When given an inverse variation, how do you find k?

After 18 years, the account balance will be calculated based on a $5000 deposit with a 4.7% interest compounded.

To calculate the account balance after 18 years, we will use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Final account balance
P = Principal amount (initial deposit)
r = Annual interest rate (in decimal form)
n = Number of times interest is compounded per year
t = Number of years
In this case, the principal amount is $5000, the annual interest rate is 4.7% (or 0.047 as a decimal), the interest is compounded annually (n = 1), and the time period is 18 years (t = 18).
Using the formula, we can calculate the account balance:
A = $5000(1 + 0.047/1)^(1*18)
= $5000(1 + 0.047)^18
= $5000(1.047)^18
≈ $5000 * 1.990
≈ $9949.92
Therefore, after 18 years, the account balance will be approximately $9949.92.

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The given integral can be expressed in terms of simpler integrals as:

[tex]\int (−3x^2 + 5x + 6x cos(x)) dx = -x^3 + (5/2)x^2 + 6x sin(x) + 6 cos(x) + C[/tex](

To express the given integral in terms of simpler integrals, we can use the properties of the indefinite integral, including the linearity property and integration by parts.

We can first break down the integrand using linearity:

[tex]\int (−3x^2 + 5x + 6x cos(x)) dx = \int (-3x^2) dx + \int (5x) dx + \int (6x cos(x)) dx[/tex]

Now, we can integrate each term separately:

[tex]\int (-3x^2) dx = -x^3 + C1[/tex] (where C1 is the constant of integration)

[tex]\int (5x) dx = (5/2)x^2 + C2[/tex] (where C2 is another constant of integration)

To integrate ∫(6x cos(x)) dx, we can use integration by parts with u = 6x and dv = cos(x) dx:

∫(6x cos(x)) dx = 6x sin(x) - ∫(6 sin(x)) dx

= 6x sin(x) + 6 cos(x) + C3 (where C3 is another constant of integration)

Putting everything together, we have:

[tex]\int (−3x^2 + 5x + 6x cos(x)) dx = -x^3 + C1 + (5/2)x^2 + C2 + 6x sin(x) + 6 cos(x) + C3[/tex]

So the given integral can be expressed in terms of simpler integrals as:

[tex]\int (−3x^2 + 5x + 6x cos(x)) dx = -x^3 + (5/2)x^2 + 6x sin(x) + 6 cos(x) + C[/tex](where C = C1 + C2 + C3 is the overall constant of integration)

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The procedure that should be conducted to directly address this research question is the one sample proportion z test. This is because the research question is about the proportion of ring-tailed lemurs that are left hand dominant, which is a categorical variable. The sample size is greater than 30, so the central limit theorem can be applied and the distribution of the sample proportion can be assumed to be approximately normal. Therefore, a one sample proportion z test can be used to test whether the proportion of left hand dominant ring-tailed lemurs is greater than 0.5.

The one sample proportion z test is a statistical test used to determine whether a sample proportion is significantly different from a hypothesized population proportion. This test requires a categorical variable and a sample size greater than 30 in order to apply the central limit theorem and assume normality of the distribution of the sample proportion. The test statistic is calculated by subtracting the hypothesized population proportion from the sample proportion and dividing by the standard error of the sample proportion.

To directly address the research question of whether more than half of all ring-tailed lemurs are left hand dominant, a one sample proportion z test should be conducted. This test is appropriate for a categorical variable with a sample size greater than 30 and assumes normality of the distribution of the sample proportion. The test will determine whether the proportion of left hand dominant ring-tailed lemurs is significantly different from 0.5, which is the null hypothesis.

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The predicted clutch engagement time is approximately 1.81 seconds when the starting speed is 18 m/s, the maximum drive torque is 17 N.m, the system inertia is 0.006 kg•m2, and the applied force rate is 10 kN/s.

