Use partial fractions to find the indefinite integral. (Use C for the constant of integration. Remember to use absolute values where appropriate.) dx x2 - 4x + 4 4 x - 4 ln x+2-x+2+ - In 2 +Cx

The equation of the tangent plane to the surface z = e^(-4x) / (17 ln(4y)) at the point (3, 1, 0.684376) is given by:

z - 0.684376 = (-4e^(-12) / (17 ln(4)))(x - 3) + (e^(-12) / (17 ln(4))) * (1/ln(4) - 1)(y - 1)

the given point to obtain the slope of the surface in the x and y directions, using the point-normal form of a plane equation, the equation of the tangent plane can be determined.

In this case, the partial derivatives are calculated as follows:

∂z/∂x = -4e^(-4x) / (17 ln(4y))

∂z/∂y = (e^(-4x) / (17y ln(4y))) * (1/ln(4y) - 1)

Substituting the values x = 3 and y = 1 into these partial derivatives gives:

∂z/∂x = -4e^(-12) / (17 ln(4))

∂z/∂y = (e^(-12) / (17 ln(4))) * (1/ln(4) - 1)

Now we have the slope of the surface in the x and y directions at the point (3, 1, 0.684376), we can determine the equation of the tangent plane using the point-normal form:

z - z0 = (∂z/∂x)(x - x0) + (∂z/∂y)(y - y0)

Plugging in the values, the equation of the tangent plane is:

z - 0.684376 = (-4e^(-12) / (17 ln(4)))(x - 3) + (e^(-12) / (17 ln(4))) * (1/ln(4) - 1)(y - 1)

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Answer: 15 - 6i

Step-by-step explanation:

3(5-2i)

3×5+3×(−2i)

Do the multiplications.

15−6i

The answers will be :

a. The equation of perpendicular bisector of QR is y = \(\frac{-5}{3}\) x + 3.

b. The slope of median of triangle passing through point R is 1/4.

c. The slope of the altitude of the triangle that passes through point R doesn't exists.

What is slope of a line segment ?

Slope is the ratio of the difference in y-coordinates over the equivalent x-coordinates between two different locations on a line.

It is given that the vertices of a triangle are P(-6,1), Q(-2,-5) and R(8,1).

a.

The midpoint (M) of QR will be equal to

( \(\frac{-2 + 8}{2} , \frac{-5 + 1}{2}\) )

or

M = (-3 , 2)

Now , slope of line QR will be ;

\(m_{QR}\) = \(\frac{1+5}{8+2}\)

\(m_{QR}\) = 6 / 10

We know that :

m × \(m_{QR}\) = -1

m = -10 / 6

or

m = -5 / 3

Equation of line will be given as :

y + 2 = \(\frac{-5}{3}\) ( x - 3 )

y = \(\frac{-5}{3}\) x + 3

b.

The midpoint of PQ will be :

= (\(\frac{-6 - 2}{2} , \frac{1 - 5}{2}\))

= ( -4 , -2)

The slope of median of triangle that passes through point R will be ;

m = \(\frac{-2 - 1}{-4-8}\)

m = -3 / -12

m = 1/4

or

m = 0.25

c.

Slope of altitude of triangle that passes through point Q will be :

m = \(\frac{1 - 1}{-6-8}\)

m = 0 / -14

m = 0

The slope in this case doesn't exist.

Therefore the answers will be :

a. The equation of perpendicular bisector of QR is y = \(\frac{-5}{3}\) x + 3.

b. The slope of median of triangle passing through point R is 1/4.

c. The slope of the altitude of the triangle that passes through point R doesn't exists.

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Answer:

\(\boxed{c = \sqrt{39} / c= - \sqrt{39} }\)

This " / " means or, btw!

Step-by-step explanation:

\(c^2 = 39\)

⇒Take square root.

\(c =\) ± \(\sqrt{39}\)

\(c = \sqrt{39}\) or \(c = -\sqrt{39}\)

Hope this helped you!

The probability of having one white ball is 1/20 or approximately 0.05.

To find the probability of selecting one white ball out of three balls, you need to know the number of white balls in the box and the total number of balls in the box. You also need to know whether the balls are being selected randomly or not.

Assuming that there is only one white ball in the box and the rest are some other color, and that the balls are being selected randomly.

The probability of selecting one white ball out of three would be:

Probability = (number of ways to select one white ball out of three) / (total number of ways to select three balls)

Since there is only one white ball, there is only one way to select one white ball out of three. The total number of ways to select three balls out of six is 6 choose 3, which is equal to 20.

Therefore, the probability of selecting one white ball out of three would be

Probability = (number of ways to select one white ball out of three) / (total number of ways to select three balls)

Probability = 1/20

Probability = approximately 0.05.

