Find sin(B) in the triangle.

Answer:

a. -2.5--->-1

b. n---> 2n/5

The solution to the equation [tex]\sqrt{3r^2} = 3[/tex] is given as follows:

[tex]r = \pm \sqrt{3}[/tex]

The equation in the context of this problem is defined as follows:

[tex]\sqrt{3r^2} = 3[/tex]

To solve the equation, we must isolate the variable r. The variable r is inside the square root, hence to isolate, we must obtain the square of each side, as follows:

3r² = 9.

Now we solve it as a quadratic equation as follows:

r² = 3.

[tex]r = \pm \sqrt{3}[/tex]

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The approximate reciprocal of 0.72 is 1.3889.

The ✓1.7 depicted as a fraction is (√170)/10.

It should be noted that to calculate the reciprocal of 0.72, we simply divide 1 by 0.72:

1/0.72 = 1.388888888888889

Thus, the approximate reciprocal of 0.72 is 1.3889.

Also, to determine the fractional equivalent for √1.7, we may again rationalize the denominator through multiplying both numerator and denominator with the expression contained beneath the radical:

√1.7=√(17/10)=(√17)/(√10)

Multiplying every entity within the adjoined numerator and denominator with (√10) offers:

(√17)/(√10)*(√10)/(√10)= (√170)/10

Therefore, √1.7 depicted as a fraction is:

√1.7=(√170)/10

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To find the volume of the solid, whose base is the region between the curve y=20 sin x.

We know that the base of the solid is the region between the curve y=20 sin x. We also know that the solid is bounded by the x-axis and the plane z=0.

Therefore, the height of the solid is the distance between the curve and the plane z=0. This distance is simply given by the function y=20 sin x.

To find the volume of the solid, we need to integrate the area of each cross-sectional slice of the solid as we move along the x-axis. The area of each slice is simply the area of the base times the height.

The area of the base is given by the integral of y=20 sin x over the region of interest. This integral is:

∫ y=20 sin x dx from x=0 to x=π

= -cos(x) * 20 from x=0 to x=π

= 40

Therefore, the area of the base is 40 square units.

The height of the solid is given by y=20 sin x. Therefore, the volume of each slice is:

dV = (area of base) * (height)

= 40 * (20 sin x) dx

Integrating this expression from x=0 to x=π, we get:

V = ∫ dV from x=0 to x=π

= ∫ 40 * (20 sin x) dx from x=0 to x=π

= 800 [cos(x)] from x=0 to x=π

= 1600

Therefore, the volume of the solid is 1600 cubic units.

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To find the volume of the solid, whose base is the region between the curve y=20 sin x.

We know that the base of the solid is the region between the curve y=20 sin x. We also know that the solid is bounded by the x-axis and the plane z=0.

Therefore, the height of the solid is the distance between the curve and the plane z=0. This distance is simply given by the function y=20 sin x.

To find the volume of the solid, we need to integrate the area of each cross-sectional slice of the solid as we move along the x-axis. The area of each slice is simply the area of the base times the height.

The area of the base is given by the integral of y=20 sin x over the region of interest. This integral is:

∫ y=20 sin x dx from x=0 to x=π

= -cos(x) * 20 from x=0 to x=π

= 40

Therefore, the area of the base is 40 square units.

The height of the solid is given by y=20 sin x. Therefore, the volume of each slice is:

dV = (area of base) * (height)

= 40 * (20 sin x) dx

Integrating this expression from x=0 to x=π, we get:

V = ∫ dV from x=0 to x=π

= ∫ 40 * (20 sin x) dx from x=0 to x=π

= 800 [cos(x)] from x=0 to x=π

= 1600

Therefore, the volume of the solid is 1600 cubic units.

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Answer:It is the one where the subway is the biggest

Step-by-step explanation:

because in the graph  it shows the subway is the biggest number

A function f: A--B is injective if each element in A maps to a unique element in B. It is surjective if every element in B has a corresponding element in A.

