Draw the following angle in standard position and nane the reference angle. 240 2. Find the exact value for each of the following: a) bin 330 b) cos(-240 ) or -0.5 tor-os 3. Use the given informati

In this number theory problem, we are given that the product of 36 and the square of a number is equal to 121. Let the number be x, so the equation is 36 * x^2 = 121. To solve for x, divide both sides by 36: x^2 = 121/36.

In number theory, we are given that the product of 36 and the square of a number is equal to 121. We can solve for the unknown number by using algebraic equations. Let the number be represented by x. Therefore, we can write the equation 36x^2 = 121. By dividing both sides by 36, we get x^2 = 121/36. Taking the square root of both sides, we obtain x = ±11/6. Thus, the two possible numbers are 11/6 and -11/6. To write the numbers from least to greatest, we can use the fact that negative numbers are smaller than positive numbers. Therefore, the numbers from least to greatest are -11/6 and 11/6. In conclusion, the product of 36 and the square of a number can be solved using algebraic equations to find the possible numbers and they can be ordered from least to greatest. Taking the square root of both sides gives us x = ±(11/6). The two numbers are -11/6 and 11/6. Writing these numbers from least to greatest, we have -11/6 and 11/6. In summary, the two numbers whose product with 36 equals 121 are -11/6 and 11/6, ordered from least to greatest.

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The function is not a solution of the differential equation xy' - 7y - xe*, x > 0. y = x

To determine if y = x(15+ e^x) is a solution of the differential equation xy' - 7y - xe^x = 0, we need to substitute y and y' into the left-hand side of the equation and see if it simplifies to 0.

First, we find y':

y' = (15 + e^x) + xe^x

Next, we substitute y and y' into the equation and simplify:

x(15 + e^x) + x(15 + e^x) - 7x(15 + e^x) - x^2 e^x

= x(30 + 2e^x - 105 - 7e^x - xe^x)

= x(-75 - 6e^x - xe^x)

Since this expression is not equal to 0 for all x > 0, y = x(15 + e^x) is not a solution of the differential equation xy' - 7y - xe^x = 0.

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Step-by-step explanation:

Perimeter of first one = 2 X (   ( 2x-5   +    3) = 4x - 4

Perimeter of second one = 2 X ( 5   +   x )    = 10 + 2x

  and these are equal

    4x - 4 = 10 + 2x

       2x = 14

    x=7

Answer:

x = 7

Step-by-step explanation:

for rectangle

perimeter (p) = 2(l+b)

having same perimeter both figures have so,

fig 1:                  fig 2:

2*((2x-5) +3) = 2*(5+x)

2*(2x-2) = 10+2x

4x-4 = 10 +2x

4x-2x = 10+4

2x = 14

x = 7

The simplified form of the given trigonometric expression is √6/2·( cos(5π/12) + i·sin(5π/12)).

Given that, 12(cos(7π)/6 +isin(7π)/6))/(4√6(cos(3π/4) +isin(3π/4)).

= (12((-0.866)+i(-0.5))/(4√6(-0.7071+i0.7071)

= 12(-0.866-0.5i)/(4√6(-0.7071+i0.7071))

= (-10.392-6i)/9.8(-0.7071+i0.7071)

= (-10.392-6i)/(-6.9+9.8i)

If you have a problem such as   a·cos(A) / b·cos(B)

you can solve it as (a/b)·cos(A - B)

For this problem a = 12 and b = 4√(6) so a/b =√6/2

and A = 7π/6 and B = 3π/4 so A - B = 5π/12

Therefore, the simplified form of the given trigonometric expression is √6/2·( cos(5π/12) + i·sin(5π/12)).

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The Taylor series expansion of the function sinh(7x) centered at 0 involves finding the first four nonzero terms. The series can be written as a polynomial expression, which allows for approximating the value of sinh(7x) near the point x = 0.

The Taylor series expansion of a function represents the function as an infinite sum of terms involving the function's derivatives evaluated at a specific point. For the function sinh(7x), we can find its Taylor series centered at 0 by evaluating its derivatives.

To find the first four nonzero terms, we start by calculating the derivatives of sinh(7x) with respect to x. The derivatives of sinh(7x) are 7, 49, 343, and 2401, respectively, for the first four terms. We also need to consider the powers of x, which are x, x^3, x^5, and x^7 for the first four terms.