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Heckscher-Ohlin Theory trade theories argues that a country should specialize in goods that demand the least from ts scarce production factors.

The Heckscher-Ohlin Theory, also known as the Factor Proportions Theory, argues that a country should specialize in producing and exporting goods that require the least amount of its scarce production factors.

According to the Heckscher-Ohlin Theory, a country will specialize in and export goods that use its abundant factors of production more intensively, while importing goods that use its scarce factors of production more intensively. This specialization and trade pattern lead to greater efficiency and overall economic welfare for the countries involved.

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Answer:

It falls in the integer category. The attached picture is a chart if you want to specify a number in a category.

Hope this helps :)

(a) Out of a population of 1 million, we can expect 1000 people to have this form of cancer, as the rate of the disease is 0.001 (0.1% of the population) multiplied by 1 million.
(b) Out of a population of 1 million, 5000 people would test positive for this form of cancer, as the false positive rate is 0.005 (0.5% of the population) multiplied by 1 million.
(c) The probability that a person who tested positive actually has this form of cancer is 16.69%.

(a) Out of a population of 1 million, how many people are expected to have this form of cancer?

To calculate this, we'll multiply the population by the rate of cancer occurrence:
1,000,000 (population) × 0.001 (rate) = 1,000 people with cancer.

(b) Out of a population of 1 million, how many people would test positive for this form of cancer?

First, we need to find out how many false positives there would be:
1,000,000 (population) × 0.005 (false positive rate) = 5,000 false positives.

Next, since the test has a perfect detection rate (1) for individuals with cancer, all 1,000 people with cancer would test positive.

Total positive tests = 1,000 (true positives) + 5,000 (false positives) = 6,000 positive tests.

(c) Bayes' theorem. The probability of having the disease given a positive test result can be calculated as follows:

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

where A is the event of having the disease, B is the event of testing positive, P(A) is the prior probability of having the disease (0.001), P(B|A) is the probability of testing positive given that you have the disease (1), and P(B) is the probability of testing positive, which can be calculated as the sum of the true positive and false positive rates:

P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)
P(B) = 1 * 0.001 + 0.005 * 0.999
P(B) = 0.005995

Now we can plug in the values and solve:

P(A|B) = 1 * 0.001 / 0.005995
P(A|B) = 0.1669

So the probability that a person who tested positive actually has this form of cancer is 16.69%.

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Answer:

x = 2, y = 4

Step-by-step explanation:

Given

(x + 1, 8 ) = (3, 2y )

Equate the values in the x and y positions , that is

x + 1 = 3 ( subtract 1 from both sides )

x = 2

and

2y = 8 ( divide both sides by 2 )

y = 4

Answer:

(X+1,8)=(3,2y)

equating corresponding value we get

x+1=3

x=3-1=2

again

8=2y

y=8/2=4

x=2

y=4

Answer:

Should have taken the absolute value

Actual length = 3.5 centimeters

Step-by-step explanation:

Length is always a positive value or zero

So one should take the absolute value which is always positive

|-3.5| = 3.5 where the | | denotes absolute value of what is enclosed

The solution to the initial value problem (I.V.P.) (1+x)dy/dx = y + 4x(1+x)², y(0) = 3 is y(x) = (x³ + 2x² + 6x + 3) / (1 + x).

To solve this I.V.P., we'll use the method of integrating factors. First, let's rewrite the equation in the standard form: dy/dx - y/(1+x) = 4x(1+x)²/(1+x). Notice that (1+x) is a factor of both the coefficient of dy/dx and the right-hand side.

To find the integrating factor, we multiply both sides of the equation by the integrating factor, which is given by e^(∫-1/(1+x)dx). Integrating -1/(1+x) with respect to x gives us -ln(1+x). Therefore, the integrating factor is e^(-ln(1+x)) = 1/(1+x).

Multiplying the original equation by the integrating factor, we get (1+x)dy/dx - y = 4x(1+x)³/(1+x) = 4x(1+x)².

Now, we can rewrite the left side of the equation as d[(1+x)y]/dx and simplify the right side to 4x(1+x)². Integrating both sides with respect to x, we obtain (1+x)y = ∫4x(1+x)² dx.

Evaluating the integral on the right side, we have (1+x)y = x²(1+x)³ + C, where C is the constant of integration. Solving for y, we get y(x) = (x³ + 2x² + 6x + 3) / (1 + x), which is the solution to the I.V.P.

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Answer:

the domain of f(x) is set of all real numbers except 3/2

Step-by-step explanation:

Since the denominator cannot equal 0,  2x - 3 ≠ 0

                                                                        2x ≠ 3

                                                                          x ≠ 3/2

Any real number can replace x in the numerator.  

