Alex scored 40 marks out of 60 in a spelling test pet scored 65% in the same test who did better

We obtain the expression for i(t) as i(t) = [tex]10[/tex][tex]e^{(-t/2)}[/tex] [(5/3)sin(√3t/2) + (5/3)cos(√3t/2)] and A = 10, B = 1, C = 5/3, and D = 1/2.

To find i(t) using Laplace transform, we first need to find the Laplace transform of the given circuit elements.

The Laplace transform of the voltage source is:

L{10u(t)} = 10/s

The Laplace transform of the inductor is:

L{L(di/dt)} = sL(I(s)) - L(i(0))

Since the initial current is zero, L(i(0)) = 0. Therefore:

L{L(di/dt)} = sLI(s)

The Laplace transform of the resistor is:

L{Ri} = R * I(s)

The Laplace transform of the capacitor is:

L{(1/C)∫i dt} = I(s)/(sC)

Using Kirchhoff's voltage law, we can write:

10 = L(di/dt) + Ri + (1/C)∫i dt

Substituting the Laplace transforms, we get:

10/s = sLI(s) + RI(s) + (1/C)(I(s)/s)

Solving for I(s), we get:

I(s) = 10/([tex]s^{2L}[/tex] + Rs + 1/CS)

Substituting the given values, we get:

Using partial fraction decomposition, we can write:

I(s) = A/(s + 1/2 - i√3/2) + B/(s + 1/2 + i√3/2)

where A and B are constants. Solving for A and B, we get:

Therefore, we can write:

I(s) = (5 + 5i√3/3)/(s + 1/2 - i√3/2) + (5 - 5i√3/3)/(s + 1/2 + i√3/2)

Taking the inverse Laplace transform, we get:

i(t) =[tex]10[/tex][tex]e^{(-t/2)}[/tex] [(5/3)sin(√3t/2) + (5/3)cos(√3t/2)]

Therefore, A = 10, B = 1, C = 5/3, and D = 1/2.

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The definite integral 3 ∫ X / √(16+3x) Dx is -16/15.

To evaluate the definite integral:

3 ∫ x / √(16+3x) dx from 0 to 3,

we can use the substitution method:

Let u = 16 + 3x
Then, du/dx = 3 and dx = du/3

Substituting in the integral, we get:

∫ 3 ∫ x / √(16+3x) dx = ∫ 3 ∫[tex]\frac{(u-16)}{3u^{\frac{1}{2} } }[/tex]du

= (1/3) ∫ 3 ∫ [[tex]\frac{(u-16)}{3u^{\frac{1}{2} } }[/tex]] du

= (1/3) ∫ 3 [(2/3)[tex]u^{\frac{3}{2} }[/tex] - 8[tex]u^{\frac{1}{2} }[/tex]] du

= (1/3) [(2/5)[tex]u^{\frac{5}{2} }[/tex] - (16/2)[tex]u^{\frac{3}{2} }[/tex])] from 16 to 25

= (1/3) [(2/5)[tex]25^{\frac{5}{2} }[/tex] - (16/2)[tex]25^{\frac{3}{2} }[/tex] - (2/5)[tex]16^{\frac{5}{2} }[/tex] + (16/2)[tex]16^{\frac{3}{2} }[/tex])]

= (1/3) [(2/5)(125) - (16/2)(25) - (2/5)(32) + (16/2)(64)]

= -16/15

Therefore, the definite integral is -16/15.

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The tree if the point in the ground is 52 feet from the tree is 81.25 feet tall.

Using the tangent function to solve this problem.

Let h be the height of the tree.

Then, using the angle of elevation of a nearby tree from a point on the ground measured to be 54° and the height of the tree if the point in the ground is 52 feet from the tree:

tan(54°) = h/52

Solving for h:

h = 52 × tan(54°)

Using a calculator:

h ≈ 81.25 feet

Therefore, the height of the tree is approximately 81.25 feet.

