In a theatre, two attached spotlights make an angle of 100'. One shines on Ben, who is 30.6 feet away. The other shines on Mariko, who is 41.1 feet away. How far apart are Ben and Mariko?

Answer:

  3308.1 m³

Step-by-step explanation:

You want the volume of a cylinder with diameter 18 m and height 13 m.

The volume can be found using the formula ...

  V = (π/4)d²h

Using the given dimensions, this is ...

  V = (π/4)(18 m)²(13 m) ≈ 3308.1 m³

The volume of the cylinder is about 3308.1 cubic meters.

__

Additional comment

If you use 3.14 for π, the volume computes to 3306.4 m³. The 5 significant figures in the answer tell you that a 3 significant figure value for π is not appropriate.

<95141404393>

For the given exercise, the curve is a line with a positive slope that passes through the point (21, 1).

The given parametric equations represent a line in the Cartesian plane. To eliminate the parameter t, we can solve the first equation for t: t = (x - 21) / 12. Substituting this expression into the second equation, we have y = ((x - 21) / 12) + 1.

Simplifying further, we get y = (x/12) + 1/4. This equation represents a linear function with a slope of 1/12 and a y-intercept of 1/4. Thus, the curve is a line that passes through the point (21, 1) and has a positive slope, meaning it increases as x increases.

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To determine the Laplace transform of the voltage v(t) = 0.435(1 - e^(-t/RC)), where R = 212 ohms and C = 3 µFarads, we can apply the standard Laplace transform formulas.

The Laplace transform of a function f(t) is given by:

F(s) = ∫[0,∞] f(t) * e^(-st) dt

Let's calculate the Laplace transform of v(t) step by step:

1. Apply the linearity property of the Laplace transform:

L[a * f(t)] = a * F(s)

v(t) = 0.435(1 - e^(-t/RC))

v(t) = 0.435 - 0.435e^(-t/RC)

Taking the Laplace transform of each term separately:

L[0.435] = 0.435 * L[1] = 0.435/s

2. Use the exponential function property of the Laplace transform:

L[e^(-at)] = 1 / (s + a)

L[e^(-t/RC)] = 1 / (s + 1/(RC))

             = RC / (sRC + 1)

3. Apply the scaling property of the Laplace transform:

L[f(at)] = 1 / |a| * F(s/a)

L[v(t)] = 0.435/s - 0.435 / (sRC + 1)

Finally, substitute the values R = 212 ohms and C = 3 µFarads:

L[v(t)] = 0.435/s - 0.435 / (s(212 * 3 * 10^(-6)) + 1)

        = 0.435/s - 0.435 / (0.000636s + 1)

Therefore, the Laplace transform of the given voltage function v(t) is:

V(s) = 0.435/s - 0.435 / (0.000636s + 1)

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At $x=2$, where $y''=0$, the graph of $y=(x-2)^3$ has an inflection point.

To find when $y'$ is zero, we need to find the values of $x$ that make the derivative $y'$ equal to zero.

First, let's find the derivative of $y=(x-2)^3$ with respect to $x$:

$y' = 3(x-2)^2$

Setting $y'$ equal to zero and solving for $x$:

$3(x-2)^2 = 0$

$(x-2)^2 = 0$

Taking the square root of both sides:

$x-2 = 0$

$x = 2$

Therefore, $y'$ is equal to zero when $x=2$.

Now, let's sketch the graph of $y=(x-2)^3$ over the interval $-4 \leq x \leq 4$:

We can start by finding the $x$-intercept and $y$-intercept of the graph:

$x$-intercept: When $y=0$, we have $(x-2)^3=0$, which means $x-2=0$, and thus $x=2$. So the graph cuts the $x$-axis at $(2, 0)$.

$y$-intercept: When $x=0$, we have $y=(-2)^3=-8$. So the graph cuts the $y$-axis at $(0, -8)$.

Based on this information, we can plot these points on the graph.

Now, let's analyze the point where $y''=0$:

To find $y''$, we need to take the derivative of $y' = 3(x-2)^2$:

$y'' = 6(x-2)$

Setting $y''$ equal to zero and solving for $x$:

$6(x-2) = 0$

$x-2 = 0$

$x = 2$

Therefore, at $x=2$, where $y''=0$, the graph of $y=(x-2)^3$ has an inflection point.