Combining the derivatives and powers of x, we obtain the following series expansion: 7x + (49/3)x^3 + (343/5)x^5 + (2401/7)x^7. These terms represent an approximation of the function sinh(7x) near x = 0. The higher-order terms, which are not considered in this approximation, would further improve the accuracy of the approximation.

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The cost per day for each brand are: Brand A: $0.01161, Brand B: $0.01300, Brand C: $0.00931. The best value brand is Brand C.

To calculate the cost per day for each brand, we divide the cost by the number of days:

Cost per day for Brand A = Cost of Brand A bulb / Number of days for Brand A

Cost per day for Brand B = Cost of Brand B bulb / Number of days for Brand B

Cost per day for Brand C = Cost of Brand C bulb / Number of days for Brand C

To determine the best value brand, we compare the cost per day for each brand and select the brand with the lowest cost.

Let's assume the costs of the bulbs are as follows:

Cost of Brand A bulb = $6.50

Cost of Brand B bulb = $7.80

Cost of Brand C bulb = $5.40

Calculating the cost per day for each brand:

Cost per day for Brand A = $6.50 / 560

≈ $0.01161

Cost per day for Brand B = $7.80 / 600

≈ $0.01300

Cost per day for Brand C = $5.40 / 580

≈ $0.00931

Comparing the costs, we see that Brand C has the lowest cost per day. Therefore, Brand C provides the best value among the three brands.

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E. NO correct choices. The volume of the solid obtained by rotating the region R about the horizontal line y = 1 is (64π/3) cubic units.

To find the volume of the solid obtained by rotating the region R about the horizontal line y = 1, we can use the method of cylindrical shells.

The region R is bounded by the curve y = [tex]5 - x^2[/tex] and the horizontal line y = 1. Let's first find the intersection points of these two curves:

[tex]5 - x^2[/tex]  = 1

[tex]x^2[/tex] = 4

x = ±2

So, the region R is bounded by x = -2 and x = 2.

Now, consider a vertical strip within R with width Δx. The height of the strip is the difference between the two curves: ( [tex]5 - x^2[/tex] ) - 1 = 4 - [tex]x^2[/tex]. The thickness of the strip is Δx.

The volume of this strip can be approximated as V = (height) * (thickness) * (circumference) = (4 - [tex]x^2[/tex]) * Δx * (2πy), where y represents the distance between the line y = 1 and the curve ( [tex]5 - x^2[/tex] ).

To find the volume, we integrate this expression over the interval [-2, 2]:

V = ∫[-2,2] (4 - [tex]x^2[/tex]) * (2πy) * dx

To express y in terms of x, we rewrite the equation y =  [tex]5 - x^2[/tex]  as x^2 = 5 - y, and then solve for x:

x = ±√(5 - y)

Now, substitute this expression for y in terms of x into the integral:

V = ∫[-2,2] (4 - [tex]x^2[/tex]) * (2π(1 + x)) * dx

Evaluating this integral:

V = 2π ∫[-2,2] (4 - [tex]x^2[/tex])(1 + x) dx

Now, expand the expression inside the integral:

V = 2π ∫[-2,2] (4 + 4x - [tex]x^2[/tex] - [tex]x^3[/tex]) dx

V = 2π [8 + 8 - (8/3) - 4] - [-8 + 8 - (-8/3) - 4]

V = 2π [24/3 - 4/3] - [-8/3 - 4/3]

V = 2π [20/3] - [-12/3]

V = 2π [32/3]

V = (64π/3)

Therefore, the volume of the solid obtained by rotating the region R about the horizontal line y = 1 is (64π/3) cubic units.

None of the given answer choices match this result, so the correct choice is E. NO correct choices.

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The problem involves finding the area of the intersection between two polar curves , r = sin(theta) and r = sqrt(3)cos(theta). The task is to determine the region where these curves intersect and calculate the area of that region.

To find the area of the intersection, we need to determine the values of theta where the two curves intersect. Let's set the equations equal to each other and solve for theta: sin(theta) = sqrt(3)cos(theta)

Dividing both sides by cos(theta), we get: tan(theta) = sqrt(3)

Taking the inverse tangent (arctan) of both sides, we find: theta = arctan(sqrt(3))

Since the intersection occurs at this specific value of theta, we can calculate the area by integrating the curves within the range of theta where they intersect. However, it's important to note that without specifying the limits of theta, we cannot determine the exact area.

