Solve for x,y,and z. 2x+3y-z =2 -6x-4y-4z=-12 3x-3y+10z=10

The slope of the tangent line to the parabola y = 3x^2 + 5x + 3 at the point (3, 45) is 23 that can be found by calculating the first derivative of the function with respect to x and then evaluating it at the given point.

First, let's find the first derivative of y with respect to x:
y = 3x^2 + 5x + 3
dy/dx = (d/dx)(3x^2) + (d/dx)(5x) + (d/dx)(3)
dy/dx = 6x + 5
Now that we have the first derivative, we can find the slope of the tangent line at the point (3, 45) by plugging in x = 3:
dy/dx = 6(3) + 5
dy/dx = 18 + 5
dy/dx = 23

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The equation y = 5x/2 represents a proportional relationship with a constant of 5/2.

A proportional relationship is a type of relationship between two quantities in which they maintain a constant ratio to each other.

The equation that defines the proportional relationship is given as follows:

y = kx.

In which k is the constant of proportionality, representing the increase in the output variable y when the constant variable x is increased by one.

The equation for this problem is given as follows:

y = 5x/2.

Which is a proportional relationship, as it has an intercept of zero, along with a constant of k = 5/2.

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The time taken by the object to reach the highest point is 0.91 seconds.

The given equation for the function h (t) = − 16t² + 29t + 6683 gives the height of an object that is thrown upward from the top of the mountain at an initial upward velocity of 29 feet per second.
To determine the time taken by the object to reach the highest point, we need to find the vertex of the function h (t). The vertex of a quadratic function is given by (-b/2a, f(-b/2a)) where a, b, c are coefficients of the quadratic equation ax² + bx + c = 0. In the given function h (t) = − 16t² + 29t + 6683, we have a = -16, b = 29, and c = 6683.

Therefore, the time taken by the object to reach the highest point is 0.91 seconds

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The estimated unit rate in miles per minute is about 0.13 miles per minute and the estimated unit rate in minutes per mile is about 8.0 minutes per mile

The unit rate is the rate of an occurrence of an event or activity for a unit quantity of something else. To calculate the unit rate in miles per minute, divide the total miles covered by the runner by the time he took to run it;26.2 miles/210 minutes≈0.125miles/minute≈0.13 miles/minute (rounded to the nearest hundredth of a mile).
Therefore, the unit rate is about 0.13 miles per minute
To calculate the unit rate in minutes per mile, divide the time taken by the runner by the total miles covered;210 minutes/26.2 miles≈8.0152447658 minutes/mile≈8.0 minutes/mile (rounded to the nearest tenth of a minute).
Therefore, the unit rate is about 8.0 minutes per mile.

The estimated unit rate in miles per minute is about 0.13 miles per minute, rounded to the nearest hundredth of a mile, and the estimated unit rate in minutes per mile is about 8.0 minutes per mile, rounded to the nearest tenth of a minute.

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The derivative of the function g(x) is g'(x) = 28x².

The derivative of the function g(x) can use the Fundamental of Calculus states that if f(x) is continuous on [a, b] then:

∫aˣ f(t) dt is differentiable on (a, b) and its derivative is f(x)

Integral with respect to x by differentiating the integrand with respect to u and then multiplying by the derivative of the upper limit of integration.

We can simplify the given integral using the provided hint:

g(x) = ∫4x²x (u² - 5u²/5)/5 du

g(x) = ∫0²x (u² - 5u²/5)/5 du - ∫0⁴x (u² - 5u²/5)/5 du

The first term on the right-hand side can be integrated as:

∫0²x (u² - 5u²/5)/5 du

= ∫0²x (u²/5 - u²) du

= [tex][(u^3/15) - (u^3/3)]_0^2x[/tex]

= (8x³/15) - (8x³/3)

= -4x³/3

The second term on the right-hand side can be integrated as:

∫0⁴x (u² - 5u²/5)/5 du

= ∫0⁴x (u²/5 - u²) du

=[tex][(u^3/15) - (u^3/3)]_0^4x[/tex]

= (64x³/15) - (64x³/3)

= -32x³

g(x) = -4x³/3 - (-32x³)

= 28x^³/3.