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The value of y when x equals 14 is 70 as x and y are in directly proportional.

Direct proportionality equation is a linear equation in two variables.

It is expressed as;

x ∝ y

then

x = ky

Where k is the proportionality constant.

First we determine the constant of proportionality.

In this case, when x is 6, y is 30. So constant of proportionality  is:

x = ky

k = x/y

k = 6/30

k = 1/5

Now, we can use constant of proportionality to find y when x is 14.

Let's substitute x = 14 into equation:

x = ky

14 = (1/5) × y

14 = y/5

y = 14 × 5

y = 70

Therefore, the value of y is 4.

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The given function is f(x) = (3)/(8)(4)^x where the base is 4, and the exponent is x. Hence, we can say that it is an exponential function of the form f(x) = a(b)^x.

Here, a = 3/8 and b = 4.

The function is an exponential function as it is of the form f(x) = a(b)^x.

It is an exponential growth function as its base is greater than 1. Since the base is 4 which is greater than 1, we can say that it is an exponential growth function.

An exponential growth function is one in which the value of the function increases as the input increases.

In this case, as the value of x increases, the value of f(x) will keep increasing more and more rapidly, as the base is greater than 1.

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Let us assume that Maya reads the entire newspaper in "x" minutes. Then the fraction of the newspaper she reads in one minute is given as 1/x. Maya reads 1/8 of a newspaper in 1/20 of a minute.

Therefore, Maya reads 1/8 of a newspaper in 3/60 of a minute => 1/20 of a minute Hence, the fraction of the newspaper she reads in one minute is given as: 1/x = 1/ (3/60) => 1/x = 20/3Therefore, she can read the entire newspaper in 20/3 minutes. We can simplify this further as follows:20/3 = 6 2/3 minutes Hence, Maya will take 6 2/3 minutes to read the entire newspaper.

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To find the composition of relations r ◦ s, we need to determine the set of ordered pairs that satisfy the composition.

The composition r ◦ s is defined as follows:

r ◦ s = {(x, z) | there exists y such that (x, y) ∈ s and (y, z) ∈ r}

Let's calculate the composition:

For each pair (x, y) ∈ s, we check if there exists a pair (y, z) ∈ r that satisfies the condition. If so, we include (x, z) in the composition.

For (a, c) ∈ s:

There is no pair (y, z) ∈ r where (c, y) and (y, z) hold simultaneously. Therefore, (a, c) does not contribute to the composition.

For (b, d) ∈ s:

There is no pair (y, z) ∈ r where (d, y) and (y, z) hold simultaneously. Therefore, (b, d) does not contribute to the composition.

For (d, a) ∈ s:

There exists a pair (y, z) = (a, b) in r, where (a, y) and (y, z) hold simultaneously. Therefore, (d, b) contributes to the composition: (d, b).

Hence, the composition r ◦ s is {(d, b)}.

Therefore, the composition of relations r ◦ s is {(d, b)}.

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Answer:

Step-by-step explanation:

To find the area of the region under the graph of the function f(x) = 2x + 5 on the interval [-1, 4], we need to integrate the function over that interval.

The integral of f(x) with respect to x over the interval [-1, 4] gives us the area under the curve.

∫[a,b] f(x) dx denotes the integral of f(x) with respect to x over the interval [a,b].

In this case, we have:

∫[-1,4] (2x + 5) dx

Evaluating this integral, we get:

∫[-1,4] (2x + 5) dx = [x^2 + 5x] evaluated from -1 to 4

Plugging in the upper and lower limits, we have:

= (4^2 + 5(4)) - ((-1)^2 + 5(-1))

= (16 + 20) - (1 - 5)

= 36 + 4

= 40

Therefore, the area of the region under the graph of the function f(x) = 2x + 5 on the interval [-1, 4] is 40 square units.