In conclusion, to find the area of the intersection between the given curves, we need to specify the limits of theta within which the curves intersect. Once the limits are defined, we can integrate the curves with respect to theta to find the area of the intersection region.

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Rules used in the above solution are: Power Rule, Sum Rule, Constant Rule, and Subtraction Rule.

Given function: f(x) = 5x¹ - 6x² + 4x - 2We are supposed to find f'(x) and f'(2).f'(x) is the derivative of the function f(x). The derivative of any polynomial is found by differentiating each of its terms.

Now, let us find f'(x):f'(x) = d/dx (5x¹) - d/dx (6x²) + d/dx (4x) - d/dx (2)f'(x) = 5 - 12x + 4f'(x) = 9 - 12x

Now, we have f'(x) = 9 - 12x.

We have to find f'(2) which means we substitute x = 2 in f'(x):f'(2) = 9 - 12(2)f'(2) = 9 - 24f'(2) = -15

Therefore, the derivative of the given function is 9 - 12x and the value of f'(2) is -15. Rules used in the above solution are: Power Rule, Sum Rule, Constant Rule, and Subtraction Rule.

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No, z is not a function of x and y in the given equation [tex]xz^2 + 3xy - y^2 = 4[/tex].

In the summary, we can state that z is not a function of x and y in the equation.

In the explanation, we can elaborate on why z is not a function of x and y.

To determine if z is a function of x and y, we need to check if for every combination of x and y, there is a unique value of z. In the given equation, we have a quadratic term [tex]xz^2[/tex], which means that for each value of x and y, there are two possible values of z that satisfy the equation. Therefore, z is not uniquely determined by x and y, and we cannot consider z as a function of x and y in this equation. The presence of the quadratic term [tex]xz^2[/tex] indicates that there are multiple solutions for z for a given x and y, violating the definition of a function.

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the following series converge absolutely, conditionally or diverge. 00 k2 Σ(-1)*. 16+1 k=1  converges absolutely.

To determine whether the series Σ((-1)^(k+1))/k^2 converges absolutely, conditionally, or diverges, we need to analyze its convergence behavior.

First, let's consider the absolute convergence by taking the absolute value of each term in the series

Σ |((-1)^(k+1))/k^2|

The series |((-1)^(k+1))/k^2| can be rewritten as Σ(1/k^2), since the absolute value of (-1)^(k+1) is always 1.

The series Σ(1/k^2) is a well-known series called the p-series with p = 2. For a p-series, the series converges if p > 1, and diverges if p ≤ 1.

In this case, p = 2, which is greater than 1. Therefore, the series Σ(1/k^2) converges.

Since the absolute value of each term in the original series converges, we can conclude that the original series Σ((-1)^(k+1))/k^2 converges absolutely. To determine whether the series converges conditionally, we would need to analyze the convergence of the original series without taking the absolute value. However, since we have already determined that the series converges absolutely, there is no need to evaluate its conditional convergence. In summary, the series Σ((-1)^(k+1))/k^2 converges absolutely.

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The price of the 10-year bond issued by Briar Corp is approximately $1,127.15.

To calculate the price of the 10-year bond issued by Briar Corp, we can use the present value of a bond formula. The formula is as follows:

Price = (Coupon Payment / Interest Rate) * (1 - (1 / (1 + Interest Rate)ⁿ) + (Face Value / (1 + Interest Rate) ⁿ)

In this case, the coupon rate is 9% (0.09), the face value is $1,000, and the interest rate for similar bonds is 6% (0.06). The bond has a 10-year maturity, so the number of periods is 10.

Plugging in these values into the formula, we can calculate the price:

Price = (0.09 * $1,000 / 0.06) * (1 - (1 / (1 + 0.06)¹⁰)) + ($1,000 / (1 + 0.06) ¹⁰)

Simplifying the equation and performing the calculations, we find the price of the bond to be approximately $1,127.15.

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Given differential equation is y" + 3y + 2y' + e^(-x) = 0. Particular solution of the given differential equation is given asyp(x) = e^(-u1(x)) + e^(-2u2(x)).  Let us substitute this particular solution into the given differential equation y" + 3y + 2y' + e^(-x) = (-u1''(x) e^(-u1(x)) - 2u2''(x) e^(-2u2(x))) + 2u1'(x) e^(-u1(x)) + 4u2'(x) e^(-2u2(x)) + e^(-x).

