In this problem you will use variation of parameters to solve the nonhomogeneous equation y" – 4y' + 4y = -6e2t A. Write the characteristic equation for the associated homogeneous equation. (User for your variable.) B. Write the fundamental solutions for the associated homogeneous equation and their Wronskian. y2 = 41 = W(91, y2) = C. Compute the following integrals. ✓ W dt = 929 dt = и D. Write the general solution. (Use c1 and c2 for cy and c2). y= (Note: Your general solution will only be correct if it is a general solution to the differential equation.)

There were 8-1 = 7 more students who slept for 4-7 hours than for 12-15 hours.

In a survey conducted by Mr. Hamilton, he asked his class about the total number of hours they slept the previous night. The histogram provided in the question shows the distribution of their responses. To find out how many more students slept for 4-7 hours than 12-15 hours, we need to count the number of bars in the 4-7 hours range and subtract it from the number of bars in the 12-15 hours range. By counting the bars in the histogram, we can see that there are 9 bars in the 4-7 hours range and 4 bars in the 12-15 hours range. Therefore, there were 5 more students who slept for 4-7 hours than 12-15 hours.

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Complete question:

Mr. Hamilton surveyed his class to find the total number of hours each of his students slept the previous night. The histogram shows the results of the survey. How many more students slept 4 – 7 hours than 12 – 15 hours? Amount of Sleep Mr: Hamilton surveyed his class to find the total number of hours each of his students slept the previous night The histogram shows the results of the survey: How many more students slept 4 ~ 7 hours than 12 15 hours? L 0 -3 8 _ [1 12 - 15 submit Time (hours)'

The possible values of n are 1 and 2, as there can be two non-integer positive numbers whose product is equal to 12. Option B is correct.

All the possible ways to express 12 as a product of three positive numbers:

1 x 2 x 6

1 x 3 x 4

To identify which products have two non-integer factors.

From the above list, we see that only the first product, 1 x 2 x 6, has two non-integer factors (2 and 6).

From the  following complete list of the possible values of n if the product of 3 positive numbers is equal to 12 and 3 positive numbers of n are not integers then answer is option (b) 1, 2.

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We can also check this by graphing the function, which would show us the symmetry about the y-axis.

In mathematics, a function is a relation between a set of inputs and a set of possible outputs with the property that each input is associated with exactly one output. This means that for every value of the input, there is a unique corresponding value of the output.

by question.

To prove that a function is even, we need to show that it satisfies the property:

g(-x) = g(x) for all x in the domain of g.

So, let's substitute -x for x in g(x) and simplify:

g(-x) = 2[tex](-x)^{6}[/tex] - [tex](-x)^{2}[/tex] + 4

= 2[tex]x^{6}[/tex] -[tex]x^{2}[/tex] + 4

Notice that we obtained the same expression as g(x), which means that g(x) is even. Therefore, we can write:

g(x) = g(-x)

This means that the graph of the function is symmetric with respect to the y-axis, since for any value of x, the value of g(x) is the same as the value of g(-x).

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In calculus, the derivative is a fundamental concept that measures how a function changes with respect to its input variable. It is a mathematical tool used to study the behavior of functions and is used in many areas of mathematics, physics, engineering, and economics.

When t = 3, the velocity of the object is 27 meters per second, and its acceleration is -2 meters per second squared.

When the velocity and acceleration have opposite signs, the speed of the object is decreasing. The acceleration of an object can be calculated using the derivative of the velocity of the object with respect to time.

v(t) = 36 - t², 0 ≤ t ≤ 6

Therefore, when t = 3, the velocity of the object is 27 meters per second. When t = 3, a(3) = -2(3) = -6 meters per second squared. The object's acceleration is -6 meters per second squared.

The speed of the object is decreasing when the velocity and acceleration have opposite signs.

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Answer:

Step-by-step explanation:

To solve this problem, we need to find out how many seats there are in total, and then multiply that by the price per ticket.

To find the total number of seats, we need to add up the number of seats in each row. We can use the formula for an arithmetic sequence to do this:

S = n/2 * (a + l)

where S is the sum of the sequence, n is the number of terms, a is the first term, and l is the last term.

In this case, we have:

n = 20 (since there are 20 rows)

a = 25 (since there are 25 seats in the first row)

d = 2 (since the difference between each row is 2 seats, the common difference is 2)

We can use d to find the last term as well:

l = a + (n-1)*d

l = 25 + (20-1)*2

l = 25 + 38

l = 63

Now we can plug these values into the formula:

S = 20/2 * (25 + 63)

S = 10 * 88

S = 880

So there are 880 seats in total.