Answer:

4. m∠5 + m∠12 = 180°

Step-by-step explanation:

5 & 13 are equal

12 & 4 are equal

So when you add them together you get a 180°

(straight line)

The volume of the composite figure is equal to 860.6 cubic meters to the nearest tenth

The composite figure is a cuboid with a cylinderical open space within, so the volume is derived by subtracting the volume of the cylinderical open space from the volume of the cuboid as follows:

Volume of cuboid = length × width × height

Volume of the cuboid = 10m × 10m × 12m

Volume of the cuboid = 1200m³

Volume of cylinder is calculated using:

V = π × r² × h

Volume of the cylinder = 22/7 × (3m)² × 12m

Volume of the cylinder = 339.4m³

Volume of the composite figure = 1200m³ - 339.4m³

Volume of the composite figure = 860.6 m³

Therefore, the volume of the composite figure is equal to 860.6 cubic meters to the nearest tenth

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I'm glad to help you with your question. Let's consider y1, ..., yn as independent uniform(0, 2) random variables. We want to find P[y(n) < 1.9], where y(n) represents a statistic used to estimate the upper limit of the uniform distribution.

First, we need to understand the properties of uniform distribution. In a uniform distribution, all values within a given range have an equal probability of occurrence. In our case, the range is [0, 2]. Therefore, the probability density function (pdf) of a uniform(0, 2) random variable Y is given by:
f(y) = 1/2, for 0 <= y <= 2
      0, otherwise
Now, let's consider the probability of a single random variable yi being less than 1.9:
P[yi < 1.9] = ∫(1/2) dy from 0 to 1.9 = (1/2) * (1.9 - 0) = 0.95
Since y1, ..., yn are independent random variables, we can calculate the probability of all of them being less than 1.9 by taking the product of their individual probabilities:
P[y(n) < 1.9] = P[y1 < 1.9] * ... * P[yn < 1.9] = (0.95)^n
So, the probability that y(n) is less than 1.9 is (0.95)^n, where n is the number of independent uniform(0, 2) random variables.

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The equivalent resultant force and couple moment acting at point O are {70i - 80j + 190k} N and {180i - 440j + 270k} N.m, respectively.

To replace the loading system by an equivalent resultant force and couple moment acting at point O, we need to find the moment of each force about point O and then sum them up.

Let's assume that the position vector of the point of application of F1 is given by r1.
F1 = {−270i, 150j, 190k} N
Find the cross product of r1 and F1.
Moment = r1 x F1 = (r1xi, r1yj, r1zk) x (−270i, 150j, 190k)
Calculate the individual components of the cross product.
[tex]Moment_x = r1y(190) - r1z(150)[/tex]
[tex]Moment_y = r1z(-270) - r1x(190)[/tex]
[tex]Moment_z = r1x(150) - r1y(-270)[/tex]
Sum up the individual components to find the total moment at point O.
[tex]Total Moment = (Moment_x)i + (Moment_y)j + (Moment_z)k[/tex]
Unfortunately, we do not have the position vector r1 given in the question.

Once we have the values for r1x, r1y, and r1z, you can plug them into the above equations to find the x, y, and z components of the couple moment acting at point O.

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To replace the loading system by an equivalent resultant force and couple moment at point O, we first need to calculate the resultant force. This can be done by taking the vector sum of all the forces acting on the system. In this case, we are given that f1 = {−270i, 150j, 190k} N.

To calculate the resultant force, we simply add up the x, y, and z components of all the forces. In this case, there is only one force, so the resultant force is simply f1.
Next, we need to determine the couple moment acting at point O. A couple moment is a pair of forces that are equal in magnitude, opposite in direction, and separated by a distance. The moment created by this pair of forces is equal to the magnitude of one of the forces multiplied by the distance between them.

In this case, we are given that the couple moment is acting at point O. We don't have enough information to calculate the distance between the forces, so we can't determine the magnitude of the moment. Therefore, we can't enter the x, y, and z components of the couple moment separated by commas.

