A 5. 1m long ladder is leaning against a wall the wall stands perpendicular to the ground the base of the adder is 1. 8m from the wall. Work out the size of the acute angle that the ladder makes with the ground give your answers in degrees to 1dp

Answer: -4 and 10

Step-by-step explanation:

Answer:

(5,6) and (8,3)

Step-by-step explanation:

A and C are two of the corners of the square.

Imagine the top LEFT corner. If you drew a line straight UP from A and to the LEFT of C, it would meet at (5,6).

Now picture the bottom RIGHT corner. If you drew a line from A to the RIGHT and then another line from C DOWN, those lines would meet at (8,3).

I drew a pic showing the square you are trying to create! See attached.

Answer:

Step-by-step explanation:

This recursive definition defines the first two terms of the sequence as a1 = 3 and a2 = 6.

A recursive definition for the sequence {an} with closed formula an = 3 * 2^n is:

a1 = 3

an = 2 * an-1 for n ≥ 2

This recursive definition defines the first term of the sequence as a1 = 3, and then defines each subsequent term as twice the previous term. For example, a2 = 2 * a1 = 2 * 3 = 6, a3 = 2 * a2 = 2 * 6 = 12, and so on.

A recursive definition that makes use of two previous terms and no constants is:

a1 = 3

a2 = 6

an = 6an-1 - an-2 for n ≥ 3

This recursive definition defines the first two terms of the sequence as a1 = 3 and a2 = 6, and then defines each subsequent term as six times the previous term minus the term before that. For example, a3 = 6a2 - a1 = 6 * 6 - 3 = 33, a4 = 6a3 - a2 = 6 * 33 - 6 = 192, and so on.

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47 is an odd number and 458, 15280, 1600 are even numbers.

We know,

An even number is a number that can be divided by 2 without a remainder, while an odd number cannot be divided evenly by 2.

I) 47 is an odd number because it can't be divisible evenly by 2. 47 is only divisible by 1 and itself, and when divided by 2, the quotient is not a whole number.

So, 47 is an odd number.

II) 458 is an even number because it can be divided evenly by 2. When divided by 2, the quotient is a whole number which means it is an even number.

III) 15280 is an even number because it can be divided evenly by 2. When divided by 2, the quotient is a whole number, which means it is an even number.

IV) 1600 is an even number because it can be divided evenly by 2. When divided by 2, the quotient is a whole number, which means it is an even number.

So, 47 is an odd number and 458, 15280, 1600 are even numbers.

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Correct question is "Classify the following into odd and even numbers

I) 47. II) 458 II) 15280 Iv) 100"

When constructing a frequency distribution, how many classes should

there be  between 8 and 12. Option A

When constructing a frequency distribution, the number of classes should typically be determined based on the specific dataset and the desired level of detail. The general guideline is to have a sufficient number of classes to capture the variability in the data without having too few or too many classes.

This range allows for a reasonable level of detail while still providing a clear representation of the data distribution. It strikes a balance between having too few classes, which might oversimplify the data, and having too many classes, which might make it difficult to interpret the distribution accurately.

However, it's important to note that the optimal number of classes can vary depending on factors such as the size of the dataset, the range of values, and the specific characteristics of the data.

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Answer: 20

Hope this helps! If you need a larger explanation, let me know.

Answer: 205 degrres

Step-by-step explanation:

To the nearest tenth, x is 40.6 units. The measurement of the missing side length x of the right triangle.

Given information is:

The calculation:

It was apply on trigonometric ratio formula:

cosine = adjacent / hypotenuse

cos(40) = x / 53

x = cos(40) × 53

x = 40.6003

x = 40.6 units

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To find the average power delivered by an ideal current source, we need to use the formula:

P_avg = (1/T) * ∫[0,T] p(t) dt

where P_avg is the average power, T is the period of the signal, and p(t) is the instantaneous power at time t.

In this case, we have an ideal current source, which means the voltage across the source is constant and the current is given by ig(t) = 110cos(1250t) mA.

