find the length of the hypotenuse in the figure shown below.

To find the inverse of a Laplace transform, one typically uses partial fraction decomposition and partial fraction expansion. Partial fraction decomposition involves breaking down a Laplace transformed function into simpler terms, each with a corresponding inverse Laplace transform. Partial fraction expansion involves using known formulas to find the inverse Laplace transform of these simpler terms.

The "inverse of a Laplace transform" is a mathematical operation that transforms a Laplace transformed function back into its original time domain form. It is a useful tool for solving linear differential equations, as well as for analyzing signals and systems.

The process of finding the inverse of a Laplace transform can also involve using convolution theorem, which states that the inverse Laplace transform of the product of two Laplace transformed functions is equal to the convolution of their respective inverse Laplace transforms. This theorem can be useful for solving systems of differential equations and for understanding the response of linear systems to different inputs.

It's also important to note that not all Laplace transformed functions have an inverse Laplace transform, as some functions may have poles in the right half of the complex plane, meaning they do not converge. In such cases, the inverse of a Laplace transform cannot be found using these methods.

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The inequality to represent the constraints will be:

x + y ≥ 325

x + 2y = 600

Inequalities are simply created through the connection of two expressions. In this case, it should be noted that the expressions in an inequality are not always equal. Inequalities implies that the expressions are not equal. Here, they are denoted by the symbols ≥ < > ≤.

Let x = number of vegetarian meals

Let y = number of meats.

The inequality to represent the constraints will be:

x + y ≥ 325

x + 2y = 600

In this case, the daily budget is 600.

Subtract the equations

y = 275

Number of meats = 275

Number of vegetarian meals = 325 - 275 = 50

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A.  These equations will give us the optimal values of x and y.

B. (b) To determine which exam you would be better prepared for, we compare the Math_grade and Physics_grade using the optimal values obtained in part (a).

C. We can follow the same steps as in part (a) to set up the equations and find the optimal number of hours spent studying for the two exams.

D. the specific impact would depend on the magnitude of the increase and how it interacts with the other variables and constraints in the problem.

(a) Let's define the variables:

Let x represent the number of hours spent studying math.

Let y represent the number of hours spent studying physics.

We want to maximize the average grade, which is proportional to the product of the hours spent studying and the ability to absorb the material.

The math grade is proportional to x * (1 - 0.04x) * (1 - 0.02y)

The physics grade is proportional to y * (1 - 0.03y) * (1 - 0.05x)

We want to maximize the average grade:

Average grade = (math grade + physics grade) / 2

Now, let's set up the optimization problem:

Maximize: (x * (1 - 0.04x) * (1 - 0.02y) + y * (1 - 0.03y) * (1 - 0.05x)) / 2

Subject to:

0 ≤ x ≤ 24 (maximum hours per day)

0 ≤ y ≤ 24 (maximum hours per day)

To solve this optimization problem, we can use calculus or numerical optimization methods.

(b) To determine which exam you would be better prepared for, calculate the average grade for the optimal number of hours obtained in part (a). Compare the average grade of math and physics to determine which one is higher.

(c) Now, assuming the exam scores improve by an additional 10% for every hour not studying, we need to modify the objective function in the optimization problem. Let's redefine the variables:

Let x represent the number of hours spent studying math.

Let y represent the number of hours spent studying physics.

Let z represent the number of hours not spent studying (24 - x - y).

The math grade is proportional to x * (1 - 0.04x) * (1 - 0.02y)

The physics grade is proportional to y * (1 - 0.03y) * (1 - 0.05x)

The additional benefit is 10% * z.