A bijective function is both injective and surjective, meaning that every element in A maps to a unique element in B, and every element in B has a corresponding element in A.  If IAI = IBI, then there are the same number of elements in A and B. Therefore, if f is injective, every element in A must map to a unique element in B, leaving no elements in B without a corresponding element in A. This means that f is also surjective. Similarly, if f is surjective, then every element in B has a corresponding element in A, which means that f is also injective. Thus, f is bijective if and only if it is injective and surjective.

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Answer: 4, 2

Step-by-step explanation:

This is a sine/cosine wave.

we can see one full revolution from 0 to 4; this means that the period is 4.

the amplitude refers to how "high" or "low" the graph goes from its center.

we can see it hits a maximum of 2, (and a minimum of -2). Since the amplitude is the absolute value of this high/low value, it will always be positive. so the amplitude is 2

In conclusion:

Period = 4

Amplitude = 2

Answer:

f^(-1)(x) = sqrt(x/49)

Step-by-step explanation:

To find the inverse function of f(x) = 49x^2, we need to solve for x in terms of f(x) and then interchange x and f(x).

f(x) = 49x^2

f(x)/49 = x^2

sqrt(f(x)/49) = x (since x > 0)

So, the inverse function of f(x) is:

f^(-1)(x) = sqrt(x/49)

Note that the domain of f^(-1) is x ≥ 0, since x must be positive for the inverse function to be defined. Also, note that f(f^(-1)(x)) = f(sqrt(x/49)) = 49(sqrt(x/49))^2 = 49(x/49) = x, and f^(-1)(f(x)) = sqrt(f(x)/49) = sqrt(49x^2/49) = x. Therefore, f^(-1) is the inverse function of f.

The 95% confidence interval for the mean price of regular gasoline in that region is $3.396 to $3.584.The 90% confidence interval for the mean price of regular gasoline in that region is $3.413 to $3.567 3and 95% confidence interval for the mean price of regular gasoline in that region with a sample size of 80 would be $3.427 to $3.55

a) The 95% confidence interval for the mean price of regular gasoline in that region can be calculated as:

[tex]x ± z(\frac{s}{\sqrt{n} } )[/tex]

where X is the sample mean, s is the sample standard deviation, n is the sample size, and z is the critical value for the desired confidence level. For a 95% confidence level, z is 1.96.

Plugging in the given values, we get:

[tex]3.149 ± 1.96(\frac{0.29}{\sqrt{40} } )[/tex]

= 3.49 ± 0.094

So the 95% confidence interval for the mean price of regular gasoline in that region is $3.396 to $3.584.

b) Similarly, the 90% confidence interval for the mean can be calculated by using z = 1.645 (the critical value for a 90% confidence level):

3.49 ± 1.645(0.29/√40)

= 3.49 ± 0.077

So the 90% confidence interval for the mean price of regular gasoline in that region is $3.413 to $3.567.

c) If we had the same statistics from 80 stations, the standard error would decrease because the sample size is larger. The new standard error would be:

s/√80 = 0.29/√80 ≈ 0.032

Using the same formula as in part (a), but with the new standard error and z = 1.96, we get:

3.49 ± 1.96(0.032)

= 3.49 ± 0.063

So the 95% confidence interval for the mean price of regular gasoline in that region with a sample size of 80 would be $3.427 to $3.553.

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The equation that shows the relationship between x, the number of months, and y, the amount still owed on the loan, is y = -400x + 1200. The correct option is C.

The graph shows that the initial amount borrowed is 1200 and the loan payments reduce the amount owed by 400 pesos per month.

The amount still owed on the loan decreases linearly over time, so we can use the point-slope form of the equation for a line to express the relationship between x (the number of months) and y (the amount still owed on the loan):

y - y₁ = m(x - x₁)

where y₁ is the y-coordinate of a point on the line (in this case, the initial amount borrowed, which is 1200), m is the slope of the line (the rate at which the amount owed decreases, which is -400), and x₁ is the x-coordinate of the same point on the line (in this case, the first month, which is 1).

Substituting the values we have, we get:

y - 1200 = -400(x - 1)

Simplifying:

y = -400x + 1600

Therefore, the equation that shows the relationship between x, the number of months, and y, the amount still owed on the loan, is y = -400x + 1200. The correct option is C.