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N = Vg(X, y, 2) at the normal vector N at (3, 2, 1) is (2, 3, 6) . The symmetric equation of the line I passing through (3, 2, 1) in the direction of N is x - 3/2 = y - 2/3 = z - 1/6. The canonical equation of the plane through (3, 2, 1) is 2x + 3y + 6z = 20.

The function g(X, Y, 2) is equal to xyz - 6. By substituting X = 3, Y = 2, and Z = 1, we find that g(3, 2, 1) = 0. The normal vector N of the function at (3, 2, 1) is (2, 3, 6). The symmetric equation of the line I passing through (3, 2, 1) in the direction of N is x - 3/2 = y - 2/3 = z - 1/6. The canonical equation of the plane through (3, 2, 1) that is normal to M is 2x + 3y + 6z = 20. Given the function g(X, Y, 2) = xyz - 6, we can substitute X = 3, Y = 2, and Z = 1 to find g(3, 2, 1). Plugging in these values gives us 3 * 2 * 1 - 6 = 0. Therefore, g(3, 2, 1) equals 0.

To find the normal vector N at (3, 2, 1), we take the partial derivatives of g with respect to each variable: ∂g/∂X = YZ, ∂g/∂Y = XZ, and ∂g/∂Z = XY. Substituting X = 3, Y = 2, and Z = 1, we obtain ∂g/∂X = 2, ∂g/∂Y = 3, and ∂g/∂Z = 6. Therefore, the normal vector N at (3, 2, 1) is (2, 3, 6). The symmetric equation of a line passing through a point (3, 2, 1) in the direction of the normal vector N can be written as follows: x - 3/2 = y - 2/3 = z - 1/6.

To find the canonical equation of the plane through (3, 2, 1) that is normal to the normal vector N, we use the point-normal form of a plane equation: N · (P - P0) = 0, where N is the normal vector, P is a point on the plane, and P0 is the given point (3, 2, 1). Substituting the values, we have 2(x - 3) + 3(y - 2) + 6(z - 1) = 0, which simplifies to 2x + 3y + 6z = 20. This is the canonical equation of the desired plane.

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The solution to the initial value problem is:

r(t) = [(1/3)t^3 + (3/2)t^2 + C1]i + (81t + C2)j + (51t + C3)k

where C1, C2, and C3 are constants determined by the initial condition.

To solve the initial value problem, we need to integrate the given differential equation with respect to t and apply the initial condition.

The differential equation is:

dr/dt = (t^2 + 3t)i + 81j + 51k

To solve this, we integrate each component of the equation separately:

∫dr/dt dt = ∫(t^2 + 3t)i dt + ∫81j dt + ∫51k dt

Integrating the first component:

∫dr/dt dt = ∫(t^2 + 3t)i dt

=> r(t) = ∫(t^2 + 3t)i dt

Using the power rule of integration, we have:

r(t) = [(1/3)t^3 + (3/2)t^2 + C1]i

Here, C1 is the constant of integration.

Integrating the second component:

∫81j dt = 81t + C2

Here, C2 is another constant of integration.

Integrating the third component:

∫51k dt = 51t + C3

Here, C3 is another constant of integration.

Combining all the components, we get the general solution:

r(t) = [(1/3)t^3 + (3/2)t^2 + C1]i + (81t + C2)j + (51t + C3)k

To apply the initial condition, we substitute t = 0 and set r(0) equal to the given initial condition:

r(0) = [(1/3)(0)^3 + (3/2)(0)^2 + C1]i + (81(0) + C2)j + (51(0) + C3)k

= C1i + C2j + C3k

Since r(0) is given as 7, we have:

C1i + C2j + C3k = 7

Therefore, the solution to the initial value problem is:

r(t) = [(1/3)t^3 + (3/2)t^2 + C1]i + (81t + C2)j + (51t + C3)k

where C1, C2, and C3 are constants determined by the initial condition.

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The probability of being dealt a specific hand consisting of the 8, 9, 10, jack, and queen, all of the same suit, from a standard 52-card deck can be calculated as follows:

First, we determine the number of ways this hand can be obtained. There are four suits in a deck, so we have four options for the suit. Within each suit, there is only one combination of the 8, 9, 10, jack, and queen. Therefore, there is a total of 4 possible combinations.

Next, we calculate the total number of possible 5-card hands that can be dealt from a 52-card deck. This can be calculated using combinations, denoted as "52 choose 5." The formula for combinations is given by nCr = n! / (r!(n-r)!), where n represents the total number of items and r represents the number of items to be chosen. For this case, we have 52 cards to choose from, and we want to select 5 cards.