In summary, to replace the loading system by an equivalent resultant force and couple moment at point O, we first calculated the resultant force by taking the vector sum of all the forces. We then determined that the couple moment is acting at point O, but we don't have enough information to calculate its magnitude.
We'll follow these steps:

1. Calculate the resultant force by summing up the individual forces. In this case, there's only one force F1 = {-270i, 150j, 190k} N. So, the equivalent resultant force acting at point O is also F1.

2. Calculate the position vector from point O to the point of application of F1. Let's denote this vector as R.

3. Find the couple moment acting at point O by computing the cross product of the position vector R and the force F1: M = R x F1.

4. Enter the x, y, and z components of the couple moment separated by commas.

Without information about the position vector R, it's impossible to calculate the exact couple moment. Please provide the coordinates of the point of separated of F1 to determine the couple moment acting at point O.

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Since t cannot be infinity in this case, we conclude that there is no maximum number of items that the worker can produce according to the function.

The number of items produced on the first day can be found by substituting t = 1 into the function P(t):

P(1) = 24 - 24e^(-0.3*1) = 13.24 (rounded to two decimal places)

To find the maximum number of items that the worker can produce, we can take the derivative of the function P(t) with respect to t and set it equal to zero:

P'(t) = 24e^(-0.3t)(0.3) = 7.2e^(-0.3t)

7.2e^(-0.3t) = 0

e^(-0.3t) = 0

t = infinity

However, we can see that as t approaches infinity, P(t) approaches 24. So, we can say that the worker can approach but never exceed 24 items.

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We have a parallelogram CDEA whose perimeter is  20 inches.

An isoceles triangle is given with a leg of 5 inches.

Two lines are drawn through some point on the base, each parallel to one of the legs.

The perimeter of the constructed quadrilateral is to be found.An isosceles triangle has two sides equal in length.

Let's draw a diagram that looks like this:

Given an isoceles triangle:The two lines drawn through some point on the base are parallel to one of the legs.

Hence, the parallelogram so formed has equal sides in the form of legs of the triangle.

The perimeter of the parallelogram can be found as the sum of the opposite sides of the parallelogram.

As seen in the diagram, the parallel lines DE and BC are the same length. Hence, we know that the parallel lines CD and AE are also the same length.

Therefore, we have a parallelogram CDEA whose perimeter is

2*(CD+CE) = 2*(5+5) = 20 inches

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The solution to the expression is x = ±√6i.

We have,

To solve x² + 6 = 0,

We can subtract 6 from both sides.

x = -6

Now,

We can take the square root of both sides, remembering to include both the positive and negative square roots:

x = ±√(-6)

Since the square root of a negative number is not a real number, we cannot simplify this any further without using complex numbers.

The solution:

x = ±√6i, where i is the imaginary unit

(i.e., i^2 = -1).

Thus,

The solution to the expression is x = ±√6i.

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We have shown that the row sum s is an eigenvalue of the matrix A with eigenvector x = (1, 1, ..., 1)T.

To show that s is an eigenvalue of the nxn matrix A, we need to find a nonzero vector x such that Ax = sx, where s is the row sum of A. One way to find such a vector is to take the vector x = (1, 1, ..., 1)T, where T denotes transpose.

Using this choice of x, we have

Ax = (s, s, ..., s)T = sx,

which shows that s is indeed an eigenvalue of A with eigenvector x.

To see why this works, consider the kth entry of Av for any nonzero vector v in R^n. We have

(Av)_k = ∑ A_ki v_i, i=1 to n

where A_ki denotes the entry in the kth row and ith column of A. Since the row sums of A all equal s, we can write

(Av)_k = ∑ A_ki v_i = s ∑ v_i

where the sum on the right-hand side is taken over all i such that A_ki is nonzero.

If we take v = x, then we have ∑ v_i = nx, and hence

(Ax)_k = s(nx) = (ns)x_k,

which shows that x is an eigenvector of A with eigenvalue s.

