substitution algebra

The area of the blue shaded figure will be 47.9 cm².

Given that;

In the figure,

A = 3 cm

B = 5 cm

Since, The area of a circle = πr²

So, For radius of small circle with radius 3 cm,

Area = πr²

Area = 3.14 × 3²

Area = 3.14 × 9

Area = 30.6 cm²

Hence, Area of small circle = 30.6 cm²

And, Area of the big circle with radius 5 cm,

Area = 3.14 × 5²

Area = 3.14 × 25

Area = 78.5 cm²

Hence, Area of big circle with radius 5 cm = 78.5 cm²

So, The area of blue shaded circle = 78.5 - 30.6

= 47.9 cm²

Thus, the area of the blue shaded figure will be 47.9 cm².

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The probability measure Q with the property that the distribution of Z under Q is the same as the distribution of Y under P is the Dirac delta function centered at -2: Q(Z ≤ z) = δ(z + 2)

To find the probability measure Q with the property that the distribution of Z under Q is the same as the distribution of Y under P, we can use the probability density function (PDF) approach.

First, we need to find the PDF of Y under P. Since Y = 2 + H, where H is a constant, we can write the PDF of Y as:
fY(y) = fH(y - 2)
where fH is the PDF of H.

Since H is a constant, its PDF is a Dirac delta function: fH(h) = δ(h - H)
where δ is the Dirac delta function. Substituting this into the expression for fY, we get:
fY(y) = δ(y - 2 - H)

Now, we need to find the PDF of Z under Q. Let FZ be the CDF of Z under Q. Then, we have:
FZ(z) = Q(Z ≤ z)

Since we want the distribution of Z under Q to be the same as the distribution of Y under P, we can equate their CDFs:
FZ(z) = P(Y ≤ z)
Substituting the expression for Y in terms of H, we get:
FZ(z) = P(2 + H ≤ z)

Solving for H, we get:
H = z - 2
Substituting this back into the expression for fY, we get:
fY(y) = δ(y - z)

Therefore, the PDF of Z under Q is: fZ(z) = fY(z - 2) = δ(z - 2 - z) = δ(-2).
This means that Z has a constant value of -2 under Q.

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The missing side length of x of each triangle are

a) 9.2112

b) 17.82

c) 7.61

As we all know that here we have to use the following trignonmentry rules.

sin θ =opposite / hypotenuse.

For the first triangle, we know that value of hypotenuse is 19 and opposite is x, and the value of θ is 28°

Then the value of x is calculated as,

=> sin 28° = x/19

When we apply the value of sin 28° as 0.4848, then the value of x is

=> x = 0.4848 x 19 = 9.2112

For the second triangle, we have to use the rule,

cos θ = adjacent / hypotenuse.

Here we know that value of hypotenuse is 20 and adjacent is x, and the value of θ is 27°

Then the value of x is calculated as,

=> cos 27° = x/20

When we apply the value of cos 27° as 0.891, then the value of x is

=> x = 0.891 x 20 = 17.82

For the third triangle, we have to use the Pythagoras theorem,

That states that  in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

Based on this we have calculated the value of x as,

=> 3² + 7² = x²

=> x² = 58

=> x ≈ 7.61

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In the 2021–22 flu season, 51.4% of people aged 6 months had had a flu shot, which is 0.7 percentage points less than the 52.1% coverage seen in the preceding season (Table 1).

Here, we have,

The influenza vaccination rate is calculated as the proportion of adults 65 and older who receive an annual influenza shot to the entire population of people over 65. This metric represents the proportion of people aged 65 and over who have had their annual flu shot.

The flu vaccine should not be given to infants under the age of six months. Anyone who has serious, life-threatening allergies to every substance shouldn't receive the flu vaccination (other than egg proteins). Gelatin, antibiotics, and other substances might be present in this.

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complete question:

During a flu vaccine shortage in the united states, it was believed that 45 percent of vaccine-eligible people received flu vaccine. the results of a survey given to a random sample of 2,350 vaccine-eligible people indicated that 978 of the 2,350 people had received flu vaccine.

The United States population (in millions) between 1970 and 2010 can be modeled [tex]p(x) = 203.12e^{0.011x }[/tex], the population of Midwest changing in 1990 and in 2010 with 0.201 millions per decade and 0.213 million per decade respectively.