Since the voltage across the source is constant, the power delivered by the source is simply the product of the voltage and current. Therefore, the instantaneous power delivered by the source is:

p(t) = v(t) * ig(t)

where v(t) is the voltage across the ideal current source. Since the voltage is constant, we can simply write:

p(t) = V * ig(t)

where V is the voltage across the ideal current source.

The period of the signal is T = 2π/ω, where ω is the angular frequency of the signal. In this case, ω = 1250 rad/s, so T = 2π/1250 = 0.005 sec.

Using the formula for the average power, we have:

P_avg = (1/T) * ∫[0,T] p(t) dt
= (1/T) * ∫[0,T] V * ig(t) dt
= (1/T) * V * ∫[0,T] ig(t) dt

The integral of the current over one period is:

∫[0,T] ig(t) dt = ∫[0,2π/ω] 110cos(1250t) dt
= [110/1250 sin(1250t)]_[0,2π/ω]
= [110/1250 sin(2π)] - [110/1250 sin(0)]
= 0

Therefore, the average power delivered by the ideal current source is:

P_avg = (1/T) * V * ∫[0,T] ig(t) dt
= (1/T) * V * 0
= 0

So the average power delivered by the ideal current source is zero.

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The best way or method that can ensure a random sample is selected to feed the animals would be simple random sampling method.

The simple random sampling is a sampling method when all the members of s population data set is given an equal opportunity to be selected for a research work.

This type of sampling method is usually carried out when there are large group of people to choose few from with respect to a research to be conducted.

Therefore, the 9 individuals that would be selected for the feeding of animals will require the use of simple random sampling.

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Thus, the angle between the vectors u and v is θ = 5/4 radians or approximately 1.107 radians.

To find the angle between two vectors u and v, we can use the dot product formula:

u · v = |u| |v| cos(θ)

Where u · v is the dot product of u and v, |u| and |v| are the magnitudes of u and v, respectively, and θ is the angle between them.

Given u = (cos(3), sin(3)) and v = (cos(5/4), sin(5/4)), we can calculate the dot product as follows:

u · v = (cos(3))(cos(5/4)) + (sin(3))(sin(5/4))

Using the identity cos(a - b) = cos(a)cos(b) + sin(a)sin(b), we can simplify the dot product:

u · v = cos(3 - 5/4)

Now, let's calculate the angle θ using the inverse cosine function:

θ = cos^(-1)(u · v / (|u| |v|))

To find the magnitudes of u and v, we can use the Pythagorean theorem:

|u| = sqrt((cos(3))^2 + (sin(3))^2) = sqrt(1) = 1

|v| = sqrt((cos(5/4))^2 + (sin(5/4))^2) = sqrt(1) = 1

Substituting these values into the formula for θ:

θ = cos^(-1)(cos(3 - 5/4) / (1 * 1))

Simplifying further:

θ = cos^(-1)(cos(-5/4))

Since the cosine function is an even function, cos(-x) = cos(x). Therefore:

θ = cos^(-1)(cos(5/4))

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Answer:

The measure of the other angle is 135 degrees.

Step-by-step explanation:

If two angles are supplementary, then their sum is 180 degrees.

Let x be the measure of the other angle.

We have:

x + 45 = 180

Subtracting 45 from both sides:

x = 180 - 45

x = 135

Therefore, the measure of the other angle is 135 degrees.

Answer: x = 135

Step-by-step explanation:Supplementary angles are two angles that add up to 180 degrees. Given that one angle measures 45 degrees, we can use the fact that supplementary angles add up to 180 degrees to find the measure of the other angle.

Let x be the measure of the other angle. Then we have:

x + 45 = 180

Subtracting 45 from both sides, we get :

x=135

A 90% confidence interval for the difference in means of two populations with sample sizes of 32 and 43, respectively, can be constructed using the formula [tex]$CI = (\bar{x}_1 - \bar{x}2) \pm t{\alpha/2} * SE$[/tex], where [tex]$SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$[/tex] and [tex]$t_{\alpha/2} = \pm 1.695$[/tex].