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At 5% rate the account been earning interest.

Given that Ruby has saved $4072.24, and she initially invested $2500, we can plug in these values into the formula:

4072.24 = 2500(1 + r/1[tex])^{(1 )(10)[/tex]

Simplifying the equation, we get:

(1 + r)¹⁰ = 4072.24/2500

Taking the 10th root of both sides, we have:

1 + r = (4072.24/2500[tex])^{(1/10)[/tex]

Subtracting 1 from both sides, we find:

r = (4072.24/2500[tex])^{(1/10)[/tex]- 1

r = 1.05000008852 - 1

r = 0.05000008852

r = 5%

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To find g(y), we first need to solve the differential equation g'(y) = 12y - 8.

We can integrate both sides of the equation to obtain the solution:

∫g'(y) dy = ∫(12y - 8) dy

Integrating, we have:

g(y) = 6y^2 - 8y + C

where C is the constant of integration.

Since we are given that g(y) = k, where k is a constant, we can set k equal to the expression we obtained for g(y):

k = 6y^2 - 8y + C

Since k is a constant, we can rewrite the equation as:

6y^2 - 8y + C - k = 0

This equation represents a quadratic equation in terms of y. To satisfy the given condition, the quadratic equation must have a single repeated root. This occurs when the discriminant of the quadratic equation is zero.

The discriminant is given by:

b^2 - 4ac = (-8)^2 - 4(6)(C - k)

Setting the discriminant to zero:

64 - 24(C - k) = 0

Simplifying the equation:

24k - 24C + 64 = 0

This equation relates the constants k and C. However, since we do not have any additional information or constraints, we cannot determine the specific values of k and C. Therefore, we cannot find the exact expression for g(y) in terms of k.

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Answer:

I think this is the answer

Step-by-step explanation:

To solve for when y=1, we can use the slope-intercept form of a linear equation, which is y = mx + b. First, we need to find the slope (m) using the two given points:

m = (10 - 100) / (8 - 4)
m = -90 / 4
m = -22.5

Now we can use the point-slope form of a linear equation, which is y - y1 = m(x - x1), where (x1, y1) is one of the given points. Let's use (4, 100):

y - 100 = -22.5(x - 4)

Simplifying this equation, we get:

y = -22.5x + 202.5

To find when y=1, we can substitute that into the equation and solve for x:

1 = -22.5x + 202.5
-22.5x = -201.5
x = 8.96

Therefore, y=1 at approximately t=8.96.

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Answer:

The expression 4-3 can be expressed as a power with base 2 by using the rule of exponentiation: 2^(4-3) = 2^1.

Thus, as d^2y/dx^2 is negative for all values of t, the curve is concave down everywhere.

Parametric equations are a way of expressing a curve in terms of two separate functions, usually denoted as x(t) and y(t).

In this case, we are given the following parametric equations: x(t) = 9cos(t) and y(t) = -6sec(t).

To find dy/dt, we simply take the derivative of y(t) with respect to t: dy/dt = -6sec(t)tan(t).

To find dx/dt, we take the derivative of x(t) with respect to t: dx/dt = -9sin(t).

Now, we can express the slope of the curve as dy/dx, which is simply dy/dt divided by dx/dt:

dy/dx = (-6sec(t)tan(t))/(-9sin(t)) = 2/3tan(t)sec(t).

To find when the curve is concave up or concave down, we need to take the second derivative of y(t) with respect to x(t): d^2y/dx^2 = (d/dt)(dy/dx)/(dx/dt) = (d/dt)((2/3tan(t)sec(t)))/(-9sin(t)) = -2/27(sec(t))^3.

Since d^2y/dx^2 is negative for all values of t, the curve is concave down everywhere.

In summary, the function for dy/dt is -6sec(t)tan(t), and the curve is concave down everywhere.