Using the formula, we have 52! / (5!(52-5)!), which simplifies to 52! / (5!47!). After evaluating this expression, we find that there are 2,598,960 possible 5-card hands.

Finally, we calculate the probability by dividing the number of ways the specific hand can be obtained by the total number of possible 5-card hands. In this case, the probability is 4 / 2,598,960, which can be further simplified if necessary.

In summary, the probability of being dealt the specific hand of the 8, 9, 10, jack, and queen, all of the same suit, from a standard 52-card deck is 4/2,598,960. This probability is calculated by determining the number of ways the hand can be obtained and dividing it by the total number of possible 5-card hands from the deck.

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a. The sum 5 + 6 + 7 + 8 + 9 can be expressed using sigma notation as:∑(k = 5 to 9) k

b. The sum 6 + 12 + 18 + 24 + 30 + 36 can be expressed using sigma notation as:

∑(k = 1 to 6) (6k)

c. The sum 10 + 20 + 30 + ... + 280 + 380 + 480 can be expressed using sigma notation as:

∑(k = 1 to 8) (10k)

d. The sum 1/4 + 1/5 + 1/6 + 1/7 + ... + 1/9 can be expressed using sigma notation as:

∑(k = 4 to 9) (1/k)

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Answer:

2.45 in^2

Step-by-step explanation:

So first, we need to find the area of circle.

A = π(r)^2 is the formula

The radius is 1/2 the diameter, so 5/2 = 2.5 in. Plug that bad boy in:

A = π(2.5)^2

(2.5)^2 = 6.25 in

A = π x 6.25 = 19.63 in^2 (Rounded to the hundredths place)

Now since we have 8 equal pieces, divide the total area by 8.

19.63/8 = 2.45 in^2

 The given information states that the derivative function f'(a) = -3r², and based on this derivative, the original function f(x) must have been of the form f(x) = r³ + c, where c is a constant. This is because the derivative of r³ is 3r². In other words, if f(x) = r³ + c, then f'(x) = 3x².

The derivative function, f'(a) = -3r², suggests that the original function, f(x), must have been obtained by taking the derivative of r³ with respect to x. By applying the power rule of differentiation, we find that the derivative of r³ is 3r².Therefore, the original function f(x) is of the form f(x) = r³ + c, where c is a constant. Adding a constant term c to the function does not change its derivative, as constants have a derivative of zero. So, by adding the constant c to the function, we still have the same derivative as given, which is f'(x) = 3x².
In summary, based on the given derivative function f'(a) = -3r², we can conclude that the original function f(x) must have been of the form f(x) = r³ + c, where c is a constant. This is because the derivative of r³ is 3r². The addition of the constant term does not affect the derivative.

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The price p of a clock radio can be expressed as [tex]p = (5000 - x) / 50[/tex] in terms of the demand x. The domain of this function represents the possible values for the demand x, which is [tex]x \leq 5000[/tex] .

To express the price p in terms of the demand x, we rearrange the given equation [tex]x = 5000 - 50p[/tex] . First, we isolate the term [tex]-50p[/tex] by subtracting 5000 from both sides, resulting in [tex]-50p = -x + 5000[/tex]. Next, we divide both sides of the equation by -50 to solve for p, which gives [tex]p = (5000 - x) / 50[/tex].

This expression allows us to find the price p for a given demand x. It indicates that the price is determined by subtracting the demand from 5000 and then dividing the result by 50.

As for the domain of this function, it represents the possible values for the demand x. Since the demand cannot exceed the total available quantity of clock radios (5000 units), the domain of the function is [tex]x \leq 5000[/tex] . Thus, the function is defined for demand values up to and including 5000.

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The correct statement is c) The full outer natural join has 4 rows.

What is join?

A join is performed by specifying a join condition that determines how the tables are connected.

To compute the full outer natural join, left outer natural join, and right outer natural join between relations R(A, B) and S(B, C, D), we need to compare the values in the common attribute B and combine the matching rows from both relations.

Here are the computations for each join:

Full Outer Natural Join on B:

The full outer natural join combines all rows from both relations R and S, including matching and non-matching rows on attribute B.

Result:

A | B | C | D

1 | 2 | NULL | NULL

3 | 4 | 5 | 1

5 | 6 | 7 | 2

NULL | 8 | 9 | 3

Number of rows: 4

Number of NULL's: 2

Left Outer Natural Join on B:

The left outer natural join combines all rows from relation R with matching rows from relation S on attribute B.

