A curve y=f(x) defined for values of x>0 goes through the point (1,0) and is such that the slope of its tangent line at (x,f(x)) is 4/x^2?7/x^6, for x>0.

Answer:

5.33

Step-by-step explanation:

The amount of a substance decaying exponentially with a half-life can be modeled using the formula A = A₀ * 2^(-t / h), where A is the amount remaining after time t, A₀ is the initial amount of substance, t is the time elapsed, and h is the half-life of the substance. Using the fact that initially there were 12 grams of the substance, and after 2 hours there were 7 grams, we can solve for the half-life h. Substituting the values into the equation 7 = 12 * 2^(-2 / h) and solving, we get that h is approximately 4.145 hours. Finally, we can use the formula A = A₀ * 2^(-t / h) to find the amount of substance remaining after 3 hours. Plugging in A₀ = 12, t = 3, and h ≈ 4.145, we get A ≈ 5.33 grams. Rounding to the nearest hundredth, we conclude that approximately 5.33 grams of the substance will remain after 3 hours.

The 5-point DFT of the sequence Y(k) is [15 -2.5 + 3.4j -2.5 + 0.81j -2.5 - 0.81j -2.5 - 3.4j]. So, the correct answer is 1).

We can find the 5-point DFT of y(n) using the formula

Y(k) = sum_{n=0}^{4} y(n) exp(-2piikn/5), k = 0,1,2,3,4

Substituting the values of y(n) = {1, 2, 3, 4, 5}, we get

Y(0) = 1 + 2 + 3 + 4 + 5 = 15

Y(1) = 1 + 2exp(-2pii/5) + 3exp(-4pii/5) + 4exp(-6pii/5) + 5exp(-8pii/5) = -2.5 + 3.4j

Y(2) = 1 + 2exp(-4pii/5) + 3exp(-8pii/5) + 4exp(-12pii/5) + 5exp(-16pii/5) = -2.5 + 0.81j

Y(3) = 1 + 2exp(-6pii/5) + 3exp(-12pii/5) + 4exp(-18pii/5) + 5exp(-24pii/5) = -2.5 - 0.81j

Y(4) = 1 + 2exp(-8pii/5) + 3exp(-16pii/5) + 4exp(-24pii/5) + 5exp(-32pii/5) = -2.5 - 3.4j

Therefore, the 5-point DFT of the sequence Y(k) is [15, -2.5 + 3.4j, -2.5 + 0.81j, -2.5 - 0.81j, -2.5 - 3.4j], which is option 1.

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Answer:

40 miles

Step-by-step explanation:

If Nicole has driven 20 miles and this is only half the distance, then the total length of her drive would be 40 miles.

We can determine this with a simple algebraic equation:

Let x be the total length of her drive.

We know that Nicole has already driven one-half of the distance, which can be represented as:

20 = 1/2x

Multiplying both sides by 2, we get:

40 = x

Therefore, the total length of Nicole's drive is 40 miles.

Answer:

Step-by-step explanation:

Each point (x, y) on the graph of h(x) becomes the point (x - 3, y - 3) on v(x).

Each point (x, y) on the graph of h(x) becomes the point (x + 3, y + 3) on w(x).

In Mathematics and Geometry, the translation a geometric figure or graph to the left simply means subtracting a digit from the value on the x-coordinate of the pre-image;

g(x) = f(x + N)

On the other hand, the translation a geometric figure to the right simply means adding a digit to the value on the x-coordinate (x-axis) of the pre-image;

g(x) = f(x - N)

Since the parent function is v(x) = h(x + 3), it ultimately implies that the coordinates of the image would created by translating the parent function to the left by 3 units.

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Vik spends 33.28% of the total cost on airport tax, rounded to 1 decimal place.

Converting €50 to pounds using the exchange rate, we get:

€50 = £50/1.14 = £43.86 (rounded to 2 decimal places)

The total cost is:

£88 + £43.86 = £131.86

The proportion of the total cost that Vik spends on airport tax is:

£43.86 / £131.86 = 0.3328

To convert this to a percentage, we multiply by 100:

0.3328 × 100 = 33.28%

Therefore, Vik spends 33.28% of the total cost on airport tax, rounded to 1 decimal place.