So, the domain of f(x) is all real numbers except 3/2

The domain of the given function is required.

The domain is (-∞,3/2),(3/2,∞)

The given function is \(f(x)=\dfrac{x-5}{2x-3}\)

The domain of a function are the values of x for which the given function is defined.

The function is undefined when \(x=\dfrac{3}{2}\)

\(f(3/2)=\dfrac{\dfrac{3}{2}-5}{2\times \dfrac{3}{2}-3}=\infty\)

So, \(x\neq \dfrac{3}{2}\)

The domain is \(\left(-\infty,\dfrac{3}{2}\right),\left(\dfrac{3}{2},\infty\right)\)

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The requried combined length of both trains is 900 meters or 900 km.

Speed is defined as when an object is in motion, the distance covered by that object per unit of time is called speed.

Here,
Let's first convert the given speeds from km/h to m/s to make the calculations easier.

Speed of the first train = 102 km/h = (102 x 1000) m/3600 s = 28.33 m/s

Speed of the second train = 30 km/h = (30 x 1000) m/3600 s = 8.33 m/s

Now, let's consider the relative speed of the two trains since they are moving in the same direction:

Relative speed = Speed of first train - Speed of second train

= 28.33 - 8.33

= 20 m/s

Let's assume that the length of the first train is x m and the length of the second train is y m.

Distance covered by first train in 45 seconds = (x + y) m

Speed of first train = Distance covered by first train / Time taken

= (x + y) / 45 km/s

We know that the speed of the first train is 20 m/s. So, we can write:

20 = [(x + y) / 45]

Solving for (x + y),

x + y = 900 m

Therefore, the combined length of both trains is 900 meters or 900 km.

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Answer:

-2.46

Step-by-step explanation:

Subtract -9.95 from -12.41. You can visualize this by starting at -12.41 on a number line and moving 9.95 units to the left. You will land on the integer -2.46.

An expression that has the same value could be..

12x12, 3x48, and 9x16

Answer:

1) 12 x 12

2) 72 x 2

Step-by-step explanation:

This is the answer because:

1) 12^2 equals 144

2) 12 x 12 equals to 144, as well as 72 x 2 because this expression also equals to 144

Hope this helps!

James will fully clear the loan after approximately 12 years when the remaining balance reaches zero.

To determine the number of years it will take for James to fully clear the loan, we need to calculate the remaining balance after each payment and divide the initial loan amount by the annual payment until the remaining balance reaches zero.

The loan amount is 9000 euros, and James pays off 770 euros each year. Since the interest is charged at a rate of 3% of the original amount per annum, the interest for each year will be \(0.03 \times 9000 = 270\) euros.

In the first year, James pays off 770 euros, and the interest on the remaining balance of 9000 - 770 = 8230 euros is \(8230 \times 0.03 = 246.9\)euros. Therefore, the remaining balance after the first year is 8230 + 246.9 = 8476.9 euros.

In the second year, James again pays off 770 euros, and the interest on the remaining balance of 8476.9 - 770 = 7706.9 euros is \(7706.9 \times 0.03 = 231.21\) euros. The remaining balance after the second year is 7706.9 + 231.21 = 7938.11 euros.

This process continues until the remaining balance reaches zero. We can set up the equation \((9000 - x) + 0.03 \times (9000 - x) = x\), where x represents the remaining balance.

Simplifying the equation, we get 9000 - x + 270 - 0.03x = x.

Combining like terms, we have 9000 + 270 = 1.04x.

Solving for x, we find x = 9270 / 1.04 = 8913.46 euros.

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Answer:

-------------------------

Each coordinate of the midpoint is the average of endpoints:

Therefore M is (13, 14).

Generally, using a protractor we see x to be an angle 32 degrees

Hence, we solve

since the angle in point is given by 360 and x=35

Therefore

y=360-x

y=360-32

y=328 degrees

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The measure of angle x would be 35° and the measure of angle y would be 325°

From the question, we are to determine the measures of angle x and angle y.

The measure of angle x can be determined with the aid of a protractor.

From the given diagram, we can observe that angle x is an acute angle. That is, the measure of angle x is less than 90°.

After determining the measure of angle x by using a protractor, for example, let us say the measure of angle x is 35°.

We can determine the measure of angle y by using the sum of angles at a point theorem.

Sum of angles at a point is 360°.

Thus, we can write that

x + y = 360°

Then,

35° + y = 360°

y = 360° - 35°

y = 325°

Hence, the measure of x = 35° and the y = 325°

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Answer:

Side length of a square game board is  

11.916

inches.

Explanation:

If the area is  

142

∈

2

, as area is square of side length of a square,

Side length is  

√

142

=

11.916

inches.