Now, we can differentiate g(x) with respect to x using the power rule:

g'(x) = d/dx [28x³/3]

= 28x²

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1)  The percentage of cups that contain between 10 and 11 ounces of liquid is approximately 34%.

2) The percentage of cups that contain between 8 and 10 ounces of liquid is approximately 81.5%.

3) The percentage of cups that spill over is approximately 0.3%.

4) The percentage of cups that contain between 8 and 9 ounces of liquid is approximately 2.5%.

To use the Empirical Rule, we need to assume that the distribution of the amount of liquid dispensed by the soft drink machine follows a normal distribution.

(a) To find the percentage of cups that contain between 10 and 11 ounces of liquid, we need to find the area under the normal curve between 10 and 11 standard deviations from the mean, which is represented by the interval (x - s, x + s).

According to the Empirical Rule, we know that approximately 68% of the data falls within one standard deviation of the mean. Therefore, the percentage of cups that contain between 10 and 11 ounces of liquid is approximately 68%/2 = 34%.

(b) To find the percentage of cups that contain between 8 and 10 ounces of liquid, we need to find the area under the normal curve between 8 and 10 standard deviations from the mean, which is represented by the interval (x - 2s, x + s).

According to the Empirical Rule, we know that approximately 95% of the data falls within two standard deviations of the mean. Therefore, the percentage of cups that contain between 8 and 10 ounces of liquid is approximately (95%-68%)/2 + 68% = 81.5%.

(c) To find the percentage of cups that spill over because 12 ounces of liquid or more is dispensed, we need to find the area under the normal curve to the right of 12 standard deviations from the mean, which is represented by the interval (x + 2s, ∞). According to the Empirical Rule, we know that approximately 99.7% of the data falls within three standard deviations of the mean. Therefore, the percentage of cups that spill over is approximately 0.3%.

(d) To find the percentage of cups that contain between 8 and 9 ounces of liquid, we need to find the area under the normal curve between 8 and 9 standard deviations from the mean, which is represented by the interval (x - 2s, x - s).

This interval is equivalent to the complement of the interval (x + s, x + 2s), which we can find using the Empirical Rule. The percentage of data falling outside of two standard deviations of the mean is (100% - 95%) / 2 = 2.5%.

Therefore, the percentage of cups that contain between 8 and 9 ounces of liquid is approximately 2.5%.

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To calculate the standard deviation of errors for the regression of y on x, we need to determine the residuals, which are the differences between the observed values of y and the predicted values of y based on the regression line.

Using the given data, we can calculate the residuals and then calculate the standard deviation of these residuals to find the standard deviation of errors for the regression. The observed ages (x) are 27, 15, 3, 35, 14, and 18, and the corresponding observed prices (y) are 165, 182, 205, 178, 180, and 161. We can use these data points to calculate the predicted values of y based on the regression line. After finding the residuals, we can calculate their standard deviation. Performing the calculations, we find the residuals to be -5.83, 4.39, 5.47, -5.83, -2.52, and -2.68 (rounded to two decimal places). To find the standard deviation of these residuals, we take the square root of the mean of the squared residuals. After calculating this, we find that the standard deviation of errors for the regression of y on x is approximately 4.550 (rounded to three decimal places). Therefore, the standard deviation of errors for the regression of y on x is 4.550 (rounded to three decimal places). This value represents the typical amount by which the predicted values of y differ from the observed values of y in the regression model.

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The number more of Michael's friends that like pizza compared to his neighbors are 2 more of his friends.

First, let's calculate how many of Michael's friends and neighbors like pizza:

55% of his 40 friends like pizza, so the number of his friends who like pizza is:

=  55 / 100 x 40

= 22

80% of his 25 neighbors like pizza, so the number of his neighbors who like pizza is :

= 80 / 100 x 25

= 20

Therefore, 2 more of Michael's friends like pizza compared to his neighbors.