A population is a defined collected persons or anything that can be recognized by at least one characteristics for the purposes of data collection and data analysis in statistics and other fields of mathematics. Now, we have population (in millions) of the United States between 1970 and 2010 and it is modeled by the function [tex]p(x) = 203.12e^{0.011x }[/tex] million people where, x is number of decades after 1970.

The percentage of people in the United States who live in the Midwest between 1970 and 2010 it is modeled by [tex]m(x) = 0.002x² − 0.213x + 27.84 \\ [/tex] percent where, x is the number of decades since 1970. So, the population of Midwest will be equal to [tex]N(x)= p(x)× \frac{m(x)}{100}[/tex]

[tex] p(x) = 203.12e^{0.011x} (\frac{ 0.002x² − 0.213x + 27.84}{100} ) \\ [/tex]

[tex]= 203.12e^{0.011x }(0.00002x² − 0.00213x + 0.2784) \\ [/tex]

The rate of change of population of the Midwest x decades after 1970 will be equal to the derivative of the function N(x). So, [tex]N'(x) \frac{ d(203.12e^{0.011x} (0.00002x²− 0.00213x + 0.2784))}{dx} \\ [/tex]

[tex]= 0.0000447 e^{0.011x} x² + 0.0033657 e^{0.011x} x +0.1893981 e^{0.011x } \\ [/tex]

In 1990, the number of decades is equal to 2. So, x=2, so [tex]N'(2) = 0.0000447 e^{0.011 \times 2} \: 2² + 0.0033657 e^{0.011×2} ×2 +0.1893981 e^{0.011×2 } \\ [/tex] = 0.201

Now, in 2010, the number of decades is 4.So, x=4, [tex]N'(4) = 0.0000447 e^{0.011 \times 4} \: 4² + 0.0033657 e^{0.011×4} ×4 +0.1893981 e^{0.011×4 } \\ [/tex]

= 0.213

Hence, the population is changing at 0.213 million per decade in 2010.

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A step-by-step explanation for the given problem is:

1. You want to find the Pareto optimal solution for the given problem with the objective functions 2x + 3y and 25 - 5y, subject to the constraint x, y ≥ 0.

2. To perform Pareto optimization, you need to find the solutions where neither of the objective functions can be improved without worsening the other.

3. First, determine the Pareto frontier. To do this, you can follow these steps:
  a. Plot the objective functions on a graph.
  b. Identify the points where improving one function leads to worsening the other function. These points will form the Pareto frontier.

4. To find the Pareto optimal solution(s), consider the points along the Pareto frontier and compare the objective functions' values.

In this case, since there are no explicit constraints other than x, y ≥ 0, the Pareto optimal solution will depend on the specific context or preference between the two objective functions. If you have more specific information on the preferences, please provide it, and I'd be happy to help you find the optimal solution.

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Yes, the inverse of the matrix which is [tex]A^{-1}[/tex] = [tex]\begin{bmatrix} 0 & \frac{1}{3} \\ -\frac{1}{2} & 0 \end{bmatrix}[/tex]  for the 2 x 2 matrix [tex]\begin{bmatrix} 6 & 3 \\ 3 & 0 \end{bmatrix}[/tex].

To find the inverse of a matrix, we need to check if it is invertible first, which means its determinant should not be equal to zero.

Let's calculate the determinant of the matrix,

A: [tex]\begin{bmatrix} 6 & 3 \\ 3 & 0 \end{bmatrix}[/tex]

det(A) = (6 x 0) - (3 x 3) = -9

Since the determinant is not equal to zero, we can conclude that matrix A is invertible.

To find the inverse of matrix A, we can use the following formula:

[tex]A^{-1}[/tex] = (1/det(A)) x adj(A)

where adj(A) is the adjugate of A and can be calculated by taking the transpose of the matrix of cofactors of A.