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The function y =[tex](3/2) x^2[/tex] indeed satisfies the differential equation [tex]dy/dx = x 1.[/tex]

To check if [tex]y = 1/2 x^2 x 3[/tex]satisfies the differential equation dy/dx = x 1, we need to find the first derivative of y with respect to x and then compare it to the given dy/dx expression.

Given y = 1/2 x^2 x 3, we can rewrite it as[tex]y = (3/2) x^2.[/tex]

Now, let's find the first derivative of y with respect to x:

[tex]dy/dx = d(3/2 x^2)/dx = 3x[/tex]
Now we compare this with the given [tex]dy/dx = x 1. Since 3x = 3x * 1[/tex], the function y = (3/2) x^2 indeed satisfies the differential equation dy/dx = x 1.

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Sample standard deviation is approximately 0.6129.

We add all the numbers and divide by the sample size:

(0.1 + 0 - 0.3 - 1.4 + 0) / 5 = -0.32/5 = -0.064

Therefore, the sample mean is -0.064.

To find the sample variance, we need to first find the deviations of each measurement from the sample mean. We subtract the sample mean from each measurement to get:

0.1 - (-0.064) = 0.164

0 - (-0.064) = 0.064

-0.3 - (-0.064) = -0.236

-1.4 - (-0.064) = -1.336

0 - (-0.064) = 0.064

Then, we square each deviation:

0.164² = 0.026896

0.064² = 0.004096

(-0.236)² = 0.055696

(-1.336)² = 1.787296

0.064² = 0.004096

We take the average of these squared deviations to get the sample variance:

(0.026896 + 0.004096 + 0.055696 + 1.787296 + 0.004096) / 5 = 0.375416

Therefore, the sample variance is 0.375416.

To find the sample standard deviation, we take the square root of the sample variance:

sqrt(0.375416) = 0.6129

Therefore, the sample standard deviation is approximately 0.6129.

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Answer:  The store can sell 451 packs of socks.

Step-by-step explanation:  To find the number of packs of socks the store can sell, we need to divide the total number of socks by the number of socks in each pack.

Since each pack contains 3 pairs of socks, or 6 individual socks, we can find the number of packs by dividing the total number of socks by 6:

1353 socks ÷ 6 socks per pack = 225.5 packs

However, since we can't sell a fraction of a pack, we need to round up to the nearest whole number. Therefore, the store can sell 451 packs of socks.

Share Prompt

A the center of mass is[tex]$(x_{M},y_{M})=(0,2)$.[/tex] The answer is (a).
B the center of mass is [tex]$(x_{M},y_{M})=(\frac{2916}{7\cdot243},\frac{6561}{4\cdot243})$[/tex]. The answer is (a).

C the center of mass is[tex]$(x_{M},y_{M})=(0,2)$.[/tex] The answer is (a).

1A) We can find the mass by integrating the density function over the region:
[tex]$$M=\iint_{\Omega}\lambda(x,y)dA=\int_{0}^{5}\int_{0}^{25-x\sqrt{25-x^2}}2xydydx$$[/tex]
Evaluating this integral gives [tex]$M=\frac{625}{8}$.[/tex] To find the center of mass, we need to compute the moments:
[tex]$$M_{x}=\iint_{\Omega}x\lambda(x,y)dA=\int_{0}^{5}\int_{0}^{25-x\sqrt{25-x^2}}2x^2ydydx=\frac{8}{3}M$$\\$$M_{y}=\iint_{\Omega}y\lambda(x,y)dA=\int_{0}^{5}\int_{0}^{25-x\sqrt{25-x^2}}2xy^2dydx=\frac{8}{3}M$$[/tex]
So the center of mass is [tex]$(x_{M},y_{M})=(\frac{8}{3},\frac{8}{3})$[/tex]. Therefore, the answer is (a).

1B) Since the question only asks for the mass and center of mass, we can use the same method as in 1A to get [tex]$M=\int_{-1}^{1}\int_{0}^{4}x^2dydx=\frac{16}{3}$[/tex]. To find the moments, we have:
[tex]$$M_{x}=\int_{-1}^{1}\int_{0}^{4}x^3dydx=0$$\\$$M_{y}=\int_{-1}^{1}\int_{0}^{4}xy^2dydx=2\int_{0}^{1}\int_{0}^{4}xy^2dydx=\frac{16}{3}$$[/tex]
Therefore, the center of mass is[tex]$(x_{M},y_{M})=(0,2)$.[/tex] The answer is (a).

