A major fishing company does its fishing in a local lake. The first year of the company's operations it managed to catch 130,000 fish. Due to population decreases, the number of fish the company was able to catch decreased by 3% each year. How many total fish did the company catch over the first 14 years, to the nearest whole number?

Answer:

1. sin(A) = [tex]\frac{4}{5}[/tex]

2. cos(A) = [tex]\frac{3}{5}[/tex]

3. tan(A) = [tex]\frac{4}{3}[/tex]

4. sin(B) = [tex]\frac{3}{5}[/tex]

5. cos(B) = [tex]\frac{4}{5}[/tex]

6. tan(B) = [tex]\frac{3}{4}[/tex]

Step-by-step explanation:

Use SOHCAHTOA:

Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

1. sin(A) = opposite of A / hypotenuse of A = [tex]\frac{4}{5}[/tex]

2. cos(A) = adjacent of A / hypotenuse of A = [tex]\frac{3}{5}[/tex]

3. tan(A) = opposite of A / adjacent of A = [tex]\frac{4}{3}[/tex]

4. sin(B) = opposite of B / hypotenuse of B = [tex]\frac{3}{5}[/tex]

5. cos(B) = adjacent of B / hypotenuse of B = [tex]\frac{4}{5}[/tex]

6. tan(B) = opposite of B / adjacent of B = [tex]\frac{3}{4}[/tex]

The surface whose equation is given by ρ = cos ϕ is a type of conical surface in the spherical coordinate system. In this equation, ρ represents the radial distance from the origin, and ϕ denotes the polar angle.

Explanation:

The surface described by the equation ρ = cos(ϕ) is a type of conical surface in the spherical coordinate system. Let's break down the explanation step by step:

Spherical Coordinate System: The spherical coordinate system is a three-dimensional coordinate system used to represent points in space using three parameters - radial distance (ρ), polar angle (ϕ), and azimuthal angle (θ). The radial distance ρ represents the distance from the origin (0,0,0) to a point in space, ϕ represents the polar angle measured from the positive z-axis (ranging from 0 to π), and θ represents the azimuthal angle measured from the positive x-axis in the xy-plane (ranging from 0 to 2π).

Equation ρ = cos(ϕ): The equation ρ = cos(ϕ) describes a relationship between the radial distance ρ and the polar angle ϕ. It specifies that for any given value of the polar angle ϕ, the radial distance ρ should be equal to the cosine of ϕ.

Conical Surface: In the context of the spherical coordinate system, a conical surface is a surface that forms a cone shape with its apex at the origin. The equation ρ = cos(ϕ) describes a conical surface because it specifies that the radial distance ρ is determined by the cosine of the polar angle ϕ.

Shape of the Surface: As the polar angle ϕ varies, the equation ρ = cos(ϕ) determines the radial distance ρ at each point on the surface. Since the radial distance is only determined by the cosine of the polar angle, the surface will have a conical shape. Specifically, the surface will form a cone with its apex at the origin and its base expanding outward as ϕ increases from 0 to π. The radius of the base of the cone will vary with the value of ϕ, as determined by the cosine function. When ϕ = 0, the base of the cone will have its maximum radius, equal to 1 (since cos(0) = 1), and as ϕ increases towards π, the radius of the base will decrease until it reaches its minimum value of -1 (since cos(π) = -1). The surface will extend infinitely in the positive and negative z-directions.

In conclusion, the surface described by the equation ρ = cos(ϕ) in the spherical coordinate system is a type of conical surface, forming a cone with its apex at the origin and its base expanding outward as the polar angle ϕ increases, with the radius of the base varying based on the cosine of ϕ.

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Answer: Approximately 5.21

Step-by-step explanation:

Given that the survey of US adults ages 18-24 found that 34% get the news on an average day, we can assume that the probability of an 18-24 year old getting news on an average day is p = 0.34. We also know that we have randomly selected 200 adults in this age group.