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The answer is (b) T is a one-to-one function in situation 2, and the other situations do not provide enough information to determine whether T is one-to-one.

We can determine whether or not T is one-to-one in each of the following situations using the definition of a one-to-one function, which says that T is one-to-one if and only if T(x) = T (y) means that x = y for all x , y in the domain T .

T(0, -2, -4) = u, T(-3, -4,1) = v, T(-3, -5, -3) = u v:

Since T(-3,-4,1) = v and T(-3, -5, -3) = u v, we can write T(-3,-4,1) T(0, -2, -4 ) = T(-3, -5, -3), which means that T(-3, -4,1) T(0, -2, -4) = T(-3, -4,1) y. Therefore, we have T(0, -2, -4) = v. This means that the vectors (0, -2, -4) and (-3, -4,1) both correspond to the same vector v under T , which means that T is not one-to-one.

T (a) = u, T (b) = v, T (c) = u + v:

Suppose that T(x) = T(y) for some x, y in the domain  T. Then we have T(x) - T(y) = 0, which means that T(x-y) = 0. Since T is inside, there exists a vector z in R3 such that T(z) = x - y. Therefore, we have T(z) = 0, which means that z = 0 by the definition of a linear transformation. So x - y = T(z) = 0, which means that x = y. Therefore, T is one-to-one.  T is a hollow function:

If T is on, every vector in R3 is the image of some vector in the domain of T. Therefore, if T(x) = T(y) for any two vectors x and y in the domain  T,  x and y must be the same vectors. Therefore, T is one-to-one.  

Therefore, the answer is (b) T is a one-to-one function in situation 2, and the other situations do not provide enough information to determine whether T is one-to-one.

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the parametric equations for the sphere of radius 3 centered at the origin are: x = 3sinθcosφ,y = 3sinθsinφ,z = 3cosθ, where 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π.

The parametric equations for a sphere of radius 3 centered at the origin can be given by:

x = 3sinθcosφ

y = 3sinθsinφ

z = 3cosθ

where θ is the polar angle (measured from the positive z-axis), and φ is the azimuthal angle (measured from the positive x-axis).

These equations describe a point on the sphere in terms of two parameters, θ and φ. For any given values of θ and φ, the equations will give the corresponding x, y, and z coordinates of a point on the sphere.

The parameter θ varies from 0 to π (or 0 to 180 degrees), while φ varies from 0 to 2π (or 0 to 360 degrees), so the sphere can be fully parameterized by the values of θ and φ within these ranges.

So, the parametric equations for the sphere of radius 3 centered at the origin are:

x = 3sinθcosφ

y = 3sinθsinφ

z = 3cosθ

where 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π.

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the 4th partial sum of the series is approximately 1.4236.

The general term of the series is given by an = n^(-2), for n >= 1.

Therefore, the first four terms are:

a1 = 1^(-2) = 1

a2 = 2^(-2) = 1/4

a3 = 3^(-2) = 1/9

a4 = 4^(-2) = 1/16

The 4th partial sum, s4, is given by:

s4 = a1 + a2 + a3 + a4 = 1 + 1/4 + 1/9 + 1/16 ≈ 1.4236

what is series?

In mathematics, a series is the sum of the terms of a sequence of numbers. It is the result of adding the terms of a sequence and is written using sigma notation as Σan, where n ranges from 1 to infinity and an is the nth term of the sequence.

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y-hat would be 29.3 when the intercept equals 12.34, the slope (b) equals 2.12, and x equals 8.

To find the value of y-hat when the intercept equals 12.34 and the slope (b) equals 2.12 for an x of 8, you can use the linear regression equation:

y-hat = intercept + (slope × x)

Step 1: Substitute the given values into the equation:
y-hat = 12.34 + (2.12 × 8)

Step 2: Multiply the slope by x:
y-hat = 12.34 + (16.96)

Step 3: Add the intercept and the product from Step 2:
y-hat = 29.3

So, y-hat would be 29.3 when the intercept equals 12.34, the slope (b) equals 2.12, and x equals 8.