Comparing the coefficients of like terms we get-u1''(x) e^(-u1(x)) - 2u2''(x) e^(-2u2(x)) = 0 [As there is no e^(-x) term in the particular solution]2u1'(x) e^(-u1(x)) + 4u2'(x) e^(-2u2(x)) = 0 [Coefficient of e^(-x) should be 1, which gives (2u1'(x) e^(-u1(x)) + 4u2'(x) e^(-2u2(x))) = e^(-x)].

Let us solve the first equation-u1''(x) e^(-u1(x)) - 2u2''(x) e^(-2u2(x)) = 0u1''(x) e^(-u1(x)) = - 2u2''(x) e^(-2u2(x)).

Integrating w.r.t x u1'(x) e^(-u1(x)) = - u2'(x) e^(-2u2(x)).

Dividing second equation by 2 we getu1'(x) e^(-u1(x)) + 2u2'(x) e^(-2u2(x)) = 0.

We can rewrite above equation asu1'(x) e^(-u1(x)) = - 2u2'(x) e^(-2u2(x)).

Substitute the value of u1'(x) in the equation obtained from dividing second equation by 2-u2'(x) e^(-2u2(x)) = 0u2'(x) e^(-2u2(x)) = - 1/2 e^(-x).

Integrating w.r.t xu2(x) = 1/4 e^(-2x) + C1.

Let us differentiate the second equation obtained from dividing the second equation by 2w.r.t xu1'(x) e^(-u1(x)) - 4u2'(x) e^(-2u2(x)) = 0u1'(x) e^(-u1(x)) = 4u2'(x) e^(-2u2(x)).

Substitute the value of u2'(x) obtained aboveu1'(x) e^(-u1(x)) = - 2( - 1/2 e^(-x)) = e^(-x).

Integrating w.r.t xu1(x) = - e^(-x) + C2.

We need to find u1(0)As u1(x) = - ln|e^(-u1(x))| + C2u1(0) = - ln|e^(-u1(0))| + C2As given u1(0) = ln2u1(0) = - ln2 + C2.

Now substitute the values of u1(0) and u2(x) obtained above into the particular solutionyp(x) = e^(-u1(x)) + e^(-2u2(x))yp(x) = e^(ln2 - ln|e^(-u1(x))|) + e^(-2 (1/4 e^(-2x) + C1))yp(x) = 2 e^(-u1(x)) + e^(-1/2 e^(-2x) - 2C1).

Therefore option A, i.e. -ln2, is the correct answer.

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For y = 5sin(5x), P5(x) = 5sin(15) + 25cos(15)(x-3) - (125sin(15)/2)(x-3)^2 - (625cos(15)/6)(x-3)^3 + (3125sin(15)/24)(x-3)^4 + (15625cos(15)/120)(x-3)^5 For y = √(2x + 1), P3(x) = √1 + (1/2√1)(2x+1) - (1/8√1)(2x+1)^2 + (1/16√1)(2x+1)^3. This polynomial is obtained by evaluating the function and its derivatives at x = 0 and using the Taylor Polynomial series formula.

For the function y = 5sin(5x), the Taylor polynomial of degree 5 near x = 3 is given by:

P5(x) = 5sin(53) + 25cos(53)(x-3) - (125sin(53)/2)(x-3)^2 - (625cos(53)/6)(x-3)^3 + (3125sin(53)/24)(x-3)^4 + (15625cos(53)/120)(x-3)^5

This polynomial is obtained by evaluating the function and its derivatives at x = 3 and using the Taylor series formula.

For the function y = √(2x + 1), the Taylor polynomial of degree 3 near x = 0 is given by:

P3(x) = √(20 + 1) + (1/2√(20 + 1))(2x+1) - (1/8√(20 + 1))(2x+1)^2 + (1/16√(20 + 1))(2x+1)^3

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The particular solution y = f(x) that satisfies the differential equation f'(x) = (3x - 4)(3x + 4) and the initial condition f(9) = 0 is f(x) = x³ - 4x² - 11x + 36.

To find the particular solution, we integrate the right-hand side of the differential equation to obtain f(x).

Integrating (3x - 4)(3x + 4), we expand the expression and integrate term by term:

∫ (3x - 4)(3x + 4) dx = ∫ (9x² - 16) dx = 3∫ x² dx - 4∫ dx = x³ - 4x + C

where C is the constant of integration.

Next, we apply the initial condition f(9) = 0 to find the value of C. Substituting x = 9 and f(9) = 0 into the particular solution, we get:

0 = (9)³ - 4(9)² - 11(9) + 36

Solving this equation, we find C = 81 - 324 - 99 + 36 = -306.