To find the total sales, we just need to multiply by the price per ticket:

total sales = 880 * $32

total sales = $28,160

Therefore, the total sales for a one-night concert with all seats taken would be $28,160.

Answer:

15/x * 100 %

Step-by-step explanation:

Since we don't know the total, we have 15/x * 100 % as the percent of how many people passed the test. (x is amount of people, 15 or greater.)

The number of minutes needed to complete a job, m, varies inversely with the number of workers, w.

Three workers can complete a job in 30 minutes.

To find, out how many minutes would it take 6 workers to complete the job.

The formula used for inverse variation is, m1w1 = m2w2

Where, m1 = 30,

w1 = 3,

m2 = ?

and w2 = 6

Substitute the given values in the above formula, 30 × 3 = m2 × 6

Simplify the above expression,90 = 6m2

Divide both sides by 6,90 / 6 = m2m2 = 15

Hence, it will take 15 minutes for 6 workers to complete the job.

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If Joel and Duncan split the 15 ounces of hot chocolate evenly, then they each got half of the container.

To find out how much hot chocolate each boy got, we need to divide the total amount of hot chocolate by the number of people sharing it. Since there are two boys, we can divide 15 by 2 to get the amount of hot chocolate each boy received: 15 ÷ 2 = 7.5

Therefore, each boy got 7.5 ounces of hot chocolate to drink while they went sledding.It's important to note that when dividing something equally between two people, we divide by two since there are two individuals.

In this case, each boy received half of the container, or half of the total amount of hot chocolate. This type of division is known as "fair" or "equal" sharing, as both parties receive an equal portion.

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The solution to the system of equations is (15, 30).

A group of two or more equations that must be solved all at once is known as a system of equations. The system's equations each show how two or more variables relate to one another. The variables' values that satisfy every equation in the system can be discovered using algebraic techniques.

Algebraic systems of equations can be solved using a variety of techniques, such as substitution, elimination, and graphing. The substitution approach involves solving one equation for one variable in terms of the other variable, and then substituting the result for that variable's expression into the other equation.

The given system of equations are y=x + 15, y= 2x.

Substitute the value of y from equation 2 in equation 1:

x + 15 = 2x

15 = x

Substitute the value of x to get the value of y:

y = 2x = 2(15) = 30

Hence, the solution to the system of equations is (15, 30).

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Answer:

Step-by-step explanation:

i assume the figure he needs to "make" is the turnover and not the marginal profit - no indication of the cost of the toys he sells is given so it is impossible to work out the marginal profit of the each toy sold. (Though how the toymaker worked out the turnover required without knowing the cost of the toy is beyond me - though that didn't stop British Leyland pricing their best selling Mini in the 1970s.)

In this case divide the income required by the price of each toy, but as the income must be at least the required amount, any fractional part of a toy must be rounded *UP* to the next whole number.

17,235 ÷ 45 = 383 (so no rounding is needed).

He needs to sell 383 toys (made at zero marginal cost).

Deion now has 2.30 grammes of pepper remaining.

pre-warnt g The g the l e the l the l a n no. If you entered x=0 into your equation, you would receive this result. Your original amount is the part of your equation that is NOT increased to the x power.

We must take the amount of pepper Deion used from the total amount he had in order to determine how much pepper he still has.

Initial amount of pepper = 4.22 grams

Amount of pepper used = 1.92 grams

Pepper remaining = Initial amount - Amount used

Pepper remaining = 4.22 grams - 1.92 grams

Pepper remaining = 2.30 grams

Therefore, Deion has 2.30 grams of pepper left.

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TU is a perpendicular bisector.

A line that divides a line segment into two equal halves and forms a 90-degree angle at the point of intersection is called a perpendicular bisector.

To put it another way, a perpendicular bisector separates a line segment at its midpoint, creating a 90-degree angle.

A line or line segment that divides a given line segment into two equal portions is known as a perpendicular bisector.

The word "bisect" refers to dividing equally.

The line segment that perpendicular bisectors overlap forms four 90° angles on either side.

A line or line segment is said to be perpendicular when it forms a 90° angle with another line or line segment.

Hence, TU is a perpendicular bisector.

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The sum of the decimal fraction equation is:

seventy-six hundredths

The equation we are given is sixteen hundredths plus six-tenths

Now, when we talk about the decimal hundredths. Hundredths after decimal points shows 1/100th part of a whole. It represents the place value of the digit that occurs two places after the decimal.