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The claim is not correct. The fact that all cars are either Volkswagens or not does not mean that there is an equal probability of selecting a Volkswagen or not.

If we assume that there are only two types of cars: Volkswagens and non-Volkswagens, and that there are an equal number of each type registered at the tag agency, then the probability of selecting a Volkswagen would indeed be 1/2. However, this assumption may not hold in reality.

In general, the probability of selecting a Volkswagen depends on the proportion of Volkswagens among all registered cars at the tag agency. Without additional information about this proportion, we cannot conclude that the probability of selecting a Volkswagen is 1/2.

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Answer: a) When the particle moves along the x-axis to (a, 0), the y-coordinate is 0. Therefore, the force F⃗ only has an x-component and is given by:

F⃗ = (2axy ı^ + x^2 ȷ^) N

The displacement of the particle is Δr⃗ = (a ı^) m, since the particle moves only in the x-direction. The work done by the force is given by:

W = ∫ F⃗ · d r⃗

where the integral is taken along the path of the particle. Along the x-axis, the force is constant and parallel to the displacement, so the work done is:

W1 = Fx ∫ dx = Fx Δx = (2ab)(a) = 2a^2 b

When the particle moves from (a, 0) to (a, b) along the y-axis, the force F⃗ only has a y-component and is given by:

F⃗ = (a^2 ȷ^) N

The displacement of the particle is Δr⃗ = (b ȷ^) m, since the particle moves only in the y-direction. The work done by the force is:

W2 = Fy ∫ dy = Fy Δy = (a^2)(b) = ab^2

Therefore, the total work done by the force is:

W = W1 + W2 = 2a^2 b + ab^2

b) When the particle moves along the y-axis to (0, b), the x-coordinate is 0. Therefore, the force F⃗ only has a y-component and is given by:

F⃗ = (a^2 ȷ^) N

The displacement of the particle is Δr⃗ = (b ȷ^) m, since the particle moves only in the y-direction. The work done by the force is given by:

W1 = Fy ∫ dy = Fy Δy = (a^2)(b) = ab^2

When the particle moves from (0, b) to (a, b) along the x-axis, the force F⃗ only has an x-component and is given by:

F⃗ = (2ab ı^) N

The displacement of the particle is Δr⃗ = (a ı^) m, since the particle moves only in the x-direction. The work done by the force is:

W2 = Fx ∫ dx = Fx Δx = (2ab)(a) = 2a^2 b

Therefore, the total work done by the force is:

W = W1 + W2 = ab^2 + 2a^2 b

The series converges for values of x such that |x-4| < 1, since the series for ln(1+x) converges for |x| < 1.

To find the Maclaurin series for f(x) = ln(x-4), we can use the formula for the Maclaurin series of ln(1+x), which is:

ln(1+x) = ∑ n=1[infinity] ((-1)^ⁿ⁺ / n) * xⁿ

We can apply this formula by replacing x with (x-4), which gives us:
ln(x-3) = ln(1 + (x-4)) = ∑ n=1[infinity] ((-1)^(n+1) / n) * (x-4)ⁿ

Therefore, the Maclaurin series for f(x) = ln(x-4) is:
f(x) = ∑ n=1[infinity] ((-1)^ⁿ⁺¹ / n) * (x-4)ⁿ

This series converges for values of x such that |x-4| < 1, since the series for ln(1+x) converges for |x| < 1.

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a) The statement "Let F be a field. If x,y∈F are nonzero, then x⎮y." is False. For a counterexample, consider the field F = ℝ (the set of real numbers).

Let x = 2 and y = 3, both of which are nonzero elements in F. However, x does not divide y since there is no integer k such that y = kx. In general, the statement is false for any field, because fields do not necessarily have a concept of divisibility like integers do.

b) The statement "The ring Z×Z has exactly two units." is False. The ring Z×Z actually has four units. Units are elements that have multiplicative inverses. The four units in Z×Z are (1, 1), (1, -1), (-1, 1), and (-1, -1). To show this, we can verify that their products with their inverses result in the multiplicative identity (1, 1):
- (1, 1) × (1, 1) = (1, 1)
- (1, -1) × (-1, 1) = (1, 1)
- (-1, 1) × (1, -1) = (1, 1)
- (-1, -1) × (-1, -1) = (1, 1)

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In order to determine how many x values will be listed in the first column of a frequency distribution table for a sample of n = 12 scores that ranges from a high of x = 7 to a low of x = 4, we need to first determine the range of the data.

