuse AB = 15, and side BC = 8.

The mnemonic SOH CAH TOA reminds you of the relationships between side lengths and trig functions in a right triangle:

  Sin = Opposite/Hypotenuse

  Cos = Adjacent/Hypotenuse

Here, the hypotenuse is given as AB=15. The side opposite angle A is given as BC=8, so we have ...

  sin(A) = 8/15   ⇒   A = arcsin(8/15) ≈ 32°

The side adjacent to angle B is given, so we have ...

  cos(B) = 8/15   ⇒   B = arccos(8/15) ≈ 58°

Of course, angles A and B are complementary, so we can find the other after we know one of them.

  B = 90° -A = 90° -32° = 58°

The measures of the angles are A = 32°, B = 58°.

__

Additional comment

The inverse trig functions can also be called arcsine, arccosine, arctangent, and so on. On a calculator these inverse functions are indicated by a "-1" exponent on the function name—the conventional way an inverse function is indicated when suitable fonts are available.

You will note the calculator is set to DEG mode so the angles are given in degrees.

Answer: The given statement "if every column of an augmented matrix contains a pivot then the corresponding system is consistent" is true. This is because when every column of an augmented matrix contains a pivot, it implies that there are no free variables in the system of equations represented by the matrix.

Step-by-step explanation: Since every variable has a pivot in the augmented matrix, there is a unique solution to the system of equations. This is the definition of a consistent system - one that has at least one solution.                                                                                                                  In summary, the statement is true because the presence of a pivot in every column of an augmented matrix guarantees a unique solution to the system of equations, which is the definition of a consistent system.

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Answer:

Step-by-step explanation:

The mean of a discrete uniform distribution is the average of the minimum and maximum values of the distribution.

In this case, X ranges from 12 to 24, so the minimum value is 12 and the maximum value is 24. Therefore, the mean is:

Mean = (12 + 24) / 2 = 18

So the answer is c. 18.0.

When writing a rational number as a decimal, the decimal may either terminate or repeat indefinitely.

If the decimal repeats, there is a pattern of digits that repeat after a certain point. To indicate the repeating pattern, a bar is placed over the digits that repeat. This bar is typically placed over the smallest repeating pattern, which may be one or more digits.

For example, in the decimal representation of 1/3, the digit 3 repeats indefinitely, so the number is written as 0.333... with a bar over the 3. In the decimal representation of 2/7, the pattern 142857 repeats indefinitely, so the number is written as 0.285714285714... with a bar over the repeating pattern.

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Complete Question:

Writing Rational Numbers as Repeating Decimals. Highlight the number that repeats.

$15

$15 because they are asking you to do a subtraction the initial amount used to be $55then you have to substract $40 because they are saying $40 OFF

$15 because they are asking you to do a subtraction the initial amount used to be $55then you have to substract $40 because they are saying $40 OFF55 - 40= 15don't forget to add the $ sign

$15 because they are asking you to do a subtraction the initial amount used to be $55then you have to substract $40 because they are saying $40 OFF55 - 40= 15don't forget to add the $ sign !Hope I helped you

The value of the car 20 years after it was purchased is approximately $4,100.

Depreciation refers to the decrease in the value of an asset over time due to wear and tear, obsolescence, or other factors. In the context of a car, depreciation means that its value decreases as it is used and ages.

To calculate the value of the car 20 years after it was purchased, we need to find out how many times the value is halved in 20 years. Since 3.5 years is the time it takes for the value to be halved, we can divide 20 by 3.5 to get the number of times the value is halved.

20 / 3.5 = 5.71 (rounded to two decimal places)

So, the value of the car after 20 years would be:

$29,000 / (2^5.71) = $4,090 (rounded to the nearest hundred dollars)

Therefore, the value of the car 20 years after it was purchased is approximately $4,100.

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To analyze how the stability of the system depends on k, simply substitute k for any of the coefficients in the characteristic equation and construct a new Routh array. By analyzing the Routh array for each value of k, you can determine the range of values of k for which the system is stable.

The Routh-Hurwitz criterion is a mathematical tool used to determine the stability of a system. The criterion relies on constructing a table called the Routh array, which consists of rows and columns of coefficients from the system's characteristic equation. The coefficients in the Routh array are used to determine the number of roots of the characteristic equation that lie in the left half of the complex plane, which is a necessary condition for stability.

If you have a system with a characteristic equation of the form:

[tex]a_n s^n + a_{n-1} s^{n-1} + ... + a_1 s + a_0 = 0[/tex]

and you want to analyze how the stability of the system depends on a constant parameter k, you can do so by constructing a series of Routh arrays, each corresponding to a different value of k.