The given regression model for predicting clutch engagement time (y) based on four predictor variables (x1, x2, x3, x4) is:

[tex]y = -0.83 + 0.017x1 + 0.0895x2 + 42.771x3 + 0.027x4 - 0.0043x2x4[/tex]

To predict the clutch engagement time when x1 = 18 m/s, x2 = 17 N.m, x3 = 0.006 kg•m2, and x4 = 10 kN/s, we simply substitute these values into the regression equation:

[tex]y = -0.83 + 0.017(18) + 0.0895(17) + 42.771(0.006) + 0.027(10) - 0.0043(17)(10)\\y = -0.83 + 0.306 + 1.5215 + 0.256626 + 0.27 - 0.731[/tex]

y = 1.809126

Therefore, the predicted clutch engagement time is approximately 1.81 seconds when the starting speed is 18 m/s, the maximum drive torque is 17 N.m, the system inertia is 0.006 kg•m2, and the applied force rate is 10 kN/s.

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The product as a sum or difference is:

1) 16 sin(28x) sin(11x) = 8[cos(17x) - cos(39x)]
2) sin(-x) sin(9x) = ([tex]\frac{1}{2}[/tex])[cos(-10x) - cos(8x)]

1) 16 sin(28x) sin(11x)
We can use the Product-to-Sum identity: sin(A)sin(B) = (1/2)[cos(A-B) - cos(A+B)]
So, 16 sin(28x) sin(11x) can be rewritten as:
8[cos(28x - 11x) - cos(28x + 11x)] = 8[cos(17x) - cos(39x)]
2) sin(-x) sin(9x)
Again, we use the Product-to-Sum identity: sin(A)sin(B) = ([tex]\frac{1}{2}[/tex])[cos(A-B) - cos(A+B)]
So, sin(-x) sin(9x) can be rewritten as:
([tex]\frac{1}{2}[/tex])[cos(-x - 9x) - cos(-x + 9x)] = ([tex]\frac{1}{2}[/tex])[cos(-10x) - cos(8x)]

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the right answer on this question is 7,9

Thus, list the elements in the set (d ∪ e) ∩ F is {4, 6, 7, 9}.

To find the elements in the set (d ∪ e) ∩ F, we first need to determine what the union of d and e is.

Given that:

d={4,7,9}, e={4,6,7,8} and f={3,5,6,7,9}.

The union of two sets, denoted by the symbol ∪, is the set of all elements that are in either one or both of the sets.

So, in this case, d ∪ e would be the set {4, 6, 7, 8, 9}.

Next, we need to find the intersection of the set {4, 6, 7, 8, 9} and f.

The intersection of two sets, denoted by the symbol ∩, is the set of all elements that are in both sets.

So, the elements in the set (d ∪ e) ∩ F would be the elements that are common to both {4, 6, 7, 8, 9} and {3, 5, 6, 7, 9}. These elements are 4, 6, 7, and 9.

Therefore, the answer to the question is (d ∪ e) ∩ F = {4, 6, 7, 9}.

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The triangle in the image is a right triangle. We are given a side and an angle, and asked to find another side. Therefore, we should use a trigonometric function.

Trigonometric Functions: SOH-CAH-TOA

---sin = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent

In this problem, looking from the angle, we are given the adjacent side and want to find the opposite side. This means we should use the tangent function.

tan(40) = x / 202

x = tan(40) * 202

x = 169.498

x (rounded) = 169 meters

Answer: the tower is 169 meters tall

Hope this helps!

If `f(x)` is a polynomial, then `f(x²)` is also a polynomial. Polynomials are mathematical expressions that consist of variables and coefficients with only the operations of addition, subtraction, multiplication, and non-negative integer exponents. We can prove this statement using the definition of a polynomial. Definition of a polynomial polynomial is an expression that can be written as follows:$$f(x)= a_nx^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdot\cdot\cdot +a_1x+a_0$$where `a0, a1, …, an` are constants, and `n` is a non-negative integer. This definition of the polynomial can be used to show that `f(x²)` is also a polynomial. Using the definition of a polynomial, we can write:$$f(x²)= a_n(x²)^n+a_{n-1}(x²)^{n-1}+a_{n-2}(x²)^{n-2}+\cdot\cdot\cdot +a_1(x²)+a_0$$Simplifying the terms of the expression, we get:$$f(x²)= a_nx^{2n}+a_{n-1}x^{2(n-1)}+a_{n-2}x^{2(n-2)}+\cdot\cdot\cdot +a_1x^2+a_0$$This proves that `f(x²)` is also a polynomial. Therefore, if `f(x)` is a polynomial, then `f(x²)` is also a polynomial.

Yes, if f(x) is a polynomial, then f(x²) is also a polynomial.