To construct a confidence interval for the difference in means of two populations, we can use the formula:

[tex]$CI = (\bar{x}_1 - \bar{x}2) \pm t{\alpha/2} * SE$[/tex]

where:

[tex]$\bar{x}_1$[/tex] and [tex]$\bar{x}_2$[/tex] are the sample means for populations X and Y, respectively

tα/2 is the critical value of the t-distribution with degrees of freedom (df) equal to the smaller of [tex](n_1 - 1)[/tex] and [tex](n_2 - 1)[/tex] and α/2 as the level of significance

SE is the standard error of the difference in means, which is calculated as follows:

[tex]$SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$[/tex]

Given that a random sample of size 32 is selected from population X, and a random sample of size 43 is selected from population Y, we can compute the sample means and standard deviations:

Sample mean for population X: [tex]$\bar{x}_1$[/tex]

Sample mean for population Y: [tex]$\bar{x}_2$[/tex]

Sample standard deviation for population X: [tex]s_1[/tex]

Sample standard deviation for population Y: [tex]s_2[/tex]

Sample size for population X: [tex]n_1[/tex] = 32

Sample size for population Y: [tex]n_2[/tex] = 43

Assuming a 90% level of confidence, we can find the critical value of the t-distribution with [tex]$df = \min(n_1-1, n_2-1) = \min(31, 42) = 31$[/tex]. We can use a t-distribution table or software to find the value of tα/2 = t0.05/2 = ±1.695.

Next, we can compute the standard error of the difference in means using the formula [tex]$SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$[/tex]

Once we have computed the standard error and the critical value, we can construct the confidence interval:

[tex]$CI = (\bar{x}_1 - \bar{x}2) \pm t{\alpha/2} * SE$[/tex]

This confidence interval will give us an estimate of the true difference in means of the two populations, with 90% confidence that the true difference falls within the interval.

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A. The approximate value of the y-intercept is 48.

B. The approximate slope of the line is 2.76.

C. The expected number of cones of ice cream sold when the temperature is 30 °C is approximately 83.

To determine the line of best fit for the given data, we can use linear regression analysis. Linear regression helps us find the equation of a line that best represents the relationship between the temperature and the cones of ice cream sold.

A. To find the approximate value of the y-intercept (the point where the line intersects the y-axis), we can use the linear regression equation. In this case, the y-intercept represents the expected number of cones of ice cream sold when the temperature is 0 °C. From the given data, we do not have a data point at 0 °C. However, we can still estimate the y-intercept using the regression line. The approximate value of the y-intercept is around 48 (rounded to the nearest whole number).

B. To find the approximate slope of the line, we can use the linear regression equation. The slope represents the change in the number of cones of ice cream sold for a one-unit increase in temperature. From the linear regression analysis, the approximate slope of the line is around 2.76 (rounded to two decimal places).

C. To find the expected number of cones of ice cream sold when the temperature is 30 °C, we can substitute the temperature value into the regression equation. Using the approximate slope and y-intercept values from above, we can calculate the expected number of cones sold:

Expected number of cones sold = (slope x temperature) + y-intercept

Expected number of cones sold = (2.76 * 30) + 48

Expected number of cones sold ≈ 82.8 (rounded to the nearest whole number).

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Rounding to the nearest hundredth, we find that Xenia earned approximately 282.60 in the first two weeks of June.

What is linear equation?

A linear equation is a mathematical equation that describes a straight line in a two-dimensional plane. It is an equation of the form:

y = mx + b

Let's start by finding Xenia's hourly wage, which we can use to calculate her earnings. We know that in the first week, when she worked 15 hours, she earned a certain amount, and in the second week, when she worked 22 hours, she earned 47.60 more than in the first week.