Step-by-step explanation:

Answer:

13.7 inches

Step-by-step explanation

Area = base*height

(since it's a square, the lengths of all four sides of the board are equal. therefore, we can write the equation above like the following)

Area of square = side length^2

(square root both sides of the equation, and you get the following)

\(\sqrt{Area\\}\) = side length

(now, substitute the info)

\(\sqrt{188\\}\) = x

(solve)

x = 13.7 inches

Answer:

52%

Step-by-step explanation:

What percent of 200 is 104?

104/200 = 0.52

0.52 = 52%

Answer: 5n²-13n+6=(n-2)(5n-3)=0

Step-by-step explanation:

\(5n^2-13n+6=0\\\\5n^2-10x-3n+6=0\\\\5n(n-2)-3(n-2)=0\\\\(n-2)(5n-3)=0\)

Answer:

true

Step-by-step explanation:

because there's less humidity

Two two numbers whose difference is 88 and whose product is a minimum are -44 and 44.

Let the numbers be x and y where x is larger value and y is smaller number.

According to ques,

x - y =88

and we need to make product of numbers i.e 'xy' minimum.

So we can take x = 88 + y ------(2)

substituting value of x in xy, we get

(88 + y) (y) = y² + 88y ------(1) to make it minimum we need to equate its derivative equal to zero.

derivative of  (1) is 2y + 88

Equating it to zero, we get

2y + 88 = 0

subtracting 88 from both sides, we get

2y = -88

Dividing 2 from both sides, we get

y = -44 ------(3)

Substituting value of y from (3) in (2), we get

x = 88 - 44

x = 44

So, we get value of x as 44 and y as -44 to get their product as minimum.

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The statement suggests that there is likely to be an interaction between nitrogen (N) and phosphorus (P), which are two fertilizer ingredients used to achieve optimum levels in plant growth. However, without further context or specific details, it is uncertain to determine the nature and extent of this interaction.

In agriculture and plant nutrition, both nitrogen and phosphorus are essential elements for plant growth and development. They play crucial roles in various physiological processes and are often supplied through fertilizers to optimize crop yields. The interaction between nitrogen and phosphorus can have significant implications for plant growth and nutrient utilization.

The ratio of nitrogen to phosphorus in the soil can affect nutrient uptake, plant growth, and overall productivity. Different plant species and soil conditions may exhibit varying responses to the interaction between these two fertilizer ingredients. Factors such as soil type, pH, organic matter content, and crop-specific nutrient requirements further contribute to the complexity of this interaction. To fully understand and assess the interaction between nitrogen and phosphorus, detailed research and experimentation specific to the particular context are necessary.

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The volume of the remaining piece of ice cube is 6911.5 cubic cm

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

The figure

Where, we have

Radius, r = 20/2  = 10 cm

Height, h = 33 cm

The volume of the remaining piece is calculated is

V = 2/3πr²h

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

V = 2/3 * 22/7 * 10² * 33

Evaluate

V = 6911.5

Hence, the volume of the remaining piece is 6911.5 cubic cm

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In linear regression, the variance around the regression line represents the variability of the dependent variable (response variable) that is not explained by the regression model.

It measures the dispersion of the actual data points around the predicted values from the regression line.

The variance around the regression line can vary with different values of the predictor variable. This is because the relationship between the predictor variable and the response variable may not be constant across the entire range of the predictor variable. In other words, the spread or dispersion of the data points around the regression line may change as the predictor variable changes.

By examining the residuals (the differences between the observed values and the predicted values from the regression line) and calculating their variances, you can assess the variability of the data points around the regression line. This variability is an important aspect of understanding the goodness of fit of the regression model and the accuracy of the predictions.

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Answer:  it lies between 2 and 3.

Step-by-step explanation:

You equation should be like this because what you wrote is not an equation.

15/6= p  where P is the number of pounds  each student received.

p=2.5

First turn any mixed numbers into improper fractions. 1 3/4 would be 7/4. Becuase you multiply the whole number by the denominator and then add the numerator.

So now you have 7/4 divided by 2/5. Whenever you divide, you flip the 2nd fraction upside down... known as the reciprocal. And after that you multiply the first number by the upside down 2nd number.

So now you have 7/4 X 5/2. When multiplying, just multiply the numerators across and the denominators across. As your answer, you get 35/8 or 4 3/8.

The quotient obtained by dividing two given fractions is 25/12.

The given fractions are \(1\frac{2}{3}\) and 4/5.

In arithmetic, a quotient is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a division, or as a fraction or a ratio.

Here, the mixed fraction \(1\frac{2}{3}\) can be written as 5/3.

Now, 5/3 ÷ 4/5

= 5/3 × 5/4

= 25/12

Hence, the quotient obtained by dividing two given fractions is 25/12.

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