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The first few terms of the power series for the product h(x) = f(x) · g(x) are co = 12, C1 = 44, C2 = 31, C3 = 69.

Given information: Suppose that f(x) and g(x) are given by the power series f(x) = 2 + 7x + 7x2 + 2x3 +...andg(x) = 6 + 2x + 5x2 + 2x3 + ...

Product of two power series means taking the product of each term of one power series with each term of another power series. Then we add all those products whose power of x is the same. Therefore, we can get the first few terms of the product h(x) = f(x) · g(x) as follows:

The product of the constant terms of f(x) and g(x) is the constant term of h(x) as follows:co = f(0) * g(0) = 2 * 6 = 12The product of the first term of f(x) with the constant term of g(x) and the product of the constant term of f(x) with the first term of g(x) is the coefficient of x in the second term of h(x) as follows:

C1 = f(0) * g(1) + f(1) * g(0) = 2 * 2 + 7 * 6 = 44The product of the first term of g(x) with the constant term of f(x), the product of the second term of f(x) with the second term of g(x), and the product of the constant term of f(x) with the first term of g(x) is the coefficient of x2 in the third term of h(x) as follows:

C2 = f(0) * g(2) + f(1) * g(1) + f(2) * g(0) = 2 * 5 + 7 * 2 + 7 * 2 = 31The product of the first term of g(x) with the second term of f(x), the product of the second term of g(x) with the first term of f(x), and the product of the third term of f(x) with the constant term of g(x) is the coefficient of x3 in the fourth term of h(x) as follows:

C3 = f(0) * g(3) + f(1) * g(2) + f(2) * g(1) + f(3) * g(0) = 2 * 2 + 7 * 5 + 7 * 2 + 2 * 6 = 69

Therefore, the first few terms of the series for the product h(x) = f(x) · g(x) are co = 12, C1 = 44, C2 = 31, C3 = 69.

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The work done by the vector field F = (-2z, 3y, 2) in moving an object along C in the positive direction is 4π - 3.

To find the work done by the vector field F = (-2z, 3y, 2) in moving an object along C in the positive direction, where C is given by r(t) = (t, sin(t), cos(t)) for 0 ≤ t ≤ 2π, we can use the line integral formula:

Work = ∫[F(r(t)) · r'(t)] dt

where F(r(t)) is the vector field evaluated at the position vector r(t) and r'(t) is the derivative of the position vector with respect to t.

First, let's find the derivative of the position vector:

r'(t) = (1, cos(t), -sin(t))

Next, evaluate F(r(t)):

F(r(t)) = (-2cos(t), 3sin(t), 2)

Now, calculate the dot product:

F(r(t)) · r'(t) = (-2cos(t), 3sin(t), 2) · (1, cos(t), -sin(t))

              = -2cos(t) + 3sin(t) + 2

Finally, evaluate the line integral:

Work = ∫[-2cos(t) + 3sin(t) + 2] dt

To calculate the definite integral over the given interval [0, 2π], we integrate term by term:

Work = ∫[-2cos(t)] dt + ∫[3sin(t)] dt + ∫[2] dt

     = -2sin(t) - 3cos(t) + 2t

Evaluate the definite integral:

Work = [-2sin(t) - 3cos(t) + 2t] evaluated from t = 0 to t = 2π

Plugging in the values:

Work = [-2sin(2π) - 3cos(2π) + 2(2π)] - [-2sin(0) - 3cos(0) + 2(0)]

Since sin(2π) = sin(0) = 0 and cos(2π) = cos(0) = 1, we have:

Work = [0 - 3(1) + 4π] - [0 - 3(1) + 0]

     = 4π - 3

Therefore, the work done by the vector field F = (-2z, 3y, 2) in moving an object along C in the positive direction is 4π - 3.

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The distance from the origin to the point is 5, and the angle between the positive x-axis and the line connecting the origin to the point is 57 degrees.

To plot the point, start at the origin (0, 0) and move 5 units in the direction of the angle, which is 57 degrees counterclockwise from the positive x-axis. This will take us to the point (5, 57) in polar coordinates. The corresponding Cartesian coordinates can be found by converting from polar coordinates to rectangular coordinates. Using the formulas x = r * cos(theta) and y = r * sin(theta), where r is the distance from the origin and theta is the angle, we have x = 5 * cos(57 degrees) and y = 5 * sin(57 degrees). Evaluating these expressions, we find x ≈ 2.694 and y ≈ 4.016. Therefore, the corresponding Cartesian coordinates are approximately (2.694, 4.016).