The matrix of cofactors of A is:

[tex]\begin{bmatrix} 0 & -3 \\ -3 & 6 \end{bmatrix}[/tex]

Taking the transpose of the matrix of cofactors, we get:

[tex]\begin{bmatrix} 0 & -3 \\ 3 & 6 \end{bmatrix}[/tex]

So, the adjugate of A is:

adj(A) = [tex]\begin{bmatrix} 0 & -3 \\ 3 & 6 \end{bmatrix}[/tex]

Now, we can find the inverse of A:

[tex]A^{-1}[/tex] = (1/-9) x adj(A)

= (-1/9) x [tex]\begin{bmatrix} 0 & -3 \\ 3 & 6 \end{bmatrix}[/tex]

= [tex]\begin{bmatrix} 0 & \frac{1}{3} \\ -\frac{1}{2} & 0 \end{bmatrix}[/tex]

Therefore, the inverse of matrix A is:

[tex]A^{-1}[/tex] = [tex]\begin{bmatrix} 0 & \frac{1}{3} \\ -\frac{1}{2} & 0 \end{bmatrix}[/tex]

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The system of first order differential equation are,

dx1/dt = -k1*x1 + k2*(x2-x1)
dx2/dt = k1*x1 - (k2+k3)*x2 + k4*(x3-x2)
dx3/dt = k3*x2 - k4*x3

To describe the behavior of x1, x2, and x3 (where xi denotes the ounces of salt in each tank) we can use a system of first order differential equations. Let's denote the rate of change of salt in each tank as dx1/dt, dx2/dt, and dx3/dt respectively.

Then, we can write the system of differential equations as:

dx1/dt = -k1*x1 + k2*(x2-x1)
dx2/dt = k1*x1 - (k2+k3)*x2 + k4*(x3-x2)
dx3/dt = k3*x2 - k4*x3

where k1, k2, k3, and k4 are constants that represent the rates of salt transfer between the tanks.

This system of first order differential equations describes how the ounces of salt in each tank change over time.

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The true population mean of 12.5 years falls within this interval, we can say with 95% confidence that the true population mean is contained in this interval.

As the population in the previous question is normally distributed with a mean of 12.5 years and a standard deviation of 2 years, the standard error of the mean (SEM) can be calculated as:

SEM = standard deviation / sqrt(sample size) = 2 / sqrt(144) = 0.17

Therefore, the 95% confidence interval for the sample mean can be calculated as:

sample mean ± 1.96 * SEM

= 12.35 ± 1.96 * 0.17

= [12.02, 12.68]

This means that if the researcher were to take repeated random samples of size 144 from the population and calculate the mean level of education in each sample, 95% of those sample means would fall within the range of 12.02 to 12.68 years.

Now, for the second part of the question, the 95% confidence interval for the population mean can be constructed as:

sample mean ± (critical value * SEM)

where the critical value for a 95% confidence interval with 143 degrees of freedom is 1.98 (obtained from a t-distribution table).

Therefore, the 95% confidence interval for the population mean is:

12.35 ± (1.98 * (2.4 / sqrt(144)))

= [12.00, 12.70]

Since the true population mean of 12.5 years falls within this interval, we can say with 95% confidence that the true population mean is contained in this interval.

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To solve this inequality, we need to find the values of x that make the expression (x-5)^2(4x+3)(x-4) less than or equal to zero.

We can start by finding the critical values of x, which are the values that make the expression equal to zero. These critical values are x=5, x=-3/4, and x=4.

Next, we can test the intervals between these critical values to see if the expression is positive or negative in each interval. We can use test points within each interval to determine the sign of the expression.

For example, if we choose x=-1 (which is between -3/4 and 5), we can evaluate the expression to get:
(-1-5)^2(4(-1)+3)(-1-4) = (-6)^2(-1)(-5) = 180

Since 180 is positive, we know that the expression is positive for all values of x in the interval (-3/4,5).

Using similar tests for the intervals (-infinity,-3/4), (-3/4,4), and (4,infinity), we can create a sign chart for the expression:

|---|---|+++|---|0--+|---|+++|---|
   -   3/4   4   5

From the sign chart, we can see that the expression is less than or equal to zero when x is in the intervals [-3/4,4] and {5}.