The new objective function to maximize the average grade is:

(x * (1 - 0.04x) * (1 - 0.02y) + y * (1 - 0.03y) * (1 - 0.05x) + 0.1 * z) / 2

Subject to:

0 ≤ x ≤ 24 (maximum hours per day)

0 ≤ y ≤ 24 (maximum hours per day)

z = 24 - x - y

(d) To examine the sensitivity of the answer to the additional 10% benefit from not studying, you can perform sensitivity analysis by changing the percentage value and observing the effect on the average grade. Increasing the effect of this benefit would likely result in a higher average grade as more emphasis is placed on the additional benefit gained from not studying. However, the exact impact would depend on the specific values used and the relationships between the variables and grades.

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Answer:

f(x), g(x), h(x)

Step-by-step explanation:

f(x) = x² + 6x - 1 = (x+3)² - 10          (-3 , 10)         y=-3

g(x) = -x² + 2 = -(x-0)² + 2               (0 , 2)          y=0

h(x) = 2x² - 4x + 3 = 2 (x-1)² + 1       (1 , 1)            y=1

Answer:

6x

Step-by-step explanation:

Step 1: Write equation

x + 2x + 3x

Step 2: Combine like terms

3x + 3x

6x

Answer:

12x   -10

Step-by-step explanation:

15x - 10 – 3x

Combine like terms

15x -3x    -10

12x   -10

Answer: 12x-10

Explanation:

I assume you want it simplified

15x-10-3x

Using the communicative property we can switch 10 and 3x making it

15x-3x-10

12x-10

Answer:

t+b≤94

Step-by-step explanation:

The most specific name for the figure is an isosceles trapezoid

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

The figure

The properties of the given figure are

Using the above as a guide, we have the following:

The figure is a trapezoid

Because the nonparallel sides are congruent, then the most specific name for the figure is an isosceles trapezoid

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By simplifying  45y/30y (7x+49)  this  we get 3 / 2(7x + 49).

45y/30y (7x+49)

45y / 30y(7x +49) : 3 / 2(7x + 49)

45y / 30y(7x + 49)

45y / 30(7x + 49)

Factor the number :45 = 15 . 3

=15.3 / 30(7x + 49)

Factor the Number : 30 = 15 . 2

15.3 / 15.2(7x + 49)

Cancel the common factor : 15

= 3 / 2(7x + 49).

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Answer:

\(11\dfrac{3}{5}$ miles\)

Step-by-step explanation:

Miguel ran \(2\dfrac{9}{10}\) miles on Monday.

On Friday, he ran 5 times as far as he did on Monday.

Miles run on Friday

\(=5 X 2\dfrac{9}{10}\)

\(=5 X \dfrac{29}{10}\\=\dfrac{29}{2}$ miles\)

Difference in Number of Miles Run

\(=\dfrac{29}{2}-\dfrac{29}{10}\\=\dfrac{145-29}{10}\\=\dfrac{116}{10}\\=11\dfrac{3}{5}$ miles\)

Therefore, Miguel ran \(11\dfrac{3}{5}$ miles\) farther on Friday.

The missing Value, x, that when multiplied by 5/3^4 gives the result of 5^4 is 13125.

The missing value that, when multiplied by 5/3^4, gives the result of 5^4, we can set up the equation:

(5/3^4) * x = 5^4

To solve for x, we can simplify both sides of the equation. First, let's simplify the right side:

5^4 = 5 * 5 * 5 * 5 = 625

Now, let's simplify the left side:

5/3^4 = 5/(3 * 3 * 3 * 3) = 5/81

Now we have:

(5/81) * x = 625

To solve for x, we can multiply both sides of the equation by the reciprocal of 5/81, which is 81/5:

(81/5) * (5/81) * x = (81/5) * 625

On the left side, the fraction (81/5) * (5/81) simplifies to 1, leaving us with:

1 * x = (81/5) * 625

Simplifying the right side:

(81/5) * 625 = 13125

Therefore, the missing value, x, that when multiplied by 5/3^4 gives the result of 5^4 is 13125.

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Open questions:

Closed questions:

Open questions allow individuals to express their thoughts, feelings, and opinions freely. These questions promote exploration and encourage the respondent to provide unique and personal insights.

