1. What is the solution of the system of equations shown below?
y = 3x + 7
y = x-9

\(y(t)\) is a step function that jumps from \(0\) to \(1\) at \(t = 0\).

To solve the initial value problem (IVP) \(\frac{{dy}}{{dt}} = \delta_1(t)\), \(y(0) = 0\), where \(\delta_1(t)\) is the Dirac delta function, we can proceed as follows:

Since the Dirac delta function is defined as \(\delta_1(t) = 0\) for \(t \neq 0\) and \(\int_{-\infty}^{\infty} \delta_1(t) \, dt = 1\), we can treat the equation as a piecewise function.

For \(t < 0\), the derivative \(\frac{{dy}}{{dt}}\) is zero since \(\delta_1(t) = 0\) for \(t < 0\). Therefore, \(y(t)\) remains constant and equal to \(0\) for \(t < 0\).

At \(t = 0\), we have a jump discontinuity due to the Dirac delta function. The derivative of the Heaviside step function \(H(t)\) with respect to \(t\) is the Dirac delta function, which allows us to rewrite the equation as \(\frac{{dy}}{{dt}} = \frac{{dH(t)}}{{dt}}\).

Integrating both sides of the equation with respect to \(t\) over the interval \([-a, a]\), we obtain:

\(\int_{-a}^{a} \frac{{dy}}{{dt}} \, dt = \int_{-a}^{a} \frac{{dH(t)}}{{dt}} \, dt\)

Applying the fundamental theorem of calculus, the integral on the left side gives \(y(a) - y(-a)\), and the integral on the right side gives \(H(a) - H(-a)\).

Since \(y(t)\) is constant for \(t < 0\) (as mentioned earlier), we have \(y(-a) = 0\). Therefore, the equation simplifies to:

\(y(a) = H(a) - H(-a)\)

For \(t > 0\), the Heaviside step function evaluates to \(1\), so \(H(a) = 1\) and \(H(-a) = 0\). Thus, the equation becomes:

\(y(a) = 1 - 0 = 1\)

In conclusion, the solution to the IVP \(\frac{{dy}}{{dt}} = \delta_1(t)\), \(y(0) = 0\) is given by:

\(y(t) = \begin{cases} 0 & \text{for } t < 0 \\ 1 & \text{for } t > 0 \end{cases}\)

This means that \(y(t)\) is a step function that jumps from \(0\) to \(1\) at \(t = 0\).

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Check the picture below.

\(\cfrac{17.5+14}{17.5}~~ = ~~\cfrac{x}{12.5}\implies \cfrac{(17.5+14)(12.5)}{17.5}~~ = ~~x\implies 22.5=x\)

Answer:

Octagon

Step-by-step explanation:

A polygon refers to a two-dimensional closed shape consists of straight sides.

Perimeter of a polygon = 24 inches

Length of each side of the polygon = 3 inches

Number of sides of a polygon = Perimeter of a polygon ÷ Length of each side of the polygon = \(\frac{24}{3} =8\)

An octagon is a polygon consisting of 8 sides of same dimension.

So,

Name of polygon is octagon.

The distance between the pair of parallel lines with the equations y = -2x + 4 and y = -2x + 14 is 10 units.

To find the distance between two parallel lines, we can consider a perpendicular line that intersects both of them. In this case, the slopes of the given lines are equal (-2), indicating that they are parallel. The difference in the y-intercepts of the two lines is 14 - 4 = 10.

Since the lines are parallel, any line perpendicular to one line will be perpendicular to the other as well. We can take any point on one line, find the equation of a line perpendicular to it, and calculate the point of intersection with the other line. The distance between these points of intersection will be the distance between the parallel lines.

In this case, let's take the first line y = -2x + 4. The slope of a line perpendicular to it will be 1/2. Using this slope and the point (0, 4) from the first line, we can find the equation of the perpendicular line: y = (1/2)x + 4.

Next, we solve the system of equations formed by the perpendicular line and the second line:

(1/2)x + 4 = -2x + 14

Simplifying, we get:

(5/2)x = 10

x = 4

Substituting this value of x back into the equation of the perpendicular line, we find the y-coordinate:

y = (1/2)(4) + 4 = 6

The points of intersection are (4, 6) on the perpendicular line and (4, 6) on the second line. The distance between these points is given by the formula:

√((x₂ - x₁)² + (y₂ - y₁)²) = √((4 - 4)² + (6 - 6)²) = √(0² + 0²) = √0 = 0

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If we use alpha as the criteria or critical value for the maximum probability of incorrectly rejecting the null hypothesis, then when the p-value is less than alpha, the null hypothesis is rejected.