Thus:

sixteen hundredths = 16/100

Similarly, six-tenths = 6/10

Thus:

The sum is:

16/100 + 6/10

= (16 + 60)/100

= 76/100 or seventy-six hundredths

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there is no polygon that satisfies the given conditions.

Why it is and what is a Polygon?

Let the number of sides of the polygon be n.

We know that the sum of angles of an n-sided polygon is (n-2) × 180 degrees.

In this polygon, two angles are 100 degrees each, and the remaining (n-2) angles are 130 degrees each.

So, we can set up an equation:

2 × 100 + (n-2) × 130 = (n-2) × 180

Simplifying this equation, we get:

200 + 130n - 260 = 180n - 360

Solving for n, we get:

50n = 100

n = 2

However, a polygon with only 2 sides is not possible.

Therefore, there is no polygon that satisfies the given conditions.

A polygon is a 2-dimensional geometric figure that is formed by connecting a finite number of line segments to create a closed shape. These line segments are called sides of the polygon. The sides of a polygon only intersect at their endpoints, and the endpoints of each side are called vertices. A polygon with three sides is called a triangle, a polygon with four sides is called a quadrilateral, and a polygon with more than four sides is generally referred to by its number of sides (e.g. a pentagon has five sides, a hexagon has six sides, etc.).

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Answer:

no lol im too lazy

Step-by-step explanation:

give me brain

Answer:

x = 9/10.

Step-by-step explanation:

0.5(4x + 9) = 3x * 7/3

2x + 9/2 = 3x * 7/3

2x + 9/2 = 7x

5x = 9/2

x = 9/10.

A 2-liter container costs R0.032 per milliliter, which is less than a 750-milliliter bottle, which costs R0.037 per milliliter.

a volumetric measurement using the metric system. One litre is equivalent to 1,000 milliliters. Additionally known as cc, mL, and cubic centimetre.

To evaluate which option is the most cost-effective, we need to calculate the cost per milliliter of each bottle of cooking oil.

For the 750ml bottle, the cost per milliliter is:

R28 / 750ml = R0.037 per ml

For the 2-liter bottle, the cost per milliliter is:

R63 / 2000ml = R0.032 per ml

Therefore, the 2-liter bottle is the more cost-effective option, as it has a lower cost per milliliter than the 750ml bottle. The cost per milliliter of the 2-liter bottle is R0.032, which is cheaper than the cost per milliliter of the 750ml bottle, which is R0.037.

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Option 1 with the discount and 4% simple interest has a total cost of $972, while Option 2 with no discount and no interest has a total cost of $1000. Option 1 is the better choice as it has a lower total cost.

Simple interest is a type of interest that is calculated on the original principal amount of a loan or investment. It is a fixed percentage of the principal amount that is paid by the borrower or earned by the lender over a specific period of time.

To compare the two options, we need to calculate the total cost of each option and compare them.

Option 1: 10% discount and pay off with 4% simple interest

The discount reduces the price of the television to $1000 x 0.9 = $900. If the customer chooses to pay it off at 4% simple interest, the total cost would be:

Total cost = $900 + ($900 x 0.04 x 3) = $972

Option 2: Full price and pay off in 3 years with no interest

The total cost of this option would be simply the full price of $1000 paid over 3 years, so:

Total cost = $1000 / 3 = $333.33 per year x 3 years = $1000

Comparing the two options, we see that Option 1 with the discount and 4% simple interest has a total cost of $972, while Option 2 with no discount and no interest has a total cost of $1000. Therefore, Option 1 is the better choice as it has a lower total cost.

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The complete question is: The store sells a television for $1000. customers can choose to receive 10% discount and pay it off at a simple interest rate of 4% or they can choose to pay the full price and pay it off in 3 years with no interest. which option is better?

Option 1 with the discount and 4% simple interest.

Option 2 with no discount and no interest.

To construct the graph of the hyperbola, the procedure is as follows:

The equation of a vertical hyperbola with center (x*, y*) is given according to the following equation:

(y - y*)²/a² - (x - x*)²/b² = 1.

A vertical hyperbola opens up and down.

In this problem, the equation is:

(y + 2)²/36 - (x + 5)²/64 = 1.

Hence the center is:

(-5, 2).

The value of coefficient a is:

a² = 36 -> a = 6.

Then the vertices of the hyperbola are:

The slopes of the asymptotes of the parabola are given as follows:

±a/b.

Hence:

Since the asymptotes pass through the center, the equation is:

y - 2 = ± 3/4(x + 5).

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Answer:

700

Step-by-step explanation:

28 % of n = 28/100 x n = 0.28n

If 28% of n = 196 that means

0.28n = 196

Divide both sides by 0.28

0.28n/0.28 = 196/0.28

n = 700

The dimensions of the gray rectangle are approximately 61.9 km by 218.1 km.

The distance around a rectangle's outside edge is known as its perimeter. It is computed by summing the measurements of the rectangle's four sides. The perimeter P of a rectangle with sides of length l and width w is given by:

P = 2l + 2w.

By understanding that a rectangle contains two pairs of equal sides (length and breadth), and that the distance around the rectangle is equal to the total of these four sides, one may deduce this formula.

Let us suppose the dimensions of the gray triangle as x and y.

The perimeter of rectangle is given as:

P = 2(1 + b)

For the blue triangle we have:

P = 2(72 + 68)

For the gray triangle we have;

P = 2(x + y)

Equating the two since they are equal we have:

2(72 + 68) = 2(x + y)

280 = x + y

We also know the area of the gray triangle thus:

xy = 3876

Substituting y = 280 - x into the second equation, we get:

x(280 - x) = 3876

x² - 280x + 3876 = 0

x = (280 ± √(280² - 4(1)(3876))) / 2

x ≈ 62.7 or x ≈ 61.9

We may discard the negative answer since a rectangle's dimensions cannot be negative.

Substituting the value of x we have:

y = 280 - 61.9 = 218.1 km.

Hence, the dimensions of the gray rectangle are approximately 61.9 km by 218.1 km.

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The probability that less than 70% of the people sampled voted is 0.9992.

Step 1: Calculate the mean of the sample. µ = p = 0.65 (since the percentage of eligible voters who actually voted in the recent election is 65%).

Step 2: Calculate the standard deviation of the sample.σ = sqrt{[p(1 - p)]/n} = sqrt{[(0.65)(0.35)]/1000} = 0.01578

Step 3: Calculate the z-score for the given probability. z = (0.7 - 0.65)/0.01578 = 3.16Step 4: Use the standard normal distribution table to find the area to the left of the z-score (since we want to find the probability that less than 70% of the people sampled voted).P(z < 3.16) ≈ 0.9992Therefore, the probability that less than 70% of the people sampled voted is approximately 0.9992 (or 99.92%).

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The expression of total number of hours worked by the 15 employees is 505 .

An expression is a mathematical statement that combines numbers, variables, and/or operators to represent a quantity or relationship. It can be evaluated or simplified, but does not have an equals sign and is not a complete equation.

To find the total number of hours worked by the 15 employees, we need to add up the hours worked by each individual employee.

Let's say that employee 1 worked x1 hours, employee 2 worked x2 hours, and so on, up to employee 15 who worked x15 hours. Then, the total number of hours worked by the 15 employees can be expressed as:

Total number of hours = x1 + x2 + ... + x15

To solve this expression, we need to know the specific number of hours worked by each employee. If we are given this information, we can simply substitute the values into the expression and add them up to find the total number of hours.

we can calculate the total number of hours worked as:

Total number of hours = 40 + 30 + 35 + ... + 20

= 505

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The probability that a person treated for an energy drink-related emergency was under 40 given that the person was at least 18 years old is approximately 0.550 or 55.0%.

In general, probability is defined as the ratio of the number of favourable outcomes to the total number of outcomes in a sample space. Event probability P(E) = (number of favourable outcomes) (Sample space).

We must use conditional probability to determine the likelihood that a person treated for an energy drink-related emergency was under 40, given that the person was at least 18 years old.

We can use the following formula:

P(A | B) = P(A and B) / P(A and B) (B)

where A is the event "the person is under 40" and B is the event "the person is over 18".

According to the table, 11,146 people under the age of 40 and over the age of 18 were treated for energy drink-related emergencies. The total number of people over the age of 18 who have been treated for energy drink-related emergencies is 20,244.

So,

P(A, B) = 11,146

P(B) = 20,244

Therefore,

P(A and B) / P(B) = 11,146 / 20,244 0.550

So the probability that a person treated for an energy drink-related emergency was under 40 is approximately 0.550 or 55.0% if the person was at least 18 years old.

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Answer:

B.6 pages per day

There are 1124 ways to choose a committee of 7 members with at least one member from each group and at least 3 artists.

Here, we have to solve this problem, we can use the concept of combinations, which involves counting the ways to choose a specific number of items from a larger set without regard to the order of selection.