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By considering the given integral, the substitution to rewrite the integrand as dx X = dx = 1) ou du is -e((4 - x) / 8) + C.

To provide a clear answer, let's use the provided information:

1. First, we'll rewrite the integral using substitution. Let x = 4 - 8u, then dx = -8 du.

2. Next, we need to solve for u in terms of x. Since x = 4 - 8u, we get u = (4 - x) / 8.

3. Now, we can substitute x and dx back into the integral:

∫ e(u) du = ∫ e((4 - x) / 8) x (-1/8) dx.

4. We can now evaluate the integral:

∫ e((4 - x) / 8) x (-1/8) dx = (-1/8) ∫ e((4 - x) / 8) dx.

5. Integrating e((4 - x) / 8) with respect to x, we get:

(-1/8) x 8 x e((4 - x) / 8) + C = -e((4 - x) / 8) + C.

So, the final answer is:

-e((4 - x) / 8) + C

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The number of teachers who teach at both Albers Elementary School and Bothel Elementary School is 19.

Let's assume the number of teachers who teach at both schools is 'x'. According to the given information, Albers Elementary School has 39 teachers and Bothel Elementary School has 84 teachers. The total number of teachers at both schools combined is 104.

We can set up an equation to solve for 'x'. The sum of the number of teachers at Albers and Bothel should be equal to the total number of teachers: 39 + 84 - x = 104. Simplifying the equation, we get 123 - x = 104. By subtracting 123 from both sides, we find -x = -19. Multiplying both sides by -1 gives us x = 19.

Therefore, the number of teachers who teach at both Albers Elementary School and Bothel Elementary School is 19.

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Step-by-step explanation:

73  and 496/1000   =   73 . 496

The missing term in the geometric sequence with a = 7,211 and r = 7340032 can be determined as -1977326741256416.

In a geometric sequence, each term is obtained by multiplying the previous term by a common ratio (r). Given the first term (a) as 7,211 and the common ratio (r) as 7340032, we can find any term in the sequence using the formula:

Tn = a * r^(n-1)

Since the missing term is denoted as T4, we substitute n = 4 into the formula and calculate:

T4 = 7211 * 7340032^(4-1)

= 7211 * 7340032^3

= -1977326741256416

Therefore, the missing term in the sequence is -1977326741256416.

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The expression ∫(x² + sin x) cos a dr can be simplified to x² sin x - 2x cos x - 2 sin x + C, where C is the constant of integration.

To find the integral of the expression ∫(x² + sin x) cos a dr, we can break it down into two separate integrals using the linearity property of integration.

The integral of x² cos a dr can be calculated by treating a as a constant and integrating term by term. The integral of x² with respect to r is (1/3) x³, and the integral of cos a with respect to r is sin a multiplied by r. Therefore, the integral of x² cos a dr is (1/3) x³ sin a.

Similarly, the integral of sin x cos a dr can be calculated by treating a as a constant. The integral of sin x with respect to r is -cos x, and multiplying it by cos a gives -cos x cos a.

Combining both integrals, we have (1/3) x³ sin a - cos x cos a. Since the constant of integration can be added to the result, we denote it as C. Therefore, the final answer is x² sin x - 2x cos x - 2 sin x + C.

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the antiderivative of 3x^2 + 4x + 12 is x^3 + 2x^2 + 12x + C.

To label the given functions and find the antiderivative, let's break down the problem as follows:

1. Label the functions as f(x), f'(x), f''(x), and f'''(x):

- f(x) refers to the original function.

- f'(x) represents the first derivative of f(x).

- f''(x) represents the second derivative of f(x).

- f'''(x) represents the third derivative of f(x).

Since the specific functions are not provided in your question, I cannot label them without more information. Please provide the functions, and I'll be happy to help you label them accordingly.

2. Find the antiderivative of 3x^2 + 4x + 12:

To find the antiderivative, we use the power rule of integration. Each term is integrated separately, applying the power rule:

∫(3x^2 + 4x + 12)dx = ∫3x^2 dx + ∫4x dx + ∫12 dx

                    = x^3 + 2x^2 + 12x + C,

where C is the constant of integration.