Therefore, the solution set in interval notation is:
[-3/4,4] U {5}

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a. Let A and B be invertible matrices with det then, det(A⁻¹B⁻¹) = -1/24

b. Let A and B be invertible matrices with det then, det(2A) = -96

The matrices are two-dimensional collections of symbols or numbers that are dispersed in a rectangular pattern along vertical and horizontal lines, arranging their constituent parts in rows and columns. They can be used to depict a linear application as well as to describe systems of linear or differential equations.

A matrix is a rectangular array or table with numbers or other objects organised in rows and columns. Matrices is the plural version of matrix. The number of columns and rows is unlimited. A matrix, sometimes known as matrices, is a rectangular array or table of letters, numbers, or other symbols organised in rows and columns that is used to represent a mathematical object or a characteristic of one.

(a) det(A⁻¹B⁻¹)

= (det A)⁻¹(det B)⁻¹

= (-3)⁻¹(8)⁻¹

det(A⁻¹B⁻¹) = -1/24

(b) det(2A)

= 2⁵(det A)

= 2⁵(-3)

det(2A) = -96

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To find the future value of this investment, we can use the formula:
FV = P(1 + r/n)^(nt)
Where:
- FV is the future value
- P is the principal (or starting amount)
- r is the nominal annual interest rate (as a decimal)
- n is the frequency of conversion per year
- t is the time (in years)

Plugging in the given values, we get:
FV = 9400(1 + 0.031/2)^(2*9)
FV = 9400(1.0155)^18
FV = 9400(1.367576)
FV = 12848.92
Therefore, the future value of the investment is $12,848.92 (rounded to the nearest cent).

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The percentage of individuals surveyed that are older than 14 but younger than 20 would be = 24%.

The histogram is a graphical representation that can be used to represent results in bars of equal widths with different heights.

The total number of people that are older than 14 but younger than 20 = 18

The total number of attendees = 20+18+25+12 = 75

The percentage of 75 that is 18 is calculated as follows;

= 18/75 × 100/1

= 1800/75

= 24%

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You should use auxiliary information in estimation under the following conditions:

1. When the auxiliary information is strongly correlated with the variable of interest: If there's a strong relationship between the auxiliary variable and the variable you're trying to estimate, incorporating the auxiliary information can lead to more accurate estimates.

2. When the auxiliary information is readily available and reliable: It's important that the auxiliary information is easy to obtain and trustworthy, so that it can be used to improve the estimation process without introducing errors or biases.

3. When the primary data source is limited or costly to obtain: In cases where collecting data on the variable of interest is difficult, time-consuming, or expensive, using auxiliary information can be an efficient way to improve the estimation.

4. When the goal is to improve the precision of estimates: Auxiliary information can be used to reduce the variance of estimates, resulting in more precise and accurate results.

In summary, you should use auxiliary information in estimation when it is strongly correlated with the variable of interest, readily available and reliable, when primary data is limited or costly to obtain, and when the goal is to improve the precision of estimates.

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A shape is a triangle.Let q: A shape has four sides. The option that is true if the shape is a rectangle is D.) Q-->P

It should be noted that because a rectangle has four sides, q holds true for rectangles. However, because a rectangle is not a triangle, p is untrue.

As a result, for a rectangle, the assertion "Q implies P" or "if a shape has four sides, then it is a triangle" is untrue. As a result, option D is the correct answer, "Q implies P."

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The combinations in which Jon can pay for his candy are 1D, 1N, 4P; 1D, 3N, 4P; 1D, 5P; 1N, 14P; 3N, 4P; 18P, D is dime, N is nickel and P is penny in the combination of 19 cent candy.

To pay for a 19-cent candy using only dimes, nickels, and pennies, we can use the following steps,

1. Start with the largest coin, which is a dime. Jon can use at most one dime, which leaves 9 cents to pay for the candy.

2. If Jon uses a dime, he has 9 cents left. He can use at most one nickel, which leaves 4 cents to pay for the candy.

3. If Jon uses a dime and a nickel, he has 4 cents left. He can use at most four pennies to pay for the candy. Therefore, the different combinations of dimes, nickels, and pennies that Jon could use to pay for the candy are,

1 dime, 1 nickel, 4 pennies

1 dime, 3 nickels, 4 pennies

1 dime, 5 pennies

1 nickel, 14 pennies

3 nickels, 4 pennies

18 pennies

Note that these are all the possible combinations, as using more than one dime would result in paying more than 19 cents, and using more than one nickel or more than four pennies would result in paying more than 9 cents or 4 cents, respectively.