A polynomial is a mathematical expression consisting of variables, coefficients, and non-negative integer exponents. It can include addition, subtraction, and multiplication operations. The terms in a polynomial can be in the form of axⁿ, where a is the coefficient, x is the variable, and n is a non-negative integer exponent.

When we substitute x² into f(x), each occurrence of x in the polynomial f(x) is replaced by x². Since x² is still a variable with a non-negative integer exponent, the resulting expression f(x²) remains a polynomial. The coefficients and exponents may change, but the essential structure of a polynomial is preserved.

Therefore, if f(x) is a polynomial, then f(x²) is also a polynomial.

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Answer:

Step-by-step explanation:

From top to bottom:  T (true), F (false)

T

F

T  51/109 x 100 = 47%

F  (49 + 58)/221 x 100 = 48%

F  109 < 112

The number of distinct ways to arrange 3 yellow marbles, 5 blue marbles, and 5 green marbles in a row will be 5625.

A permutation is an act of arranging items or elements in the correct order.

There are 3 yellow marbles, 5 blue marbles, and 5 green marbles.

The number of distinct ways to arrange 3 yellow marbles, 5 blue marbles, and 5 green marbles in a row will be

[tex]\Rightarrow (3 \times 5 \times 5)^2[/tex]

[tex]\Rightarrow 75^2[/tex]

[tex]\Rightarrow 5625[/tex]

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(a) The change is -5.403 million enplaned passengers.

The number of enplaned passengers on the given airline decreased from 98.175 million in 2007 to 92.772 million in 2008, resulting in a decrease of 5.403 million enplaned passengers.

(b) The percentage change is -5.51%.

The percentage change is calculated using the formula: ((new value - old value) / old value) x 100%. In this case, the percentage change is ((92.772 - 98.175) / 98.175) x 100% = -5.51%. This indicates a 5.51% decrease in the number of paying passengers on the given airline between 2007 and 2008.

(c) The average rate of change is -2.702 million enplaned passengers per year.

The average rate of change is calculated by dividing the total change in the number of enplaned passengers by the number of years between 2007 and 2008. In this case, the average rate of change is (-5.403 / 2) = -2.702 million enplaned passengers per year.

This means that the number of paying passengers on the given airline decreased by an average of 2.702 million per year between 2007 and 2008.

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Here are the masses of some planets and moons expressed in both standard and scientific notation:

Planet Mass in Standard NotationMass in Scientific Notation:

Venus = 4,870,000,000,000,000,000,000,000 kg4.87 × 10²⁴ kg

Earth = 5,970,000,000,000,000,000,000,000 kg5.97 × 10²⁴ kg

Mars = 6,420,000,000,000,000,000,000,000 kg6.42 × 10²⁴ kg

Jupiter = 1,898,000,000,000,000,000,000,000,000 kg1.90 × 10²⁷ kg

Saturn = 568,000,000,000,000,000,000,000,000 kg5.68 × 10²⁶ kg

Uranus = 86,800,000,000,000,000,000,000 kg8.68 × 10²⁵ kg

Neptune = 102,000,000,000,000,000,000,000 kg1.02 × 10²⁶ kg

Moon = 7,340,000,000,000,000,000 kg7.34 × 10²² kg

Io = 8,930,000,000,000,000,000 kg8.93 × 10²² kg

Ganymede = 1,480,000,000,000,000,000,000 kg1.48 × 10²³ kg

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The statement given "5 is in {1, 2, 3, 4, 5}" is true because 5 is included in  the set given {1, 2, 3, 4, 5}.

In set notation, the curly brackets {} represent a set. The set {1, 2, 3, 4, 5} contains the elements 1, 2, 3, 4, and 5. So, when we check if 5 is in this set, we find that it is indeed present. Therefore, the statement is true. Option A is the correct answer.

A set is an unordered collection of unique elements. In this case, the set {1, 2, 3, 4, 5} includes the numbers 1, 2, 3, 4, and 5. When we check if the number 5 is in this set, we find that it is one of the elements in the set. Thus, the statement "5 is in {1, 2, 3, 4, 5}" is true.

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The probability that the mean of the sample will be less than 35 is approximately 0.0205, or 2.05%.

To solve this problem, we'll use the central limit theorem, which states that for a large enough sample size, the distribution of sample means approximates a normal distribution, regardless of the shape of the population distribution.