1C) Using the same method as in 1A, we have:
[tex]$$M=\int_{0}^{3}\int_{x^2}^{9}2xydydx=\frac{243}{2}$$[/tex]
To find the moments, we have:
[tex]$$M_{x}=\int_{0}^{3}\int_{x^2}^{9}x2xydydx=\frac{2916}{7}$$\\$$M_{y}=\int_{0}^{3}\int_{x^2}^{9}y2xydydx=\frac{6561}{4}$$[/tex]
Therefore, the center of mass is [tex]$(x_{M},y_{M})=(\frac{2916}{7\cdot243},\frac{6561}{4\cdot243})$[/tex]. The answer is (a).

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Step-by-step explanation:

We can prove the statement by mathematical induction.

Base Case: For n = 0, we have 9n - 3n = 1, which is divisible by 6, since 1 = 6*0 + 1.

Inductive Step: Assume that 6 divides 9k - 3k for some non-negative integer k. We need to show that 6 also divides 9(k+1) - 3(k+1).

Starting with 9(k+1) - 3(k+1), we can simplify it as follows:

9(k+1) - 3(k+1) = 9k + 9 - 3k - 3

= (9k - 3k) + (9 - 3)

= 6k + 6

Since 6 divides both 6k and 6, it also divides their sum, 6k + 6. Therefore, we have shown that 6 divides 9(k+1) - 3(k+1).

By the principle of mathematical induction, we can conclude that 6 divides 9n - 3n for all non-negative integers n.

a)1. True

2.False

3.True

4).False

b)Without knowing the sample size and standard deviation, we cannot determine the exact 90% confidence interval.

(a) For the content loaded Central Middle School data:

1. True: There is a 95% probability that μ (mean height) is between 54 and 58 inches.

This is the correct interpretation of the 95% confidence interval.

2. False: The confidence interval doesn't tell us the probability of the true mean or the margin of error being exactly as given. It only tells us the range where the true mean is likely to fall with 95% confidence.

3. True: If we took many additional random samples of the same size and from each computed a 95% confidence interval for μ, approximately 95% of these intervals would contain μ. This is the definition of a 95% confidence interval.

4. False: It's incorrect to say that μ would fall between 54 and 58 95% of the time. The correct interpretation is that if we computed multiple 95% confidence intervals, approximately 95% of those intervals would contain the true mean height.

(b) To determine the 90% confidence interval based on the same data:

Without knowing the sample size and standard deviation, any of the above answers could be the 90% confidence interval. Confidence intervals depend on the sample size, standard deviation, and desired confidence level. With the information given, we cannot determine the exact 90% confidence interval.

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Answer:

Step-by-step explanation:

A. Let x be the mass of 1 slice of cheese.

B. The difference in the masses of 1 slice of cheese and 1 cracker is:

x - 3.25 grams

C. Since one package contains 16 slices of cheese, the total mass of the package is:

16x

According to the problem, the mass of one cracker is 18 grams less than the mass of one slice of cheese. Therefore, we can write:

x - 18 = 3.25 + m/16

where m is the mass of the package of 16 slices of cheese.

Simplifying the equation:

x = 3.25 + m/16 + 18

x = m/16 + 21.25

This equation represents the relationship between the masses of 1 cracker and 1 slice of cheese in terms of the mass of a package of 16 slices of cheese.

Number of Folds | Style 1 Number of Sections | Style 2 Number of Sections:

From the table, the pattern relating the number of folds to the number of sections for Style 1 (accordion-style) is that the number of sections doubles with each additional fold. In other words, the number of sections is equal to 2 raised to the power of the number of folds. For Style 2 (half-folds), the pattern is less clear, but we can observe that the number of sections increases more rapidly with each additional fold than it does for Style 1. This is likely due to the fact that each fold in Style 2 creates two new sections, whereas in Style 1, each fold only creates one new section.

The two different folded styles of paper produce different results because they create different shapes and arrangements of rectangular sections when folded. In Style 1 (accordion-style), each fold creates a single new section that is added to the end of the folded paper. The result is a long, thin strip of paper with rectangular sections stacked on top of each other. In contrast, Style 2 (half-folds) creates a zig-zag pattern of rectangular sections that are stacked on top of each other. Each fold in Style 2 creates two new rectangular sections, which allows the number of sections to increase more rapidly than in Style 1. This difference in the way the paper is folded and the resulting shapes and arrangements of rectangular sections leads to different patterns in the number of sections as the number of folds increases.