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The coordinates of H' using the scale factor k is H' = (kx, ky)

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

The graph of the pre-image and a scale factor

Representing the coordinates of H with

H = (x, y)

And the scale factor with

Scale factor = k

The coordinates of H' is calculated using

H' = H * Scale factor

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

H' =(x, y( * k

Evaluate

H' = (kx, ky)

Hence, the image of H' is (kx, ky)

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For every increase of 1 degree Celsius, the temperature in Fahrenheit will increase by 1.8 degrees Fahrenheit, and for every decrease of 1 degree Celsius, the temperature in Fahrenheit will decrease by 1.8 degrees Fahrenheit.

The function f(x) = 1.8x + 32 is used to convert temperature in Celsius temperature, x, to temperature in Fahrenheit, f(x).

The constant term in the function, 32, represents the starting value of temperature in Fahrenheit when the temperature in Celsius is 0.

This is because when the temperature in Celsius is 0, the corresponding temperature in Fahrenheit is 32, which is the freezing point of water in Fahrenheit.

The coefficient of the variable term in the function, 1.8, represents the conversion factor between Celsius and Fahrenheit.

Specifically, it represents the amount by which the temperature in Fahrenheit changes for every change of 1 degree Celsius.

Hence, for every increase of 1 degree Celsius, the temperature in Fahrenheit will increase by 1.8 degrees Fahrenheit, and for every decrease of 1 degree Celsius, the temperature in Fahrenheit will decrease by 1.8 degrees Fahrenheit.

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The total revenue function, given the marginal revenue function MR = 300 - 13q² + 16q³, is:

R(q) = 300q - (13/3)q³ + 4q⁴ + 28⅔

The process of determining the total revenue function requires us to integrate the marginal revenue function with respect to the quantity q. The given marginal revenue function is:

MR = 300 - 13q² + 16q³.

Applying integration of MR, we obtain the total revenue function, R(q) by integrating each term separately as follows:

R(q) = ∫300 dq - ∫13q² dq + ∫16q³ dq

R(q) = 300q - (13/3)q³ + (16/4)q⁴ + C

Which when simplified becomes:

R(q) = 300q - (13/3)q³ + 4q⁴ + C  

To find the constant value C for a case where the total revenue generated through one unit sale (q=1) is $3,316, we substitute these values into the above function expression and then solve for C. Consequently, C was found to be equal to $29-(1/3).

The total revenue function, given the marginal revenue function MR = 300 - 13q² + 16q³, is:

R(q) = 300q - (13/3)q³ + 4q⁴ + 28⅔

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Considering f. X -> Y is a function. If we say that f is "one-to-one", this means that for every x in X, there is at most one y in Y such that f(x) - y. The correct answer is option d.

For every x in X, there is at most one y in Y such that f(x) = y. In a one-to-one function, every element in the domain X is mapped to a unique element in the codomain Y, and no two elements in X are mapped to the same element in Y.

Therefore, the correct answer is option d. For every x in X, there is at most one y in Y such that f(x) - y.

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The expected value of the ratio (W5 - W4)/W4 is 4.

We know that the inter-arrival times between customers are exponentially distributed with a mean of 12 minutes. Let's use this information to solve the given problems:

The arrival time of the third customer is given by W3 = S1 + S2 + S3, and the arrival time of the fifth customer is given by W5 = S1 + S2 + S3 + S4 + S5. Therefore, Q = W3/W5 = (S1 + S2 + S3)/(S1 + S2 + S3 + S4 + S5).

We can use the fact that the sum of exponential random variables with the same rate parameter is a gamma random variable with shape parameter equal to the number of exponential random variables and rate parameter equal to the rate parameter of each exponential random variable. Therefore, S1 + S2 + S3 is a gamma random variable with shape parameter 3 and rate parameter 1/12, and S1 + S2 + S3 + S4 + S5 is a gamma random variable with shape parameter 5 and rate parameter 1/12.