To do this, first write the characteristic equation as:

[tex]s^n + (a_{n-1}/a_n) s^{n-1} + ... + (a_1/a_n) s + (a_0/a_n) = 0[/tex]

Then, construct the first two rows of the Routh array as follows:

[tex]Row 1: a_n a_{n-2} a_{n-4} ...[/tex]

[tex]Row 2: a_{n-1} a_{n-3} a_{n-5} ...[/tex]

For each subsequent row, calculate the coefficients using the following formula:

[tex]a_{i-1} = (1/a_{n-1}) [a_{n-i} a_{n-1} - a_{n-i-1} a_n][/tex]

If at any point in the construction of the Routh array a zero entry is encountered, it indicates that there is at least one root of the characteristic equation with positive real part, and therefore the system is unstable. If all entries in the first column of the Routh array are nonzero and have the same sign, the system is stable.

To analyze how the stability of the system depends on k, simply substitute k for any of the coefficients in the characteristic equation and construct a new Routh array. By analyzing the Routh array for each value of k, you can determine the range of values of k for which the system is stable.

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The decision variables for this problem are:

This problem has the following constraints:

Production time cannot exceed the available time on each machine:

5x1,1 + 8x2,1 ≤ 600

4x1,2 + 5x2,2 ≤ 600

Production cannot be negative:

x1,1 ≥ 0

x1,2 ≥ 0

x2,1 ≥ 0

x2,2 ≥ 0

Demand must be met for each product:

x1,1 + x1,2 ≥ 120

x2,1 + x2,2 ≥ 150

Demand cannot exceed the maximum demand for each product:

x1,1 + x1,2 ≤ 200

x2,1 + x2,2 ≤ 130

Therefore, this problem has 4 constraints.

The decision variables for this problem are x1,1 (the number of units of product 1 produced on machine 1), x1,2 (the number of units of product 1 produced on machine 2), x2,1 (the number of units of product 2 produced on machine 1), and x2,2 (the number of units of product 2 produced on machine 2).

Therefore, this problem has 4 decision variables.

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The polar coordinates are (4√3, 5π/6). The correct answer is D. (4√3, 5π/6). The other given options are incorrect.

To convert Cartesian coordinates (-6, 2√3) to polar coordinates, we use the formulas:

r = √(x^2 + y^2)
θ = tan^-1 (y/x)

Plugging in the values, we get:

r = √((-6)^2 + (2√3)^2) = √(36 + 12) = 2√13
θ = tan^-1 (2√3/-6) = -π/3

However, since the point is in the second quadrant, we need to add π to the angle, giving us:

θ = -π/3 + π = 2π/3

Therefore, the polar coordinates of (-6, 2√3) can be expressed in two ways:

A. (4 √3, 3 π/4)
B. (3 √3, 3 π/4)

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The location of all points of inflection of the function f'(x) = (x - 2)(x-4)(x - 5) are option (d) 11- √7/3,  11+ √7/3.

To find the points of inflection of the function f, we need to find its second derivative and set it equal to zero, and then solve for x. If the second derivative changes sign at x, then x is a point of inflection.

Taking the derivative of f'(x), we get

f''(x) = 3x^2 - 22x + 32

Setting f''(x) = 0, we get

3x^2 - 22x + 32 = 0

We can solve this quadratic equation using the quadratic formula

x = [22 ± sqrt(22^2 - 4(3)(32))] / (2*3)

x = [22 ± sqrt(244)] / 6

x = (11 ± sqrt(61))/3

Therefore, the points of inflection of the function f are

x = (11 - sqrt(61))/3 ≈ 0.207

x = (11 + sqrt(61))/3 ≈ 3.793

So the answer is (d) 11- √7/3,  11+ √7/3.

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The location of all points of inflection of the function f'(x) = (x - 2)(x-4)(x - 5) are option (d) 11- √7/3,  11+ √7/3.

To find the points of inflection of the function f, we need to find its second derivative and set it equal to zero, and then solve for x. If the second derivative changes sign at x, then x is a point of inflection.

Taking the derivative of f'(x), we get

f''(x) = 3x^2 - 22x + 32

Setting f''(x) = 0, we get

3x^2 - 22x + 32 = 0

We can solve this quadratic equation using the quadratic formula

x = [22 ± sqrt(22^2 - 4(3)(32))] / (2*3)

x = [22 ± sqrt(244)] / 6

x = (11 ± sqrt(61))/3

Therefore, the points of inflection of the function f are

x = (11 - sqrt(61))/3 ≈ 0.207

x = (11 + sqrt(61))/3 ≈ 3.793

So the answer is (d) 11- √7/3,  11+ √7/3.