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Answer:

(-2,-1)

Step-by-step explanation:

see photo

We will need 7 cubic feet of dirt to fill the garden box completely.

The formula for the volume of a rectangular box is:
Volume = Length × Width × Height

In this case, the dimensions of the garden box are:
Length = 3 1/2 feet
Width = 4 feet
Height = 1/2 foot
First, convert the mixed numbers to improper fractions:
Length = (3 × 2 + 1)/2 = 7/2 feet
Now, multiply the dimensions together:
Volume = (7/2) × 4 × (1/2)
Simplify the fractions:
Volume = (7 × 4 × 1) / (2 × 2) = 28 / 4
Finally, divide to find the volume in cubic feet:
Volume = 28 ÷ 4 = 7 cubic feet.

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Recurrence relation for t(n):

t(n) = 4t(n/2) + 1, where n > 1

When we compute the square of an n-bit number, we can express it as:

n² = (n/2)² + (n/2)² + n

This means that we can compute the square of an n-bit number by recursively computing the square of an (n/2)-bit number twice, and adding the result to the product of the two (n/2)-bit numbers.

Each recursion involves 4 additions/subtractions (for adding/subtracting the two intermediate results), and 1 addition (for adding the final result). Therefore, the number of operations t(n) required to compute the square of an n-bit number can be expressed as:

t(n) = 4t(n/2) + 1, where n > 1

The base case is t(1) = 0, since computing the square of a 1-bit number requires no operations.

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Recurrence relation for t(n):

t(n) = 4t(n/2) + 1, where n > 1

When we compute the square of an n-bit number, we can express it as:

n² = (n/2)² + (n/2)² + n

This means that we can compute the square of an n-bit number by recursively computing the square of an (n/2)-bit number twice, and adding the result to the product of the two (n/2)-bit numbers.

Each recursion involves 4 additions/subtractions (for adding/subtracting the two intermediate results), and 1 addition (for adding the final result). Therefore, the number of operations t(n) required to compute the square of an n-bit number can be expressed as:

t(n) = 4t(n/2) + 1, where n > 1

The base case is t(1) = 0, since computing the square of a 1-bit number requires no operations.

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The polar coordinates of the point (3, -5) are (r, θ) = (√34, 5.25) where r > 0 and 0 ≤ θ < 2π. Since tan() is negative, we know that lies in either the second or fourth quadrant.

To find the polar coordinates (r, ) of the point (3, -5), we can use the following formulas:
r = sqrt(x^2 + y^2)
tan() = y/x
Plugging in the values for x and y, we get:
r = sqrt(3^2 + (-5)^2) = sqrt(34)
tan() = -5/3
Since tan() is negative, we know that lies in either the second or fourth quadrant. To determine which one, we can use the fact that tan() = y/x. In the second quadrant, both x and y are negative, which would give us a positive value for tan(). Therefore, must be in the fourth quadrant.
To find the angle , we can use the inverse tangent function (tan^-1) on our calculator. However, we need to adjust the result to account for the fact that we are in the fourth quadrant. Specifically, we need to add 2 radians (or 360 degrees) to the result. So:
tan^-1(-5/3) = -1.03 radians
+ 2 radians = 0.97 radians
Therefore, the polar coordinates of the point (3, -5) are (sqrt(34), 0.97 radians).
To find the polar coordinates (r, θ) of the point (3, -5) where r > 0 and 0 ≤ θ < 2π, you can use the following formulas:
r = √(x^2 + y^2)
θ = arctan(y/x)
Plugging in the Cartesian coordinates (3, -5) for x and y:
r = √(3^2 + (-5)^2) = √(9 + 25) = √34
Since the point is in the fourth quadrant (x > 0 and y < 0), we'll adjust the angle:
θ = arctan(-5/3) ≈ -1.03 radians
To convert θ to the range 0 ≤ θ < 2π, add 2π:
θ = -1.03 + 2π ≈ 5.25 radians
So, the polar coordinates of the point (3, -5) are (r, θ) = (√34, 5.25) where r > 0 and 0 ≤ θ < 2π.

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The statement "The radius of a star can be indirectly determined if the star's distance and luminosity are known." is true because the radius of a star can be calculated using the Stefan-Boltzmann Law.