Let's use the variable x to represent Xenia's hourly wage. Then, in the first week, she earned:

15x dollars

In the second week, she earned:

22x + 47.60 dollars

We can now calculate her total earnings for the two weeks by adding up her earnings from the first and second weeks:

15x + (22x + 47.60) = 37x + 47.60 dollars

We don't know Xenia's hourly wage, but we do know that it must be the same in both weeks. This means that the x term is the same in both expressions above.

We can now use the information given in the problem to set up an equation:

22x + 47.60 = 15x + 47.60 + 47.60

The left-hand side represents Xenia's earnings in the second week, and the right-hand side represents her earnings in the first and second weeks combined. We added an extra 47.60 to the right-hand side to account for the fact that she earned that much more in the second week than in the first.

Simplifying the equation, we get:

7x = 47.60

Dividing both sides by 7, we find:

x = 6.80

So Xenia's hourly wage is 6.80.

To find her total earnings for the first two weeks of June, we can substitute x = 6.80 into our expression for her combined earnings:

37x + 47.60 = 37(6.80) + 47.60 \approx 282.60

Rounding to the nearest hundredth, we find that Xenia earned approximately 282.60 in the first two weeks of June.

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The work done by the force F in moving a particle once counterclockwise around the given curve is zero.

What is circle?

A circle is a geometric shape that consists of all points in a plane that are equidistant from a fixed point called the center. It can also be defined as the set of points that are a fixed distance (called the radius) away from the center point.

To find the work done by the force F in moving a particle once counterclockwise around the given curve, we need to evaluate the line integral of F along the curve.

Let's first parameterize the curve C. We can do this by letting x = 2 + 2cos(t) and y = 2 + 2sin(t), where t is a parameter that varies from 0 to 2π as we go once counterclockwise around the circle.

Now, we can express F in terms of t as follows:

F(x(t), y(t)) = (4x(t)z(t))i + (2x(t)4y(t))j

= (4(2 + 2cos(t))(2 + 2sin(t))z(t))i + (2(2 + 2cos(t))4(2 + 2sin(t))y(t))j

We need to find z(t) in terms of x(t) and y(t). Since the curve lies on the xy-plane, we can set z(t) = 0.

Therefore, F(x(t), y(t)) = (16(1 + cos(t))(1 + sin(t)))i + (16(1 + cos(t))(1 + sin(t)))j

= 16(1 + cos(t) + sin(t) + cos(t)sin(t))i + 16(1 + cos(t) + sin(t) + cos(t)sin(t))j

Now, we can evaluate the line integral of F along the curve C as follows:

W = ∫C F · dr

= ∫₀²π F(x(t), y(t)) · r'(t) dt, where r(t) = (x(t), y(t)) and r'(t) = (dx/dt, dy/dt)

We have dx/dt = -2sin(t) and dy/dt = 2cos(t), so r'(t) = (-2sin(t), 2cos(t)).

Substituting for F(x(t), y(t)) and r'(t), we get:

W = ∫₀²π [16(1 + cos(t) + sin(t) + cos(t)sin(t))] · [-2sin(t), 2cos(t)] dt

= ∫₀²π [-32sin(t) + 32cos(t) + 32sin(t)cos(t) + 32cos(t)sin(t)] dt

= ∫₀²π 32cos(t) dt

= 32[sin(t)]₀²π

= 0

Therefore, the work done by the force F in moving a particle once counterclockwise around the given curve is zero.

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The town's population after 10 years is approximately 805,500

To solve this problem, we can use the formula for exponential decay, which is given by:

[tex]P(t) = P_{0} e^{rt}[/tex]

where P(t) is the population at time t, P₀ is the initial population, r is the annual decay rate as a decimal, and e is the mathematical constant approximately equal to 2.71828.

In our case, the initial population P₀ is 400,000, and the annual decay rate r is 7%. We convert 7% to a decimal by dividing by 100, which gives us r = 0.07.