Given that the population follows a uniform distribution with α = 24 and β = 48, we know that the mean (μ) of the population is given by the formula:

μ = (α + β) / 2

Substituting the values, we have:

μ = (24 + 48) / 2 = 72 / 2 = 36

The standard deviation (σ) of the population is given by the formula:

σ = (β - α) / √12

Substituting the values, we have:

σ = (48 - 24) / √12 = 24 / √12 = 24 / 3.464 = 6.928

According to the central limit theorem, the distribution of sample means follows a normal distribution with a mean equal to the population mean (μ) and a standard deviation equal to the population standard deviation (σ) divided by the square root of the sample size (n). Therefore:

μ_s = μ = 36

σ_s = σ / √n = 6.928 / √200 ≈ 0.490

To find the probability that the mean of the sample will be less than 35, we need to find the area under the normal distribution curve to the left of 35. We'll use a standard normal distribution with a mean of 0 and a standard deviation of 1, and then transform it using the mean and standard deviation of the sample distribution.

Let's calculate the z-score for 35:

z = (x - μ_s) / σ_s = (35 - 36) / 0.490 ≈ -2.041

Using a standard normal distribution table or a calculator, we can find the probability corresponding to a z-score of -2.041. The probability that the mean of the sample will be less than 35 is approximately 0.0205, or 2.05%.

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The value of tobt for the test of the significance of r is 3.10 option B.

To find the value of tobt for the test of the significance of r, we can use the formula:

tobt = (r * √(N - 2)) / √(1 - r²)

Given r = 0.84 and N = 6, we can plug the values into the formula:

tobt = (0.84 * √(6 - 2)) / √(1 - 0.84²)

tobt = (0.84 * √4) / √(1 - 0.7056)

tobt = (0.84 * 2) / √0.2944

tobt = 1.68 / 0.542

tobt ≈ 3.10

The answer is (B) 3.10.

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First you need to identify the type of equation in the table, then you can set up the correspondent equation or system of equations to find your equation.

To find an equation from a table, you will need to identify the pattern or relationship between the given inputs and outputs (so the first thing you need to do, is identify which type of equation is represented by the table)

There are different methods depending on the type of relationship and the data provided. Here are a few common approaches:

Linear Relationship (y = ax + b)

If the table data suggests a linear relationship between the inputs (x-values) and outputs (y-values), you can use the method of finding the equation of a straight line. This can be done by calculating the slope (m) and the y-intercept (b) using two data points from the table.

Quadratic Relationship (y = ax² + bx + c)

If the table data suggests a quadratic relationship, meaning the outputs change according to a quadratic function of the inputs, you can use the method of finding the equation of a quadratic function. This involves using three data points from the table and solving a system of equations to determine the coefficients of the quadratic equation.

Exponential Relationship (y = A*bˣ)

If the table data suggests an exponential relationship, where the outputs change exponentially with respect to the inputs, you can use the method of finding the equation of an exponential function. This involves determining the base and exponent of the exponential function by examining the ratios between the outputs.

Please notice that these are only 3 types of equations, but there are a lot more, like logarithmic functions, trigonometric functions, cubic functions.

And each one will have a different way of setting up equations to find the equation represented in the table.

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Sophie needs approximately 14.82 ounces of flour to bake her cake .

To convert grams to ounces, we can use the conversion factor that 1 ounce is approximately equal to 28.35 grams . The mass m in grams (g) is equal to the mass m in ounces (oz) times 28.34952

1 ounces = 28.35 gram

So, to find the number of ounces of flour Sophie needs, we can divide the weight in grams by the conversion factor .

420 g × 1 ounces / 28.35 g

420 g / 28.35 g = 14.82 ounces

Therefore, Sophie needs approximately 14.82 ounces of flour to bake her cake .

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A median in a right triangle joining the right angle to the hypothenuse has the same length as the segment connecting the midpoints of the legs.