Closed questions provide limited options for responses and are useful when seeking specific information or clarifying details. They are used in surveys, assessments etc.

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the standard deviation is not less than 13 hours based on the given sample data.

To calculate the standard deviation of the sample mean, we can use the formula:

Standard deviation of the sample mean = Standard deviation of the population / √(sample size)

Given that the population standard deviation is 180 and the sample size is 345, we can plug in these values:

Standard deviation of the sample mean = 180 / √345 ≈ 9.693

The standard deviation of the sample mean represents the average variability or spread of sample means that could be obtained from repeated sampling. In this case, it indicates how much the sample means would typically vary from the true population mean. A smaller standard deviation of the sample mean suggests that the sample means are closer to the population mean, leading to greater precision and accuracy in estimating the population mean.

Regarding the assumptions for the confidence interval at 95%, they typically include:

1. Random Sample: The sample should be selected randomly from the population to ensure its representativeness.

2. Independence: The observations within the sample should be independent of each other. Each data point should not be influenced by others.

3. Normality or Large Sample Size: The population distribution should be approximately normal, or the sample size should be large enough to satisfy the Central Limit Theorem. The CLT states that for a large sample size, the distribution of the sample mean becomes approximately normal, regardless of the population distribution.

4. Unbiasedness: The sample should be unbiased and free from selection or measurement bias.

As for testing the claim that the standard deviation is at least 13 hours, we can use a one-sample t-test with the null and alternative hypotheses:

Null hypothesis (H0): The standard deviation is less than 13 hours.

Alternative hypothesis (Ha): The standard deviation is at least 13 hours.

Using the sample data, we can calculate the test statistic:

t = (sample standard deviation - claimed standard deviation) / (standard deviation of the sample mean / √sample size)

t = (10.4 - 13) / (10.4 / √32) ≈ -2.035

Next, we determine the critical value at a significance level (α) of 0.05 for a one-tailed test with degrees of freedom (df) = sample size - 1 = 32 - 1 = 31. Checking a t-distribution table or using a statistical calculator, we find the critical value to be approximately -1.697.

Since the calculated t-value (-2.035) is less than the critical value (-1.697), we have evidence to reject the null hypothesis. Therefore, we can conclude that the standard deviation is not less than 13 hours based on the given sample data.

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a) The 90% confidence interval for the population mean is approximately (21.52, 23.48).

b) The 95% confidence interval for the population mean is approximately (21.322, 23.678).

c) The 99% confidence interval for the population mean is approximately (20.926, 24.074).

To develop confidence intervals for the population mean, we can use the formula:

Confidence Interval = sample mean ± (critical value * standard error)

where the standard error is equal to the sample standard deviation divided by the square root of the sample size.

a) For a 90% confidence interval, we need to find the critical value corresponding to a confidence level of 90%. The critical value can be obtained from the t-distribution table with (n-1) degrees of freedom. Since the sample size is 56, the degrees of freedom is 56-1 = 55.

From the t-distribution table, the critical value for a 90% confidence interval with 55 degrees of freedom is approximately 1.671.

The standard error can be calculated as:

Standard Error = sample standard deviation / sqrt(sample size)

Standard Error = 4.4 / sqrt(56)

Standard Error ≈ 0.5882

Now we can calculate the confidence interval:

Confidence Interval = 22.5 ± (1.671 * 0.5882)

Confidence Interval = 22.5 ± 0.9816

Confidence Interval ≈ (21.52, 23.48)

b) For a 95% confidence interval, the critical value for 55 degrees of freedom is approximately 2.004 (obtained from the t-distribution table).

Standard Error = 4.4 / sqrt(56) ≈ 0.5882

Confidence Interval = 22.5 ± (2.004 * 0.5882)

Confidence Interval = 22.5 ± 1.178

Confidence Interval ≈ (21.322, 23.678)

c) For a 99% confidence interval, the critical value for 55 degrees of freedom is approximately 2.678 (obtained from the t-distribution table).