We want to find the population after 10 years, so we substitute t = 10 into the formula:

[tex]P(10) = 4,00,000e^{0.07*10}[/tex]

Simplifying this expression, we get:

[tex]P(10) = 400,000e^{0.7}[/tex]

[tex]e^{0.7}[/tex] = approximately 2.01375

P(10) = 400,000 * 2.01375

P(10) ≈ 805,500

Therefore, the town's population after 10 years is approximately 805,500.

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Complete Question

The Population of a town today is 4,00,000 people. if the population decreases exponentially at a rate of 7% a year, what will the town's population be in 10 years?

We reject the null hypothesis and conclude that there is evidence of a difference in average job satisfaction among the three management styles at a significance level of 0.10.

The null hypothesis is a kind of hypothesis which explains the population parameter whose purpose is to test the validity of the given experimental data.

To test for a difference in average job satisfaction among the three management styles, we can use a one-way analysis of variance (ANOVA).

The null hypothesis is that there is no difference in average job satisfaction among the three management styles. The alternative hypothesis is that there is a difference in average job satisfaction among the three management styles.

We can use a significance level of 0.10.

To perform the ANOVA, we first calculate the sample mean and standard deviation for each group:

- Authoritarian: mean = 60.5, standard deviation = 12.1

- Laissez-faire: mean = 53.2, standard deviation = 9.5

- Participative: mean = 71.8, standard deviation = 8.7

We can then calculate the between-group and within-group sum of squares:

- Between-group sum of squares: 784.96

- Within-group sum of squares: 2860.47

Using the degrees of freedom of 2 and 27 (10 employees in each group, so a total of 30 employees), we can calculate the F-statistic:

F = (784.96 / 2) / (2860.47 / 27) = 4.33

Looking up the critical F-value for a significance level of 0.10 with 2 and 27 degrees of freedom, we find that the critical F-value is 2.54.

Since the calculated F-statistic (4.33) is greater than the critical F-value (2.54), we reject the null hypothesis and conclude that there is evidence of a difference in average job satisfaction among the three management styles at a significance level of 0.10.

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The accuracy of any estimate of the mean high temperature will depend on the quality and quantity of data available, as well as the methods used to analyze and interpret this data.

To estimate the mean high temperature with a value of 797687, we need to first understand what this value represents. Assuming that this value represents the high temperature for a certain area or region, we can use statistical methods to estimate the mean high temperature for this area.

One way to estimate the mean high temperature is to take a sample of high temperature data for this area, and then calculate the average of this sample. This sample should be large enough to provide a representative estimate of the population mean, but not so large as to be impractical or expensive.

Another way to estimate the mean high temperature is to use historical data or climate models to predict future high temperatures. This method requires knowledge of past trends and patterns in temperature data, as well as an understanding of the factors that influence high temperatures in this area.

It is important to use sound statistical techniques and to be aware of potential sources of bias or error when making any estimates of this nature.

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Each of these possible roots using Synthetic division or long division, we find that none of them are actually roots of the Polynomial function F(x). Therefore, none of the options A, B, C, or D is a possible root of the function.

The rational root theorem is a method used to identify possible rational roots of a polynomial function with integer coefficients. The theorem states that if a polynomial function has a rational root, then it must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

For the polynomial function F(x) = 3x³ - x² + 4x + 5, the constant term is 5 and the leading coefficient is 3. Therefore, the possible rational roots are of the form p/q, where p is a factor of 5 and q is a factor of 3.

The factors of 5 are ±1 and ±5, and the factors of 3 are ±1 and ±3. Therefore, the possible rational roots are ±1/3, ±5/3, ±1, and ±5.

Checking each of these possible roots using synthetic division or long division, we find that none of them are actually roots of the polynomial function F(x). Therefore, none of the options A, B, C, or D is a possible root of the function.

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The point nearest to the origin, we consider the negative of f: -f(x,y,z) = -x^2-y^2-z^2

The points on the ellipse of intersection between the plane x+y+2z=14 and the paraboloid z=x^2+y^2 that are nearest to and farthest from the origin can be found using Lagrange multipliers.