The median equals half the hypotenuse

In triangle ABC where ∠B = 90° BD is median

AD = DC median divides into two equal part

DX ⊥ BC

BX = XC = BC/2

DX = AB/2

By Pythagorean theorem

BD² = DX² + BX²

BD² = BC²/4 + AB²/4

BD² = AC²/4

BD = AC/2

Now in triangles BXD and DXC

DX = DX ( common )

AB║ DX

∠BXD = ∠DXC (as corresponding angles )

BX = XC (corresponding side)

By SAS congruency

ΔBXD ≅ ΔDXC

BD = DC

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Answer:

Step-by-step explanation:

Answer:

A. 1,800≤250+2i .

Step-by-step explanation:

Answer:

$0.16

Step-by-step explanation:

$4.88 is how much 8 apple juice boxes cost

to find out how many 1 costs we divide by 8

$4.88÷8=$0.61

so 1 apple juice box costs $0.61

This is closest to option d) 343 cm³,  The volume of the cube is 343 cm³. which is the correct answer.

The volume of a cube is given by the formula [tex]V = s^3,[/tex] where s is the length of any side of the cube. In this case, the height, length, and width are all equal to 7 cm. Thus, the length of any side of the cube is also 7 cm.

Substituting s = 7 cm into the formula for the volume of a cube, we get:

V = s^3 = 7^3 = 343 cm³

Therefore, the volume of the cube is 343 cm³. This is closest to option d) 343 cm³, which is the correct answer.

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The length of the enlarged pencil will be 29.75 units.The original length of the pencil is 7 units, and the width is 1.5 units. The scale factor is 425%.

We need to find the new length of the pencil after it is enlarged by the given scale factor of 425%.

The formula for calculating the new length of the pencil is:New Length of Pencil = Original Length × Scale Factor/100 Adding the given values in the above formula,

To find the length of the enlarged pencil, we need to multiply the original length by the scale factor.

The scale factor is given as 425%, which can be written as a decimal as 4.25.

Length of enlarged pencil = Original length * Scale factor

= 7 units * 4.25

= 29.75 units

Therefore, the length of the enlarged pencil will be 29.75 units.

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The answer of the given question based on the triangle is , - 15.75 ,  this is not possible as the length cannot be negative.

We are given:

In ΔCDE, the measure of ∠E = 90°, CD = 9.2 feet, and DE = 8.3 feet.

To find:

The measure of ∠C to the nearest tenth of a degree.

Solution:

In ΔCDE, applying Pythagoras theorem:

CE² + CD² = DE²CE² + (9.2)² = (8.3)²

CE² = (8.3)² - (9.2)²CE²

= 68.89 - 84.64CE²

= - 15.75

This is not possible as the length cannot be negative.

Hence, the given values are not possible.

So, there is no such triangle ΔCDE, which satisfies the given conditions.

Hence, we cannot find the measure of ∠C.

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The limit as a definite integral on the given interval is lim n→∞ nΣi=1n exi* Δxi = ∫0⁹ ex dx = e⁹ - 1.

To express the limit as a definite integral on the given interval, use the definition of a Riemann sum:

lim n→∞ Σi=1n f(xi*) Δxi = ∫aᵇ f(x) dx

where f(x) = ex, a = 0, b = 9, and Δx = (b - a)/n = 9/n. Also, xi* = point in the i-th subinterval [xi-1, xi], where xi = a + iΔx.

Substituting the values:

lim n→∞ Σi=1n exi* Δxi = ∫0⁹ ex dx

Integrating:

lim n→∞ Σi=1n exi* Δxi = [ex]0⁹ = e⁹ - 1

Therefore, the limit as a definite integral on the given interval is:

lim n→∞ nΣi=1n exi* Δxi = ∫0⁹ ex dx = e⁹ - 1

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Answer:

To eliminate x, you need a positive coefficient in front of x for one equation and its negative counterpart in front of the other equation as a positive number plus its negative opposite equals 0 (e.g., -4 + 4 = 0 and -80 + 80 = 0)

Step 1:  Therefore, we can eliminate x by first determining the least common multiple (LCM) between 5 and 4.  We know that 5 * 4 = 20 and 4 * 5, so the LCM between 5 and 4 is 20.

Step 2:  In order to have 20 as coefficient for x in one equation and -20 for x as a coefficient in the other equation, we can multiply the entire first equation by 4 and the entire second equation by -5:

Equation 1 multiplied by 4:  4 * (5x - 2y = 10) = 20x - 8y = 40

Equation 2 multiplied by -5:  -5* (4x - 3y = 15) = -20x + 15y = -75

Step 3:  Adding the two equations shows that the xs cancel as 20x - 20x = 0, leaving us with 15y - 8y = 40 - 75, which simplifies to 7y = -35

Answer: See below.