Result:

A | B | C | D

1 | 2 | NULL | NULL

3 | 4 | 5 | 1

5 | 6 | 7 | 2

Number of rows: 3

Number of NULL's: 1

Right Outer Natural Join on B:

The right outer natural join combines all rows from relation S with matching rows from relation R on attribute B.

Result:

A | B | C | D

1 | 2 | NULL | NULL

3 | 4 | 5 | 1

5 | 6 | 7 | 2

NULL | 8 | 9 | 3

Number of rows: 4

Number of NULL's: 2

Now let's determine the correct statement:

a) The left outer natural join has 5 rows. - False, the left outer natural join has 3 rows.

b) The right outer natural join has 3 NULL's. - False, the right outer natural join has 2 NULL's.

c) The full outer natural join has 4 rows. - True, the full outer natural join has 4 rows.

d) The right outer natural join has 2 NULL's. - False, the right outer natural join has 2 NULL's.

Therefore, the correct statement is c) The full outer natural join has 4 rows.

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Answer:

  f'(x) = 1/10

Step-by-step explanation:

You want the derivative of the function f(x) = 1/10x -1/4.

The derivative is the limit ...

  [tex]\displaystyle f'(x)=\lim_{h\to0}{\dfrac{f(x+h)-f(x)}{h}}\\\\\\f'(x)=\lim_{h\to0}{\dfrac{\left(\dfrac{1}{10}(x+h)-\dfrac{1}{4}\right)-\left(\dfrac{1}{10}(x)-\dfrac{1}{4}\right)}{h}}\\\\\\f'(x)=\lim_{h\to0}{\dfrac{\dfrac{1}{10}h}{h}}\\\\\\\boxed{f'(x)=\dfrac{1}{10}}[/tex]

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The statement that is not true is D. To find the image of a vector x under T: x → Ax, we calculate the product Ax.

The given options are related to properties of the linear transformation T: R^n → R^m defined by T(x) = Ax, where A is an m × n matrix with columns V1, ..., Vn.

Option A is true because the domain of T is R^n, which means T can accept any vector x in R^n as input.

Option B is true because the range of T is the set of all possible outputs of T, which is R^m.

Option C is true because a vector b is in the range of T if and only if the equation Ax = b has a solution, which means T can map some vector x to b.

Option D is not true. The image of a vector x under T is the result of applying the transformation T to x, which is Ax. Thus, to find the image of x under T, we calculate the product Ax.

Option E is true. The range of T: x → Ax is the set of all possible outputs, which is the set of all linear combinations of the columns of A or, equivalently, the span of {V1, ..., Vn}.

Therefore, the statement that is not true is D.

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The volume of the rectangular prism would be = 385 in³.

To calculate the volume of the prism, the formula that should be used would be given below as follows:

Volume of rectangular prism;

Volume of rectangular prism;= length×width×height.

But length×width = base area

Volume = Base area × height.

where;

base area = 35in²

height = 11in

Volume = 35×11= 385 in³

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Equation of the path of the particle: y = (x-2)^2 + 2. Velocity vector at t=2: v = (4i + 4j). Acceleration vector at t=2: a = (2i + 0j)

The position of the particle is given by the vector-valued function r(t) = (t-2) i + (x^2+2) j. To find the equation of the path of the particle, we need to eliminate the parameter t. We can do this by completing the square in the y-coordinate.

The y-coordinate of r(t) is given by y = x^2 + 2. Completing the square, we get y = (x-1)^2 + 1. Therefore, the equation of the path of the particle is y = (x-2)^2 + 2.

To find the velocity vector of the particle, we need to take the derivative of r(t). The derivative of r(t) is v(t) = i + 2x j. Therefore, the velocity vector at t=2 is v = (4i + 4j). To find the acceleration vector of the particle, we need to take the derivative of v(t). The derivative of v(t) is a(t) = 2i. Therefore, the acceleration vector at t=2 is a = (2i + 0j).

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The marginal profit function is P'(x) = 2.2 - 0.1x, indicating the rate of change of profit with respect to the quantity produced.

To find the marginal profit function, we need to calculate the derivative of the profit function P(x), which is given by P(x) = R(x) - C(x).

First, we substitute the given cost and revenue functions into the profit function: P(x) = (3x - 0.05x²) - (293 + 0.8x).