Standard Error = 4.4 / sqrt(56) ≈ 0.5882

Confidence Interval = 22.5 ± (2.678 * 0.5882)

Confidence Interval = 22.5 ± 1.574

Confidence Interval ≈ (20.926, 24.074)

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The value of y is 24.

Non of the given option is correct.

To find the value of y in the given triangle, we can apply the Pythagorean theorem.

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In the given triangle, we have a right angle and one leg of length 12. The other leg has a length of 12√3. Let's assume y represents the length of the hypotenuse. Applying the Pythagorean theorem, we have:

(12)^2 + (12√3)^2 = y^2

144 + 432 = y^2

576 = y^2

Taking the square root of both sides, we get:

y = √576

y = 24

Non of the given option is correct.

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Answer:

I need the answer

Step-by-step explanation:

Answer:

Step-by-step explanation:

The sample is the part of the population that somebody wants to study. therefore, the sample is the 100 male students interviewed

The equation of a parabola with vertex at the origin and directrix y = 3 is given by the equation x^2 = 4py, where p is the distance from the vertex to the directrix. In this case, the distance from the vertex to the directrix is 3 units, so p = 3. Substituting this value into the equation, we will get the required solution. The equation of a parabola whose vertex is at the origin and whose directrix is y = 3 is x^2 = 12y.

To find the equation, we need to use the standard form of a parabolic equation, which is (x - h)^2 = 4p(y - k), where (h, k) represents the vertex and p represents the distance between the vertex and the directrix.

Since the vertex is at the origin (0, 0), we have h = 0 and k = 0. The directrix is y = 3, which means p = 3.

Plugging these values into the standard form equation, we get:
(x - 0)^2 = 4(3)(y - 0)
x^2 = 12y

Therefore, the correct answer is d. x^2 = 12y.

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Answer:

4.372×10^13

Step-by-step explanation: The question is asking "How much more does this one thing weigh than this other thing?" Like how much more does my cat weigh than this bowling ball? Just that, the things are really, really, really BIG! But, we're just going to subtract the two numbers, that's all. So, unfortunately, they are asking you to "show work" bc literally, you could punch these two numbers into a calculator and hit enter. But, let's see how we can "show work"!

The iceberg is the bigger number because of the exponent 13.

4.55×10^13 is

45500000000000.

That's a mess to work with but thought you might want to know/see it.

The Rock is 1.78×10^12 which is

1780000000000.

Again, not easy to work with, but there it is.

(4.55×10^13) - (1.78×10^12)

Change both to 10^13.

(4.55×10^13)-(.178×10^13)

factor out the 10^13.

10^13(4.55 - 1.78)

subtract.

10^13(4.372)

write in scientific notation.

4.372 × 10^13

You could subtract those two crazy long numbers with all the zeros. And then convert to scientific notation. It will come out the same.

The dimensions required are 125 feet and 50 feet.

The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables.

The total length will be x and the height will be y.

Perimeter = 500 = 2x + 5y

Total Area = A = xy

Solve for y using the equation of perimeter:

5y = 500 - 2x

y = 100 - 2x/5

Put y in the equation of the Area

A = x(100 - 2x/5)

A = -2x²/5 +100x

To find the maximum area, find the derivative:

A' = -4x/5 + 100

To maximise the area, we equate A' to zero.

A' = 0

-4x/5 +100 = 0

∴ x = 125

A'  is positive when x < 125 and A' is negative when x >125, therefore meaning that x=125 is a maximum. Since this value is a maximum, the area is maximized when the total length is 125ft.

Put x = 125 in equation of perimeter:

500 = 2(125) + 5y

500 = 250 + 5y

5y = 250

∴ y = 50 ft

Thus, the dimensions required are 125 feet and 50 feet.