The range is simply the difference between the highest and lowest scores in the sample, which in this case is 7 - 4 = 3.
Next, we need to determine the width of the intervals that will be used in the frequency distribution table. A common rule of thumb is to use intervals that are approximately equal to the square root of the sample size. For a sample size of 12, this would suggest using intervals that are approximately 3 wide (since the square root of 12 is 3.464).Based on this information, we can create intervals that range from 4-6, 7-9, etc. There will be 2 intervals (4-6 and 7-9), which means that there will be 2 x values listed in the first column of the frequency distribution table.Alternatively, we could use narrower intervals, such as 4-4.9, 5-5.9, 6-6.9, 7-7.9, 8-8.9, and 9-9.9. In this case, there would be 6 intervals and 6 x values listed in the first column of the frequency distribution table. However, the intervals would be quite narrow and may not provide a very useful summary of the data.

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To find the nth term of the sequence, we add 4 to the (n-1)th term.

(a) To give a recursive definition of the sequence {an} where an = 4n - 2, we can define it as follows:

a1 = 2

an = an-1 + 4 for n > 1

This means that to find the nth term of the sequence, we add 4 to the (n-1)th term.

(b) To give a recursive definition of the sequence {an} where an = 1 (-1)^n, we can define it as follows:

a1 = 1

an = -an-1 for n > 1

This means that to find the nth term of the sequence, we multiply the (n-1)th term by -1.

(c) To give a recursive definition of the sequence {an} where an = n(n+1), we can define it as follows:

a1 = 2

an = an-1 + 2n + 1 for n > 1

This means that to find the nth term of the sequence, we add 2n+1 to the (n-1)th term.

(d) To give a recursive definition of the sequence {an} where an = n^2, we can define it as follows:

a1 = 1

an = an-1 + 2n - 1 for n > 1

This means that to find the nth term of the sequence, we add 2n-1 to the (n-1)th term.

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To find out how many hours it will take for Sue to catch up to Sam, we can set up an equation based on their relative speeds and the time difference.

Let's denote the time it takes for Sue to catch up to Sam as t hours.

In that time, Sam will have traveled a distance of 4 km/h * (t + 2) hours (since he started 2 hours earlier).

Sue, on the other hand, will have traveled a distance of 6 km/h * t hours.

Since they meet at the same point, the distances traveled by Sam and Sue must be equal.

Therefore, we can set up the equation:

4 km/h * (t + 2) = 6 km/h * t

Now we can solve for t:

4t + 8 = 6t

8 = 6t - 4t = 2t

t = 8/2 = 4

Therefore, it will take Sue 4 hours to catch up to Sam.

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After applying the ratio test to the given geometric series, the answer is option a: the series is convergent.

The given series is: 10 - 6 + 18/5 - 54/25 + ...

To determine whether this series is convergent or divergent, we can use the ratio test.

The ratio test states that a series of the form ∑aₙ is convergent if the limit of the absolute value of the ratio of successive terms is less than 1, and divergent if the limit is greater than 1. If the limit is equal to 1, then the ratio test is inconclusive.

So, let's apply the ratio test to our series:

|ax₊₁ / ax| = |(18/5) * (-25/54)| = 15/20.24 ≈ 0.74

As the limit of the absolute value of the ratio of successive terms is less than 1, we can conclude that the series is convergent.

Therefore, the answer is (a) convergent.

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Proposition: Suppose n ∈ Z (n is an integer). If n^2 is not divisible by 4, then n is not even.

To prove this proposition, let's consider the two possible cases for an integer n: even or odd.