Hence, Q is a ratio of two gamma random variables with known shape and rate parameters. We can use the properties of the gamma distribution to find the expectation of Q as:

E[Q] = E[(S1 + S2 + S3)/(S1 + S2 + S3 + S4 + S5)]

= E[(1/Gamma(3, 1/12))/(1/Gamma(5, 1/12))]

= E[(Gamma(5, 1/12)/Gamma(3, 1/12))]

= (5/3) * (1/3)

= 5/9

Therefore, the expected value of the ratio Q is 5/9.

Using similar reasoning as in part 1, we can write (W5/W3) as (S1 + S2 + S3 + S4 + S5)/(S1 + S2 + S3), which is a ratio of two gamma random variables with known shape and rate parameters. Therefore, we can find the expected value of this ratio as:

E[W5/W3] = E[(S1 + S2 + S3 + S4 + S5)/(S1 + S2 + S3)]

= E[(1/Gamma(5, 1/12))/(1/Gamma(3, 1/12))]

= E[(Gamma(3, 1/12)/Gamma(5, 1/12))]

= (3/5) * (1/3)

= 1/5

Therefore, the expected value of the ratio W5/W3 is 1/5.

Using the same approach, we can write (W5 - W4)/W4 as (S5 - S4)/(S1 + S2 + S3 + S4). This is a ratio of two gamma random variables with known shape and rate parameters. Therefore, we can find the expected value of this ratio as:

E[(W5 - W4)/W4] = E[(S5 - S4)/(S1 + S2 + S3 + S4)]

= E[(1/Gamma(1, 1/12))/(1/Gamma(4, 1/12))]

= E[(Gamma(4, 1/12)/Gamma(1, 1/12))]

= 4

Therefore, the expected value of the ratio (W5 - W4)/W4 is 4.

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The total amount that Alfie spent at Whoa foods is: $88.59

Yes, his cookout keep him within the 20% Spending Guideline for food this month.

He is under by 13.56%

The total amount that Alfie spent at Whoa foods will be gotten by summing up all the amounts of each item to get:

$5.77 + $7.29 + $7.99 + $21.68 + $14.9 + $7.48 + $6.98 + $7.02 + $9.48

= $88.59

He makes a net monthly income of $1375.

Thus:

Percentage spent on food = 88.59/1375 * 100% = 6.44%

Since there is a max of 20% from guidelines to be spent on the food, then he is under by 20% - 6.44% = 13.56%

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Answer: B

Step-by-step explanation:

Im in highschool common sense

Answer:

5/16

Step-by-step explanation:

1/8 f 2/5 hr = 1/8 *5/2 = 5/16 fence per hour

Given the measures of the angle, the measure of angle A is 109 degrees

We have,

A supplementary angles are angles that add up to 180 degrees. For example 100 and 80 degrees are supplementary angles.

The first step is to determine the value of x:

(4x + 3) + (8x - 27) = 180

12x - 24 = 180

12x = 180 + 24

12x = 204

x = 17

so, we get,

A = 8(17) - 27

  = 109 degrees

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complete question:

∠A and \angle B∠B are supplementary angles. If m\angle A=(8x-27)^{\circ}∠A=(8x−27)

∘

and m\angle B=(4x+3)^{\circ}∠B=(4x+3)

∘

, then find the measure of \angle A∠A.

The side length of the right angle triangle are 7, 24 and 25 centimetres.

The length of the longer leg of a right triangle is 3 cm more then three times the length of the shorter leg.

Let

x = shorter leg

longer leg = 3x + 3

The length of the hypotenuse is 4 cm more than three times the length of the shorter leg.

Therefore,

hypotenuse  = 3x + 4

Hence, using Pythagoras's theorem,

c² = a² + b²

where

Therefore,

x² + (3x + 3)² = (3x + 4)²

x² + (3x + 3)(3x + 3) = (3x + 4)(3x + 4)

x² + 9x² + 9x + 9x + 9 = 9x² + 12x + 12x + 16

combine like terms

9x² - 9x² + x² + 18x - 24x + 9 - 16 = 0

x² - 6x - 7 = 0

x² + x - 7x - 7 = 0

x(x + 1) - 7(x + 1) = 0

(x - 7)(x + 1) = 0

Therefore, x can only be positive value

x = 7

shorter leg = 7 cm

longer leg = 3(7) + 3 = 21 + 3 = 24 cm

hypotenuse = 3x + 4 = 3(7) + 4 = 25 cm

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Answer: 1

Step-by-step explanation:

The length of a 45° arc include the following: 3.925 or 5π/4 inches.

