What is a fundamental solution in differential equations?

The gradiant of the line which passes through the given points is 1

A straight line's steepness may be determined by looking at its gradient. A line's gradient does not have to be a whole integer and might be positive or negative.

Any two different points on a line may be used to compute the slope of any line. The ratio of "vertical change" to "horizontal change" between two separate locations on a line is calculated using the slope of a line formula.

The increase divided by the run, or the ratio of the rise to the run, is known as the line's slope. In the coordinate plane, it describes the slope of the line. Finding the slope between two separate locations and calculating the slope of a line are related tasks. In general, we require the values of any two separate coordinates of a line in order to determine its slope.

The points on the line :

x: -5, -4, -3, -2, -1

y : 1,2,3,4,5

The gradiant of the line :

(y₂ - y₁)/(x₂ - x₁) = (5 - 4)/(-1 + 2) = 1

The gradiant of the line which passes through the given points is 1

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Answer:

When you rotate a figure around a center point other than the origin, the resulting image is translated or shifted from the original figure's position. This happens because the center of rotation affects the position of the figure after the rotation is performed.

If you rotate a figure around a center point that is not the origin, the figure will still undergo a rotation, but the position of the rotated figure will depend on the location of the center point. The center point of rotation acts as a pivot point and determines the new location of the rotated figure.

In summary, rotating a figure around a center point other than the origin will result in a translated image that is shifted from the original position of the figure. The amount of translation will depend on the location of the center of rotation.

Because he multiplied the triangle's base by its height, Kelvin's solution is incorrect

Use the equation area = 1/2 * bases * altitude to determine a triangle's surface area Choose which side will be the triangle's base, and then measure the height of the triangle from that base.. After that, enter the height and base measurements you have into the formula.

Area of triangle= 1/2 base × height

The base of the triangle= 5 in

The height of the triangle = 8 in

Area of triangle= 1/2 base × height

=  1/2 × 5 in × 8 in

=  1/2 × 40 square inches

= 20  square inches

As a result, Kelvin's estimation of the triangle's area is incorrect.

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As the sample size rises, the width of a 95% confidence interval for a sample proportion decreases.

If the sample size is four times greater than the initial sample size but the sample percentage remains constant, the breadth of the 95% confidence interval is reduced by around a factor of two.

This is due to the fact that the standard error, which defines the width of the confidence interval, is proportional to the sample size squared. When the sample size is multiplied by four, the square root of the sample size multiplies by two, resulting in a lower standard error and a smaller confidence range.

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Answer: 120 inches cubed

Step-by-step explanation:

Answer:

[tex]a_1=-4, \,a_n=a_{n-1}+4[/tex]

Step-by-step explanation:

Clearly, our common difference is [tex]d=4[/tex], and we can see that the first term if we work backward from [tex]a_4[/tex] is [tex]a_1=-4[/tex].

Hence, by using the recursive formula [tex]a_n=a_{n-1}+d[/tex], we have our final formula as [tex]a_n=a_{n-1}+4[/tex]

The probability that a 4 or an odd number is rolled be 4/7.

The area of mathematics known as probability deals with numerical descriptions of how likely it is for an event to happen or for a claim to be true. A number between 0 and 1 is the probability of an event, where, broadly speaking, 0 denotes the event's impossibility and 1 denotes its certainty.

Simply put, probability measures how probable something is to occur. We can discuss the probabilities of various outcomes, or how likely they are, whenever we are unsure of how an event will turn out. Statistics is the study of events subject to probability.

Let the given data be  1, 2, 3, 4, 5, 6, 7

The odd numbers: {1, 3, 5, 7}

The values that are 7, which is just the set { 7 }

Add the two sets of 1, 3, 5, and 7 to get 1, 3, 5, and 7. Due to the overlap, the first set does not change.

The set has 4 things (1, 3, 5, 7), out of the total of 7 items (1, 2, 3, 4, 5, 6, 7).

Therefore, 4/7 is the probability stated as a fraction.

This is nearly equivalent to 0.5714, or 57.14 percent.

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Answer:

76

Step-by-step explanation:

m∠N : y

4x+36+y=180

6x-2+y=180

=>x=17

y=76

The system of linear equations 2 · x + 4 · y = 7 and 2 · x + 4 · y = 9 has no solutions.

In this case we find a representation of two linear equations with two variables, each represented by two orthogonal axes, whose mathematical representation is described below:

a · x + b · y = c

e · x + f · y = g

Where a, b, c, d, e, f, g are real coefficients.