Simplifying, we have P(x) = 2.2x - 0.05x² - 293.

Taking the derivative with respect to x, we get P'(x) = 2.2 - 0.1x.

Therefore, the marginal profit function is P'(x) = 2.2 - 0.1x.

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The question is -

Find the marginal profit function if cost and revenue are given by C(x) = 293 +0.8x and R(x) = 3x - 0.05x²

P'(x) = ?

The function y = (3x^2)(2^2) represents a quadratic equation, and we need to find the extreme points within the given interval. By evaluating the function at the critical points and endpoints, we can determine the absolute maximum and minimum values.

To find the extreme points of the function y = (3x^2)(2^2), we start by calculating its derivative. Taking the derivative with respect to x, we get dy/dx = 12x(2^2) = 48x. To find critical points, we set the derivative equal to zero: 48x = 0. This gives us x = 0 as the only critical point.

Next, we evaluate the function at the critical point and the endpoints of the given interval. When x = -0.5, y = (3(-0.5)^2)(2^2) = 1.5. When x = 0, y = (3(0)^2)(2^2) = 0. Finally, when x = 0.5, y = (3(0.5)^2)(2^2) = 1.5.

Comparing these values, we can conclude that the function reaches its absolute maximum of 1.5 at both x = -0.5 and x = 0.5, and its absolute minimum of 0 at x = 0 within the given interval.

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The line integral R = Scy?dx + xdy, where C is the arc of the parabola x = 4 – 42 from (-5, -3) to (0,2) is 28.

Let's have detailed explanation:

1. Rewrite the line integral:

                          R = ∫C (4 - y2)dx + xdy

2. Substitute the equations of the line segment C into the line integral:

                          R = ∫(-5,-3)->(0,2) (4 - y2)dx + xdy

3. Solve the line integral:

            R = ∫(-5,-3)->(0,2) 4dx - ∫(-5,-3)->(0,2) y2dx + ∫(-5,-3)->(0,2) xdy

            R = 4(0-(-5)) – ∫(-5,-3)->(0,2) y2dx + ∫(-5,-3)->(0,2) xdy

            R = 20 – ∫(-5,-3)->(0,2) y2dx + ∫(-5,-3)->(0,2) xdy

4. Use the Fundamental Theorem of Calculus to solve the line integrals:

                R = 20 – [y2] (−5,2) + [x] (−5,0)

                R = 20 – (−22 + 32) + (0 – (−5))

                R = 28

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To find a power series representation for the integral of x * cos(x)dx, we can use an established series such as the Taylor series expansion of cos(x).

The Taylor series expansion for cos(x) is given by: cos(x) = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + ... We can integrate term by term to obtain a power series representation for the integral of x * cos(x)dx. Integrating each term of the Taylor series for cos(x), we have: ∫ (x * cos(x))dx = ∫ (x - (x^3)/2! + (x^5)/4! - (x^7)/6! + ...)dx. Integrating term by term, we get:∫ (x * cos(x))dx = ∫ (x)dx - ∫ ((x^3)/2!)dx + ∫ ((x^5)/4!)dx - ∫ ((x^7)/6!)dx + ...

Simplifying the integrals, we have: ∫ (x * cos(x))dx = (x^2)/2 - (x^4)/4! + (x^6)/6! - (x^8)/8! + ... Therefore, the power series representation for the integral of x * cos(x)dx is: ∫ (x * cos(x))dx = (x^2)/2 - (x^4)/4! + (x^6)/6! - (x^8)/8! + ...

This power series representation provides an expression for the integral of x * cos(x)dx as an infinite series involving powers of x.

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Answer:

3m + 10 + 2m + 15 (expansion)

3m + 2m + 10 + 15 (group like terms)

5m + 25

The series ∑[tex](-1)^n[/tex](n+4)/(n(n+3)) is divergent because it does not satisfy the conditions for convergence.

To determine whether the series ∑[tex](-1)^n[/tex](n+4)/(n(n+3)) is conditionally convergent, absolutely convergent, or divergent, we need to analyze its convergence behavior.

First, we can examine the absolute convergence by taking the absolute value of each term in the series. This gives us ∑ |[tex](-1)^n[/tex](n+4)/(n(n+3))|. Simplifying further, we have ∑ (n+4)/(n(n+3)).

Next, we can use a convergence test, such as the comparison test or the ratio test, to evaluate the convergence behavior. Applying the ratio test, we find that the limit of the ratio of consecutive terms is 1.