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D > 0 and (∂²f/∂x²)(∂²f/∂y²) > 0, the critical point (10/3, 5/3) is a local minimum. B. A local minimum

To find the critical point of the function f(x, y) = x² + y² - xy - 1 - 5x, we need to find the values of x and y where the gradient of the function is equal to zero.

First, let's find the partial derivatives of the function with respect to x and y:

∂f/∂x = 2x - y - 5

∂f/∂y = 2y - x

To find the critical point, we set both partial derivatives equal to zero and solve the system of equations:

2x - y - 5 = 0 -- (1)

2y - x = 0 -- (2)

From equation (2), we can rearrange it to solve for x:

x = 2y -- (3)

Substituting equation (3) into equation (1), we have:

2(2y) - y - 5 = 0

4y - y - 5 = 0

3y - 5 = 0

3y = 5

y = 5/3

Substituting y = 5/3 into equation (3):

x = 2(5/3) = 10/3

Therefore, the critical point is (10/3, 5/3).

To determine the nature of the critical point, we need to use the Second Derivative Test. We need to find the second partial derivatives of f(x, y) and evaluate them at the critical point (10/3, 5/3).

The second partial derivatives are:

∂²f/∂x² = 2

∂²f/∂y² = 2

∂²f/∂x∂y = -1

Now let's evaluate the second partial derivatives at the critical point:

∂²f/∂x² = 2 (evaluated at (10/3, 5/3))

∂²f/∂y² = 2 (evaluated at (10/3, 5/3))

∂²f/∂x∂y = -1 (evaluated at (10/3, 5/3))

To determine the nature of the critical point, we'll use the discriminant:

D = (∂²f/∂x²)(∂²f/∂y²) - (∂²f/∂x∂y)²

D = (2)(2) - (-1)² = 4 - 1 = 3

Since D > 0 and (∂²f/∂x²)(∂²f/∂y²) > 0, the critical point (10/3, 5/3) is a local minimum. Therefore, the correct answer is:

B. A local minimum

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There is a 76% chance that a student chosen at random from this class will have both a cat and a dog.

There are 15 students who have cats and 16 who have dogs. Thus, if a student is chosen at random, there are 15 + 16 = 31 students who could have either a cat or a dog. And the remaining 3 students have neither a cat nor a dog. Thus, there are 25 – 3 = 22 students in total who have either a cat or a dog. To find the probability that a student chosen at random has both a cat and a dog, we can use the formula:P(cat and dog) = (number of students with both cat and dog) / (total number of students)Therefore, we need to find the number of students who have both a cat and a dog. This can be done by subtracting the number of students who don’t have either a cat or a dog (3) from the total number of students who have either a cat or a dog (22).number of students who have both cat and dog = 22 – 3 = 19Therefore, the probability that a student chosen at random has both a cat and a dog is:P(cat and dog) = 19/25 = 0.76 or 76%Thus, there is a 76% chance that a student chosen at random from this class will have both a cat and a dog.

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The possible values of the number are all real numbers except for zero.

In the given problem, we have the inequality:

|x - 1.5| < |20 - 5|

Simplifying the inequality:

|x - 1.5| < 1

To solve this inequality, we consider two cases:

Case 1: x - 1.5 > 0

In this case, the absolute value becomes:

x - 1.5 < 15

Solving for x:

x < 16.5

Case 2: x - 1.5 < 0

In this case, the absolute value becomes:

-(x - 1.5) < 15

Simplifying and solving for x:

x > -13.

Combining the solutions from both cases, we find that the possible values of x are any real numbers greater than -13.5 and less than 16.5, excluding zero.

Therefore, all real numbers except zero are possible values of the number that satisfy the given inequality.

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The first plant produces 2x + 21 more items daily than the second plant.

Here's the solution:

Let the number of items produced by the first plant be represented by 5x + 14, and the number of items produced by the second plant be represented by 3x - 7.

The first plant produces how many more items daily than the second plant we will calculate here.