A system has no solution when a = e and b = f but c ≠ g. Graphically speaking, this case is represented by two parallel lines. One example is the following system:

2 · x + 4 · y = 7

2 · x + 4 · y = 9

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For the function y = ( x² + 5 )² and x = 2 and dx = 0.3 then the value of differential dy is equal to 21.6.

y = (x² + 5)²

Calculate the differential dy with respect to x  we get,

dy / dx = d /dx (x² + 5)²

⇒ dy / dx = 2 ( x² + 5 )²⁻¹ × ( 2x )

⇒ dy = 4x ( x² + 5 ) dx

Now substitute the value of  x = 2, and dx = 0.3 to get the value of differential dy we get,

dy = 4( 2 ) ( 2² + 5 ) ( 0.3 )

⇒ dy = 8 × 0.3 × ( 9 )

⇒ dy = 21.6

Therefore, the value of the differential function dy for the function          y = ( x² + 5 )²  is equal to 21.6.

The above question is incomplete, the complete question is :

Let y = (x² + 5)². find the differential dy when the value of x = 2 and dx = 0.3.

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A simple method for adding and subtracting fractions with unlike denominators:

Find a common denominator: The first step is to find a common denominator that is a multiple of both denominators. The smallest common denominator is usually the least common multiple (LCM) of the two denominators.

Convert the fractions to equivalent fractions with the common denominator: Once you have a common denominator, you can convert each fraction to an equivalent fraction with the same denominator by multiplying the numerator and denominator by the same number.

Add or subtract the numerators: Once both fractions have the same denominator, you can add or subtract the numerators to get the numerator of the resulting fraction.

Simplify the resulting fraction: Finally, simplify the resulting fraction by dividing both the numerator and denominator by their greatest common factor, if possible.

Here is an example:

To add 1/3 + 1/4, we first find a common denominator:

The LCM of 3 and 4 is 12, so we multiply both the numerator and denominator of each fraction by 4 to get 12/12 as the common denominator:

1/3 * 4/4 = 4/12

1/4 * 3/3 = 3/12

Next, we add the numerators:

4/12 + 3/12 = 7/12

Finally, we simplify the fraction:

7/12 = (7 ÷ 1) / (12 ÷ 1) = 7/12.

So the result is 7/12.

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Answer:

second option

Step-by-step explanation:

vertical angles are situated opposite each other at a vertex. For this reason they are often called vertically opposite angles.

from the diagram the vertical angles opposite to each other at vertex E are

∠ 1 ≅ ∠ 3 and ∠ 2 ≅ ∠ 4

Answer:

∡1 ≅ ∡3 and ∡2 ≅ ∡4

Explanation:

Definition of vertical angles: Each of the pairs of opposite angles made by 2 intersecting lines.

By definition, vertical angles are on opposite sides of the intersection, and are always equal.

Since the pairs ∡1,∡3 and ∡2,∡4 are on opposite sides of the intersection, they are equal.

Proof for the Vertical Angles Theorem:  https://www.cuemath.com/geometry/vertical-angles/

Based on the information given about the opportunity cost, the correct values will be:

Local Department Store 56 100 156

Across Town 84 86 170

Neighboring City 140 63 203

An opportunity cost simply means the forgone benefit when an individual chooses another alternative.

here, we have,

From the complete question, it should be noted that when one wants to evaluate opportunity cost, the benefits and costs of each decision will be considered.

The opportunity cost of Juanita's time and the total cost of shopping at each location is illustrated above.

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All of the above (a, b, c) are measures of dispersion.

If a data set has an even number of observations, the median:

b. is the average value of the two middle items when all items are arranged in ascending order.

The median is a measure of central tendency that indicates the middle value of a set of data. In a data set with an even number of observations, the median is found by taking the average of the two middle items. For example, if a data set has 8 observations, the median is the average of the 4th and 5th observations when the data is sorted in ascending order.

c. must be equal to the mean.

This statement is not always true. The median and the mean are two different measures of central tendency, and they can be equal or not equal depending on the distribution of the data. In a symmetrical data distribution, the median and the mean are equal, while in a skewed data distribution, they are not equal.

Which of the following is a measure of dispersion?

d. all of the above are measures of dispersion.

Dispersion is a statistical term that refers to the extent to which the values in a data set are spread out or dispersed. It measures how far the values are from the central tendency of the data.

a. Percentiles are a measure of dispersion that indicates the value below which a certain percentage of the data falls. For example, the 50th percentile (or median) is the value below which 50% of the data falls.

b. Quartiles are a measure of dispersion that divides a data set into four equal parts. The first quartile (Q1) is the 25th percentile, the second quartile (Q2) is the 50th percentile (or median), and the third quartile (Q3) is the 75th percentile.

c. The interquartile range (IQR) is the range between the first and third quartiles and is a measure of dispersion that indicates the spread of the middle 50% of the data.

Therefore, all of the above (a, b, c) are measures of dispersion.

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The perimeter of shed is, (6x+ 4)

The area of the shed is, (2x² + 5x + 3)

A rectangle is a two dimension figure with 4 sides, 4 corners and 4 right angles. The opposite sides of the rectangle are equal and parallel to each other.

We have to given that;

Shirley is building a shed. The length is 2x + 3 feet long and the width is

x + 1 feet long.

Now, We know that;

The perimeter of rectangle = 2 (Length + Width)

Substitute all the values, we get;

⇒ Perimeter = 2 (2x + 1 + x + 1)

⇒ Perimeter = 2 (3x + 2)

⇒ Perimeter = (6x + 4)

And, Area of rectangle = Length × Width

Substitute all the values , we get;

⇒ Area = (2x + 3) × (x + 1)

⇒ Area = (2x² + 2x + 3x + 3)

⇒ Area = (2x² + 5x + 3)

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In a hypothesis test, YOU presumptively believe the alternative hypothesis, a statistical hypothesis is a claim regarding a sample, If you choose to reject the null hypothesis, you can then decide whether or not the alternative hypothesis is correct.

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The limit of this expression as δt approaches zero is 7t5

The limit is a concept in mathematics that represents the value that a function approaches as the input approaches a certain value.

To find the limit, we will simplify the expression and cancel out δt wherever possible.

First, we can simplify the expression inside the parentheses,

=> (t+δt)2 − 7(t+δt) 5.

We can rewrite this as

=> (t2x δt2) - (7 x t5 x δt).

Then, we can simplify the second part of the expression,

=> (δt x t2) - (δt * 7t5)

Finally, we can subtract these two expressions and simplify further,

=> (t2 x δ t2) - (7 x t5 x δ t) - (δt x t2) + (δt * 7t5).

Since δt is a small value that approaches zero, we can assume that delta t2 approaches zero.

This means that we can simplify the expression to

=> δt * 7t5.

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The output volume is 40 x 40 x 32

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

Input volume = 121 x 121 x 16

Filters = 32 with 4 x 4

Stride = 3

Padding = False

The output element (n) is calculated as

n = (nH + 2 x p - f)/s + 1

Using the given parameters, we have

nH = 121, p = 0, f = 4 and s = 3

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

n = (121 + 2 x 0 - 4)/3 + 1

Evaluate

n = 40

This means that

Output volume = 40 x 40 x 32

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Using decimals, 4/3 equals 1.33.

In this handbook, we'll show you how to use the division method to convert a fraction to a decimal. There are many other ways to convert numbers into decimals, percentages, and other units.

Solution: The decimal equivalent of 4/3 is 1.33.

The divide approach to explaining:

The numerator and denominator of a fraction are the numbers above and below the line, respectively. A fraction is essentially represented by two pieces separated by a line.

To obtain a decimal, use the division method to respond to the following query:

To get the decimal, just divide the numerator (4), by the denominator (3). This gives you 1.33.

The result of translating 4/3 to is 1.33.

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The rate for a ratio of the two different quantities is; ¹⁸/₃ puppies per liter

Two quantities can be referred to as proportional provided that they can be represented in the general form y = kx,

where k is a constant of proportionality

Now, we have the parameters as;

Number of Puppies = 18 puppies

Number of Litters = 3 litters

Thus, the rate for the ratio between the two quantities is expressed as;

18 puppies/3 Litters

= ¹⁸/₃ puppies per liter

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The amount that he will have after 4 years is given as follows:

$14,983.85.

The amount of money earned, in compound interest, after t years, is given by the equation presented as follows:

[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]

In which the parameters are listed as follows:

The parameters for this problem are given as follows:

P = 8000, r = 0.16, n = 4, t = 4.

Hence the balance will be given as follows:

P = 8000 x (1 + 0.16/4)^(4 x 4)

P = $14,983.85.

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The number of 3-digit numbers that can be formed is: 216.

The numbers that are less than 500 is: 120.

The number of 3-digit numbers that can be formed from the digits 2, 3, 4,5, 6 and 9 is calculated as:

6 * 6 * 6 = 216

This is because each digit can be chosen from the six available digits in 6 ways.

Out of these 216 numbers, those that are less than 500 have the hundreds digit equal to 2, 3, 4, or 5, and the first two digits can be chosen from any of the available digits in any order.

So, there are 4 * 6 * 5 = 120 numbers less than 500.

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Sergeant Carlton's monthly pay of $1856.40, using the conversion of the units of time measurement, indicates that his equivalent weekly rate of pay is $464.1

The factors of the units of time, to an accuracy of a day can be presented as follows;

12 months is equivalent to 1 year

4 weeks are equivalent to 1 month

7 days are equivalent to 1 week

However, there are;

52 weeks in a year

365 days in a year

366 days in a leap year.

