Given x=e−t and y=te8t, find the following derivatives as functions of t .
dy/dx = ________
d²y/dx² = ________

The solution to the differential equation is [tex]y = e^{-\frac{x^2}{2} + \ln(50)}[/tex]

The value of y is decreasing for x > 0.

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

dy/dx = -xy

Rewrite as

dy/y = -x dx

Differentiate both sides

So, we have

ln|y| = -x²/2 + c

Take the exponent of both sides

[tex]|y| = e^{-\frac{x^2}{2} + c}[/tex]

Assume y > 0

So, we have

[tex]y = e^{-\frac{x^2}{2} + c}[/tex]

When x = 0, y = 50

So, we have

[tex]50 = e^{-\frac{0^2}{2} + c}[/tex]

Evaluate

c = ln(50)

So, we have

[tex]y = e^{-\frac{x^2}{2} + \ln(50)}[/tex]

Recall that dy/dx = -xy.

This means that if dy/dx < 0, then y is decreasing.

This means that, y is decreasing for x > 0.

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Answer:

x = -3 and x = 2

Step-by-step explanation:

The expression for the quadratic equation is as follows:

(x+3)(x-2)

For (x+3)(x-2) to equal 0, either (x+3) or (x-2) must be equal to 0. So,

(x+3) = 0 and (x-2) = 0

It implies,

x = -3 and x = 2

So, the values of x are -3 and 2.

The function  [tex]f(x) = x^3 / (x^2 - 49)[/tex]has a vertical asymptote at x = 7 and x = -7. There are no slant asymptotes. To find the vertical asymptotes of the function f(x), we need to identify the values of x for which the denominator becomes zero, as division by zero is undefined.

In this case, the denominator is (x² - 49). To determine when it equals zero, we set it equal to zero and solve for x:

x² - 49 = 0

This equation can be factored as the difference of squares:

(x - 7)(x + 7) = 0

Setting each factor equal to zero, we find that x = 7 and x = -7. Therefore, the function has vertical asymptotes at x = 7 and x = -7.

On the other hand, to determine if there are any slant asymptotes, we need to check if the degree of the numerator (x³) is greater than the degree of the denominator [tex](x^2 - 49)[/tex]. In this case, the degree of the numerator is 3, while the degree of the denominator is 2. Since the degree of the numerator is greater, there are no slant asymptotes for this function.

In summary, the function [tex]f(x) = x^3 / (x^2 - 49)[/tex] has vertical asymptotes at x = 7 and x = -7. There are no slant asymptotes.

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[tex]\boxed{\large{\bold{\blue{ANSWER~:) }}}}[/tex]

Here,

The two lines l and m are parallel,

cut by a transversat line t

So The two angles are supplementary each other.

we know that,

The sum of two supplementary angles are 180°

According to the question,

Therefore,

The value of x is 8°

Answer:

y - 6

Step-by-step explanation:

y is the number and  it is less than 6 so 6 has to be negative:

y - 6

Answer: good luck

Step-by-step explanation:

Answer:$15 per yard

Step-by-step explanation: $825 divided by 55 = $15

The cross product of

[tex](i+j) \times k[/tex]= 0.

To calculate the cross product of (i+j) and k,

Step 1: Assign unit vectors to the given vectors:

(i+j) = i + j + 0k

k = 0i + 0j + k

Step 2: Apply the cross product formula:

(i+j) × k = (i × 0i) + (i × 0j) + (i × k) + (j × 0i) + (j × 0j) + (j × k) + (0k × 0i) + (0k × 0j) + (0k × k)

Step 3: Simplify the cross product using the properties of the cross product:

(i × 0i) = (j × 0j) = (0k × 0i) = (0k × 0j) = 0

(i × k) = - (k × i)

(j × k) = - (k × j)

(k × k) = 0

Step 4: Substitute the simplified cross products into the formula:

(i+j) × k = 0 + 0 + (i × k) + 0 + 0 + (j × k) + 0 + 0 + 0

= 0 + 0 + (i × k) + 0 + 0 + (j × k) + 0 + 0 + 0

= (i × k) + (j × k)

Step 5: Calculate the cross products:

(i × k) = (0 - 0)k - (0 - 0)j = 0k - 0j = 0k

(j × k) = (0 - 0)i - (0 - 0)k = 0i - 0k = 0i

Step 6: Substitute the calculated cross products into the formula:

(i+j) × k = 0k + 0i

= 0k + 0

= 0

Therefore k=0.