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To find f(pi), we need to use the fundamental theorem of calculus which states that if the integral of f(x) is F(x), then the derivative of F(x) with respect to x is f(x).

Given that the integral of f(x) is xsin2x, we can use this theorem to find f(x).Taking the derivative of xsin2x with respect to x gives: f(x) = d/dx (xsin2x), f(x) = sin2x + 2xcos2x, Now, to find f(pi), we simply substitute pi for x in the expression we just found: f(pi) = sin2(pi) + 2(pi)cos2(pi) , f(pi) = 0 + 2(pi)(-1) , f(pi) = -2pi .Therefore, f(pi) = -2pi. To find f(π) when the integral of f(x) is x*sin(2x),

we need to differentiate the given integral with respect to x. So, let's find the derivative of x*sin(2x) using the product rule: f(x) = d/dx(x*sin(2x)), f(x) = x * d/dx(sin(2x)) + sin(2x) * d/dx(x), f(x) = x * (cos(2x) * 2) + sin(2x) * 1, f(x) = 2x * cos(2x) + sin(2x), Now, to find f(π), we simply substitute x with π: f(π) = 2π * cos(2π) + sin(2π), Since cos(2π) = 1 and sin(2π) = 0, f(π) = 2π * 1 + 0 = 2π.

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The absolute maximum value of f(x) on the interval [1, 4] is 1.5, which occurs at x = 1, and the absolute minimum value is 0.176, which occurs at x = 4.

To find the absolute maximum and minimum values of the function f(x) = 3/(x^2 + 1) on the interval [1, 4], we need to first find the critical points and then evaluate the function at the endpoints of the interval.

Critical points occur where the derivative of the function is equal to 0 or is undefined.

First, find the derivative of f(x):
f'(x) = -6x / (x^2 + 1)^2

To find the absolute maximum and minimum values of the function f(x) = 3/(x^2 + 1) on the interval [1, 4], we need to first find the critical points and the endpoints of the interval.
f'(x) = -6x/(x^2 + 1)^2 = 0

Next, we evaluate the function at the endpoints of the interval:
f(1) = 3/(1^2 + 1) = 1.5
f(4) = 3/(4^2 + 1) = 0.176
Set f'(x) to 0 and solve for x:
-6x / (x^2 + 1)^2 = 0

Since the denominator can never be 0, the only way this equation can be true is if the numerator is 0:
-6x = 0
x = 0

However, x = 0 is not in the interval [1, 4], so there are no critical points in the interval.

Now, evaluate the function at the endpoints of the interval:
f(1) = 3/(1^2 + 1) = 3/2 = 1.5
f(4) = 3/(4^2 + 1) = 3/17 ≈ 0.2

Since there are no critical points in the interval, the absolute maximum and minimum values occur at the endpoints. Thus, the absolute maximum value is 1.5 at x = 1, and the absolute minimum value is approximately 0.2 at x = 4.

Answer: Absolute maximum = 1.5 at x = 1
Absolute minimum ≈ 0.2 at x = 4

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Answer:

Let's assume that the cost of a box of candy is "x" dollars.

According to the problem, Karen bought 3 boxes of candy and 2 small bags of popcorn, and paid $18.35. So we can write the equation:

3x + 2y = 18.35

Similarly, Holly bought 4 boxes of candy and 3 small bags of popcorn, and paid $26.05. So we can write the equation:

4x + 3y = 26.05

We want to find the cost of a box of candy, so we can solve for "x" using these two equations. One way to do this is to use elimination. If we multiply the first equation by 3 and the second equation by -2, we can eliminate the "y" term:

9x + 6y = 55.05

-8x - 6y = -52.10

Adding these two equations gives:

x = 2.95

So the cost of a box of candy is $2.95.

Answer:

$2.95

Step-by-step explanation:

Let x be the cost of a box of candy while y be the cost of a small bag of popcorn.

Out of the given data, two equations is formulated.