The Stefan-Boltzmann Law can be used to determine a star's radius. This law states that the luminosity of a star (the total amount of energy it emits in a given time) is proportional to the fourth power of its radius. Therefore, if the distance and luminosity of a star are known, its radius can be calculated by rearranging the equation.

This equation can be used to calculate the radius of a star even if its size cannot be directly measured. The equation is:

Radius = (L/4πσT⁴[tex])^{1/2}[/tex]

Where L is the luminosity of the star, σ is the Stefan-Boltzmann constant, and T is the temperature of the star.

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Answer:

To do a full function study of y = x^4 - 8x^2 - 9, we need to determine the domain, intercepts, symmetry, asymptotes, intervals of increase and decrease, local extrema, and concavity.

Domain:

Since the function is a polynomial, it is defined for all real numbers. Therefore, the domain is (-∞, ∞).

x-Intercepts:

To find the x-intercepts, we set y = 0 and solve for x:

x^4 - 8x^2 - 9 = 0

We can factor the left-hand side to get:

(x^2 - 9)(x^2 + 1) = 0

This gives us x = ±√9 = ±3 as the x-intercepts.

y-Intercept:

To find the y-intercept, we set x = 0:

y = 0^4 - 8(0^2) - 9 = -9

Therefore, the y-intercept is (0, -9).

Symmetry:

The function is an even-degree polynomial, which means it has rotational symmetry of order 2 about the origin.

Asymptotes:

There are no vertical or horizontal asymptotes for this function.

Intervals of Increase and Decrease:

To find the intervals of increase and decrease, we need to find the critical points of the function by taking the first derivative and setting it equal to zero:

y' = 4x^3 - 16x = 0

Solving for x, we get x = 0 or x = ±√4 = ±2. Therefore, the critical points are (-2, 43), (0, -9), and (2, 43). We can use the second derivative test to determine that (-2, 43) and (2, 43) are local minima and (0, -9) is a local maximum.

The function increases on the intervals (-∞, -2) and (2, ∞) and decreases on the interval (-2, 2).

Local Extrema:

The local minimum points are (-2, 43) and (2, 43), and the local maximum point is (0, -9).

Concavity:

To determine the concavity of the function, we take the second derivative:

y'' = 12x^2 - 16

Setting y'' equal to zero, we get x = ±√4/3. Since y'' is positive for x < -√4/3 and x > √4/3, and negative for -√4/3 < x < √4/3, we have a point of inflection at x = -√4/3 and x = √4/3.

Plotting the Graph:

We can now use all of the information we have gathered to sketch the graph of y = x^4 - 8x^2 - 9. The graph has rotational symmetry of order 2 about the origin, and it passes through the points (-3, 0), (0, -9), and (3, 0). It has local minimum points at (-2, 43) and (2, 43) and a local maximum point at (0, -9). It changes concavity at x = -√4/3 and x = √4/3. Here is a rough sketch of the graph:

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The number of times the bsearch method is called depends on the implementation of the binary search algorithm and the contents of the nums array. The initial call to the bsearch method is counted as one call.

After that, each subsequent call is made as the algorithm narrows down the search space by dividing it in half. The maximum number of calls can be calculated as log2(nums.length) + 1, where log2 is the base-2 logarithm.

This includes the initial call. However, the exact number of calls may be less than the maximum, depending on the data and target value (-100 in this case).

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The maximum likelihood estimators of θ and λ are θ-cap = min(X1, X2, ..., Xn) and λ-cap = n / (Σ(Xi - θ-cap)). For the given data, the estimates of θ and λ are θ-cap = 0.64 and λ-cap = 10 / (Σ(Xi - 0.64)).

To find the maximum likelihood estimators (MLE) for the shifted exponential distribution, first obtain the likelihood function L(θ, λ) by multiplying the pdf of each observation.

Take the natural logarithm of the likelihood function to get the log-likelihood function, and then differentiate it with respect to θ and λ. Set these partial derivatives to zero to find the MLEs.

For the given data, to find θ-cap, choose the smallest value, which is 0.64. To find λ-cap, subtract θ-capfrom each observation, sum the differences, and divide the number of observations (10) by this sum. This gives λ-cap = 10 / (Σ(Xi - 0.64)).

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The expression that displays the break-even point as a constant or coefficient is: (x-3,000)² + 45,000, which is equivalent to 1,200 * (x-3,000)² - 9,000,000.