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Answer:

the answer is B

Step-by-step explanation:

because you just put 100 underneath

Answer:

B, 71/100

Step-by-step explanation:

To do this, just put each fraction into a calculator, and whichever fraction results in the decimal you're looking for will be your answer.

Answer:

d (or V in the picture) = 16

Step-by-step explanation:

Because the two triangle are similar, the ratios of each of the side lengths should be equal.  Therefore, 8/1 should be equal to d/2.

1.) Set up equality of ratios: 8/1 = d/2

2.) Cross multiply: 8*2 = d*1

3.) 16 = d

Answer:

As per the conversion in the question, £40.50 = €54

Step-by-step explanation:

Here, we want to make a conversion;

from the question;

£3 = €4

£40.50 = €x

Thus;

3 * x = 4 * 40.5

x = (4 * 40.5)/3

x = €54

Answer:

A.

B.

E.

Step-by-step explanation:

hope this helps ❤️

This formula calculates the value corresponding to a 90% confidence level using the T.INV function. The first argument, 0.95, represents 1 minus the desired confidence level (0.05 for a 90% confidence level). The second argument, `COUNT(A2:A89)-1`, calculates the degrees of freedom by subtracting 1 from the count of data points.

To find the value from the appropriate probability table to construct a 90% confidence interval in cell B8, you can use the T.INV function in Excel. Here's how you can do it:

1. In cell B8, enter the formula:

  ```

  =T.INV(0.95, COUNT(A2:A89)-1)

  ```

  This formula calculates the value corresponding to a 90% confidence level using the T.INV function. The first argument, 0.95, represents 1 minus the desired confidence level (0.05 for a 90% confidence level). The second argument, `COUNT(A2:A89)-1`, calculates the degrees of freedom by subtracting 1 from the count of data points.

Make sure to adjust the cell references in the formula based on the actual location of your data in the spreadsheet.

Note: The T.INV function returns the value from the Student's t-distribution, which is used for small sample sizes or when the population standard deviation is unknown.

If you have a large sample size (usually considered more than 30), you can use the Z.INV function instead to calculate the value from the standard normal distribution.

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Answer:

The answer is a

Step-by-step explanation:

Answer:

Hey I know this isn’t an answer but your bf Jordan told me to tell you he is looking for you and he needs to talk to you. Sorry again for wasting ur time but he kinda worried

Step-by-step explanation:

The statistic provides an estimate of the parameter based on the information obtained from the sample.

In this scenario, the population refers to the entire group of customers of the local store.

The sample is a subset of this population and consists of the 82 customers who were randomly surveyed.

The variable of interest is the age of the customers, as the store owner wants to determine the mean age.

Age is a continuous numerical variable since it represents a quantitative measurement.

The parameter is the unknown mean age of all customers of the local store, while the statistic is the calculated mean age of the 82 surveyed customers.

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Answer:

x=0

Step-by-step explanation:

have a nice day or evening

21 is the right answer.

180-138=42°

and since it is given in the question that the triangle is isoceles ,so half of 42 is 21°

Answer:

the last three answers is what i think

Step-by-step explanation:

When the company insists on producing 140 units then the selling price per unit required to break even is approximately $14.60.

To calculate the selling price per unit required to break even, we need to consider the fixed costs, variable cost per unit, and the desired production quantity.

Given:

Fixed costs = $1,596.00 per month

Variable cost per unit = $3.20

Production quantity = 140 units

To break even, the total revenue should cover both the fixed costs and the variable costs.

The total cost can be calculated as follows:

Total cost = Fixed costs + (Variable cost per unit × Production quantity)

Plugging in the given values:

Total cost = $1,596.00 + ($3.20 × 140)

Total cost = $1,596.00 + $448.00

Total cost = $2,044.00

To break even, the total revenue should be equal to the total cost.

The selling price per unit required to break even can be calculated as:

Selling price per unit = Total cost / Production quantity

Plugging in the values:

Selling price per unit = $2,044.00 / 140

Selling price per unit ≈ $14.60

Therefore, the selling price per unit required to break even is approximately $14.60.

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Answer: 56

Step-by-step explanation: According to AIA(Alternate Interior Angles Congruency) they are equal. So just solve the equation that they're both equal to each other and you get x=-6. Plug this back in to find the angle measure, 56! Hope this helped!

Answer:

x = -6

56 degrees

Step-by-step explanation:

Those two angles that are labeled are equal for a reason that I forgot (but trust me they are). So, -8x+8 = -6x+20

just solve the equation to get x = -6

after plugging in -6 for x, you get that the angles that are labeled are 56 degrees.