Equation 1

Equation 2

Multiply 3 to both sides of Eq.1 to derive Eq.1'

Multiply 2 to both sides of Eq.2 to derive Eq.2'

Elimination using Eq.1' and Eq.2' to derive x

A box of candy costs $2.95

The temperature distribution in the rod at time t.

We can use the method of separation of variables to solve this problem. Let's assume that the temperature function can be written as a product of two functions: u(x,t) = X(x)T(t). Substituting this in the heat equation and dividing by XT, we get:

(1/X) d²X/dx² = (1/a) (1/T) dT/dt = -λ²

where λ² = -a is a constant. This gives us two separate differential equations:

d²X/dx² + λ² X = 0, X(0) = X(2) = 0

and

dT/dt + a/T = 0, T(0) = 1

The first equation has the general solution:

X(x) = B sin(λx)

where B is a constant determined by the boundary conditions. Since X(0) = X(2) = 0, we have:

X(x) = B sin(nπx/2)

where n is an odd integer (to satisfy X(0) = 0) and B is a normalization constant such that X(2) = 0. We have:

X(2) = B sin(nπ) = 0

which implies that nπ = 2kπ, where k is an integer. Since n is odd, we must have n = 2m + 1 for some integer m, so we get:

nπ = (2m + 1)π = 2kπ

which implies that k = m + 1/2. Therefore, the eigenvalues are:

λ² = -(nπ/2l)² = -(2m + 1)²π²/4l²

and the corresponding eigenfunctions are:

X_m(x) = B_m sin((2m + 1)πx/2l)

where B_m is a normalization constant.

The second equation has the solution:

T(t) = exp(-at)

Using the principle of superposition, the general solution of the heat equation is:

u(x,t) = Σ_m B_m sin((2m + 1)πx/2l) exp(-a(2m + 1)²π²t/4l²)

To determine the coefficients B_m, we use the initial condition:

u(x,0) = f(x) = x (0 < x < 1), f(x) = 0 (1 < x < 2)

This gives us:

Σ_m B_m sin((2m + 1)πx/2l) = x (0 < x < 1)

Σ_m B_m sin((2m + 1)πx/2l) = 0 (1 < x < 2)

Using the orthogonality of the sine functions, we can solve for B_m:

B_m = (4/l) ∫_0^l x sin((2m + 1)πx/2l) dx

B_m = (8l/(2m + 1)π)² ∫_0^1 x sin((2m + 1)πx/2) dx

B_m = (-1)^(m+1)/(2m + 1)

Therefore, the solution is:

u(x,t) = Σ_m (-1)^(m+1)/(2m + 1) sin((2m + 1)πx/2l) exp(-a(2m + 1)²π²t/4l²)

This is the temperature distribution in the rod at time t.

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We have shown that |g(x) - g(y)| < 12ε whenever |x - y| < δ. Since ε was arbitrary, this shows that g(x) is uniformly continuous on (0, 1).

A stronger version of continuity known as uniform continuity ensures that functions defined on metric spaces, such as the real numbers, only vary by a small amount when their inputs change by a small amount. Contrary to uniform continuity, continuity merely demands that the function act "locally" around each point. To clarify, this means that for any given point x, there exists a tiny neighbourhood around x such that the function behaves properly inside that neighbourhood.

For the function g(x) to be continuous we need to have any ε > 0, and  δ > 0 such that if |x - y| < δ, then |g(x) - g(y)| < ε for all x, y in (0, 1).

Now, g(x) is bounded as the parent function f(x) is bounded.

Suppose, (0, 1) such that  |x - y| < δ.

Thus, without generality we have:

|g(x) - g(y)| = |x(1-x)f(x) - y(1-y)f(y)|

= |x(1-x)(f(x) - f(y)) + y(f(y) - f(x)) + xy(f(x) - f(y))|

≤ x(1-x)|f(x) - f(y)| + y|f(y) - f(x)| + xy|f(x) - f(y)|

< x(1-x)4ε + y4ε + xy4ε (by the choice of δ)

= 4ε(x(1-x) + y + xy)

< 4ε(x + y + xy)

≤ 4ε(1 + 1 + 1) = 12ε

Hence, we have shown that |g(x) - g(y)| < 12ε whenever |x - y| < δ. Since ε was arbitrary, this shows that g(x) is uniformly continuous on (0, 1).