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Answer:

a scale factor of 1.5 means the shape expands by a factor of 1.5

Step-by-step explanation:

to draw your new expanded shape, list the 3 coordinates. Multiply each x an y value by 1.5. Your shape should stay the same just get larger

Answer:

252 ÷ 18 = 14

325 ÷ 13 = 25

3612 ÷ 42 = 86

258 ÷ 12 = 21.5

324 ÷ 15 = 21.6

3636 ÷ 12 = 303

Step-by-step explanation:

For 252 ÷ 18, the answer is 14. First, we have to see how many times 18 goes into 25 which is 1. You put the 1 above the 5 in 252 and then figure out what 1 x 18 is. The answer you should get is 18.

After, you subtract 25 - 18 which is 7. So, after you solve all that you are supposed to bring the 2 down. Then, you have your new number which is "72." You have to figure out how many times 18 goes into 72 which is 4 times. You put the 4 above the 2 in 252 and figure out 4 x 18. It would be best if you got your answer as 72. You subtract 72 by 72 and that should get you 0.

There should be no remainder, and your answer should be 14. You have to use the method for every division problem: see how many times the divisor goes into the dividend, multiply, and subtract.

Hopefully, this helps, I did the best I could!

Answer:

Option (3). x < 3

Step-by-step explanation:

Given function is a logarithmic function,

\(y=\text{ln}[\frac{(-x+3)}{2}]\)

We know that logarithm is defined for all positive real numbers except zero.

\(\frac{(-x+3)}{2}>0\)

(-x + 3) > 0

-x > -3

x < 3

That means the given logarithmic function is defined for all real values lees than 3.

And the domain of the function will be x < 3

Option (3) will be the answer.

Answer:

x < 3        Option C on E2020.

Step-by-step explanation:

The air speed of a small airplane during the first 25 seconds of takeoff and flight can be modeled by Newton's Second Law of Motion.

According to Newton's Second Law of Motion, the rate of change of the airplane's momentum is equal to the sum of all the forces acting on the airplane. As the plane takes off and accelerates, the thrust of the engines acting in the forward direction causes the plane's momentum to increase. This increase in momentum results in an increase in air speed. The air resistance acting against the motion of the plane is another force that affects the plane's speed. As the plane accelerates, the air resistance increases, and the plane's speed decreases.

In summary, the air speed of a small airplane during the first 25 seconds of takeoff and flight can be modeled by Newton's Second Law of Motion, where the thrust of the engines and air resistance act as forces that influence the plane's air speed.

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Answer:

21

Step-by-step explanation:

Since the girls have the same age, let their age be x.
Then, their average is

\(\frac{x+x}{2} = \frac{2x}{2} = x\)

Let \(S_{i}\) denote the age of 'i' girls.
Then, \(S_{8} = S_{6} + x + x - eq(1)\)

Also, we have,

\(\frac{S_{8}}{8} =15 - eq(2)\)

\(\frac{S_{6}}{6} =13 - eq(3)\)

Then eq(2):

(from eq(1) and eq(3))

\(\frac{S_{6} + 2x}{8} =15\\\\\frac{13*6 + 2x}{8} = 15\\\\78+2x = 120\\\\2x = 120-78\\\\x = 21\)

The average of the other two girls with equal age​ is 21

The value of R(x) term is 36.

We are given that;

The equation x³ + 2x² - 72x - 28

Now,

We divide the first term, 8x, by the first term of d(x), which is x. This gives us 8, which is the third term of Q(x). We write 8 above the division bar and multiply it by d(x), which gives us 8x - 64. We subtract this from the new dividend, which gives us 36.

Since we cannot divide 36 by x-8 any further, we stop here and write 36 as the remainder R(x).

   x² + 10x + 8

  _______________

x-8 | x³ + 2x² - 72x - 28

    - (x³ - 8x²)

    -------------

        10x² - 72x

      - (10x² -80x)

      -------------

             8x -28

           - (8x -64)

           ----------

                36

Q(x) = x² + 10x + 8 and R(x) = 36.

Therefore, by the equation the answer will be R(x) = 36.

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