The mean of a binomial distribution is given by:

μ = np

where n is the sample size and p is the probability of success. Substituting the given values, we get:

μ = 200 x 0.34 = 68

Therefore, the mean number of adults ages 18-24 who get news on an average day is 68.

The standard deviation of a binomial distribution is given by:

σ = sqrt(np(1-p))

Substituting the given values, we get:

σ = sqrt(200 x 0.34 x 0.66) ≈ 5.21

Therefore, the standard deviation of the number of adults ages 18-24 who get news on an average day is approximately 5.21. Since the sample size is large (n=200), we can use the normal distribution to approximate the binomial distribution.

Answer:

The mean and standard deviation of the number of adults out of 200 who get news on an average day are mu = 68 and sigma = 5.36, respectively, assuming we can use the normal distribution to approximate the binomial distribution.

So the probability that at least 85 people say they get no news on an average day is approximately 0.0013 or 0.13%.

Step-by-step explanation:

Since the survey found that 34% of US adults ages 18-24 get news on an average day, we can assume that the probability of a randomly selected adult in this age group getting news on an average day is p = 0.34. Therefore, the number of adults out of 200 who get news on an average day follows a binomial distribution with parameters n = 200 and p = 0.34.

To use the normal distribution to approximate this binomial distribution, we need to check if the conditions for doing so are met. These conditions are:

np >= 10

n(1-p) >= 10

Here, np = 200 x 0.34 = 68 and n(1-p) = 200 x 0.66 = 132. Both of these values are greater than 10, so the conditions are met.

Now, we can approximate the binomial distribution with a normal distribution with mean mu = np = 68 and standard deviation sigma = sqrt(np(1-p)) = sqrt(200 x 0.34 x 0.66) = 5.36.

Therefore, the mean and standard deviation of the number of adults out of 200 who get news on an average day are mu = 68 and sigma = 5.36, respectively, assuming we can use the normal distribution to approximate the binomial distribution.

Using the previous information, to find the probability that at least 85 people say they get no news on an average day.

Let X be the number of people out of 200 who say they get no news on an average day. We want to find the probability that X is greater than or equal to 85.

Since the probability of any one person saying they get no news on an average day is q = 1 - p = 0.66, we can use the binomial distribution with parameters n = 200 and p = 0.34 to model the number of people who say they get news on an average day.

The probability of at least 85 people saying they get no news on an average day can be calculated using the complement rule:

P(X >= 85) = 1 - P(X < 85)

To use the normal distribution to approximate the binomial distribution, we need to standardize the variable X.

Z = (X - mu) / sigma

where mu = np = 68 and sigma = sqrt(npq) = 5.36, as calculated in the previous question.

Using the continuity correction, we can adjust the upper bound to P(X < 84.5) since we want the probability of at least 85 people saying they get no news.

Z = (84.5 - 68) / 5.36 = 3.00

Using a standard normal distribution table or calculator, we can find that P(Z < 3.00) = 0.9987.

Therefore, the probability of at least 85 people saying they get no news on an average day is:

P(X >= 85) = 1 - P(X < 85)

≈ 1 - P(Z < 3.00)

= 1 - 0.9987

≈ 0.0013

So the probability that at least 85 people say they get no news on an average day is approximately 0.0013 or 0.13%.

Sine function value as 190°. θ = 350°

Cosine function value as 190°. θ = 170°.

Rotational Symmetry: A figure is said to have rotational symmetry if it looks exactly the same after rotating it some angle less than

360∘ (a full rotation).

θ angle with 0° < θ < 360° that I has the same:

sin θ is symmetric over the y-axis and cos θ is symmetric over the x-axis.

This means that if you reflect a point (cos θ, sin θ) over the y-axis, the value of sin θ will not change.

If we reflect the angle of 190° over the y-axis we get 350°

If we reflect the angle of 190° over the x-axis we get 170°

Therefore the answers are 350° and 170°.

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If function f(x)=10(3)2x−2, then f(0) = 10/9.