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If t=0 in an independent-measures hypothesis test, it indicates that b) the two sample means must be equal

An independent-measures hypothesis test compares the means of two independent samples to determine if there is a significant difference between them. The test uses a t-value to evaluate whether the difference between the two sample means is greater than what would be expected by chance.

If the calculated t-value is equal to 0, it means that the difference between the two sample means is not statistically significant. In other words, the null hypothesis cannot be rejected. Therefore, we cannot conclude that there is a significant difference between the two population means.

Therefore, option (b) is correct, as the two sample means must be equal if the t-value is equal to 0. Option (a) is incorrect because the equality of population means is not directly related to the t-value being 0.

Option (c) is also incorrect because the equality of sample variances is not a requirement for the t-value to be 0 in an independent-measures hypothesis test.

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To determine which of the two forecasts (a or b) is more accurate, you must compare their MAD values. The forecast with the lower MAD value is considered more accurate. The MAD value of forecasts a and b can be calculated by finding the absolute value of the difference between each forecasted value and its corresponding actual value, then taking the average of those differences.

Explanation:

To calculate the MAD (Mean Absolute Deviation) for each forecast, you need to find the absolute value of the difference between the forecasted values and the actual values, then take the average of those differences. To calculate the MAD (Mean Absolute Deviation) for each forecast, you'll need to follow these steps:

Step 1: Find the absolute differences between the actual data points and the forecasted values.
Step 2: Sum up these absolute differences.
Step 3: Divide the total sum by the number of data points.


a) To determine which of the two forecasts (a or b) is more accurate, you must compare their MAD values. The forecast with the lower MAD value is considered more accurate, as it signifies smaller deviations from the actual data points.

b) The MAD value of forecast a can be calculated by finding the absolute value of the difference between each forecasted value and its corresponding actual value, then taking the average of those differences.

c) Similarly, the MAD value of forecast b can be calculated by finding the absolute value of the difference between each forecasted value and its corresponding actual value, then taking the average of those differences.

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The area of region d is 4.5π.

To find the area of region d between the circles of radius 4 and radius 5 centered at the origin in the second quadrant, we can use the following steps:

Find the area of the larger circle (radius 5) and subtract the area of the smaller circle (radius 4) to find the area of the annulus (ring-shaped region) between them:

Area of larger circle = π[tex](5)^2[/tex] = 25π

Area of smaller circle = π[tex](4)^2[/tex] = 16π

Area of annulus = (25π) - (16π) = 9π

Divide the annulus into two equal parts since we are only interested in the portion of the region in the second quadrant. This gives us:

Area of region d = 1/2 (9π) = 4.5π

Therefore, the area of region d is 4.5π.

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To find the implicit solution for dy/dx = x^2/(1-y^2), we can start by separating the variables and integrating both sides.

dy/(1-y^2) = x^2 dx

To integrate the left-hand side, we can use partial fractions:

dy/(1-y^2) = (1/2) * (1/(1+y) + 1/(1-y)) dy

Integrating both sides, we get:

(1/2) * ln|1+y| - (1/2) * ln|1-y| = (1/3) * x^3 + C

Where C is the constant of integration.

We can simplify this expression by combining the natural logs:

ln|1+y| - ln|1-y| = (2/3) * x^3 + C'

Where C' is a new constant of integration.

Finally, we can use the logarithmic identity ln(a) - ln(b) = ln(a/b) to get the implicit solution:

ln|(1+y)/(1-y)| = (2/3) * x^3 + C''

Where C'' is a final constant of integration.

Therefore, the implicit solution for dy/dx = x^2/(1-y^2) is ln|(1+y)/(1-y)| = (2/3) * x^3 + C''.

Given the differential equation:

dy/dx = x^2 / (1 - y^2)

To find an implicit solution, we can use separation of variables. Rearrange the equation to separate the variables x and y:

(1 - y^2) dy = x^2 dx

Now, integrate both sides with respect to their respective variables:

∫(1 - y^2) dy = ∫x^2 dx

The result of the integrations is:

y - (1/3)y^3 = (1/3)x^3 + C

This is the implicit solution to the given differential equation, where C is the integration constant.

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The directional derivative of f(x, y) = xy at point p(5, 5) in the direction from p to q(8, 1) is -1.

To find the directional derivative of f(x, y) = xy at point p(5, 5) in the direction from p to q(8, 1), we need to first find the unit vector in the direction from p to q.

This can be done by subtracting the coordinates of p from those of q to get the vector v = <3, -4> and then dividing it by its magnitude, which is sqrt(3^2 + (-4)^2) = 5. So, the unit vector in the direction from p to q is u = v/|v| = <3/5, -4/5>.

Next, we need to compute the gradient of f at point p, which is given by the partial derivatives of f with respect to x and y evaluated at p: grad(f)(5, 5) =  evaluated at (5, 5) = <5, 5>.

Finally, we can compute the directional derivative of f at point p in the direction of u as follows:

D_u f(5, 5) = grad(f)(5, 5) · u = <5, 5> · <3/5, -4/5> = (5)(3/5) + (5)(-4/5) = -1.

Therefore, the directional derivative of f(x, y) = xy at point p(5, 5) in the direction from p to q(8, 1) is -1.

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Answer:

x=-((y-2)^2)/12  +5

Step-by-step explanation:

 B = | 1 0 0 |
      | 0 1 0 |
      | 1 1 2 |

To construct a matrix whose column space contains vectors [1, 0, 1] and [0, 1, 1], and whose nullspace contains the vector [2, 1, 2], follow these steps:

1. Form the column space by arranging the given vectors as columns of the matrix A:
  A = | 1 0 |
      | 0 1 |
      | 1 1 |

2. Determine a third column vector (C) that, when added to A, will create a matrix that has the given nullspace vector. To do this, use the equation Ax = 0, where x is the nullspace vector [2, 1, 2]. Multiply the matrix A with the nullspace vector to find the third column vector:

  A * x = | 1 0 | * |2| = |2|
          | 0 1 |   |1|   |1|
          | 1 1 |   |2|   |4|

3. Subtract the nullspace vector from the product obtained in step 2 to get the third column vector (C):

  C = Ax - x = | 2 - 2 | = | 0 |
               | 1 - 1 |   | 0 |
               | 4 - 2 |   | 2 |

4. Combine the matrix A with the third column vector C to form the final matrix B:

  B = | 1 0 0 |
      | 0 1 0 |
      | 1 1 2 |

The matrix B has the required column space containing [1, 0, 1] and [0, 1, 1] and a nullspace containing [2, 1, 2].

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Note that the values of the six trigonometric functions of θ are:

Given that sin(θ) = 0 and that π/2 ≤ θ ≤ 3π/2, we need to find the values of the six trigonometric functions of θ.

Since sin(θ) = 0, we know that θ must be an integer multiple of π. However, since θ also satisfies the condition π/2 ≤ θ ≤ 3π/2, the only possible values of θ that satisfy sin(θ) = 0 are θ = π/2 or θ = 3π/2.

Using these values, we can calculate the values of the other trigonometric functions:

cos(θ) = cos(π/2) = 0 (when θ = π/2)

cos(θ) = cos(3π/2) = 0 (when θ = 3π/2)

tan(θ) = sin(θ)/cos(θ) = UNDEFINED (since cos(θ) = 0)

csc(θ) = 1/sin(θ) = UNDEFINED (since sin(θ) = 0)

sec(θ) = 1/cos(θ) = UNDEFINED (since cos(θ) = 0)

cot(θ) = cos(θ)/sin(θ) = UNDEFINED (since sin(θ) = 0)

So the values of the six trigonometric functions of θ are:

sin(θ) = 0

cos(θ) = 0

tan(θ) = UNDEFINED

csc(θ) = UNDEFINED

sec(θ) = UNDEFINED

cot(θ) = UNDEFINED

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Answer:

see below

Step-by-step explanation:

1) adjacent angles are 2 angles right next to each other and are labeled with 3 letters, not 2.  

Examples in the picture would include <ABE, <ABD

Vertical angles are angles opposite of each other, so 2 examples are ABE and DBC

2) adjacent angles: PQT and QTR

vertical angles: PQR and SQR

3) a) adjacent

b) neither

c) vertical

d) vertical

e) adjacent

f) neither

hope this helps!

The system of inequalities that describes this situation is;

x + y ≥ 8

x*15 + y*30 ≤ 300

Let's define the variables that we need to use.

He wants at least 8 tickets, then the first inequality is:

x + y ≥ 8

And he wants to spend no more than $300, then:

x*15 + y*30 ≤ 300

Then that is the system of inequalities.

x + y ≥ 8

x*15 + y*30 ≤ 300

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The general expression for the (100p)th percentile of the distribution is :

x_p = -ln(1 - p)/λ

The median of an exponential distribution with parameter λ is :

ln(2)/λ.

An exponential distribution is a continuous probability distribution that describes the time between events in a Poisson process, where events occur continuously and independently at a constant average rate.

The probability density function (PDF) of an exponential distribution with parameter λ is given by:

f(x) = λe^(-λx)

where x ≥ 0 and λ > 0.

To derive the (100p)th percentile of the distribution, we need to find the value x_p such that P(X ≤ x_p) = p, where p is a given percentile (e.g. p = 0.5 for the median). In other words, x_p is the value of X that separates the bottom p% of the distribution from the top (100-p)%.

To find x_p, we can use the cumulative distribution function (CDF) of the exponential distribution, which is given by:

F(x) = P(X ≤ x) = 1 - e^(-λx)

Using this formula, we can solve for x_p as follows:

1 - e^(-λx_p) = p
e^(-λx_p) = 1 - p
-λx_p = ln(1 - p)
x_p = -ln(1 - p)/λ

This is the general expression for the (100p)th percentile of the exponential distribution. To obtain the median, we set p = 0.5 and simplify:

x_median = -ln(1 - 0.5)/λ = ln(2)/λ

Therefore, the median of an exponential distribution with parameter λ is ln(2)/λ.

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(a) The covariance of X and Y is 0, since X and Y are independent.

(b) The standard deviation of U is sqrt(2(3^2) + 4(-4^2)) = 2*sqrt(13).

(c) The expected value of V is 0, since E(V) = 6E(X)E(Y) + 3E(Y) = 0.

(a) Since X and Y are independent, the covariance between them is 0. The formula for covariance is Cov(X,Y) = E(XY) - E(X)E(Y). Since E(XY) = E(X)E(Y) when X and Y are independent, the covariance is 0.

(b) The formula for the standard deviation of U is SD(U) = sqrt(Var(3X) + Var(-4Y)). Since Var(aX) = a^2Var(X) for any constant a, we can calculate Var(3X) = 3^2Var(X) = 9(2) = 18 and Var(-4Y) = (-4)^2Var(Y) = 16(4) = 64. Thus, SD(U) = sqrt(18 + 64) = 2*sqrt(13).

(c) The expected value of V is E(V) = E(6XY + 3Y). Since X and Y are independent, we can calculate this as E(6XY) + E(3Y) = 6E(X)E(Y) + 3E(Y). Since E(X) = 3 and E(Y) = 0, we get E(V) = 6(3)(0) + 3(0) = 0. Therefore, the expected value of V is 0.

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The maximum and minimum values of f(x, y, z) subject to the given constraint can be found by substituting the values of x, y and z in the function f(x, y, z).

Function is an operation that takes one or more inputs and produces an output, or a set of outputs, depending on the type of function. Functions are commonly used in mathematics and computer science, and are essential for solving a wide range of problems.

We are given a function f(x, y, z) = xyz and the constraint x2 + 2y2 + 3z2 = 96. To find the maximum and minimum values of f(x, y, z) subject to the given constraint, we can use the method of Lagrange multipliers.

Let λ be the Lagrange multiplier. Then, the Lagrange function is given by:

L(x, y, z, λ) = xyz + λ (x2 + 2y2 + 3z2 - 96)

We will now calculate the partial derivatives of L with respect to x, y, z and λ.

∂L/∂x = yz + 2xλ = 0
∂L/∂y = xz + 4yλ = 0
∂L/∂z = xy + 6zλ = 0
∂L/∂λ = x2 + 2y2 + 3z2 - 96 = 0

Solving the above equations, we get:
2xλ = -yz
4yλ = -xz
6zλ = -xy
x2 + 2y2 + 3z2 = 96

Substituting the values of λ in the first three equations, we get:
2x(-xz/4y) = -yz
2x2z/4y = -yz
x2z/2y = -yz

From the fourth equation, we get:
x2 + 2y2 + 3z2 = 96

Substituting the values of x2, y2 and z2 from the above equation in the fifth equation, we get:
(96 - 2y2 - 3z2)z/2y = -yz
96z/2y - yz - 3z3/2y = 0

Solving for z, we get:
z = (96/4y) ± √(962/16y2 - 3y2)

Substituting the values of z from the above equation in the fourth equation, we get:
x2 + 2y2 + 3 (96/4y)2 ± √(962/16y2 - 3y2)2 = 96

Solving for y, we get:
y = ±√(96/14 - 3z2/2)

Substituting the values of y from the above equation in the third equation, we get:
x = ± 2z √(14z2/96 - 1/3)

Hence, the maximum and minimum values of f(x, y, z) subject to the given constraint can be found by substituting the values of x, y and z in the function f(x, y, z).

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The solution to the boundary-value problem is[tex]y(x) = (9/2)cos(x) - (9/2)cos(1)sin(x) + (1/2)x^2 - 1/2.[/tex]

To solve the boundary-value problem, we can follow these steps:

Step 1: Find the general solution of the homogeneous differential equation y'' + y = 0.

The characteristic equation is r^2 + 1 = 0, which has complex roots r = ±i. Therefore, the general solution of the homogeneous equation is y_h(x) = c_1 cos(x) + c_2 sin(x), where c_1 and c_2 are constants.

Step 2: Find a particular solution of the non-homogeneous differential equation y'' + y = x^2 + 1.

We can use the method of undetermined coefficients to find a particular solution. Since the right-hand side of the equation is a polynomial of degree 2, we can assume a particular solution of the form y_p(x) = ax^2 + bx + c. Substituting this into the equation, we get:

[tex]y_p''(x) + y_p(x) = 2a + ax^2 + bx + c + ax^2 + bx + c = 2ax^2 + 2bx + 2c + 2a[/tex]

Equating this to the right-hand side of the equation, we get:

2a = 1, 2b = 0, 2c + 2a = 1

Solving for a, b, and c, we get a = 1/2, b = 0, and c = -1/2.

Therefore, a particular solution is y_p(x) = (1/2)x^2 - 1/2.

Step 3: Find the general solution of the non-homogeneous differential equation.

The general solution of the non-homogeneous differential equation is y(x) = y_h(x) + y_p(x), where y_h(x) is the general solution of the homogeneous equation and y_p(x) is a particular solution of the non-homogeneous equation.

Substituting the values of c_1, c_2, and y_p(x) into the general solution, we get:

y(x) = c_1 cos(x) + c_2 sin(x) + (1/2)x^2 - 1/2

Step 4: Apply the boundary conditions to determine the values of the constants.

Using the first boundary condition, y(0) = 4, we get:

c_1 - 1/2 = 4

Therefore, c_1 = 9/2.

Using the second boundary condition, y(1) = 0, we get:

9/2 cos(1) + c_2 sin(1) + 1/2 - 1/2 = 0

Therefore, c_2 = -9/2 cos(1).

Step 5: Write the final solution.

Substituting the values of c_1 and c_2 into the general solution, we get:

[tex]y(x) = (9/2)cos(x) - (9/2)cos(1)sin(x) + (1/2)x^2 - 1/2[/tex]

Therefore, the solution to the boundary-value problem is[tex]y(x) = (9/2)cos(x) - (9/2)cos(1)sin(x) + (1/2)x^2 - 1/2.[/tex]

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