Given the conditions that at least one member must be chosen from each group (artists, singers, writers) and there must be at least 3 artists, we can break down the problem into cases.

Case 1: Choosing 1 artist, 1 singer, and 5 members from the remaining groups (writers).

Case 2: Choosing 2 artists, 1 singer, and 4 members from the remaining groups (writers).

Case 3: Choosing 3 artists, 1 singer, and 3 members from the remaining groups (writers).

For each case, we will calculate the number of ways to choose members and then sum up the results from all three cases to get the total number of ways.

Let's calculate the number of ways for each case:

Case 1:

Number of ways to choose 1 artist: 6C1 (6 ways)

Number of ways to choose 1 singer: 4C1 (4 ways)

Number of ways to choose 5 writers: 5C5 (1 way)

Total ways for case 1: 6C1 * 4C1 * 5C5 = 6 * 4 * 1 = 24

Case 2:

Number of ways to choose 2 artists: 6C2 (15 ways)

Number of ways to choose 1 singer: 4C1 (4 ways)

Number of ways to choose 4 writers: 5C4 (5 ways)

Total ways for case 2: 6C2 * 4C1 * 5C4 = 15 * 4 * 5 = 300

Case 3:

Number of ways to choose 3 artists: 6C3 (20 ways)

Number of ways to choose 1 singer: 4C1 (4 ways)

Number of ways to choose 3 writers: 5C3 (10 ways)

Total ways for case 3: 6C3 * 4C1 * 5C3 = 20 * 4 * 10 = 800

Now, add up the total ways from all three cases:

Total ways = 24 + 300 + 800 = 1124

So, there are 1124 ways to choose a committee of 7 members with at least one member from each group and at least 3 artists.

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Answer:

To solve the system of equations:

n + d = 21 ---(1)

0.05n + 0.10d = 1.70 ---(2)

We can use the substitution method by solving for one variable in terms of the other from equation (1) and substituting it into equation (2).

Solving equation (1) for n:

n = 21 - d

Substituting this expression for n into equation (2):

0.05(21 - d) + 0.10d = 1.70

Distributing the 0.05:

1.05 - 0.05d + 0.10d = 1.70

Combining like terms:

0.05d = 0.65

Dividing both sides by 0.05:

d = 13

Substituting this value of d into equation (1):

n + 13 = 21

Solving for n:

n = 8

Therefore, the solution to the system of equations is n = 8 and d = 13.

Answer:

  (A)  10e^(i7π/4)

Step-by-step explanation:

You want the exponential form of 5√2 -5i√2.

There are numerous ways a complex number can be written in "polar form".

The usual choices are ...

  a +bi . . . . . . . . . . . . . rectangular form

  A(cos(θ) +i·sin(θ)) . . . . a sort of hybrid form

  A·cis(θ) . . . . . . . . . . an abbreviation of the above

  A∠θ . . . . . . . . . . . . polar form

  A·e^(iθ) . . . . . . . . . using Euler's formula

The conversion from rectangular form to any of the others makes use of trig identities and the Pythagorean theorem.

  A = √(a² +b²)

  θ = arctan(b/a) . . . . . with attention to quadrant

For the given number, ...

  A = √((5√2)² +(-5√2)²) = (5√2)√(1 +1) = 5·2

  A = 10

  θ = arctan(-5√2/(5√2)) = -1 . . . in the 4th quadrant

  θ = 7π/4

Then the desired exponential form of the complex number is ...

  10e^(i7π/4)

__

Additional comment

Spreadsheets and some calculators have an ATAN2(x, y) function that performs a quadrant-sensitive angle conversion.

The total cost of metal in the pattern would be  £24.6.

Let's assume that the cost of one arc is 3x, then the cost of one straight line would be 2x.

The total number of lines and arcs is given as 20 + 14 = 34.

Since 20 straight lines cost £12, the cost of one straight line is £12/20 = £0.6.

Therefore, the cost of one arc is (3/2) * £0.6 = £0.9.

The total cost of the 20 straight lines would be 20 x £0.6 = £12.

The total cost of the 14 arcs would be 14 x £0.9 = £12.6.

So, the total cost of metal in the pattern would be £12 + £12.6 = £24.6.

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The complete question is below:

A different pattern is made using 20 straight lines

and 14 arcs.

The straight lines and arcs are made from metal.

20 straight lines cost £12

cost of one straight line : cost of one arc = 2 : 3

Work out the total cost of metal in the pattern.