If we use alpha as the criteria or critical value for the maximum probability of incorrectly rejecting the null hypothesis, then when the p-value is less than alpha, the null hypothesis is rejected. This means that there is strong evidence to suggest that the alternative hypothesis is true and that the data set supports this conclusion. Therefore, we can conclude that the null hypothesis is not supported by the data and that we can reject it.
If we use alpha as the criteria or critical value for the maximum probability of incorrectly rejecting the null hypothesis, then when the p-value is less than alpha, the null hypothesis is rejected.

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Answer: d = -1.2

Step-by-step explanation: First you combine like terms. You add 5 +2 which gives you 7, and then you subtract 9 from 5 which gives you -5d. Then you subtract 7 from 13 which gives you 6. Divide both sides by -5 and you get -1.2. Always make sure to check your answer by plugging your answer into the place of the d variable

Therefore, there are 86,870,400 possible anagrams that can be created from the word 'metamorphosis' where the new words do not need to be meaningful.

To determine the number of anagrams that can be created from the word 'metamorphosis', we need to calculate the permutations of its letters, accounting for repeated letters. Since 'metamorphosis' has 14 letters, including 2 'm's, 2 'o's, and 2 's's, the total number of anagrams can be calculated as:

14! / (2! * 2! * 2!)

where '!' represents the factorial function. Simplifying the expression:

14! = 14 * 13 * 12 * ... * 2 * 1

The calculation results in:

(14 * 13 * 12 * ... * 2 * 1) / (2 * 2 * 2) = 86,870,400

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The measurement that is closest to the value of x, in centimeters, is given as follows:

14.3.

In this problem, we have a right triangle, in which the legs, which are the sides between the angle of 90º, are of 11.9 cm and 7.9 cm.

Then the hypotenuse x, which is the segment connecting both legs, is obtained using the Pythagorean Theorem.

The Pythagorean Theorem states that the measure of the hypotenuse squared is equals to the sum of the squares of the measures of each side.

Then the length of x is calculated as follows:

x² = 11.9² + 7.9²

x = square root of (11.9² + 7.9²)

x = 14.28.

Closest to 14.3, rounding to the nearest tenth, meaning that the third option is correct.  

We suppose that 11.9 cm and 7.9 cm are the measures of the legs.

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(a) Given function is f(x) = x² − 4/x − 2; we have to fill the following table of values for f(x):Xf(x)1.93.931.9943.99943.999934.0014.014.91(b) Based on the table of values, the limit of f(x) as x approaches 2 is 4. (c) Graph of the given function is as follows:The limit of the given function f(x) as x approaches 2 is 4. Therefore, lim_x→2 x² − 4/x − 2 = 4.Also, the interval for x near 2 such that the difference between the conjectured limit and the value of the function is less than 0.01 is 1.995 <= x <= 2.005.What is the window? 3.99 <= y <= 4.01.

Using the Malthusian law of population growth, the estimated alligator population in this region in the year 2020 is approximately 61,541.

The Malthusian law of population growth can be used to determine the population of alligators in a particular region in the year 2020 given the estimated populations of alligators in the year 1980 and 2008. We can use the formula for exponential population growth given by P = P0ert, where: P = final populationP0 = initial population r = growth rate as a decimal t = time (in years)We can find r by using the following formula: r = ln(P/P0)/t Where ln is the natural logarithm.

Using the given data, we can find the growth rate: r = ln(6500/1300)/(2008-1980)= ln(5)/(28)= 0.0643 (rounded to 4 decimal places)Therefore, the formula for exponential population growth is: P = P0e^(rt)Using the growth rate we found above, we can find P for the year 2020 (40 years after 1980):P = 1300e^(0.0643*40)P ≈ 61,541.15Rounding this to the nearest whole number, we get: P ≈ 61,541

Therefore, the estimated alligator population in this region in the year 2020 is approximately 61,541.

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The examples of SSS similarity theorem (i.e If all the three sides of a triangle are in proportion to the three sides of another one) are ∆ABC is similar ∆DEF and one large triangle is similar to smaller one as seen in above figure.

SSS Triangle Similarity Examples : 1) How can you use the SSS similarity criterion to show that the following triangles are similar?

Only non-zero angles are specified. SSA is not a measure of triangle similarity, so we cannot say that these triangles are similar by SSA. However, since they are right triangles, you can use the Pythagorean theorem to find the third side of any triangle.

=> x² = 45² - 36² = 2025 - 1296 = 729

=> x = √729 = 27

=> x² = 15² - 12² = 225 - 144 = 81

=> x =√81 = 9

All three pairs of sides are proportional in ratio 3. Therefore, triangles are more similar to SSS~.