The amount Sergeant Carlton's is paid per month = $1856.40

The number of weeks in a month, using the time units = 4 weeks

The equivalent weekly pay can be obtained by dividing the amount earned monthly by the number of weeks in a month as follows;

The equivalent weekly pay = $1856.4/4 = $464.1

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The p-value is less than 0.01 so we reject the null hypothesis.

Formulating Hypotheses:

Whenever formulating the null and alternative hypotheses, we also consider that the null hypothesis is true. Therefore, it is always a statement of equality and by definition, the alternative hypothesis is contrary to the null hypothesis.

Null hypothesis

H0: σ1² ≤ σ2²

Alternative hypothesis

H1: σ1² > σ2²

We have the level of significance = 0.01

Test statistic

F = S1²/S2²

F= 170²/100²

F= 28900/10000

F = 2.89

Df = degrees of freedom

Df1 = n1 - 1

= 26-1

Df1= 25

Df2 = n2 - 1

= 25-1

DF2= 24

B. We get the p-value using the f distribution

Fdist(2.89,25,24)

P value = 0.0057

The p-value is less than 0.01 so we reject the null hypothesis.

So we can conclude that automobiles that are of 4 years have variances larger In annual repair costs than those that are of 2 years.

Reasonableness: we expect this since 4byears old automobiles are more likely to have more expenses during repair leading to greater variances.

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The independent variable is the number of weeks, and the dependent variable is the number of pencils that Flora has.

The independent variable is the one that changes freely, and the dependent variable changes in response.

Here we know that that Flora starts with 10 pencils, and each week she gets two more. Then the independent variable is the number of weeks w.

And the dependent variable is the number of pencils p that she has, so we can write:

p = 10 + 2w

That is the linear equation.

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The length of h in the given parallelogram is 0.24

To help us solve this problem, please refer to the attached picture below for each corner annotation of the parallelogram.

First, we will try to focus on triangle AOD to find the length of OD.

We know that ABCD is a parallelogram and AD is parallel to BC; then AD = BC

AD = 5

To find the length of OD we will use the phytagoras theorem:

AD² = AO² + OD²

0.5² = 0.3² + OD²

0.25 = 0.9 + OD²

OD² = 0.16

OD = 0.4

Then, we will use the congruent triangle concept between triangle AOD and triangle CPD to find the length of h or PD.

Based on the congruent triangle concept; we know that AD and DC should be congruent; then:

AD / DC = AO / DP

0.5 / 0.4 = 0.3 / h

h = 0.24

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The absolute maximum value of f(x) = ln(x)/x is f(e) = ln(e)/e.

What is a function?

A function is a relationship or expression involving one or more variables.  It has a set of inputs and outputs.  Each input has only one output.  The function is the description of how the inputs relate to the output.

The absolute maximum value of a function is the largest value that the function takes on its entire domain.

For the function f(x) = ln(x)/x, the domain is x > 0, since the natural logarithm and division by x are undefined for x <= 0.

To find the absolute maximum value of f(x), we can use calculus. Taking the first derivative of f(x) and setting it equal to zero, we get:

f'(x) = (1 - ln(x))/x^2 = 0

Solving for x, we find x = e, where e is the mathematical constant approximately equal to 2.71828.

To determine whether f(x) has an absolute maximum at x = e, we can take the second derivative of f(x) and evaluate it at x = e:

f''(x) = -(2 + 1/x)/x^3

f''(e) = -(2 + 1/e)/e^3 < 0

Since the second derivative is negative at x = e, this means that f(x) has a relative maximum at x = e, which is also the absolute maximum, since the function is defined on a closed and bounded interval (0, ∞).

Therefore, the absolute maximum value of f(x) = ln(x)/x is f(e) = ln(e)/e.

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The 0.28 in the function f(x) = 5719(0. 28)^x represents the rate of decrease in the cat population over time.

This is because the function is decreasing, meaning as time progresses, the cat population decreases. The number 0.28 is equal to the mouse-catch-rate for each cat, which is 72%. This means that each cat can catch 72% of the mice, which means that the cat population decreases by 28% each year.

The number 5719 is the initial cat population in Ellison before they are sent to Clarksville. The function f(x) is used to represent the change in the cat population over time. As the value of x increases, the value of f(x) decreases, indicating a decrease in the cat population.

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Answer: jasmine

Step-by-step explanation: this problem is actually pretty easy when you solve it!

all you have to do is divide 8 by 10 to find Joe’s rate, which would be 2 songs per min

then divide 12 by 15 to find jasmines rate which would be .8 songs per minute!

so because Jasmine download songs at a rate of .8 songs per minute, Jasmine is faster.