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Answer:

Step-by-step explanation:

1008

This text presents information about two exponential functions f and g. Function f passes through the points (-1, 5) and (2, -1.5), and intercepts the x-axis at (1, 0) and the y-axis at (0, 2). Function g is a decreasing exponential function with a y-intercept of 5 and no x-intercept. The text asks to compare the end behavior of these two functions as x approaches negative and positive infinity. End behavior refers to the behavior of the function as x approaches either positive or negative infinity.

Answer:

Total Surface Area is 20

Step-by-step explanation:

The formula for surface are with slant heigh is

SA = a^2 + 2×a×l

a = Base Edge (this case 2)

I = Slant Height (this case 4

2^2 + 2(2)(4) = 4+16=20

The trigonometric identity tan(π/4 - x) = [1 - tan(x)]/[1 + tan(x)]

Trigonometric identities are mathematical equations that contain trigonometric ratios.

Since we have the trigonometric identity

tan(π/4 - x) = [1 - tan(x)]/[1 + tan(x)]. We want to show that the left-hand-side L.H.S = right-hand-side R.H.S. We proceed as folows

Since we have L.H.S = tan(π/4 - x)

Using the trigonometric identity tan(A - B) = (tanA - tanB)/(1 + tanAtanB). So, comparing with tan(π/4 - x), we have that

So, substituting the values of the variables into the equation, we have that

tan(A - B) = (tanA - tanB)/(1 + tanAtanB)

tan(π/4 - x) = [tanπ/4 - tan(x)]/[1 + tan(π/4)tan(x)].

Since tanπ/4 = 1, we have that

tan(π/4 - x) = [tanπ/4 - tan(x)]/[1 + tan(π/4)tan(x)]

tan(π/4 - x) = [1 - tan(x)]/[1 + 1 × tan(x)]

tan(π/4 - x) = [1 - tan(x)]/[1 + 1 × tan(x)]

= R.H.S

Since L.H.S = R.H.S

So, the trigonometric identity tan(π/4 - x) = [1 - tan(x)]/[1 + tan(x)]

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Let x be a random variable with pdf f(x) = 1/13, 21 P(X > 30) = 0.31.

We are given that X is a random variable with a probability density function (pdf) of f(x) = 1/13 for the interval 21  x  34.

We are asked to find P(X > 30), which means we need to find the probability of the random variable X being greater than 30. To do this, we will calculate the area under the PDF in the interval [30, 34].

Step 1: Determine the width of the interval [30, 34].
Width = 34 - 30 = 4

Step 2: Calculate the area under the PDF in the interval [30, 34].
Since the pdf is a constant value (1/13) within the given interval, we can calculate the area as follows:
Area = f(x) * width
Area = (1/13) * 4

Step 3: Round off the result to the second decimal place.
Area ≈ 0.31 (rounded to two decimal places)

So, P(X > 30) ≈ 0.31.

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Answer:

x = 13.8 ft

The triangle is obtuse

Step-by-step explanation:

Using the cosine rule to determine x:

[tex]x=\sqrt{(11.7)^{2}+(7.4)^{2} -2(11.7)(7.4) * cos90 } \\=13.8 ft\\[/tex]

Testing whether or not the Pythagoras theorem applies

[tex]r^{2} =x^{2} +y^{2} \\(13.8)^{2} = (7.4)^{2} +(11.7)^{2} \\190.44\neq 191.65[/tex]

Therefore the triangle is obtuse

The value of the probability P(3) is 0.250 and P(More than 2) is 0.474

From the question, we have the following parameters that can be used in our computation:

n = 10 questions

x = 3 questions answered correctly

p = 1/4 i.e. the probability of getting a right answer

The probability is then calculated as

P(x = x) = nCr * p^x * (1 - p)^(n - x)

Substitute the known values in the above equation, so, we have the following representation

P(x = 3) = 10C3 * (1/4)^3 * (1 - 1/4)^7

Evaluate

P(x = 3) = 0.250

Hence, the probability is 0.250

This is represented as

P(x > 2) = 1 - P(0) - P(1)  - P(2)

Using a graphing tool, we have

P(x > 2) = 0.474

Hence, the probability is 0.474

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Answer:

B. The length of the arc is 1.5 times longer than the radius.

C. The ratio of arc length to radius is 1.5.

CD is the altitude to side AB of right [tex]\triangle[/tex]ABC, where m[tex]\angle[/tex]ACB = [tex]90^o[/tex] The value of BC is 7.28 units.

Value in math is a concept that describes the magnitude, or size, of a number. It can refer to absolute value, which is the actual number, or it can refer to relative value, which is the number compared to other numbers. Value is important in math because it is used to compare and measure different quantities. For example, in addition and subtraction, the value of the numbers being added or subtracted determines the answer. In multiplication, the value of the factors determines the product. Value is also important for performing calculations, such as finding averages, which requires knowledge of numbers and their relative values.
The given triangle is a right triangle, with ∠acb as the right angle. Using the Pythagorean Theorem, we can find the length of the side BC. The Pythagorean Theorem states that the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Therefore, BC² = AC² + BD²
Substituting the given values in the equation,
BC² = 52 + (5 1/3)²
Simplifying the equation,
BC² = 25 + 27.69
Therefore, BC² = 52.69
Taking the square root of both sides,
BC = √52.69
Therefore, BC = 7.28 units.
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Using proportions, the equation in terms of Tim is given by:

T(t) = 17t.

We have,

A proportion is a fraction of a total amount, and the measures are related using a rule of three. Due to this, relations between variables, either direct(when both increase or both decrease) or inverse proportional(when one increases and the other decreases, or vice versa), can be built to find the desired measures in the problem, or equations to find these measures.

For this problem, we have that:

Isaac sells four times as much as Tim, hence I = 4t.

Hannah sells three times as much as Isaac, hence H = 3I = 3 x 4t = 12t.

Hence the total amount, as a function of Isaac's amount, is given by:

T(t) = I + H + t

T(t) = 4t + 12t + t

T(t) = 17t.

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complete question:

Tim (t), isaac (i), and hannah (h) all sell individual insurance policies. isaac sells four times as much as tim, and hannah sells three times as much as isaac. create an equation in terms of tim (t) in order to find the portion he sells.

The probability of spinning a 5 first and then a 1 is:

(1/6) * (1/6) = 1/36

Expressed as a percent, this is:

(1/36) * 100% = 2.78%

So the probability of landing on a 5 and then a 1 is 2.78%

Answer:

Step-by-step explanation:

Expanding the expression (g+h)(p+q-r) using the distributive property, we get:

(g+h)(p+q-r) = g(p+q-r) + h(p+q-r)

Now, applying the distributive property again, we can simplify this expression to:

(g+h)(p+q-r) = gp + gq - gr + hp + hq - hr

Therefore, the expression (g+h)(p+q-r) is equivalent to:

gp + gq - gr + hp + hq - hr

From the given choices, only P[{clubs}) = 0.7, P{{diamonds}) = 0.3 satisfies the requirements of the definition of probability.

The choice that fulfills the requirements of the definition of probability is:

P[{clubs}) = 0.7, P{{diamonds}) = 0.3.

For an event A in a sample space S, the probability of A, denoted by P(A), must satisfy the following conditions:

P(A) is a non-negative real number: This means that the probability of an event cannot be negative.

P(S) = 1: The probability of the sample space is always equal to 1. This implies that at least one of the events in the sample space must occur.

If A and B are two mutually exclusive events, then P(A or B) = P(A) + P(B): This means that the probability of either event occurring is equal to the sum of their individual probabilities.

In the given sample space S = {clubs, diamonds}, the probabilities of the two events must add up to 1, since there are only two possible outcomes.

Therefore, the probabilities of the events cannot be negative or greater than 1.

From the given choices, only P[{clubs}) = 0.7, P{{diamonds}) = 0.3 satisfies the requirements of the definition of probability.

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Answer:

X° = 72.6459

Step-by-step explanation:

To solve x we must use tan b/c it contain both side,

which is opposite and adjecent

tan ( x°) =16/5

tan ( x°) =16/5tan ( x°) = 3.2

tan ( x°) =16/5tan ( x°) = 3.2X °= tan^-1(3.2)

tan ( x°) =16/5tan ( x°) = 3.2X °= tan^-1(3.2)X° = 72.6459 round to 72.65°

The function y(x) = 2e⁻³ˣ + 8e⁻⁶ˣ + 16xe⁻⁶ˣ + 20x²e⁻⁶ˣ satisfies the given conditions.

To find y(x), we first solve the differential equation y''' - 15y'' + 54y' = 40e^x. The characteristic equation r³ - 15r² + 54r = 0 has roots r1 = 3, r2 = 6, and r3 = 6.

The general solution is y(x) = Ae³ˣ + Be⁶ˣ + Cxe⁶ˣ.

Using the initial conditions y(0) = 26, y'(0) = 18, and y''(0) = 26, we can find the values of A, B, and C. After substituting the initial conditions and solving the system of equations, we obtain A = 2, B = 8, and C = 16. Thus, y(x) = 2e⁻³ˣ + 8e⁻⁶ˣ + 16xe⁻⁶ˣ + 20x²e⁻⁶ˣ.

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