2) Are the triangles (that is, ΔABC and ΔDEF) similar?

All triangles have the same three sides. The ratio between each pair of sides is 4x/x = 4. The triangles through SSS~ are similar because the three pairs of sides are proportional.

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The compound interest comes into the picture when interest adds on the principal amount. The total value of $4080 after 4 years with a rate of interest of 5.7% is $4124.83.

Compound interest is applicable when there will be a change in principal amount after the given time period.

For example, if you give anyone $500 at the rate of 10% annually then $500 is your principle amount but after 1 year it will convert into $550.

As per the given,

Principle amount P = $4080

Rate of interest r = 5.7% = 0.057

Time period t = 4 years.

The formula for total value is given as,

V = P\(e^{rt}\)

V = 4080\(e^{(0.057\times4)}\)

V = $5124.83

Hence "The compound interest comes into the picture when interest adds on the principal amount. The total value of $4080 after 4 years with a rate of interest of 5.7% is $4124.83".

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The given question is incomplete complete question follows;

Answer:

A, B, F

Step-by-step explanation:

To solve this, you look at the powers of x and take either the square root (if it is to a power of 2) or the cube root (if a power of 3) of both sides to isolate x. For the first equation, you take the square root of 48, so A is correct. For the second one, you take the square root of 63, so B is correct. For the final equation, you take the cube root of 73, so F is correct.

Answer:

\(x = \sqrt{48} \\\\x = \sqrt{63} \\\\x = \sqrt[3]{73}\)

Step-by-step explanation:

\(x^2 = 48\\x^2 = 63\\x^3 = 73\)

Solve for x

Equation 1 ;

\(x^2 =48\\\\Square\:root\:both\:sides\\\\\sqrt{x^2} =\sqrt{48} \\\\Simplified \:form;\\x =4\sqrt{3}\)

Equation 2;

\(x^2 =63\\\\Square\:root\:both\:sides\\\\\sqrt{x^2} =\sqrt{63} \\\\Simplified \:form;\\x =3\sqrt{7}\)

Equation 3 ;

\(x^3 =73\\\\Cube\:root\:both\:sides\\\\\sqrt[3]{x^3}= \sqrt[3]{73} \\\\x = \sqrt[3]{73}\)

Answer:

Stop spamming

Step-by-step explanation: The number of brain cells you have is the number of gusets he can serve. 12.

The ratio of the number of sundaes with nuts to the number of sundaes without nuts is 11/14

What is ratio?

Ratio is a quantitative relationship between two values  indicating the number of times one value contains within the other value

In other words, we are expressing the number of sundaes with nuts as a fraction of the ones without nuts

There are 22 sundaes with nuts

number of sundaes without nuts=50-22

number of sundaes without nuts=28

sundaes with nuts/sundaes without nuts=22/28

2 is common to both numbers

sundaes with nuts/sundaes without nuts=11/14

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Now the given coordinates are,

(-4,10) and (2,-10)

Thus, we have,

x₁  = -4

y₁ = 10

x₂ = 2

y₂ = -10

Given ratio, m₁ : m₂ = 1 : 3

⇒ m₁ =  1; m₂ =  3

Let (x,y) be the coordinates of the point on the directed line segment from (-4,10) to (2,-10) that partitions the segment into a ratio of 1 to 3.

Since the section formula is given as,

So, x = (m₁x₁ + m₂x₂) / (m₁ + m₂)  

y =  (m₁y₁ + m₂y₂) / (m₁ + m₂)  

Put the values,

x = [1(-4) + 3*2)] / (1 + 3) = (-4+6) / 4 = 2/4

⇒ x = 1/2

Similarly y = [1(10) + 3*(-10)] / (1 + 3) = (10-30) / 4 = -20/4

⇒ y = -5

So, the required coordinates are (1/2, -5)

Thus, the coordinates of the point on the directed line segment that partitions the segment into a ratio of 1 to 3 is (1/2, -5).

Answer:(-3,-6)

Step-by-step explanation:

Answer:

$3,500 labor and $3,500 materials

Step-by-step explanation:

furnishings + labor + materials = 10,000

furnishings = 3000

3000 + labor + materials = 10,000

labor = materials

3000 + labor + labor = 10,000

2(labor) = 7,000

labor = 7,000/2

labor = 3,500

labor = materials = 3,500

Answer:

5(2x−9)

Step-by-step explanation:

hope this helps

Answer:

3(x+2)=12

Step-by-step explanation:

let the number be x

the sum=x+2

the answer

3(x+2)=12

The purpose of converting a random variable to a z-value is to Compare a normal distribution to a standard normal distribution and Express the distance from the mean in terms of the standard deviation.