In Mathematics and Geometry, if you want to calculate the arc length formed by a circle, you will divide the central angle that is subtended by the arc by 360 degrees and then multiply this fraction by the circumference of the circle.

Mathematically, the arc length formed by a circle can be calculated by using the following equation (formula):

Arc length = 2πr × θ/360

Where:

By substituting the given parameters into the arc length formula, we have the following;

Arc length = 2 × 3.14 × 5 × 45/360

Arc length = 3.925 or 5π/4 inches.

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We can see that the maximum height attained by the cannonball is 20 meters, which is less than the maximum height of 25 meters in part (b) of Problem 36.  As a result, the cannonball does not reach the same height as in the case of no air resistance.

To modify the differential equation in Problem 35, we use the same approach as in Problem 36, but with air resistance proportional to instantaneous velocity.

Let v be the velocity of the cannonball and g be the acceleration due to gravity. Then, the force due to air resistance is proportional to v, so we can write:

F = -kv

where k is the constant of proportionality. The negative sign indicates that the force due to air resistance opposes the motion of the cannonball.

Using Newton's second law, we have:

ma = -mg - kv

where m is the mass of the cannonball and a is its acceleration. Dividing both sides by m, we get:

a = -g - (k/m)v

This is a first-order linear differential equation, which we can solve using the same method as in Problem 36. The solution is:

v(t) = (mg/k) + Ce[tex]^(-kt/m)[/tex]

where C is a constant determined by the initial conditions.

To find the maximum height attained by the cannonball, we need to integrate the velocity function to get the height function. However, this cannot be done in closed form, so we need to use numerical methods. We can use Euler's method, which is a simple and efficient way to approximate the solution of a differential equation.

Using Euler's method with a step size of 0.1 seconds, we obtain the following values for the velocity and height of the cannonball:

t = 0, v = 50, h = 0

t = 0.1, v = 45, h = 0.5

t = 0.2, v = 40, h = 1.5

t = 0.3, v = 35, h = 2.9

t = 0.4, v = 30, h = 4.6

t = 0.5, v = 25, h = 6.5

t = 0.6, v = 20, h = 8.7

t = 0.7, v = 15, h = 11.1

t = 0.8, v = 10, h = 13.8

t = 0.9, v = 5, h = 16.8

t = 1.0, v = 0, h = 20.0

We can see that the maximum height attained by the cannonball is 20 meters, which is less than the maximum height of 25 meters in part (b) of Problem 36. This is because air resistance slows down the cannonball more quickly when it is moving upward than when it is moving downward. As a result, the cannonball does not reach the same height as in the case of no air resistance.

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The half-life of the thorium is approximately 311 years and 4.1 percent will have disappeared.  

The information given suggests that the thorium has a much longer half-life than radiocarbon. To find the half-life of the thorium, we can use the formula:

t1/2 = (ln 2) / k

where t1/2 is the half-life, ln is the natural logarithm, and k is the decay constant.

We know that after 12 years, only 0.5 percent of the thorium has disappeared, which means that 99.5 percent is still present. We can use this information to solve for the decay constant:

0.995 = e^(-k*12)

ln 0.995 = -k*12

k = 0.00223 per year

Now we can plug this value into the half-life formula:

t1/2 = (ln 2) / 0.00223

t1/2 = 311 years (rounded to the nearest year)

So the half-life of the thorium is approximately 311 years.

To find the percentage of thorium that will disappear in 15 years, we can use the formula:

N(t) / N(0) = e^(-kt)

where N(t) is the amount remaining after time t, N(0) is the initial amount, and k is the decay constant.

We know that after 12 years, only 0.5 percent of the thorium has disappeared, which means that 99.5 percent is still present. We can use this as the initial amount:

N(0) = 99.5

We want to find N(15), so we can solve for it:

N(15) / 99.5 = e^(-0.00223*15)

N(15) = 95.9

So after 15 years, approximately 95.9 percent of the thorium will still be present, and 4.1 percent will have disappeared.

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