Step-by-step explanation:

       First, we are already given these equations in standard form.

5x − 2y = 10

4x − 3y = 15

       Next, we need to make the coefficients of the x variables opposites (as in 5 and -5, etc), since we want to eliminate the x's. To do this, we will find a common multiple (here, the Lowest Common Multiplb is 20). Then, we will multiply every term by the number that makes the coefficient of x our common multiple.

       We will make the first equation with a coefficient of 20 for the x and the second with a coefficient of -20 for the x.

       See this visually below.

5x − 2y = 10 ➜ 4(5x) − 4(2y) = 4(10) ➜ 20x - 8y = 40

4x − 3y = 15 ➜ -5(4x) − -5(3y) = -5(15) ➜ -20x + 15y = -75

       Lastly, add these two equations together. The x's are eliminated. This also will let us solve for y.

      20x - 8y = 40

+   -20x + 15y = -75

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

7y = -35

y = -5

Starting with the given equation F sin(θ) = 24/26, we can use trigonometric identities to find expressions for cos(θ), tan(θ), and sec(θ).

First, we square both sides of the equation to get:

F^2 sin^2(θ) = (24/26)^2

Then, we use the identity sin^2(θ) + cos^2(θ) = 1 to solve for cos(θ):

cos^2(θ) = 1 - sin^2(θ)

cos^2(θ) = 1 - (24/26)^2

cos(θ) = ± √(1 - (24/26)^2)

Since 0 ≤ θ ≤ π/2, we know that cos(θ) must be positive, so we take the positive square root:

cos(θ) = √(1 - (24/26)^2)

Next, we can use the fact that tan(θ) = sin(θ)/cos(θ) to find an expression for tan(θ):

tan(θ) = sin(θ)/cos(θ)

tan(θ) = (F sin(θ))/cos(θ)

tan(θ) = (F sin(θ))/√(1 - (24/26)^2)

Finally, we can use the fact that sec(θ) = 1/cos(θ) to find an expression for sec(θ):

sec(θ) = 1/cos(θ)

sec(θ) = 1/√(1 - (24/26)^2)

So, in summary, we have:

cos(θ) = √(1 - (24/26)^2)

tan(θ) = (F sin(θ))/√(1 - (24/26)^2)

sec(θ) = 1/√(1 - (24/26)^2)

Note that we cannot simplify these expressions any further without more information about the value of F.

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The parametric equations for the ellipse x^2/2^2 + y^2/4^2 = 1, traced clockwise as the parameter increases, are:
x = 2cos(t)
y = -2sin(t)

To find the corresponding y-value for a given x-value on the ellipse, we can rearrange the equation:

x^2/2^2 + y^2/4^2 = 1
y^2/4^2 = 1 - x^2/2^2
y^2 = 4^2(1 - x^2/2^2)
y = ±2sqrt(1 - x^2/2^2)

Since the curve is traced clockwise as the parameter t increases, we can set x = 2cos(t) and y = -2sqrt(1 - x^2/2^2) to trace the lower half of the ellipse:

x = 2cos(t)
y = -2sqrt(1 - (2cos(t))^2/2^2)
y = -2sqrt(1 - cos^2(t))

Using the identity sin^2(t) + cos^2(t) = 1, we can solve for sin(t):

sin^2(t) = 1 - cos^2(t)
sin(t) = ±sqrt(1 - cos^2(t))

Since we want the negative value to trace the lower half of the ellipse, we have:
y = -2sin(t)

Therefore, the parametric equations for the ellipse x^2/2^2 + y^2/4^2 = 1, traced clockwise as the parameter increases, are:
x = 2cos(t)
y = -2sin(t)

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The matrix A:  [[0, 2], [2, 1]] has two eigen value i.e. λ1 = (1 + sqrt(17))/2,

λ2 = (1 - sqrt(17))/2 and their eigen values are [2/(1 + sqrt(17)), 1] , [2/(1 - sqrt(17)), -1] respectively and similarly the eigen value of the matrix

A^-1 is λ1 = (1 + sqrt(3))/2 ,  λ2 = (1 - sqrt(3))/2 and their eigen vector is

[2/(1 + sqrt(17)), 1] and [2/(1 - sqrt(17)), -1] respectively and the trace of the matrix  A and A-1 is 1 and 1/2 respectively.