Since the ratio test is inconclusive, we can try the comparison test. By comparing the series with the harmonic series ∑ 1/n, we observe that (n+4)/(n(n+3)) < 1/n for all n > 0.

Since the harmonic series ∑ 1/n is known to be divergent, and the given series is smaller than it, the given series must also be divergent.

Therefore, the series ∑ [tex](-1)^n[/tex](n+4)/(n(n+3)) is divergent.

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The question is -

Consider the series ∑ n = 1 to ∞ (-1)^n n+4/(n(n+3)). Is this series conditionally convergent, absolutely convergent, or divergent? Explain your answer.

To maximize its profits, Myspace.com should sell approximately 300 ads each month.The maximum point of a quadratic function P(x) = -0.04x^2 + 240x - 10,000 occurs at the vertex.

The estimated monthly profit for Myspace.com from selling advertising space is given by the equation P(x) = -0.04x^2 + 240x - 10,000, where x represents the number of ads sold each month.

To determine the number of ads that will yield maximum profit, we need to find the value of x that corresponds to the maximum point on the profit function.

To find this, we can use calculus. The maximum point of a quadratic function occurs at the vertex, which can be found using the formula x = -b / (2a), where a, b, and c are coefficients in the quadratic equation ax^2 + bx + c = 0. In our profit equation, the coefficient of x^2 is -0.04, and the coefficient of x is 240.

Using the formula, we can calculate x = -240 / (2 * -0.04) = 300. Therefore, to maximize its profits, Myspace.com should sell approximately 300 ads each month.

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The required solutions are: i) The gradient at the point (1, 2) on the curve is -4/5. ii) The equation of the tangent line to the curve going through the point (1, 2) is y = (-4/5)x + 14/5.

(i) To find the gradient at the point (1, 2) on the curve given by [tex]x^2 + xy + y^2 = 12 - 22 - y[/tex], we need to find the derivative dy/dx and evaluate it at x = 1, y = 2.

First, let's differentiate the given equation implicitly with respect to x:

[tex]d/dx (x^2 + xy + y^2) = d/dx (12 – 22 – y)[/tex]

2x + (x dy/dx + y) + (2y dy/dx) = 0

Simplifying:

2x + x dy/dx + y + 2y dy/dx = 0

Rearranging:

x dy/dx + 2y dy/dx = -2x - y

Factoring out dy/dx:

dy/dx (x + 2y) = -2x - y

Now, we can find dy/dx by dividing both sides by (x + 2y):

dy/dx = (-2x - y) / (x + 2y)

Substituting x = 1 and y = 2:

dy/dx = (-2(1) - 2) / (1 + 2(2))

      = (-4) / (1 + 4)

      = -4/5

Therefore, the gradient at the point (1, 2) on the curve is -4/5.

(ii) To find the equation of the tangent line to the curve going through the point (1, 2), we have the point (1, 2) and the slope (-4/5) from part (i).

Using the point-slope form of the equation of a line:

y - y₁ = m(x - x₁)

where (x₁, y₁) is the given point and m is the slope, we can substitute the values:

y - 2 = (-4/5)(x - 1)

Simplifying:

y - 2 = (-4/5)x + 4/5

y = (-4/5)x + 14/5

Therefore, the equation of the tangent line to the curve going through the point (1, 2) is y = (-4/5)x + 14/5.

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Answer:

Step-by-step explanation:

This is an answer.

To find the matrix P that transforms a vector from the C basis to the B basis, we need to express the vector [c]C in terms of the B basis.

We have the C basis vector[tex][c]C = [3 -2][/tex] and we want to find the coefficients x and y such that[tex][c]C = x * [2 0] + y * [0 1].[/tex]

Setting up the equations, we have:

[tex]3 = 2x-2 = y[/tex]

Solving these equations, we find x = 3/2 and y = -2.

Therefore, the matrix P is given by:

[tex]P = [3/2 0][-2 1][/tex]

This means that for any vector [c]C in R2, we can find its equivalent representation [c]B in the B basis by multiplying it with the matrix P: [c]B = P * [c]C.

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The radius of the given cylindrical tank is 82.2 centimeter.

a) Here, volume = 3500 L

We know that 1 L = 1000 cm³

Now, 3500 L = 3500000 cm³

Height (cm) = 165 cm

We know that, the volume of the cylinder = πr²h

3500000 = 3.14×r²×165

r² = 3500000/518.1

r² = 6755.45

r = √6755.45

r = 82.2 cm

Therefore, the radius of the given cylindrical tank is 82.2 centimeter.