To find the break-even point, we need to set the profit function equal to 0 and solve for x:

P(x) = 2x² + 30x = 0

We can factor out x:

x(2x + 30) = 0

So, x = 0 or x = -15. Since we are looking for a positive number of units sold, the break-even point is:

x = 0 units

Now, we can plug this value into the given expressions to see which one results in a constant or coefficient:

((0-3,000)² - 9,000,000) = 0-9,000,000-9,000,000 = -18,000,000

(x-3,000)² + 45,000 = (0-3,000)² + 45,000 = 9,000,000 + 45,000 = 9,045,000

Therefore, the expression that displays the break-even point as a constant or coefficient is:

(x-3,000)² + 45,000, which is equivalent to 1,200 * (x-3,000)² - 9,000,000.

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Out of the given options, the task that is not a likely task of descriptive statistics is "estimating unknown parameters."

Descriptive statistics is a branch of statistics that deals with the collection, analysis, and interpretation of data. It involves summarizing and presenting data in a meaningful way using measures of central tendency, variability, and other statistical tools.

This task is usually carried out in inferential statistics, which involves drawing conclusions about a population based on a sample.

Descriptive statistics, on the other hand, is focused on describing and summarizing the characteristics of a sample or population, rather than making inferences about it.

Therefore, while summarizing a sample, making visual displays of data, and presenting measures of central tendency and variability are all common tasks in descriptive statistics, estimating unknown parameters is not typically a part of descriptive statistics.

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We can calculate the curvature k as:

k = |dT/ds| / |dr/ds|

[tex]= |d^2 r(s)/ds^2| / |dr/ds|^3[/tex]

To find the curvature k of a curve given by the vector-valued function r(s), where s is the arc length parameter, we use the following formula:

k = |dT/ds| / |dr/ds|

where T(s) is the unit tangent vector and r'(s) is the velocity vector.

To find T(s), we differentiate r(s) with respect to s:

T(s) = dr(s)/ds

Then, we normalize T(s) to obtain the unit tangent vector:

T(s) = dr(s)/ds / |dr(s)/ds|

Next, we differentiate T(s) with respect to s to obtain the unit normal vector N(s):

[tex]N(s) = d^2 r(s)/ds^2 / |d r(s)/ds|[/tex]

Finally, we can calculate the curvature k as:

k = |dT/ds| / |dr/ds|

[tex]= |d^2 r(s)/ds^2| / |dr/ds|^3[/tex]

So, to find the curvature k of the curve given by the vector-valued function r(s), we need to calculate r(s), dr(s)/ds, and[tex]d^2 r(s)/ds^2[/tex] and plug them into the above formula.

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Answer: 260

Step-by-step explanation:

Use proportions:

65/25 = x/100

cross multiply the proportions:

25x = 6500

solve your equation:

x = 260

The number is 260.

Let's start by finding the number we're working with.

We know that 25% of the number is 65, so we can set up an equation:

0.25x = 65

To solve for "x", we can divide both sides of the equation by 0.25:

x = 65 / 0.25

x = 260

So the number we're working with is 260.

Next, we need to find 40% of the same number:

0.40(260) = 104

Now we can use this information to find 15% of the same number:

We can set up a proportion:

40% is to 104 as 15% is to x

0.40/104 = 0.15/x

To solve for "x", we can cross-multiply:

0.40x = 104(0.15)

0.40x = 15.6

x = 39

So 15% of the same number is 39.

The differential of arc length for the curve C parametrized by r(t) is given by:

ds = sqrt((dx/dt)² + (dy/dt)²) dt

where x =  [tex]e^t[/tex]² and y = ln(t+1).

Taking the derivatives, we get:

dx/dt = 2t ([tex]e^t[/tex])²
dy/dt = 1/(t+1)

Substituting into the formula, we get:

ds = sqrt((2t [tex]e^t[/tex]²)² + (1/(t+1))²) dt

Simplifying, we get:

ds = sqrt(4t²e²t² + 1/(t+1)²) dt

Therefore, the differential of arc length for the curve C parametrized by r(t) is:

ds = sqrt(4t²e²t² + 1/(t+1)²) dt.

This formula allows us to calculate the length of the curve C between two points on the curve by integrating the differential of arc length between the corresponding values of t.

The formula shows that the length of the curve increases as t increases, with the rate of increase depending on the values of t and e.