The difference in their production can be found by subtracting the production of the second plant from the first plant's production:

( 5x + 14 ) - ( 3x - 7 ) = 2x + 21

Thus, the first plant produces 2x + 21 more items daily than the second plant.

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The cos(α - β) is equal to (2√15 + √390)/45, rounded to four Decimals

To find cos(α - β), we can use the trigonometric identity:

cos(α - β) = cos(α)cos(β) + sin(α)sin(β)

Given that cos(α) = √3/9 and sin(β) = √5/5, we need to find sin(α) and cos(β) to evaluate the expression.

Since α is a first quadrant angle, sin(α) is positive. We can find sin(α) using the Pythagorean identity:

sin^2(α) + cos^2(α) = 1

sin^2(α) = 1 - cos^2(α)

sin(α) = √(1 - cos^2(α))

Given that cos(α) = √3/9, we can substitute the value:

sin(α) = √(1 - (√3/9)^2)

= √(1 - 3/81)

= √(78/81)

= √78/9

Now, we can evaluate cos(β):

cos^2(β) + sin^2(β) = 1

cos^2(β) = 1 - sin^2(β)

cos(β) = √(1 - sin^2(β))

Given that sin(β) = √5/5, we can substitute the value:

cos(β) = √(1 - (√5/5)^2)

= √(1 - 5/25)

= √(20/25)

= √20/5

= 2√5/5

Now we can substitute the values of sin(α), cos(β), cos(α), and sin(β) into the expression for cos(α - β):

cos(α - β) = cos(α)cos(β) + sin(α)sin(β)

= (√3/9)(2√5/5) + (√78/9)(√5/5)

= (2√15)/45 + (√390)/45

= (2√15 + √390)/45

Therefore, cos(α - β) is equal to (2√15 + √390)/45, rounded to four decimals

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cos(α - β) = (√15 + √78)/45 or approximately 0.8895.

We can use the identity cos(α - β) = cos(α)cos(β) + sin(α)sin(β) to find cos(α - β).

Given that cos(α) = √3/9, we can find sin(α) using the Pythagorean identity: sin²(α) + cos²(α) = 1.

sin²(α) + (√3/9)² = 1

sin²(α) = 1 - (√3/9)²

sin(α) = √(1 - (√3/9)²) = √(1 - 3/81) = √(78/81) = √78/9

Given that sin(β) = √5/5, we can find cos(β) using the Pythagorean identity: cos²(β) + sin²(β) = 1.

cos²(β) + (√5/5)² = 1

cos²(β) = 1 - (√5/5)²

cos(β) = √(1 - (√5/5)²) = √(5/25) = 1/√5

Now we can substitute these values into the formula for cos(α - β):

cos(α - β) = cos(α)cos(β) + sin(α)sin(β)

= (√3/9)(1/√5) + (√78/9)(√5/5)

= (√3/9√5) + (√(78/5)/9)

= (√15 + √78)/45

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Where the polygon hs been transformed, note that :

Parallel lines are coplanar infinite straight lines that do not cross at any point in geometry. Parallel planes are planes that never intersect in the same three-dimensional space.

Parallel curves are those that do not touch or intersect and maintain a constant minimum distance.

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The rule for this translation. and the coordinates of the image point are D. (x, y) = (x – 4, y + 1); (–1, 0)

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

translated to the left 4 units and up 1 unit

Mathematically, this can be expressed as

(x, y) = (x - 4, y + 1)

Given that

C = (3, -1)

And, we have

(x, y) = (x - 4, y + 1)

This means that

C' = (3 - 4, -1 + 1)

Evaluate

C' = ( -1, 0)

So, the image point is ( -1, 0)

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We have shown that 1^3 + 2^3 + ... + k^3 + (k+1)^3 = ((k+1)((k+1)+1)/2)^2, which completes the proof by induction.

(a) Basis Step: When n = 1, we have 1^3 = (1(1+1)/2)^2, which is true.