Therefore, the antiderivative of 3x^2 + 4x + 12 is x^3 + 2x^2 + 12x + C.

Note: The bonus question is worth 1 mark, and I have provided the antiderivative as requested.

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To find the average length of the skid marks for cars weighing between 2,100 and 3,000 pounds and traveling at speeds between 45 and 55 miles per hour, we need to set up a double integral and evaluate it.

Let's set up the double integral over the given range. The average length of the skid marks can be calculated by finding the average value of the function L(x, y) = 0.0000133xy^2 over the specified weight and speed ranges.

We can express the weight range as 2,100 ≤ x ≤ 3,000 pounds and the speed range as 45 ≤ y ≤ 55 miles per hour.

The double integral is given by:

∬R L(x, y) dA

Where R represents the rectangular region defined by the weight and speed ranges.

Now, we need to evaluate this double integral to find the average length of the skid marks. However, without specific limits of integration, it is not possible to provide a numerical value for the integral.

To complete the calculation and find the average length of the skid marks, we would need to evaluate the double integral using appropriate numerical methods, such as numerical integration techniques or software tools.

Please note that the specific limits of integration are missing in the given information, which prevents us from providing a precise numerical answer.

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The function f(x) defined as f(3) = 16 for x < 3 and f(3) = k for x > 3 is continuous for k = 16.

For a function to be continuous at a point, the limit of the function as x approaches that point from both sides should exist and be equal. In this case, the function is defined differently for x < 3 and x > 3, but the continuity at x = 3 depends on the value of k.

For x < 3, f(x) is defined as 16. As x approaches 3 from the left side (x < 3), the value of f(x) remains 16. Therefore, the left-hand limit of f(x) at x = 3 is 16.

For x > 3, f(x) is defined as k. As x approaches 3 from the right side (x > 3), the value of f(x) should also be k to ensure continuity. Therefore, k must be equal to 16 in order for the function to be continuous at x = 3.

Hence, the function f(x) is continuous when k = 16.

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The limit of the expression as x approaches infinity is -2. a) There are no points of discontinuity for this function and b) The points of discontinuity for the function f(x) = (x² + 5x + 6) / (2x² + 5x - 3) are x = -3/2 and x = 1/2.

To find the limit of the given expression, we need to evaluate it as x approaches a certain value. Let's calculate the limit.

lim(x->∞) (5 - 4x²) / (5x² – 3x² + 6x - 4)

First, let's simplify the expression:

lim(x->∞) (5 - 4x²) / (2x² + 6x - 4)

Next, let's divide both the numerator and denominator by the highest power of x, which is x²:

lim(x->∞) (5/x² - 4) / (2 + 6/x - 4/x²)

As x approaches infinity, the terms with 1/x or 1/x² become negligible. So we can simplify the expression further:

lim(x->∞) (0 - 4) / (2 + 0 - 0)

lim(x->∞) -4 / 2

lim(x->∞) -2

Therefore, the limit of the expression as x approaches infinity is -2.

Regarding the second part of your question, let's determine the points of discontinuity for the given functions.

a) f(x) = - (x + 3)

To find the points of discontinuity, we need to look for values of x where the function is undefined. In this case, the function is defined for all real values of x because there are no denominators or square roots involved. Therefore, there are no points of discontinuity for this function.

b) f(x) = (x² + 5x + 6) / (2x² + 5x - 3)

To find the points of discontinuity, we need to check if there are any values of x that make the denominator equal to zero, as division by zero is undefined.

For the given function, the denominator is 2x² + 5x - 3. To find the points of discontinuity, we set the denominator equal to zero and solve for x:

2x² + 5x - 3 = 0

Using factoring, quadratic formula, or any other method, we find that the solutions to this equation are x = -3/2 and x = 1/2.

Therefore, the points of discontinuity for the function f(x) = (x² + 5x + 6) / (2x² + 5x - 3) are x = -3/2 and x = 1/2.

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The graph that shows the solution to the system of equations in this problem is given as follows:

Second graph.

The equations that define the system of equations in this problem are given as follows:

Equaling both equations, the x-coordinate of the solution is given as follows:

-2x/3 + 1 = -2x - 1

4x/3 = -2

4x = -6

x = -1.5.

Hence the y-coordinate of the solution is given as follows:

y = -2(-1.5) - 1

y = 3 - 1

y = 2.

Hence the two lines intersect at the point (-1.5, 2), hence the second graph is the solution to the system of equations.

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