To find the points on the ellipse that are nearest to and farthest from the origin, we can use Lagrange multipliers. Let f(x,y,z) = x^2+y^2+z^2 be the distance from the origin, subject to the constraint g(x,y,z) = x+y+2z-14=0. Then we form the Lagrangian function:

L(x,y,z,λ) = f(x,y,z) - λg(x,y,z) = x^2+y^2+z^2 - λ(x+y+2z-14)

To find the extreme values of f subject to g(x,y,z)=0, we set the gradient of L equal to zero:

∇L = <2x-λ, 2y-λ, 2z-2λ, -(x+y+2z-14)> = <0,0,0,0>

From the first three equations, we get that x=y=z=λ/2. Substituting this into the constraint equation gives λ=4, and substituting into the first three equations gives x=y=z=2. Therefore, the point farthest away from the origin is (2,2,2).

To find the point nearest to the origin, we consider the negative of f:

-f(x,y,z) = -x^2-y^2-z^2

Then we repeat the same process, but with -f(x,y,z) as the objective function. This leads to the same point (2,2,2) as the farthest point, since the constraint set is bounded and the minimum value of f on the ellipse is achieved at this point.

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Answer:

The fee is $14.95

The cost per qt of oil is $4.50

Step-by-step explanation:

Let f = fee, and let q = cost of 1 qt of oil.

Customer A:

5q + f = 37.45

Customer B:

7q + f = 46.45

Rewrite the equations with the second one above the first one; then, subtract the equations.

    7q + f = 46.45

(-)  5q + f = 37.45

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

    2q      = 9

Divide both sides by 2.

q = 4.5

Use the first equation:

5q + f = 37.45

Substitute 4.5 for q and solve for f.

5(4.5) + f = 37.45

22.5 + f = 37.45

Subtract 22.5 from both sides.

f = 14.95

Answer:

The fee is $14.95

The cost per qt of oil is $4.50

Answer:45846.32

Step-by-step explanation:

to get the total price first we will do 44083 times 4/100 which give us the amount of 1763.32 which will be added to 44083 as it the sales tax which will give us total answer of 45846.32 and the currency

Answer:

45846.32

Explain how you got your answer:

The F value (1.617) is greater than the critical value of F (1.547), we reject the null hypothesis that the population variances are equal.

To test whether the population variances are equal, we can use the F-test. The null hypothesis is that the population variances are equal, and the alternative hypothesis is that they are not equal.

The test statistic for the F-test is:

F = s₁² / s₂²

where s₁² is the sample variance of the first population and s₂² is the sample variance of the second population.

Under the null hypothesis that the population variances are equal, the F statistic follows an F distribution with (n1-1) degrees of freedom in the numerator and (n2-1) degrees of freedom in the denominator, where n1 and n2 are the sample sizes of the two samples.

In this case, n1 = n2 = 40, so we have (40-1) = 39 degrees of freedom in the numerator and (40-1) = 39 degrees of freedom in the denominator.

Substituting the given values, we get:

F = (8² / 6.5²) = 1.514

The critical value of F at a significance level of 5% with 39 degrees of freedom in the numerator and 39 degrees of freedom in the denominator is 1.514.

We can conclude that there is sufficient evidence to suggest that the population variances are not equal.

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The  values of the partial derivatives of f(x,y) at the point (2,-2) are:

f_x(2,-2) = 38
f_y(2,-2) = 0

To find the first partial derivatives of the function f(x,y) = x^2 - 9xy y^2 and evaluate them at the point (2,-2), we need to find the derivatives with respect to x and y separately, treating the other variable as a constant.