The purpose of converting a random variable to a z-value is to:

Compare a normal distribution to a standard normal distribution: By converting a random variable to a z-value, it can be compared to the standard normal distribution, which has a mean of 0 and a standard deviation of 1. This allows for the calculation of probabilities and the interpretation of the results in a standard way.

Express the distance from the mean in terms of the standard deviation: The z-value expresses the distance of the random variable from the mean in terms of the standard deviation. This standardization makes it easier to compare values and to perform statistical analysis.

Converting a normal distribution to a z-value does not convert it to a uniform distribution or ensure that the sum of the probabilities is one. The sum of the probabilities in a normal distribution does not equal one, but rather approaches one as the values approach infinity.

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The total maximum area enclosed by both the shapes square and a circle using given wire is equal to 2.27 square feet.

As given in the question,

Let 'x' be the side length of the square

And 'r' be the radius of the circle.

Length of the wire = 4 feet

Maximum side length of the square is:

4x = 4

⇒ x = 1feet

Maximum area of the square = x²

                                                = 1²

                                                = 1

Maximum radius of the circle is:

2πr = 4

⇒ r = 2 / π

Maximum area of the circle = πr²

                                              = π × (2/π)²

                                              = 4 / π

Total maximum area enclosed by square and a circle

= 1 + (4/π)

= 1 + 1.27

= 2.27 square feet.

Therefore, the total maximum area enclosed by square and a circle from the given wire is equal to 2.27 square feet.

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the answer is 8.75 for tanya item

Answer:

28 degrees

Step-by-step explanation:

A triangle is 180 degrees. 180-152=28.

Answer:

The third angle is 28 degrees

Step-by-step explanation:

A triangle has to add up to 180 degrees so just take 180 subtract your 152 from it and you'll have your answer!. (180 - 152 = 28) Hope this helps! :)

Answer:

6/8

Step-by-step explanation:

3/4

Multiply by 2/2

3/4 *2/2 = 6/8

Answer:

Hundreth because it rounds if you were to round it the nearest hundreth

then it will give you the exact answer \

Step-by-step explanation:

Answer:

Here are six equivalent expressions to 12x - 6:

6(2x - 1)

3(4x - 2)

2(6x - 3)

1(12x - 6)

4(3x - 1)

(12x - 6) + 0

These expressions are all equivalent to 12x - 6 because they all have the same value, regardless of the value of x. This is because the terms on the right side of the equation cancel out, leaving only the constant value on the left side. For example, in the first expression, 6(2x - 1), the 2x and -1 terms cancel out, leaving only 6. This is the same for the other expressions listed above.

Step-by-step explanation:

Answer:

Step-by-step explanation:

Here are six equivalent expressions to the given expression:

Note that in each of the above expressions, the value of 12x - 6 is preserved, but the terms and the order of operations may be different. For example, in the second expression, the subtraction has been expressed as addition with a negative number, and in the fourth and sixth expressions, the subtraction has been expressed as multiplication with a fraction.

Since the sum of the numbers selected must exceed 1, we know that n must be at least 1. Therefore, the expected value of X is finite and is equal to 2.

The expected value of the count of numbers that are selected can be found using the geometric distribution, since it models the number of trials until a success occurs.

Let X be the number of numbers that are selected until the sum of all numbers exceeds 1. Then X has a geometric distribution with parameter p, where p is the probability of success on each trial, i.e. the probability that the sum of all numbers selected does not exceed 1.

Since the numbers are selected uniformly over [0, 1], the probability of success on each trial is given by the area under the curve between 0 and 1 after the first number is selected.

On the first trial, the probability of success is 1. On the second trial, the probability of success is the area under the curve between 0 and (1 - the first number selected).

On the third trial, the probability of success is the area under the curve between 0 and (1 - the sum of the first two numbers selected), and so on.

The expected value of X can be found using the formula for the expected value of a geometric distribution:

E(X) = 1/p

Since the expected value of the sum of n independent and identically distributed uniform [0, 1] random variables is n/2, we can use this to find p:

E(X_1 + X_2 + ... + X_n) = n/2

where X_1, X_2, ..., X_n are the n numbers selected.

So, p = 1 / (1 + n/2).

Finally, we can substitute this into the formula for E(X) to find the expected value of X:

E(X) = 1 / (1 - n/2) = 2 / (2 - n)

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Answer:

No they did not they ate more crab then tuna

Step-by-step explanation: