Use Greens Theorem to find the counterclockwise circulation and outward flux for the field F = (6y2 - x2)i - (x2 +6y2)j and curve C: the tsquare bounded by y = 0, x= 0, and y = x. The flux is_____ .

The second partial derivatives of the following function, so the mixed partial derivatives of f(x,y) are equal.

To find the second partial derivatives of f(x,y) = ln(1+xy), we first need to find the first partial derivatives:

f_x = (1/(1+xy)) * y

f_y = (1/(1+xy)) * x

To find the second partial derivatives, we differentiate each of these partial derivatives with respect to x and y:

f_xx = -y/(1+xy)^2

f_xy = 1/(1+xy) - y/(1+xy)^2

f_yx = 1/(1+xy) - x/(1+xy)^2

f_yy = -x/(1+xy)^2

To show that the mixed derivatives f_xy and f_yx are equal, we can compare their expressions:

f_xy = 1/(1+xy) - y/(1+xy)^2

f_yx = 1/(1+xy) - x/(1+xy)^2

We can see that these expressions are equal, so:

f_xy = f_yx

Therefore, the mixed partial derivatives of f(x,y) are equal.

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Answer:

Each side of square ACEG has length 10 since √(6^2 + 8^2) = √(36 + 64) = √100 = 10, so the area of square ACEG is 100.

The solution for the initial value problem [tex]y'' + 2y' + y = t^2 \;with \;y(0) = y'(0) = 1[/tex] is found by finding the general solution. The particular solution is found using the method of undetermined coefficients. The final solution is [tex]y(t) = e^{(-t)} + (1/2)t^2 - (1/2)t + 1/2.[/tex]

To solve the initial value problem [tex]y'' + 2y' + y = t^2 \;with \;y(0) = y'(0) = 1[/tex], we first find the characteristic equation by assuming [tex]y = e^{(rt)}[/tex] as a solution.

Plugging this into the differential equation gives us [tex]r^{2e}^{(rt)} + 2re^{(rt)} + e^{(rt)} = 0[/tex], which simplifies to [tex](r+1)^2 = 0[/tex]. Therefore, r = -1 is a repeated root, and the general solution is [tex]y(t) = (c1 + c2t)e^{(-t)}[/tex].

To find the particular solution, we use the method of undetermined coefficients and assume [tex]y(t) = At^2 + Bt + C[/tex]. Plugging this into the differential equation gives us [tex]2At + 2B + At^2 + 2At + Bt + C = t^2[/tex].

Equating coefficients, we get A = 1/2, B = -1/2, and C = 1/2. Thus, the particular solution is [tex]y(t) = (1/2)t^2 - (1/2)t + 1/2[/tex].

The general solution is [tex]y(t) = (c1 + c2t)e^{(-t)} + (1/2)t^2 - (1/2)t + 1/2[/tex]. Using the initial conditions y(0) = 1 and y'(0) = 1, we get c1 = 1 and c2 = 1. Therefore, the solution to the initial value problem is[tex]y(t) = e^{(-t)} + (1/2)t^2 - (1/2)t + 1/2.[/tex]

In summary, we solved the initial value problem [tex]y'' + 2y' + y = t^2 \;with \;y(0) = y'(0) = 1[/tex] by finding the general solution and particular solution using the method of undetermined coefficients and then applying the initial conditions to solve for the constants.

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a. To find the average angular velocity, we need to calculate the change in angle over the change in time between t=2.0s and t=3.1s.

Δθ = θ2 - θ1 = (a(3.1) - b(3.1)^2 + c(3.1)^4) - (a(2.0) - b(2.0)^2 + c(2.0)^4)

Δt = t2 - t1 = 3.1 - 2.0 = 1.1

The average angular velocity is:

ω(avg) = Δθ/Δt = ((a(3.1) - b(3.1)^2 + c(3.1)^4) - (a(2.0) - b(2.0)^2 + c(2.0)^4))/1.1

b. To find the average angular acceleration, we need to calculate the change in angular velocity over the change in time between t=2.0s and t=3.1s.

Δω = ω2 - ω1 = ((a(3.1) - b(3.1)^2 + c(3.1)^4) - (a(2.0) - b(2.0)^2 + c(2.0)^4))/1.1 - ((a(2.0) - b(2.0)^2 + c(2.0)^4) - (a(1.0) - b(1.0)^2 + c(1.0)^4))/1.0

Δt = t2 - t1 = 3.1 - 2.0 = 1.1

The average angular acceleration is:

α(avg) = Δω/Δt = (((a(3.1) - b(3.1)^2 + c(3.1)^4) - (a(2.0) - b(2.0)^2 + c(2.0)^4))/1.1 - ((a(2.0) - b(2.0)^2 + c(2.0)^4) - (a(1.0) - b(1.0)^2 + c(1.0)^4))/1.0)/1.1

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After performing the calculation, we find that there are 85,900,584 possible combinations of 7 numbers out of the 49 available in the Washington Lotto.

In the Washington Lotto, there are 49 possible numbers and you need to choose 7 of them. As you mentioned, the order does not matter, so we will use the concept of combinations to find the number of possible outcomes. In general, the number of combinations of n items taken r at a time is given by the formula:

C(n, r) = n! / (r!(n-r)!)

In this case, we have n = 49 and r = 7, so we can plug these values into the formula:

C(49, 7) = 49! / (7!(49-7)!)

Calculating the factorials and simplifying, we get:

C(49, 7) = 49! / (7!42!)

After performing the calculation, we find that there are 85,900,584 possible combinations of 7 numbers out of the 49 available in the Washington Lotto. Remember that this assumes the order of the numbers drawn does not matter, so each unique combination of 7 numbers is considered a single possibility.

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Answer:

Step-by-step explanation:

Answer:

The distance measure is 4.2 ft

Step-by-step explanation:

Here, we want to get the distance from the bottom of the ladder to the side of the house

To get this, we can see that what we have is a right-angled triangle

We need to apply the appropriate trigonometric identity to get the value of what we want

Let us call the measure we want to calculate x

Mathematically;

Tan 72 = 13/x

x = 13/tan 72

x = 4.2 ft

Answer:

Step-by-step explanation:

Have you heard of PEMDAS

If twice the sum of the number of marbles in a bag and 2 is 14 more than the number of marbles in the bag then there are 10  marbles.

we can use algebraic equations to solve problems like these by assigning a variable to represent the unknown quantity and setting up an equation based on the given information. We then simplify the equation and isolate the variable to find the solution.

To solve this problem, we can use algebraic equations. Let's start by assigning a variable to represent the number of marbles in the bag, say "x".

According to the problem, twice the sum of the number of marbles in the bag and 2 is 14 more than the number of marbles in the bag. This can be written as:

2(x + 2) = x + 14

We can simplify this equation by distributing the 2 on the left side, which gives us:

2x + 4 = x + 14

Next, we can isolate the variable on one side by subtracting x and 4 from both sides, which gives us:

x = 10

Therefore, there are 10 marbles in the bag.

Let x represent the number of marbles in the bag. According to the problem, twice the sum of the number of marbles and 2 is equal to the number of marbles plus 14. We can write this as an equation: 2(x + 2) = x + 14. To solve for x, first distribute the 2: 2x + 4 = x + 14. Then, subtract x from both sides: x + 4 = 14. Finally, subtract 4 from both sides: x = 10. Therefore, there are 10 marbles in the bag.

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Absolute humidity in conjunction with other factors, such as relative humidity and dew point temperature, can help provide a comprehensive understanding of the atmospheric conditions and their implications.

The term you are looking for is "absolute humidity." Absolute humidity is the mass of water vapor in a given amount of air expressed in grams per cubic meter (g/m3). This measurement represents the actual amount of moisture present in the air, regardless of the temperature or pressure. It is essential to understand and monitor absolute humidity for various applications, such as meteorology, environmental studies, and indoor air quality management. In contrast to relative humidity, which describes the percentage of moisture in the air compared to the maximum amount it can hold at a given temperature, absolute humidity provides a more accurate and direct representation of the water vapor content in the air. By measuring absolute humidity, scientists and professionals can better assess and predict weather patterns, manage heating, ventilation, and air conditioning (HVAC) systems, and ensure optimal conditions for health and comfort. It is important to note that absolute humidity can change as air temperature and pressure change, even if the amount of water vapor remains constant. This is because the air's capacity to hold water vapor depends on its temperature and pressure.

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The exact value of the given expression. sin 2 cos−1 12 13 is 240/169.

To find the exact value of the given expression, we need to use the identity sin(2θ) = 2sin(θ)cos(θ) and the information provided. The expression you want to find is sin(2 * cos^(-1)(12/13)).

First, let's find the angle θ, which is cos^(-1)(12/13). In a right triangle with the adjacent side = 12 and hypotenuse = 13, we can use the Pythagorean theorem to find the opposite side: Opposite side = √(13^2 - 12^2) = √(169 - 144) = √25 = 5

Now, we can find sin(θ) and cos(θ): sin(θ) = opposite/hypotenuse = 5/13 cos(θ) = adjacent/hypotenuse = 12/13 Next, use the sin(2θ) identity: sin(2θ) = 2sin(θ)cos(θ) = 2(5/13)(12/13) = (10/169)(24) = 240/169 So, the exact value of the given expression is 240/169.

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Question 3: If y = [tex]x^{(2x)[/tex], then dy/dx = (d) 2y(ln x + x).

Question 4: If y = (1 + 1 + [tex]7x^2)^4[/tex], then dy/dx at x = 1 is: (c) 126.

Question 5: (a) f is increasing on (0,∞) and the minimum value of f occurs at x = 1/e is true.

Question 6: The diagonal is increasing at (c) 2 cm/sec at the moment when it is 5V2 cm long.

Question 3:

Taking the natural logarithm of both sides, we get:

ln y = 2x ln x

Differentiating with respect to x, we get:

1/y * dy/dx = 2 ln x + 2

Multiplying both sides by y, we get:

dy/dx = y * (2 ln x + 2)

Substituting y = [tex]x^{(2x)[/tex], we get:

dy/dx = [tex]x^{(2x)[/tex] * (2 ln x + 2)

Therefore, the answer is (d) 2y(ln x + x).

Question 4:

Differentiating with respect to x, we get:

dy/dx = [tex]4(1 + 7x^2)^3[/tex] * d/dx[tex](1 + 7x^2)[/tex]

Using the chain rule, we get:

dy/dx = [tex]4(1 + 7x^2)^3[/tex] * 14x

At x = 1, we have:

dy/dx = [tex]4(1 + 7)^3 * 14[/tex] = 126

Therefore, the answer is (c) 126.

Question 5:

Taking the derivative, we get:

f'(x) = 1 + ln x

Setting f'(x) = 0, we get:

ln x = -1

Solving for x, we get:

x = 1/e

Taking the second derivative, we get:

f''(x) = 1/x > 0

Therefore, f(x) has a minimum value at x = 1/e.

Therefore, the answer is (a) f is increasing on (0,∞) and the minimum value of f occurs at x = 1/e.

Question 6:

Let s be the length of the side of the square, and let d be the length of the diagonal. Then we have:

[tex]d^2 = s^2 + s^2 = 2s^2[/tex]

Differentiating with respect to time t, we get:

2d(dd/dt) = 4s(ds/dt)

Simplifying, we get:

dd/dt = 2s(ds/dt)/d

At the moment when d = 5√2 cm, we have s = d/√2 = 5 cm.

Also, we know that ds/dt = [tex]1 cm^2/sec.[/tex]

Substituting these values, we get:

dd/dt = [tex]2(5 cm)(1 cm^2/sec)[/tex]/(5√2 cm) = √2 cm/sec

Therefore, the answer is (c) 2 cm/sec.

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The additional congruency statement(s) needed to prove ∆ABC ≅ ∆FGH for the given theorem are,

A) We want to one more side for the prove of ∆ABC ≅ ∆FGH.

As, GH = CB

B) We want to one more angle for the prove of ∆ABC ≅ ∆FGH.

As, ∠FHG = ∠ACB

We have to given that;

State the additional congruency statement(s) needed to prove ∆ABC ≅ ∆FGH for the given theorem.

Here, We have to given that;

AB = FH

∠BAC = ∠HFG

Hence, For SAS congruency theorem;

We want to one more side for the prove of ∆ABC ≅ ∆FGH.

As, GH = CB

For ASA congruency theorem;

We want to one more angle for the prove of ∆ABC ≅ ∆FGH.

As, ∠FHG = ∠ACB

Thus, The additional congruency statement(s) needed to prove ∆ABC ≅ ∆FGH for the given theorem are,

A) We want to one more side for the prove of ∆ABC ≅ ∆FGH.

As, GH = CB

B) We want to one more angle for the prove of ∆ABC ≅ ∆FGH.

As, ∠FHG = ∠ACB

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The volume of the part of the ball ρ ≤ 8 that lies between the cones ϕ = π/6 and ϕ = π/3 found using spherical coordinates is approximately 416.78.

The bounds for ρ are 0 and 8, while the bounds for θ are 0 and 2π, since we want to integrate over the entire sphere. The bounds for ϕ are π/6 and π/3, since we only want to consider the part of the sphere that lies between the two cones.b

Using the formula for the volume element in spherical coordinates, we have: dV = ρ² sin ϕ dρ dϕ dθ  Integrating over the appropriate ranges, we obtain:  ∫(θ=0 to 2π) ∫(ϕ=π/6 to π/3) ∫(ρ=0 to 8) ρ² sin ϕ dρ dϕ dθ      = 2π ∫(π/6 to π/3) ∫(0 to 8) ρ² sin ϕ dρ dϕ

= 2π [8³/3 - 0] [cos(π/6) - cos(π/3)] = 512π/3 (cos(π/6) - cos(π/3))= 416.78 Therefore, the volume of the part of the ball ρ ≤ 8 that lies between the cones ϕ = π/6 and ϕ = π/3pro is approximately 416.78.

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There are over 22 billion possible license plates with the given restrictions.

To find the number of different license plates possible in 2020 with the given restrictions, we need to use the principle of multiplication.

There are a total of 26 letters in the English alphabet, but we cannot use the letters d, t, or x in the fourth position, or the letters i, o, or q in any position.

For the first three positions, we have 26 options for each position, since we can use any letter. Therefore, the number of possible combinations for the first three positions is 26 x 26 x 26 = 17,576.

For the fourth position, we cannot use the letters d, t, or x, so we have 23 options.

For the fifth position, we have 26 options again, since we can use any letter.

For the sixth position, we cannot use the letters i, o, or q, so we have 23 options.

Therefore, the total number of possible license plates is:

17,576 x 23 x 26 x 23 = 22,075,269,568

So, there are over 22 billion possible license plates with the given restrictions.

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Left by 4 units is the right response.

The table gives details on the quadratic functions f(x) and g(x). As the values for each x are different, it is clear from the table that f(x) and g(x) are not identical to one another. One of the functions needs to be adjusted in order to match f(x) and g(x).

It is clear from looking at the table that f(x) has to be moved to the left by 4 units in order to match g(x).

This can be calculated by subtracting the g(x) values from the f(x) values for each x.

For example, at x = -2, the difference between f(x) and g(x) is -32.

This difference is the same for all x values, meaning that f(x) must be shifted left by 4 units to match g(x). Thus, the correct answer is Left by 4 units.

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A figure has a surface area of 496 m². If the dimensions are doubled by half, the surface area of the new figure is 124 m².

Firstly, we will assume it being a rectangle then calculate the new area using the formula and then we will put the values of original figure into new figure.

Assume we're working with a rectangle. We know that the area equals the length (l) multiplied by the width (w).

A = l x w

If we divide the dimensions in half, we get A = (1 / 2)l x (1 / 2)w.

A = (1 / 4) × (l  x w)

As a result, the new surface area would be one-quarter of the original:

[tex]A_{original}[/tex] = 496 m²

[tex]A_{new}[/tex] = (1/4) × [tex]A_{original}[/tex]

[tex]A_{new}[/tex] = (1 / 4) × (496)

[tex]A_{new}[/tex] = 124 m²

As a result, the new area would be 124 m².

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The specific antiderivative of [tex]F(u)= ln |u|+\frac{u^2}{2} + \frac{5}{2}[/tex]

To find the specific antiderivative [tex]$F(u)$[/tex] of [tex]$f(u)=\frac{1}{u}+u$[/tex] such that [tex]$F(1)=3$[/tex].

First, we find the antiderivative of [tex]$f(u)$[/tex]:

[tex]\int\frac{1}{u}+u du=ln|u|+\frac{u^2}{2}+C[/tex]

where [tex]$C$[/tex] is the constant of integration.

Now, we can use the initial condition. [tex]$F(1)=3$[/tex] to solve for [tex]$C$[/tex]:

[tex]F(1)=ln|1|+\frac{1^2}{2}+C=\frac{1}{2}+C=3[/tex]

Solving for [tex]$C$[/tex], we get [tex]C=\frac{5}{2}$.[/tex]

The specific antiderivative [tex]$F(u)$[/tex]of [tex]$f(u)$[/tex] is:

[tex]F(u)= ln |u|+\frac{u^2}{2} + \frac{5}{2}[/tex]

To locate the precise antiderivative [tex]$F(u)$[/tex] of  [tex]$f(u)=\frac{1}{u}+u$[/tex] like that   [tex]$F(1)=3$[/tex].

The antiderivative of[tex]$f(u)$[/tex] is first discovered:

where C is the integration constant.

We may now apply the initial condition.  [tex]$F(1)=3$[/tex] to overcome C:

[tex]F(1)=ln|1|+\frac{1^2}{2}+C=\frac{1}{2}+C=3[/tex]

The specific antiderivative

[tex]F(u)= ln |u|+\frac{u^2}{2} + \frac{5}{2}[/tex]

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(a) The Laplace Transform of f(t) = te^(at) is defined as:

Lf(t) = ∫₀^∞ f(t) e^(-st) dt

Substituting f(t) into this formula, we get:

Lf(t) = ∫₀^∞ te^(at) e^(-st) dt

= ∫₀^∞ t e^((a-s)t) dt

We can integrate this expression by parts, using u = t and dv = e^((a-s)t) dt. Then du = dt and v = (1/(a-s)) e^((a-s)t).

The integral becomes:

Lf(t) = [t (1/(a-s)) e^((a-s)t)] ∣₀^∞ - ∫₀^∞ (1/(a-s)) e^((a-s)t) dt

Since e^((a-s)t) goes to 0 as t goes to infinity, the first term evaluates to 0. The second term is an integral we can easily evaluate:

Lf(t) = [-1/(a-s)] [e^((a-s)t)] ∣₀^∞

= [1/(s-a)]

Therefore, the Laplace Transform of f(t) = te^(at) is:

Lf(t) = [1/(s-a)]

(b) Let f(t) = t cosh(at) for t > 0. Using the identity cosh(at) = (1/2) (e^(at) + e^(-at)), we can express f(t) as:

f(t) = (t/2) (e^(at) + e^(-at))

Using linearity and the result from part (a), the Laplace Transform of f(t) is:

Lf(t) = (1/2) Lt e^(at) + (1/2) Lt e^(-at)

= (1/2) [1/(s-a)^2] + (1/2) [1/(s+a)^2]

= [(s^2 + a^2)/(2s^2 - 2as + 2a^2)]

Therefore, the Laplace Transform of f(t) = t cosh(at) is:

Lf(t) = [(s^2 + a^2)/(2s^2 - 2as + 2a^2)]

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The number of cylindrical container that will fill the box is approximately 10.

Each cylindrical container has a height of 8 inches and a base with a radius of 3 inches.

The box is 24 inches long, 12 inches wide, and 8 inches high.

Therefore, let's find the number of cylindrical container that will contain the boxes.

Therefore,

volume of the box = lwh

volume of the box = 24 × 12 × 8

volume of the box = 2304 inches³

volume of the cylindrical container = πr²h

where

Therefore,

volume of the cylindrical container =  3.14 × 3² × 8

volume of the cylindrical container =  3.14 × 9 × 8

volume of the cylindrical container =  226.08 inches³

Hence,

number of cylindrical containers that will fill the boxes = 2304 / 226.08 ≈ 10

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The  integral of x^2/9 + 16x^6 dx is (x^3/27) + (16x^7/7) + C, where C is the constant of integration.

To evaluate the integral of x^2/9 + 16x^6 dx, you need to find the antiderivative of each term separately and then combine the results. Here's a step-by-step explanation:

Step 1: Separate the integral into two parts:
int(x^2/9) dx + int(16x^6) dx

Step 2: Evaluate the antiderivative of the first term, x^2/9:
Using the power rule (integral of x^n is x^(n+1)/(n+1)), we have:
(1/9) * (x^3/3)

Step 3: Evaluate the antiderivative of the second term, 16x^6:
Using the power rule again, we have:
16 * (x^7/7)

Step 4: Combine the results from steps 2 and 3:
(1/9) * (x^3/3) + 16 * (x^7/7)

Step 5: Simplify the result:
(x^3/27) + (16x^7/7)

So the evaluated integral of x^2/9 + 16x^6 dx is (x^3/27) + (16x^7/7) + C, where C is the constant of integration.

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There's no point of intersection between the line connecting (2,3,6) to (2,2,3) and the line connecting (1,2,1) to (3,0,-1)

To find the point of intersection between two lines in three-dimensional space, we need to solve a system of equations. We can set up the equations of the two lines using the parametric form:

Line 1:

x = 2 + t(0)

y = 3 + t(-1)

z = 6 + t(-3)

Line 2:

x = 1 + s(2)

y = 2 - s(2)

z = 1 - s(2)

We can set the x, y, and z values equal to each other for the point of intersection, and solve for t and s:

2 + t(0) = 1 + s(2)

3 + t(-1) = 2 - s(2)

6 + t(-3) = 1 - s(2)

Simplifying the second equation, we get:

t + s = 1

Multiplying the first equation by 2, we get:

4 = 2 + 2t + 2s

Substituting t + s = 1, we get:

4 = 2 + 2(t + s)

4 = 2 + 2(1)

4 = 4

This means that our system of equations has no unique solution, and the two lines do not intersect at a single point. Therefore, there is no point of intersection between the two lines.

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On a hot summer day, you visit an ice cream shop that offers 15 ice cream flavors, 3 types of cones, and 8 toppings. If you want one ice cream flavor, one cone, and one topping, there are a total of 360 possible combinations you can create.

To find the total number of possible combinations, you simply multiply the number of options for each category. In this case, you have 15 ice cream flavors, 3 types of cones, and 8 toppings.

So, the total combinations would be: 15 (flavors) × 3 (cones) × 8 (toppings) = 360 possible combinations. You can create 360 different ice cream combinations at the shop on a hot summer day.

This is calculated by multiplying the number of ice cream flavors (15) by the number of cone types (3) and the number of toppings (8), which equals 360. So, there are plenty of delicious options to choose from to cool off on a hot summer day.

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It takes the unpowered raft 192 hours to travel the same distance between Pier A and Pier B.

Let's assume the distance between Pier A and Pier B is D.

When the motorboat is going downstream, it benefits from the current, so its effective speed is increased.

Let's say the speed of the motorboat in still water is S.

The speed of the current is C.

Therefore, the motorboat's speed downstream is S + C.

When the motorboat is going upstream, it has to overcome the current, so its effective speed is decreased.

The speed of the motorboat upstream is S - C.

We are given that the motorboat takes 32 hours to go downstream and 48 hours to return upstream.

Downstream speed: S + C

Downstream time: 32 hours

Distance = Speed × Time

D = (S + C) × 32

Upstream speed: S - C

Upstream time: 48 hours

D = (S - C) × 48

Since the distance (D) is the same in both cases, we can equate the two equations:

(S + C) × 32 = (S - C) × 48

32S + 32C = 48S - 48C

16S = 80C

S = 5C

We know that the raft is unpowered, so its speed is equal to the speed of the current (C).

Therefore, the speed of the unpowered raft is C.

Time = Distance / Speed

Time = D / C

Since D = (S + C) × 32

we can substitute the value of S:

Time = [(5C) + C] × 32 / C

Time = 6C × 32 / C

Time = 192

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Answer:

The maximum amount of water that the rectangular container will hold can be calculated by finding its volume.

The volume of a rectangular container is given by the formula:

Volume = length x width x height

In this case, the length is 30 inches, the width is 24 inches, and the height is 25.8 inches. So we have:

Volume = 30 in x 24 in x 25.8 in

Volume = 18,576 cubic inches

Therefore, the maximum amount of water that the rectangular container can hold is 18,576 cubic inches.

So the answer is 18,576 in3.

Answer:

The answer to your problem is, 18,576 [tex]ft^{3}[/tex]

Step-by-step explanation:

In order to find the area of the rectangular container. We will use the formula down below:

l × w × h

So replace step by step:

30 × w × h

30 × 24 × h

30 × 24 × 25.8

Then that will equal:

= 18,576 represented as option a.

Thus the answer to your problem is, 18,576 [tex]ft^{3}[/tex]

The suitable equation for n in terms of t is: n(t) = [tex]Ce^{(kt)[/tex], where C and k are constants.

Since the rate of increase of n is proportional to n, we can write this as dn/dt = kn, where k is the proportionality constant. Solving this differential equation, we get n(t) =  [tex]Ce^{(kt)[/tex], where C is the constant of integration. We can determine the value of C from an initial condition, such as the number of phones sold at time t=0.

The value of k can be determined from the rate of increase of n over a given time interval. The resulting equation can be used to predict the number of phones sold at any time t after the release of the smartphone.

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The number of ways the student can answer the questions in either true or false for 8 questions is:  256 ways

Permutations and combinations are basically the various ways that we select objects from a particular set generally without any replacement, to form subsets. This selection of subsets is referred to as a permutation when the order of selection is a factor. However, it is referred to as a combination when order is not a factor.

In this question, we are to find the number of ways in which this question can be answered when there are 8 true/false questions in an examination.

So, we have the number of ways for a single question to select the answer as 2 which is true or false.

Thus,  the expression for the number of ways to answer the question in either true or false for 8 questions is:

2⁸ = 256 ways

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To find the work done by pumping out water over the top of the tank until the water level inside the tank is at 2 meters, we first need to find the volume of water in the tank when it is filled to a level of three-quarters of the height of the tank.

The volume of a pyramid is given by the formula V = (1/3)Ah, where A is the area of the base and h is the height of the pyramid. In this case, the base is a 2 meter by 2 meter square, so its area is 4 square meters. The height of the pyramid is 8 meters, so the volume of the pyramid is V = (1/3)(4)(8) = 32/3 cubic meters.

When the water is filled to a level of three-quarters of the height of the tank, the volume of water in the tank is (3/4)(32/3) = 8 cubic meters.

The mass of the water in the tank is given by m = ρV, where ρ is the mass density of water. In this case, ρ = 1000 kg/m3, so the mass of the water in the tank is m = (1000)(8) = 8000 kg.

The weight of the water in the tank is given by W = mg, where g is the gravitational acceleration. In this case, g = 9.8 m/s2, so the weight of the water in the tank is W = (8000)(9.8) = 78400 N.

To pump out water over the top of the tank until the water level inside the tank is at 2 meters, we need to remove a volume of water equal to the difference in volume between the water level at three-quarters of the height of the tank and the water level at 2 meters.

The volume of water we need to remove is (1/3)(4)(6) - (1/3)(4)(2) = 4 cubic meters.

The mass of the water we need to remove is m = ρV = (1000)(4) = 4000 kg.

The work done to pump out the water is equal to the weight of the water times the distance it is pumped, which is the height of the water that is pumped out. In this case, the height of the water that is pumped out is 8 - 2 = 6 meters.

So the work done is W = mgd = (4000)(9.8)(6) = 235200 J.

Therefore, the work done by pumping out water over the top of the tank until the water level inside the tank is at 2 meters is 235200 J.

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Hi!

Requirement: An 80meter long rope was cut into four pieces. The four pieces were distributed among Form One, Form Two, Form Three and Form Four students. In this distribution, Form One got 15 metres, Form Two got 20 metres, Form Three got 40 metres and the rest of the pieces was given to Form Four.

a) write the fraction of the piece of rope distributed into each class.

b) Which fraction is the largest of all the fractions?

c) Find the different between the largest fraction and the smallest fraction​

Answer:

a) To write the fractions of the pieces of rope distributed into each class, we need to first determine the total length of the rope distributed, which is:

15 + 20 + 40 + x = 80

Where x is the length of the rope given to Form Four. Solving for x, we get:

x = 80 - 15 - 20 - 40 = 5

Therefore, Form Four got 5 meters of the rope. Now we can write the fractions as follows:

b) To determine which fraction is the largest, we can compare the fractions using a common denominator. In this case, the common denominator is 16, so we can rewrite the fractions as:

From this, we can see that the fraction 8/16, which is equivalent to 1/2, is the largest.

c) The difference between the largest fraction and the smallest fraction is:

8/16 - 3/16 = 5/16

Therefore, the difference between the largest and smallest fractions is 5/16 of the total length of the rope, which is equivalent to:

(5/16) * 80 = 25 meters.

I hope I helped you!

from 2007 to 2012 is only 5 years, so we can see this as a compound interest with a rate of 1.2% per annum for 5 years, so

hope this helped!

The total number of scooters that use LPG is 51.

Total vehicles = 140.

The ratio bicycles to scooters = 4: 6

The equation can be created as:

4x+6x= 140

This will be further solved as:

10 x = 140

x = 14

The number of scooters present will be = 14 * 6 = 84

The number of bicycles present will be = 14 * 4 = 56

Number of scooters which utilized Electricity =84*25% = 21

Number of scooters which utilized petrol =1/7 * 84 = 12

The total number of scooters that use LPG. ​

= 84-(21+12)

=84 -33

=51

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