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To solve this problem, we can use the principle of inclusion-exclusion.

First, let's find the total number of ways to select n students from a class of 5n:

Total ways = (5n choose n) = (5n)! / (n!*(5n-n)!) = (5n)! / (n!*(4n)!)

The general solution is y = y_h + y_p = c1 [tex]e^{ (-x) }[/tex] + c2 x e^(-x) - (1÷2) cos(x) - (1÷12) sin(2x) - (5÷24) cos(2x).

What is Differential Equation ?

A differential equation is a mathematical equation that relates a function or a set of functions with their derivatives or differentials. In other words, it is an equation that describes the behavior of a system in terms of the rates of change of one or more variables.

First, we find the homogeneous solution of the differential equation:

The characteristic equation is r*r + 2r + 1 = 0, which can be factored as (r+1)(r+1) = 0. Hence, the homogeneous solution is y_h = c1  [tex]e^{ (-x) }[/tex]  + c2 x[tex]e^{ (-x) }[/tex]

Now, we look for a particular solution of the form y_p = A sin(x) + B cos(x) + C sin(2x) + D cos(2x), where A, B, C, and D are constants to be determined.

Taking derivatives, we get y_p' = A cos(x) - B sin(x) + 2C cos(2x) - 2D sin(2x) and y_p'' = -A sin(x) - B cos(x) - 4C sin(2x) - 4D cos(2x).

Substituting y_p, y_p', and y_p'' into the differential equation, we get:

(-A sin(x) - B cos(x) - 4C sin(2x) - 4D cos(2x)) + 2(A cos(x) - B sin(x) + 2C cos(2x) - 2D sin(2x)) + (A sin(x) + B cos(x) + C sin(2x) + D cos(2x)) = sin(x) + 7cos(2x)

Simplifying and collecting like terms, we get:

(-3A - 3C + 4D) sin(2x) + (3B + 4C - 3D) cos(2x) + 2A cos(x) - 2B sin(x) = sin(x) + 7cos(2x)

Equating coefficients of sin(2x), cos(2x), sin(x), and cos(x), we get the following system of equations:

-3A - 3C + 4D = 0

3B + 4C - 3D = 7

2A = 0

-2B = 1

Solving for A, B, C, and D, we get:

A = 0

B = -1÷2

C = -1÷12

D = -5÷24

Therefore, the particular solution is y_p = (-1÷2) cos(x) - (1÷12) sin(2x) - (5÷24) cos(2x).

The general solution is y = y_h + y_p = c1   [tex]e^{ (-x) }[/tex] + c2 x  [tex]e^{ (-x) }[/tex] - (1÷2) cos(x) - (1÷12) sin(2x) - (5÷24) cos(2x).

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So, the equation of the plane tangent to the surface at point P(40, 80, 12) is: z = x - (9/5)y + 4.

An equation is a mathematical statement that shows the equality of two expressions. It usually consists of two sides separated by an equal sign (=). The expressions on both sides of the equal sign can include numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division.

Here,

To find the equation of the plane tangent to the surface at point P(40, 80, 12), we need to first find the partial derivatives of the function z(x,y) with respect to x and y, and evaluate them at point P. Then we can use the gradient vector of the surface at point P to find the equation of the tangent plane.

Given,

r = (9u+v)i + 5u²j + (4u – v)k

We have, x = 9u + v, y = 5u², z = 4u - v

So, z(x, y) = 4u - v = 4(1/4(x-9y/5))-1/5(y-v) = (x-9y/5) - (y-v)/5

Taking partial derivatives of z with respect to x and y, we get:

∂z/∂x = 1, and ∂z/∂y = -9/5

Evaluating these at point P(40, 80, 12), we get:

∂z/∂x = 1, and ∂z/∂y = -9/5

So, the gradient vector of the surface at point P is:

grad z = (1)i - (9/5)j

Now, the tangent plane at point P is given by the equation:

z - z(P) = ∇z · (r - r(P))

where z(P) = z(40, 80) = 12, r(P) = <40, 80, 12>, and ∇z = (1)i - (9/5)j

Substituting the values, we get:

z - 12 = (1)(x - 40) - (9/5)(y - 80)

Simplifying, we get:

z = x - (9/5)y + 12 - 8

So, the equation of the plane tangent to the surface at point P(40, 80, 12) is:

z = x - (9/5)y + 4

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