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To compute the integral of (2x + 1) / ((x - 1)(x - 28)), we can use the method of partial fractions. The first step is to factorize the denominator of the rational function.

Factoring the denominator (x - 1)(x - 28), we have: (x - 1)(x - 28) = (2 - 1)(x - 1)(x - 28) = (2 - a)(x - 1)(x - 28), where a is a constant that we need to determine. By equating the numerators of both sides, we have: 2x + 1 = A(x - 1)(x - 28), where A is a constant that we need to determine as well.

To find the value of A, we can simplify the right side of the equation by expanding the terms: A(x - 1)(x - 28) = A(x^2 - 29x + 28) . Now, equating the coefficients of like terms on both sides of the equation, we have: 2x + 1 = Ax^2 - 29Ax + 28A. Comparing the coefficients of x^2, x, and the constant term, we get: A = 2 (coefficient of x), -29A = 0 (coefficient of x), 28A = 1 (constant term). From the second equation, we have -29A = 0, which implies A = 0 since -29 ≠ 0. However, this contradicts the third equation where 28A = 1, indicating that there is no value of A that satisfies both equations simultaneously.

Therefore, the partial fraction decomposition cannot be performed in this case, and the integral (2x + 1) / ((x - 1)(x - 28)) cannot be evaluated using partial fractions.

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The measure of angle DGE formed by the intersection of chord AG and DG is determined as 26⁰.

The value of angle DGE is calculated by applying intersecting chord theorem, which states that the angle at tangent is half of the arc angle of the two intersecting chords.

From the given diagram we can infer the following;

If point C is the center of the circle, then arc AFB = 180⁰ (sum of angles in a semi circle)

If point E is the midpoint of line DF, then arc BF = arc BD = 64⁰

arc FA = 180 - 64⁰

arc FA = 116⁰

The value of arc AD is calculated as follows;

AD + BD + BF + FA = 360 (sum of angles in a circle)

AD + 64 + 64 + 116⁰ = 360

AD + 244 = 360

AD = 360 - 244

AD = 116⁰

The measure of angle DGE is calculated as follows;

m∠DGE = ¹/₂ (arc AD - arc BD) (exterior angle of intersecting secants)

m∠DGE = ¹/₂ ( 116 - 64 )

m∠DGE = ¹/₂ ( 52 )

m∠DGE = 26⁰

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The average value of the function f(x) = 4x⁵ over the interval [25,54] is 1814437900/29.

The region bounded by y=6x², x=2, x=3, and y=0 is rotated around the x-axis. To determine the volume of the resulting solid, we'll use the washer method.

The shaded region's horizontal cross-section is shown in the figure. As a result, a washer is formed. The radius of the washer is determined by the value of x, and it is given by 6x². The washer's thickness is determined by dy, which ranges from 0 to 6x².

Volume is found by integrating from 0 to 6x² using the washer method for slicing solid formed by rotating the region bounded by y=6x², x=2, x=3

and y=0 about the x-axis.

V = π∫ from a to b [R(x)²-r(x)²]dxwhere R(x)

= Outer Radius and r(x)

= Inner RadiusV = π∫ from 2 to 3 [(6x²)²-(0)²]dx= 108π cubic units.

3. VolumeThe function y = e^1x+3, y = 0, x = 0, and x = 0.4, when rotated around the x-axis, encloses a region whose volume can be calculated using the washer method.

The region's cross-section is a washer whose inner radius is zero (since the region extends to the x-axis) and whose outer radius is e⁽¹ˣ⁺³⁾.

The volume of the solid is calculated using the following integral:

V = π ∫a to b [R(x)²-r(x)²]dx= π ∫0 to 0.4 [(e¹ˣ+3)²-0²]dx= π ∫0 to 0.4 (e⁽²ˣ⁺⁶⁾)dx= 16.516π cubic units.4. Average value of the function

The average value of a function f(x) over an interval [a,b] is given by the formula

The average value of a function f(x) over an interval [a,b] = 1/(b-a) ∫a to b f(x)dx

Given that the interval is [25,54], and the function is f(x) = 4x⁵.

The average value of the function f(x) over the interval [25,54] is given by= 1/(54-25) ∫25 to 54 (4x⁵)dx= 1/29 [(4/6) (54^6-25⁶)]

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