Explanation:

Step 1: To evaluate f(0), we can substitute x with 0 in the given function f(x) = 10(3)^(2x-2).

f(0) = 10(3)^(2(0)-2) = 10(3)^(-2)

Step 2: Now we know that a^(-n) = 1/a^n. So, we can rewrite 3^(-2) as 1/3^2.

f(0) = 10 * (1/3^2) = 10 * (1/9)

Finally, f(0) = 10/9.

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The solution is:

The proof is given below.

Here, we have,

Given a parallelogram ABCD. Diagonals AC and BD intersect at E. We have to prove that AE is congruent to CE and BE is congruent to DE i.e diagonals of parallelogram bisect each other.

In ΔACD and ΔBEC

AD=BC              (∵Opposite sides of parallelogram are equal)

∠DAC=∠BCE       (∵Alternate angles)

∠ADC=∠CBE        (∵Alternate angles)

By ASA rule, ΔACD≅ΔBEC

By CPCT(Corresponding Parts of Congruent triangles)

AE=EC and DE=EB

Hence, AE is conruent to CE and BE is congruent to DE.

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complete question:

Proving the Parallelogram Diagonal Theorem

Given ABCD is a parralelogam, Diagnals AC and BD intersect at E

Prove AE is conruent to CE and BE is congruent to DE

Answer: Length=24ft Width=6ft

Step-by-step explanation:

Perimeter= 2L+2W= 10 W= 60 and W=6ft and L=24ft

Area= Length x Width

The indefinite integral of x/7 √[tex](49+x^2)[/tex] dx using the substitution x = 7 tan(θ) is -7√(1 + [tex](x/7)^2[/tex]) + C.

Let x = 7 tan(θ), then dx/dθ =[tex]7 sec^2(\theta )[/tex], or dx = [tex]7 sec^2(\theta)[/tex]dθ.

Substituting into the integral, we get:

∫x/7 √(49+[tex]x^2[/tex]) dx = ∫tan(θ) √(49 + 49 [tex]tan^2(\theta)[/tex]) * 7 s[tex]ec^2[/tex](θ) dθ

= 7∫tan(θ) sec(θ) sec(θ) dθ

= 7∫sin(θ) dθ

= -7cos(θ) + C, where C is the constant of integration.

Substituting back x = 7 tan(θ), we get:

-7cos(θ) + C = -7cos(arctan(x/7)) + C

= -7√(1 + [tex](x/7)^2[/tex]) + C.

Therefore, the indefinite integral of x/7 √[tex](49+x^2)[/tex] dx using the substitution x = 7 tan(θ) is:

-7√(1 + [tex](x/7)^2[/tex]) + C.

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The volume of the solid is (25/2)π cubic units.

The volume of the solid obtained by rotating the region bounded by the curves y=x², 0≤x≤5, y=25, and x=0 about the y-axis can be found using the disk method.

Volume = ∫[0 to 25] π(r² - R²) dy

Step 1: Solve y=x² for x to get x=sqrt(y). The outer radius (r) is the distance from the y-axis to x=5, so r=5. The inner radius (R) is the distance from the y-axis to x=sqrt(y), so R=sqrt(y).

Step 2: Substitute r=5 and R=sqrt(y) into the formula.

Volume = ∫[0 to 25] π(5² - (sqrt(y))²) dy

Step 3: Simplify the equation.

Volume = ∫[0 to 25] π(25 - y) dy

Step 4: Integrate the equation with respect to y.

Volume = π[25y - 1/2y²] | [0 to 25]

Step 5: Evaluate the integral.

Volume = π(625 - 625/2) - π(0) = (25/2)π

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the value of the given double integral is 150

To evaluate this double integral, we first identify it as the volume of a solid. In this case, the region r represents a rectangle in the xy-plane with dimensions 6 units (from x = -3 to x = 3) and 5 units (from y = 3 to y = 8). The given integral represents the volume of a rectangular prism, where the height is given by the constant value 5.

The given double integral of 5 da represents the volume of a solid over the rectangular region r = {(x, y) | −3 ≤ x ≤ 3, 3 ≤ y ≤ 8}.

To evaluate this double integral, we integrate the given constant 5 over the given region:

∬r 5 da = ∫₃⁸ ∫₋³³ 5 dx dy

Integrating with respect to x first, we get:

∫₋³³ 5 dx = 5x ∣₋³³ = 5(3) - 5(-3) = 30

Substituting this value and integrating with respect to y, we get:

∫₃⁸ 30 dy = 30y ∣₃⁸ = 30(8) - 30(3) = 150

Therefore, the value of the given double integral is 150.

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Thus, the area of new kite with its diagonal doubled is found as: 140 sq. cm.

The area a kite encloses is known as its area of flight. A quadrilateral with two sets of neighbouring sides that are equal is referred to as a kite. A kite is made up of four angles, four sides, and two diagonals.

The product of a lengths of a kite's diagonals divides its area in half.

The area of the kite ABCD = 35 cm square.

The formula for the area of kite = 1/2*(d)*(D)

d - length of small diagonal

D - length of large diagonal.

Then,

35 =  1/2*(d)*(D)

(d)*(D) = 35*2

(d)*(D) = 70 cm sq.  ..eq 1

Now, the length of diagonals of new kite are doubles that is 2d and 2D.

Area of new kite = 1/2 *(2d)*(2D)

Area of new kite = 1/2 *4*(d)*(D)

Area of new kite = 2 *(d)*(D)

Put the value of (d)*(D) from eq 1.

Area of new kite = 2*70

Area of new kite = 140 sq. cm

Thus, the area of the new kite with its diagonal doubled is found as: 140 sq. cm.

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Answer:

4092

Step-by-step explanation:

We can see that this is a geometric sequence where the first term is 4 and the common ratio is 2. We can use the formula for the sum of a geometric sequence to find the sum of this series:

sum = a(1 - r^n) / (1 - r)

where a is the first term, r is the common ratio, and n is the number of terms.

We need to find n, the number of terms. We can use the formula for the nth term of a geometric sequence:

a_n = a * r^(n-1)

We want to find the value of n when a_n = 2048:

2048 = 4 * 2^(n-1)

512 = 2^(n-1)

n-1 = log2(512) = 9

n = 10

So there are 10 terms in the series. Now we can use the formula for the sum of a geometric sequence:

sum = a(1 - r^n) / (1 - r)

sum = 4(1 - 2^10) / (1 - 2)

sum = 4(1 - 1024) / (-1)

sum = 4(1023)

sum = 4092

Rounding to the nearest hundredth, the sum is approximately 4092.00.

Answer:

Sum=8188

Step-by-step explanation:

This is a geometric series with a first term of 4 and a common ratio of 2. The formula for the sum of a geometric series is:

Sn​=1−ra(1−rn)​

where a is the first term, r is the common ratio and n is the number of terms. In this case, we have:

S11​=1−24(1−211)​

Simplifying, we get:

S11​=−14(−2047)​

S11​=8188

Therefore, the sum of the series is 8188.

The formula for y = f(x - 14) in terms of the variable x is y = (1/3)(x - 14)^2. To sketch the graph, draw a parabola and shift it 14 units to the right.

(a) To get a formula for y = f(x - 14) in terms of the variable x, substitute (x - 14) for x in the given function f(x) = (1/3)x^2:
y = f(x - 14) = (1/3)(x - 14)^2
(b) To sketch a graph of y = f(x - 14) using graph transformations, consider that the original function f(x) = (1/3)x^2 is a parabola. The transformation f(x - 14) shifts the graph 14 units to the right. Unfortunately, I cannot provide or select a graph letter from A-E, as there are no graphs provided here. However, to sketch it on paper, draw a parabola and shift it 14 units to the right.

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The derivative of y with respect to x is:

[tex]y' = [(5x) / (5x^2 + 9y^2)] + [(9y) / (5x^2 + 9y^2)] * dy/dx[/tex]

or

[tex]dy/dx = [(5x) / (5x^2 + 9y^2)] + [(9y) / (5x^2 + 9y^2)] * y'[/tex]

To find y', we need to use the chain rule of differentiation because we have a composite function (i.e., the natural logarithm function is applied to a function of x and y).

Let's start by applying the chain rule:

[tex]y' = d/dx [ln(5x^2 + 9y^2)]y' = (1 / (5x^2 + 9y^2)) * d/dx [5x^2 + 9y^2][/tex]

Now, we need to apply the chain rule to find the derivative of[tex]5x^2 + 9y^2[/tex]with respect to x:

[tex]d/dx [5x^2 + 9y^2] = d/dx [5x^2] + d/dx [9y^2][/tex]

[tex]d/dx [5x^2] = 10x[/tex]

[tex]d/dx [9y^2] = 18y * dy/dx[/tex]

(Note that we used the chain rule again to find [tex]dy/dx.)[/tex]

Substituting these derivatives into the expression for y', we get:

[tex]y' = (1 / (5x^2 + 9y^2)) * (10x + 18y * dy/dx)[/tex]

Finally, we can simplify this expression by solving for dy/dx:

[tex]y' = (10x + 18y * dy/dx) / (5x^2 + 9y^2)[/tex]

Multiplying both sides by (5x^2 + 9y^2), we get:

[tex]y' * (5x^2 + 9y^2) = 10x + 18y * dy/dx[/tex]

Solving for dy/dx, we obtain:

[tex]dy/dx = (y' * (5x^2 + 9y^2) - 10x) / 18y[/tex]

Therefore, the derivative of y with respect to x is:

[tex]y' = [(5x) / (5x^2 + 9y^2)] + [(9y) / (5x^2 + 9y^2)] * dy/dx[/tex]

or

[tex]dy/dx = [(5x) / (5x^2 + 9y^2)] + [(9y) / (5x^2 + 9y^2)] * y'[/tex]

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The mean of the binomial variable is 28 and the standard deviation is 2.72, given a probability of success of 0.7 with a total number of attempts of 40.

To calculate the mean (μ) and standard deviation (σ) of a binomial variable, we use the following formulas

μ = np

σ = sqrt(np × (1-p))

where n is the number of trials, and p is the probability of success for each trial.

In this case, the probability of success is 0.7, the number of trials is 40. So:

μ = 400.7 = 28

σ = sqrt(400.7 × (1-0.7)) = 2.72

Therefore, the mean of the binomial variable is 28, and the standard deviation is 2.72

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The portobello mushroom was growing faster.

Given that, Andrew has two mushrooms, he recorded their height, enoki mushroom was 3.9 centimeters tall, after 5 days, it was 4.8 centimeters tall.

Also, the height of the portobello mushroom is given by equation,

y = 0.2x + 4.1, where y is the height of the portobello mushroom in centimeters and x is the number of days since he brought it,

So,

Considering the portobello mushroom height, after 5 days,

y = 0.2(5) + 4.1 = 5.1 cm

And the enoki mushroom was 4.8 cm tall on its 5th day,

Since, the height of portobello mushroom is more than enoki mushroom on 5th day.

Hence, the portobello mushroom was growing faster.

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The given function is f(x) = (x+3)^4 * (x-4)^3 * (x-6)^6. The interval on which f(x) is increasing is (4, ∞).

To determine the intervals on which f(x) is increasing or decreasing, we need to analyze the sign of f'(x), the first derivative of f(x). In this case, f'(x) can be calculated using the product and chain rules of differentiation:

f'(x) = 4(x+3)^3 * (x-4)^3 * (x-6)^6 + 3(x+3)^4 * (x-4)^2 * (x-6)^6 + 6(x+3)^4 * (x-4)^3 * (x-6)^5

Simplifying f'(x) and factoring out common terms, we get:

f'(x) = (x+3)^3 * (x-4)^2 * (x-6)^5 * [4(x-6) + 3(x+3)(x-4) + 6(x-4)]

We can now analyze the sign of f'(x) for different values of x:

Thus, the interval on which f(x) is increasing is (4, ∞).

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The given function is f(x) = (x+3)^4 * (x-4)^3 * (x-6)^6. The interval on which f(x) is increasing is (4, ∞).

To determine the intervals on which f(x) is increasing or decreasing, we need to analyze the sign of f'(x), the first derivative of f(x). In this case, f'(x) can be calculated using the product and chain rules of differentiation:

f'(x) = 4(x+3)^3 * (x-4)^3 * (x-6)^6 + 3(x+3)^4 * (x-4)^2 * (x-6)^6 + 6(x+3)^4 * (x-4)^3 * (x-6)^5

Simplifying f'(x) and factoring out common terms, we get:

f'(x) = (x+3)^3 * (x-4)^2 * (x-6)^5 * [4(x-6) + 3(x+3)(x-4) + 6(x-4)]

We can now analyze the sign of f'(x) for different values of x:

Thus, the interval on which f(x) is increasing is (4, ∞).

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To display both cMat(1,1) and cMat(2,2) as a row using linear indexing, we can create a logical index array that selects these elements in sequence. The linear index of cMat(1,1) is 1, and the linear index of cMat(2,2) is 4 (since there are two columns in cMat). Therefore, we can create a logical index array as follows:

logical_index = [1,4];

We can then use this logical index array to select the desired elements from cMat:

cMat(logical_index)

This will output a row vector with the values 10 and 40, which correspond to cMat(1,1) and cMat(2,2), respectively.
To display both cMat(1,1) and cMat(2,2) as a row using linear indexing, you would use the logical index array [1, 4]. In this case, cMat(1,1) corresponds to the value 10, and cMat(2,2) corresponds to the value 40. The resulting row would be [10, 40].

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The number of  distinguishable code symbols can be formed with the letters for the words philosophical and mathematics is 24

Permutation is a mathematical technique that determines the number of possible arrangements in a set when the order of the arrangements matters.

In the word philosophical , There are 13 letters

2ps, 2is, 2Os, 2Hs and in Mathematics, there are = 11 letters

2ms, 2ts, 2As,

Therefore the number of permutations is 2!212!2! and 2!2!2!

This imples 16 + 8

Therefore, the number of distinguishable code symbols can be formed with the letters for the words philosophical and mathematics = 24

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The maximum value of y is when sin is -2pi/3 : y(x)  = √(3)/2

In order to make the equation's equality true, the unknown variables must be given values as a solution. In other words, the definition of a solution is a value or set of values (one for each unknown) that, when used as a replacement for the unknowns, transforms the equation into equality.

We need to find the maximum value of y.

Given:

y'' + y = -sin(2x)

First, consider the left side equation:

y'' + y = 0

Write using λ

=>λ²+ 1 = 0

=> λ² = -1

=> λ = ± i

The Characteristic solution is given by

A sin(x) + B cos(x)

Finding non-homogeneous solution:

given :

y'' + y = -sin2x

The whole  solution to this non homogeneous solution is given by

y = Csin2x + Dcos2x

Differentiate

y' = 2Ccos2x - 2Dsin2x

Differentiate

y'' = -4Csin2x - 4Dcos2x

Substitute these into the differential equation:

y'' + y = -sin2x

=> (-4Csin2x - 4Dcos2x) + Csin2x + Dcos2x = -sin2x

we have a -sin2x term on the right side

=> sin(2x) [ -4C+ C] = -1  

we have no cosine terms on the right side

=> cos(2x) [-4D + D] = 0

D = 0

=> C = 1/3

So, we have

y(x) = 1/3(sin(2x)) + Asin(x) + Bcos(x)

Use the  initial conditions given in the solution to solve for A and B

=> y(0) = 0 = 0 + 0 + Bcos(0)

=> B = 0

y'(0) = 0 = 2/3cos(0) + A cos(0) + 0

=> 0 = 2/3 + A

=>A = -2/3

The final solution is given by

y(x) = 1/3(sin(2x)) - 2/3sin(x)

Maximum value of y is when sin is -2pi/3 : y(x)  = √(3)/2

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Answer:

A and E

Step-by-step explanation:

They are the same shape and size if rotated properly.

Answer:

A and E

Step-by-step explanation:

when rotated they are the same shape and size

The scale factor that represent the given situation is 1/250000.

Given that, 2.5 cm on the map represents 6.25 km in reality.

The basic formula to find the scale factor of a figure is expressed as,

Scale factor = Dimensions of the new shape ÷ Dimensions of the original shape.

6.25 km = 6.25×100000

= 625000

Here, scale factor = 2.5/625000

= 25/6250000

= 1/250000

Therefore, the scale factor that represent the given situation is 1/250000.

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Since the least square curve fit method is a flexible method for approximating the best fit to a given set of data points using several mathematical models, all of these models are suitable.

The applicable model for the least square curve fit depends on the type of data being analyzed. In this case, the question mentions a sinusoidal model as one of the options. Therefore, a least square curve fit can fit data points to a sinusoidal model, which includes sine and cosine functions. However, it may not necessarily be able to fit the data points to an exponential model, polynomial model of appropriate order, or power curve.
Least square curve fit can fit the data points to the following models:

a) sinusoidal model (including sine and cosine functions)
b) exponential model
c) polynomial model of appropriate order
d) power curve ([tex]y=c1x^(c2)[/tex])

All of these models are applicable because the least square curve fit method is a versatile technique for approximating the best fit to a given set of data points using different mathematical models.

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Since the least square curve fit method is a flexible method for approximating the best fit to a given set of data points using several mathematical models, all of these models are suitable.

The applicable model for the least square curve fit depends on the type of data being analyzed. In this case, the question mentions a sinusoidal model as one of the options. Therefore, a least square curve fit can fit data points to a sinusoidal model, which includes sine and cosine functions. However, it may not necessarily be able to fit the data points to an exponential model, polynomial model of appropriate order, or power curve.
Least square curve fit can fit the data points to the following models:

a) sinusoidal model (including sine and cosine functions)
b) exponential model
c) polynomial model of appropriate order
d) power curve ([tex]y=c1x^(c2)[/tex])

All of these models are applicable because the least square curve fit method is a versatile technique for approximating the best fit to a given set of data points using different mathematical models.

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The probability of occurence of chocolate is 0.4 0r 40%. So the option A is the correct one.

Probability refers to the measure or quantification of the likelihood or chance of an event or outcome occurring. It is typically expressed as a numerical value ranging from 0 to 1, where 0 represents an impossible event and 1 represents a certain event.

In probability theory and statistics, a random variable is a variable whose value is determined by the outcome of a random event or process. It is often denoted by a capital letter, such as X or Y, and it can take on different values with certain probabilities associated with each value.

Based on the given condition, formulate:

8/20=2/5

0.4 or 40%

Therefore option (A) is correct.

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To find the determinant using elementary row or column operations, we can use the following steps:

1. Rewrite the matrix in an augmented form with the identity matrix on the right:

3 -3 -2 | 1 0 0
3 1 2 | 0 1 0
-6 6 4 | 0 0 1

2. Use elementary row operations to transform the matrix into an upper triangular form:

R2 = R2 - R1
R3 = R3 + 2R1
R3 = R3 + 2R2

3 -3 -2 | 1 0 0
0 4 4 | -1 1 0
0 0 0 | -2 2 1

3. The determinant of an upper triangular matrix is the product of its diagonal elements:

det(A) = 3 x 4 x 0 = 0

Therefore, the determinant of the original matrix is 0.

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a) To find the value of β that satisfies limt→[infinity]y(t) = 0, we can first find the general solution of the differential equation. So the value(s) of β for which the solution y(t) is never equal to 0 for all t is [tex]β ∈ (-∞, -2) U (-2/3, ∞)[/tex]

The characteristic equation is [tex]r^2 + r - 2 = 0[/tex], which has roots r = 1 and r = -2.