Answer:

y

=

x

2

−

2

x

,  

y

=

x

Step-by-step explanation:

Glacial HVAC, Inc. had a relative market share of 54.39% in Fulton County in 2010, based on the sale of 3,295 air conditioning units, compared to their largest competitor, Estes Heating and Air.

To calculate the relative market share, we need to divide Glacial HVAC's sales by the total market sales. The formula for relative market share is:

Relative Market Share = Glacial HVAC's Sales / Total Market Sales.

In this case, Glacial HVAC sold 3,295 units, and their largest competitor, Estes Heating and Air, sold 2,759 units. So the total market sales would be the sum of these two figures, which is 3,295 + 2,759 = 6,054 units.

Now, we can calculate the relative market share:

Relative Market Share = 3,295 / 6,054 * 100%.

Evaluating the expression, we find that Glacial HVAC's relative market share is approximately 54.39% when rounded to two decimal places.

This means that Glacial HVAC had a market share of 54.39% in Fulton County in 2010, indicating their position in the market relative to their largest competitor.

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Exercise 1: When tossing 4 identical and distinct coins, the number of macrostates and microstates are given below:MoleculesMacrostatesMicrostates4 coins16 states2^4=16Microstates: The number of ways in which the particles can be distributed among different energy levels is referred to as microstates. Macrostates: The number of ways in which the total energy of the system can be divided into different energy levels is referred to as macrostates. The distribution is represented in the following table: Distribution Microstates (W) Macrostates (Ω)TTTT1111HHHHT4C4,216HHHH3C4,715

Exercise 2:When distributing two distinct particles among three cells, the number of macrostates and microstates are as follows: Molecules Macrostates Microstates 2 particles10 states3^2=9Microstates: The number of ways in which the particles can be distributed among different energy levels is referred to as microstates. Macrostates: The number of ways in which the total energy of the system can be divided into different energy levels is referred to as macrostates. The distribution is represented in the following table: Distribution Microstates (W) Macrostates (Ω)2 in 11C21,23 in 11C31,33 in 11C32,310 in total 9.

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The probability which is in a random sample of 133 people from State College, more than 50 are not college students is 0.0000313 (rounded to 4 decimal places).

Given that in State College, Pennsylvania, the proportion of college students in the population is 71.1%.

We need to find the probability that in a random sample of 133 people from State College, more than 50 are not college students.

We need to round the final answer to at least 4 decimal places and intermediate z-value calculations to 2 decimal places.

The proportion of college students in State College, Pennsylvania is 71.1%.

Therefore, the proportion of non-college students in State College, Pennsylvania is 100% - 71.1% = 28.9%.

Let X be the number of non-college students in a sample of 133 people from State College, Pennsylvania.

As the sample is random, X follows the binomial distribution with parameters n = 133 and p = 0.289.

The probability of getting more than 50 non-college students can be obtained using the normal distribution approximation to the binomial distribution.

Using the normal distribution approximation, we can convert the binomial distribution to a standard normal distribution using the following formula: Z = (X - np) / sqrt(npq)

Where q = 1 - p is the proportion of college students in the population, and np = 133 x 0.289

= 38.397 and npq = 133 x 0.289 x 0.711 = 9.728.

The probability of getting more than 50 non-college students is : P(X > 50)

= P(Z > (50 - 38.397) / sqrt(9.728))

= P(Z > 3.92)

= 0.0000313 (rounded to 4 decimal places).

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After considering the given data we conclude that
a) r = -2 + i and r = -2 - i value generated after solving the auxiliary equation
b)[tex]y_c = c_1e^{(-2t)} cos(t) + c_2e^{(-2t)} sin(t)[/tex] complementary solution
c) general solution to the nonhomogeneous equation is [tex]y = y_c + Y_p = c_1e^{(-2t)} cos(t) + c_2e^{(-2t)} sin(t) + (1/26)e^{(-21)} .[/tex]
d) initial value problem is [tex]y(t) = e^{(-2t)} (cos(t) + 2sin(t))/2 + (1/26)e^{(-21)} .[/tex]