(b) Inductive Hypothesis: Assume that for some positive integer k, the statement 1^3 + 2^3 + ... + k^3 = (k(k+1)/2)^2 is true.

(c) Inductive Conclusion: We want to show that the statement is also true for k+1, that is, 1^3 + 2^3 + ... + k^3 + (k+1)^3 = ((k+1)((k+1)+1)/2)^2.

(d) Proof of Inductive Conclusion:

Starting with the left-hand side of the equation:

1^3 + 2^3 + ... + k^3 + (k+1)^3

= (1^3 + 2^3 + ... + k^3) + (k+1)^3

Using the inductive hypothesis, we know that 1^3 + 2^3 + ... + k^3 = (k(k+1)/2)^2, so:

= (k(k+1)/2)^2 + (k+1)^3

= (k^2(k+1)^2/4) + (k+1)^3

= [(k+1)^2/4][(k^2)+(4k+4)]

= [(k+1)^2/4][(k+2)^2]

Therefore, we have shown that 1^3 + 2^3 + ... + k^3 + (k+1)^3 = ((k+1)((k+1)+1)/2)^2, which completes the proof by induction.

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Here is the answer to the question. The answer does exist if you look in to the equation properly

(a) The vertical asymptotes occur where the denominator equals zero. Therefore, we need to solve the equation x - 9[tex]x^{2}[/tex] = 0, which gives us x = 0 and x = 9[tex]x^{2}[/tex]. Therefore, the vertical asymptotes are x = 0 and x = [tex]\frac{1}{9}[/tex]. To find the horizontal asymptote, we need to look at the limit as x approaches infinity and negative infinity. As x approaches infinity, the highest power of x in the denominator dominates and the function approaches y = -9[tex]x^{-1}[/tex]. As x approaches negative infinity, the highest power of x in the denominator dominates and the function approaches y = -9[tex]x^{-1}[/tex].
(b) To find the intervals where the function is increasing and decreasing, we need to find the derivative of the function and determine the sign of the derivative on different intervals. The derivative is f'(x) = -([tex]\frac{-7}{x^{2} }[/tex]) + [tex]\frac{18}{x^{3} }[/tex]. The derivative is positive when ([tex]\frac{-7}{x^{2} }[/tex]) + [tex]\frac{18}{x^{3} }[/tex]. > 0, which occurs when x < 0 or x > [tex]\frac{7}{3}[/tex]. Therefore, the function is increasing on (-∞, 0) and (7/3, ∞) and decreasing on (0, [tex]\frac{7}{3}[/tex]).
(c) To find the local maximum and minimum values, we need to find the critical points of the function, which occur where the derivative equals zero or is undefined. The derivative is undefined at x = 0, but this is not a critical point because the function is not defined at x = 0. The derivative equals zero when -([tex]\frac{-7}{x^{2} }[/tex]) + [tex]\frac{18}{x^{3} }[/tex]. = 0, which simplifies to x = [tex]\frac{18}{7}[/tex]Therefore, the function has a local maximum at x = [tex]\frac{18}{7}[/tex]. To determine whether this is a local maximum or minimum, we can look at the sign of the second derivative, which is f''(x) =.[tex]\frac{14}{x^{3} } - \frac{54}{x^{4} }[/tex] When x = [tex]\frac{18}{7}[/tex], f''([tex]\frac{18}{7}[/tex]) < 0, so this is a local maximum.
(d) To find the intervals where the function is concave up, we need to find the second derivative of the function and determine the sign of the second derivative on different intervals. The second derivative is f''(x) = [tex]\frac{14}{x^{3} } - \frac{54}{x^{4} }[/tex]. The second derivative is positive when [tex]\frac{14}{x^{3} } - \frac{54}{x^{4} }[/tex]> 0, which occurs when x < 2.09 or x > 5.46. Therefore, the function is concave up on (-∞, 0) and (2.09, 5.46) and concave down on (0, 2.09) and (5.46, ∞).

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