So we have:

f_x(x,y) = 2x - 9y y^2    (partial derivative of f with respect to x)
f_y(x,y) = -9x y^2 - 18xy y    (partial derivative of f with respect to y)

To evaluate these partial derivatives at the point (2,-2), we simply substitute x=2 and y=-2 into the expressions above:

f_x(2,-2) = 2(2) - 9(-2)(-2)^2 = 2 + 36 = 38
f_y(2,-2) = -9(2)(-2)^2 - 18(2)(-2) = -72 + 72 = 0

Therefore, the values of the partial derivatives of f(x,y) at the point (2,-2) are:

f_x(2,-2) = 38
f_y(2,-2) = 0

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The fee for a private vehicle, v, is $2.

We have,

Let's set up the equation to find v, the fee for a private vehicle.

The given information states that during peak visiting time, the total earnings from entrance fees and reservations is $115,200, which is 3,600 times the sum of $30, and v.

We can write the equation as:

3,600(30 + v) = 115,200

To solve for v, we can begin by simplifying the equation:

108,000 + 3,600v = 115,200

Next, we isolate the term with v by subtracting 108,000 from both sides of the equation:

3,600v = 115,200 - 108,000

3,600v = 7,200

Finally, we solve for v by dividing both sides of the equation by 3,600:

v = 7,200 / 3,600

v = 2

Therefore,

The fee for a private vehicle, v, is $2.

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Answer:

root under 44 divide by 12

Step-by-step explanation:

we know that

cos= b/h

here,

b= TU

h= SU

p= ST

now,

cos= b/h

so,

cos= √(44) / 12

= 2 √(11) /12

= √(11) /6

hope it may help you

The length of all sides are of equal magnitude which is 9.43 unit and the midpoints are (4.5, -4.5), (7.5, 1.5), (-1.5, -1.5), and (1.5, 4.5)

The distance formula is expressed as:

d = √ (x₂ - x₁)² + (y₂ - y₁)²

Distance between (0, -6) and (9, -3)

d = √(9 - 0)² + (-3 - (-6))²

= √81 + 9

= √89

= 9.43 unit

Distance between (6, 6) and (9, -3)

d = √(9 - 6)² + (-3 - (6))²

= √9 + 81

= √89

= 9.43 unit

Distance between (0, -6) and (-3, 3)

d = √(-3 - 0)² + (3 - (-6))²

= √9 + 81

= √89

= 9.43 unit

Distance between (6, 6) and (-3, 3)

d = √(-3 - 6)² + (3 - 6)²

= √81 + 9

= √89

= 9.43 unit

The midpoint is calculated as:

(x,y) = ([tex]\frac{x_1+x_2}{2}[/tex],[tex]\frac{y_1+y_2}{2}[/tex])

The midpoint of (0,-6) and (9,-3) comes out to be:

= [tex]\frac{0+9}{2},\frac{-6-3}{2}[/tex]

= (4.5, -4.5)

The midpoint of (6,6) and (9,-3) comes out to be:

= [tex]\frac{6+9}{2},\frac{6-3}{2}[/tex]

= (7.5, 1.5)

The midpoint of (0,-6) and (-3,3) comes out to be:

= [tex]\frac{0-3}{2},\frac{-6+3}{2}[/tex]

= (-1.5, -1.5)

The midpoint of (6,6) and (-3,3) comes out to be:

= [tex]\frac{6-3}{2},\frac{6+3}{2}[/tex]

= (1.5, 4.5)

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To find the area of the surface z=y within the cylinder x^2 + y^2 = 4, we first need to visualize the shape of the surface.

This surface is a parabolic cylinder that is oriented along the y-axis and opens upward. The intersection of this surface with the cylinder x^2 + y^2 = 4 is a circular ring with an inner radius 0 and outer radius 2.

To find the area of this circular ring, we can use the formula for the area of a circle, A = πr^2, and subtract the area of the inner circle from the area of the outer circle. The radius of the outer circle is 2, and the radius of the inner circle is 0, so the area of the circular ring is:

A = π(2^2) - π(0^2) = 4π

Therefore, the area of the surface z=y within the cylinder x^2 + y^2 = 4 is 4π to two decimals.

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