To determine the position of the body at time t, we can apply the given differential equation to solve for y(t). To do this, we can follow the steps below:
(a) the characteristic (auxiliary) equation is
The characteristic equation is obtained by setting the coefficients of the differential equation to zero, which gives us the equation  [tex]r^2 + 4r + 5 = 0.[/tex]Solving this quadratic equation, we get r = -2 + i and r = -2 - i.
(b) evaluate the complementary solution, yc, to the corresponding homogeneous equation, [tex]y" + 4y' + 5y = 0.[/tex]
The complementary solution is given by[tex]y_c = c_1e^{(-2t)} cos(t) + c_2e^{(-2t)} sin(t),[/tex]where [tex]c_1[/tex] and [tex]c_2[/tex] are constants determined by the initial conditions.
(c) Calculate a particular solution, Yp, to the nonhomogeneous differential equation, [tex]Y" + 4y' + 5y = e^{(-21)}[/tex]. Hence state the general solution to the nonhomogeneous equation as [tex]y = y_c + y_p.[/tex]
To describe a particular solution, we can use the method of undetermined coefficients and guess a solution of the form [tex]Yp = Ae^{(-21)}[/tex], where A is a constant to be determined.
Staging this into the differential equation, we get A = 1/26. Therefore, the particular solution is [tex]Y_p = (1/26)e^{(-21)}[/tex]. The general solution to the nonhomogeneous equation is [tex]y = y_c + Y_p = c_1e^{(-2t)} cos(t) + c_2e^{(-2t)} sin(t) + (1/26)e^{(-21)}[/tex]
(d) Solve the initial value problem if the initial position of the body is 1 m and its initial velocity is zero.
Applying the initial conditions, we can solve for the constants [tex]c_1[/tex] and [tex]c_2[/tex] in the complementary solution. Since the initial velocity is zero, we have [tex]y'(0) = -2c_1 + c_2 = 0[/tex], which gives us [tex]c_2 = 2c_1.[/tex]
Applying the initial position, we have [tex]y(0) = c_1 = 1[/tex], which gives us [tex]c_2 = 2.[/tex]Therefore, the solution to the initial value problem is [tex]y(t) = e^{(-2t)} (cos(t) + 2sin(t))/2 + (1/26)e^{(-21)}[/tex]
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Answer:

58

Step-by-step explanation:

Answer: 58°

Step-by-step explanation:

We know it's a right triangle, one of them must be 90°. The sum of the three interior angles are 180°, so we know the sum of the other two angles is 180 - 90 = 90°.

If one of the angles is 32°, another angle is 90 - 32 = 58°

1. is a central angle, therefore, the arc will have the same measurement as the angle. Set the equation:

Arc = Central Angle

2x - 7 = 47

Isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS.

First, add 7 to both sides of the equation:

2x - 7 (+7) = 47 (+7)

2x = 47 + 7

2x = 54

Next, divide 2 from both sides of the equation:

(2x)/2 = (54)/2

x = 54/2 = 27

27 is your answer.

2. is a inscribed angle, meaning that the angle measurement will be half of the arc. Set the equation:

212 = 2(13x - 24)

Isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS.

First, divide 2 from both sides of the equation:

(212)/2 = (2(13x - 24))/2

106 = 13x - 24

Next, add 24 to both sides of the equation:

106 (+24) = 13x - 24 (+24)

106 + 24 = 13x

130 = 13x

Finally, divide 13 from both sides of the equation:

(130)/13 = (13x)/13

x = 130/13

x = 10

10 is your answer.

3. is a central angle. The arc and the angle will, therefore, have the same measurement:

137 = 3x + 5

First, subtract 5 from both sides of the equation:

137 (-5) = 3x + 5 (-5)

137 - 5 = 3x

132 = 3x

Next, divide 3 from both sides of the equation to isolate the variable, x:

(132)/3 = (3x)/3

x = 132/3 = 44

44 is your answer.

4. is a inscribed angle. The arc will be twice the measurement of the angle.

86 = 2(2x + 3)

First, isolate the variable x by dividing 2 from both sides of the equation.

(86)/2 = (2(2x + 3)/2

43 = 2x + 3

Next, subtract 3 from both sides of the equation:

43 (-3)  = 2x + 3 (-3)

40 = 2x

Finally, divide 2 from both sides of the equation:

(40)/2 = (2x)/2

x = 40/2 = 20

20 is your answer.

~

The true statements from the sinusoidal graph are:

From the table, we have the following parameters:

So, the radius (r) of the wheel is:

[tex]r = \frac{64 - 4}{2}[/tex]

This gives

[tex]r = \frac{60}{2}[/tex]

[tex]r = 30[/tex]

Also, from the table, the lowest point of the wheel is 4 ft;

The Ferris wheel is at this lowest point at 7.5 seconds and 17.5 seconds

Hence, the true statement is (c) and (d)

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