To compute the eigenvalues and eigenvectors of matrix A, we need to solve the characteristic equation det(A - λI) = 0, where I is the 2x2 identity matrix.

STEP 1:-This gives us:

det(A - λI) = (0 - λ)(1 - λ) - 4 = λ^2 - λ - 4 = 0

Using the quadratic formula, we can solve for the eigenvalues:

λ1 = (1 + sqrt(17))/2

λ2 = (1 - sqrt(17))/2

STEP 2 :-To find the eigenvectors, we can solve the system of equations (A - λI)x = 0 for each eigenvalue. This gives us:

For λ1:

-λ1x1 + 2x2 = 0

2x1 - (λ1 - 1)x2 = 0

Solving this system, we get the eigenvector [2/(1 + sqrt(17)), 1].

For λ2:

-λ2x1 + 2x2 = 0

2x1 - (λ2 - 1)x2 = 0

Solving this system, we get the eigenvector [2/(1 - sqrt(17)), -1].

STEP 3:-

To compute the eigenvalues and eigenvectors of matrix A^-1, we need to solve the characteristic equation det(A^-1 - λI) = 0. We can simplify this expression using the fact that det(A^-1) = 1/det(A), which gives us:

det(A^-1 - λI) = (1/2 - λ)(-λ) - (1/2)(1) = -λ^2 + (1/2)λ - (1/2) = 0

Using the quadratic formula, we can solve for the eigenvalues:

λ1 = (1 + sqrt(3))/2

λ2 = (1 - sqrt(3))/2

We can see that A^-1 has the same eigenvectors as A, since the equation (A - λI)x = 0 is equivalent to A^-1(Ax - λx) = 0. Therefore, the eigenvectors of A^-1 are [2/(1 + sqrt(17)), 1] and [2/(1 - sqrt(17)), -1].

We can also check that the trace of A is equal to the sum of its eigenvalues, and the trace of A^-1 is equal to the sum of its eigenvalues. We have:

trace(A) = 0 + 1 = 1

trace(A^-1) = 1/2 + 0 = 1/2

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The extreme values of f(x,y) can be expressed in terms of the marginal cumulative distribution functions f_x(x) and f_y(y) using the formulas above.

To express the extreme values of f(x,y) in terms of the marginal cumulative distribution functions f_x(x) and f_y(y), we can use the following formulas:

f(x,y) = (d^2/dx dy) F(x,y)

where F(x,y) is the joint cumulative distribution function of X and Y, and

f_x(x) = d/dx F(x,y)

and

f_y(y) = d/dy F(x,y)

are the marginal cumulative distribution functions of X and Y, respectively.

To find the maximum value of f(x,y), we can differentiate f(x,y) with respect to x and y and set the resulting expressions equal to zero. This will give us the critical points of f(x,y), and we can then evaluate f(x,y) at these points to find the maximum value.

To find the minimum value of f(x,y), we can use a similar approach, but instead of setting the derivatives of f(x,y) equal to zero, we can find the minimum value by evaluating f(x,y) at the corners of the rectangular region defined by the range of X and Y.

Therefore, the extreme values of f(x,y) can be expressed in terms of the marginal cumulative distribution functions f_x(x) and f_y(y) using the formulas above.

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The distance from the point (1,2) to the line 4x - 3y = 0 is 2/5 units.

To find the distance between a point and a line, we need to use the formula:

distance = |ax + by + c| / √(a^2 + b^2)

where a, b, and c are the coefficients of the equation of the line in the form ax + by + c = 0. In this case, the equation of the line is 4x - 3y = 0, so a = 4, b = -3, and c = 0.

To apply the formula, we need to find the values of x and y that correspond to the point (1,2) when they are plugged into the equation of the line. Solving for y in terms of x, we get:

4x - 3y = 0

-3y = -4x

y = (4/3)x

Now we can plug in the coordinates of the point (1,2) and find the distance:

distance = |4(1) - 3(2) + 0| / √(4^2 + (-3)^2)

= |-2| / √(16 + 9)

= 2 / √25

= 2/5

Therefore, the distance from the point (1,2) to the line 4x - 3y = 0 is 2/5 units.

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