1. If n is even, then n = 2k, where k is an integer. In this case, n^2 = (2k)^2 = 4k^2. Since 4k^2 is a multiple of 4, n^2 is divisible by 4.

2. If n is odd, then n = 2k + 1, where k is an integer. In this case, n^2 = (2k + 1)^2 = 4k^2 + 4k + 1. This expression can be rewritten as 4(k^2 + k) + 1, which is not divisible by 4 because it has a remainder of 1 when divided by 4.

Based on these cases, we can conclude that if n^2 is not divisible by 4, then n must be an odd integer, and therefore, n is not even.

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To find the difference between the length and width of the garden, we simply subtract the width from the length.

Given:

Length of the garden = 20.75 meters

Width of the garden = 15.8 meters

Difference = Length - Width

Difference = 20.75 - 15.8

Difference = 4.95 meters

Therefore, the difference between the length and width of the garden is 4.95 meters.

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A month has 4 weeks, Jose's earnings for a month would be $619.8

First, let's calculate how much Jose earns in a week:

Earnings per day = $10.33/hour * 3 hours/day = $30.99/day

Weekly earnings = $30.99/day * 5 days/week = $154.95/week

Now, let's calculate the monthly earnings by multiplying the weekly earnings by the number of weeks in a month:

Monthly earnings = $154.95/week * 4 weeks/month = $619.80/month

Therefore, Jose will have $619.80 at the end of the month if he works 3 hours a day, 5 days a week, at a rate of $10.33 per hour.

At the end of the month, Jose would have earned $619.8.

As  Jose works 3 hours a day, 5 days a week, at $10.33 an hour, he would earn:

$10.33 x 3 hours a day x 5 days a week= $154.95 per week.

Since a month has 4 weeks, Jose's earnings for a month would be:

4 weeks x $154.95 per week= $619.8

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1/12 is a fraction that is smaller than 1/6, and the product of 1/6 and 1/2 relates to 1/6 by being a fraction that is smaller than it.

The product of 1/6 and 1/2 is 1/12, which is not directly related to 1/6200.

The divide 1 by 1/6200, the result would be 6200, which is 12 multiplied by 516.67.

This shows that 1/6200 is equivalent to 1/12 of 516.67, which is a way to indirectly relate it to the product of 1/6 and 1/2.
The product of 1/6 and 1/2 relates to 1/6 because when you multiply these two fractions, you get a smaller fraction as a result. In this case, (1/6) x (1/2) = 1/12.

Which is smaller than both original fractions.

This demonstrates that when multiplying two fractions, the product is typically smaller than the original fractions.

The product of 1/6 and 1/2 which is (1/6) x (1/2) = 1/12 is smaller than 1/6.

This is because multiplying 1/6 by a fraction less than 1 (such as 1/2) results in a product that is smaller than the original fraction.

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The maximum value of function f(x) on the interval -8 < x ≤ 8 is 2.

Option D is the correct answer.

We have,

The cube root parent function is given by f(x) = ∛x.

To find the maximum value of f(x) on the interval -8 < x ≤ 8, we need to look for critical points of f(x) on this interval.

The function f(x) does not have any critical points on this interval, since its derivative f'(x) = 1/(3∛(x²)) is always positive.

The maximum value of f(x) on the interval -8 < x ≤ 8 occurs at one of the endpoints, which are -8 and 8.

Evaluating f(x) at these endpoints.

f(-8) = ∛(-8) = -2

f(8) = ∛8 = 2

Thus,

The maximum value of function f(x) on the interval -8 < x ≤ 8 is 2.

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Answer:

$1.58 per pair

Step-by-step explanation:

Unit price means the price for each pair.

So $9.49 /6 = 1.58166666667, so approx $1.58 per pair of socks.

There are 324,632 six-digit strings with a digit sum of 35.

To find the number of six-digit strings with a digit sum of 35, we'll use the "stars and bars" combinatorial method.

Since we're looking for six-digit strings, subtract the minimum possible value for each digit (1) from the total digit sum: 35 - 6 = 29. This means we need to distribute 29 units among the six digits.