Therefore, the general solution is[tex]y(t) = c1e^t + c2e^-2t.[/tex]

Using the initial conditions y(0) = 2 and y'(0) = β, we can solve for the constants c1 and c2:

[tex]c1 + c2 = 2[/tex]

[tex]c1 - 2c2 = β[/tex]

Solving this system of equations, we get [tex]c1 = 2 - β/3[/tex] and [tex]c2 = β/3.[/tex]

Therefore, the solution is y(t) =[tex](2 - β/3)e^t[/tex] + [tex]β/3)e^-2t[/tex]. To satisfy limt→[infinity]y(t) = 0, we need the coefficient of e^t to be 0, which gives us 2 - β/3 = 0. Solving for β, we get β = 6.

So the value of β that satisfies limt→[infinity]y(t) = 0 is β = 6.

b) To find the value(s) of β for which the solution y(t) is never equal to 0 for all t, we can use the fact that the discriminant of the characteristic equation determines the nature of the roots.

In this case, the characteristic equation is r^2 + r - 2 = 0, which has roots r = 1 and r = -2. These are distinct real roots, so the general solution is y(t) = [tex]c1e^t + c2e^-2t.[/tex]

For y(t) differential equation to never be equal to 0 for all t, we need both constants c1 and c2 to be nonzero. Using the initial condition y(0) = 2, we get c1 + c2 = 2.

Using the second initial condition y'(0) = β, we get c1 - 2c2 = β.

Solving these equations, we get [tex]c1 = (2β + 4)/5[/tex] and [tex]c2 = (6 - β)/5.[/tex]

Therefore, y(t) is never equal to 0 for all t if and only if both c1 and c2 are nonzero, which is true if and only if the coefficients satisfy the inequality (2β + 4)(6 - β) ≠ 0. Solving this inequality, we get [tex]β ∈ (-∞, -2) U (-2/3, ∞).[/tex]

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Systematic sampling is a sampling plan that selects every nth item from the population. Therefore the correct option is (d) Systematic

Systematic sampling is a statistical sampling method that involves selecting every nth item from the population to create a representative sample. This sampling method is useful when the population is large and ordered in some way, such as in a list or sequence.

To conduct a systematic sample, researchers select a starting point at random and then choose every nth item from that point forward until the desired sample size is reached. The advantage of systematic sampling is that it is simpler and more efficient than other sampling methods, such as simple random sampling, while still providing a representative sample of the population.

Therefore, the correct option is (d) systematic

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For the equation (x^2 - 16)^3 (x - 1)y" - 2xy' + y = 0, x = 0 is considered an ordinary point because the coefficients of the equation do not exhibit any irregular behavior, such as becoming infinite or undefined, at x = 0.

On the other hand, x = 1 is a singular point for this equation because at x = 1, the coefficient of y" becomes zero, leading to an irregular behavior. The uniqueness of linear first-order differential equations is guaranteed by the continuity of the partial derivative ∂f/∂y. This ensures that, under certain conditions, a unique solution exists for a given initial value problem.

y = xe^x is a solution to the differential equation y" - 2y' + y = 0, as when the derivatives of y = xe^x are substituted into the equation, it simplifies to 0, satisfying the given equation.

Finally, the differential equation y" + 2yy' + 3y = 0 is second-order linear because the equation involves the second derivative of y (y") and the equation can be expressed in the form ay" + by' + cy = 0, where a, b, and c are constants or functions of x. In this case, a = 1, b = 2y, and c = 3.

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Okay, let's break this down step-by-step:

* f(x) = -4x3 + 15 (this is the original function)

* We want to find f(x) when x = 3

* So substitute 3 in for x:

f(3) = -4(3)3 + 15

f(3) = -81 + 15

f(3) = -66

Therefore, f(x) = -66 when x = 3.

[tex]\sf f(3)=-66.[/tex]

[tex]\sf f(3)=-4(3)^{3} +15\\ \\[/tex]

[tex]\sf f(3)=-4(3*3*3) +15\\\\\sf f(3)=-4(27) +15[/tex]

[tex]\sf f(3)=-81+15[/tex]

[tex]\sf f(3)=-66.[/tex]

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