Use the "stars and bars" method, which involves placing "bars" between "stars" to divide them into groups. In this case, the stars represent the units to be distributed, and we need to place 5 bars to divide the 29 units into 6 groups.

Count the total number of stars and bars: 29 stars + 5 bars = 34 objects.

Calculate the number of ways to choose 5 bars from 34 objects: C(34, 5) = 34! / (5! * (34 - 5)!).

Evaluate the expression: C(34, 5) = 34! / (5! * 29!) = 324,632.

So, there are 324,632 six-digit strings with a digit sum of 35.

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The potential function for F is φ(x,y) = 2xy² + x² + z²y + C

The given vector field F = (2y, 2x+z², 2yz) is conservative on its domain. To find the potential function, we need to check if the partial derivatives of F with respect to x and y are equal.

∂F/∂x = (0, 2, 2y) and ∂F/∂y = (2, 0, 2z)

Since these partial derivatives are equal, we can integrate F with respect to x and y to get the potential function:

φ(x,y) = ∫F.dx = xy² + C1(x)

φ(x,y) = ∫F.dy = x² + z²y + C2(y)

By comparing these two expressions, we can determine that C1(x) = C2(y) = C.

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The velocity of the satellite is [tex]\sf 3.6 \times10^3 \ m / s[/tex].

Given data:

The velocity of the satellite is,

[tex]\sf Velocity =\sqrt{\dfrac{GM}{R} }[/tex]

[tex]=\sqrt{\dfrac{6.7\times10^{-11}\times4.4\times10^{24}}{2.3\times10^7} }[/tex]

[tex]\sf = 3.6 \times10^3 \ m / s[/tex].

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The solution to the initial-value problem is y = (1/(3x + 1))^2and the solution to the homogeneous equation is y = Cx^2 + x and the solution to the Bernoulli equation is y = (1 - 2Ct)^(1/2)

Solve the given initial-value problem. The DE is a Bernoulli equation.
yy' + y^(3/2) = 1, y(0) = 9

We can solve this Bernoulli equation by using the substitution v = y^(1/2). Then, y = v^2 and y' = 2v(v'). Substituting these into the equation gives:

2v(v')v^2 + v^3 = 1

Simplifying and separating the variables gives:

2v' = (1 - v)/v^2

Now, we can solve this separable equation by integrating both sides:

∫(1 - v)/v^2 dv = ∫2 dx

This gives:

1/v = -2x - 1/v + C

Simplifying and solving for v gives:

v = 1/(Cx + 1)

Substituting y = v^2 and y(0) = 9 gives:

9 = 1/(C*0 + 1)^2

Solving for C gives C = 1/3.

Solve the given differential equation by using an appropriate substitution: The DE is homogeneous.
(x - y) dx + x dy = 0

We can see that this is a homogeneous equation, since both terms have the same degree (1) and we can factor out x:

x(1 - y/x) dx + x dy = 0

Now, we can use the substitution v = y/x. Then, y = vx and y' = v + xv'. Substituting these into the equation gives:

x(1 - v) dx + x v dx + x^2 dv = 0

Simplifying and separating the variables gives:

dx/x = dv/(v - 1)

Now, we can solve this separable equation by integrating both sides:

ln|x| = ln|v - 1| + C

Simplifying and solving for v gives:

v = Cx + 1

Substituting y = vx gives:

y = Cx^2 + x


Solve the given differential equation by using an appropriate substitution: The DE is a Bernoulli equation.
2 dy/dt + y^2 = t

We can solve this Bernoulli equation by using the substitution v = y^(1 - 2) = 1/y. Then, y = 1/v and y' = -v'/v^2. Substituting these into the equation gives:

-2v' + 1/v = t

Simplifying and separating the variables gives:

v' = (-1/2)(1/v - t)

Now, we can solve this separable equation by integrating both sides:

ln|v - 1| = (-1/2)ln|v| - (1/2)t^2 + C

Simplifying and solving for v gives:

v = (C/(1 - 2Ct))^2

Substituting y = 1/v gives:

y = (1 - 2Ct)^(1/2)

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