Find the unit tangent vector, the unit normal vector, and the binormal vector of r(t) = sin(2t)i + 3tj + 2 sin2 (t) k at the point (0, 3π/2 , 2 ). Then compute the curvature at that point. Hint: Use the fact that 2 sin(θ) cos(θ) = sin(2θ).

To find the angle that maximizes the range of a projectile, you can follow these steps:

1. Determine the range formula: The range (R) of a projectile can be found using the formula R = (v² * sin(2θ)) / g, where v is the initial velocity, θ is the launch angle, and g is the acceleration due to gravity (approximately 9.81 m/s²).

2. In this case, the initial velocity (v) is 70 m/s, so the formula becomes R = (70² * sin(2θ)) / 9.81.

3. To maximize the range, you need to find the angle (θ) that results in the highest value of R. To do this, you can use a graphing utility to graph the function R(θ) = (4900 * sin(2θ)) / 9.81 and find its maximum value.

4. Using a graphing utility, you will find that the maximum range occurs when θ ≈ 45°.

5. Round your answer to one decimal place: The angle that maximizes the arc length of the trajectory is approximately 45.0°.

So, to maximize the range of a projectile launched at 70 m/s, the optimal angle is 45.0° with the horizontal.

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Answer: y=(7/8)x+6

Step-by-step explanation:

The slope is (7/8) and the y-intercept is 6. The slope-intercept form of a line is y=mx+b.

The prime factorization of 8051 is 8051 = 13 * 619.

Suppose we know that 79612 ≡ 72 (mod 8051).

We can use this information to factor 8051 as follows:

Let's subtract 72 from 79612 and get:

79540 ≡ 0 (mod 8051)

This means that 8051 divides 79540 - 0, or equivalently, 8051 divides 79540.

We can use long division to find:

79540 / 8051 = 9 with a remainder of 539

This means that 79540 = 9 * 8051 + 539.

We can repeat this process with 8051 and 539:

8051 / 539 = 14 with a remainder of 165

This means that 8051 = 14 * 539 + 165.

We can repeat again with 539 and 165:

539 / 165 = 3 with a remainder of 44

This means that 539 = 3 * 165 + 44.

We can repeat one last time with 165 and 44:

165 / 44 = 3 with a remainder of 33

This means that 165 = 3 * 44 + 33.

Now, we can write each remainder as a linear combination of 8051 and 539:

539 = 8051 - 14 * 539 + 165 - 165 = 8051 - 15 * 539 - 165

165 = 539 - 3 * 165 + 44 - 44 = -2 * 539 + 4 * 165 + 44

44 = 165 - 3 * 44 - 33 = -3 * 8051 + 44 * 539 - 7 * 165 - 33

Substituting the values of 539 and 165 in the second equation yields:

44 = -2 * (8051 - 15 * 539 - 165) + 4 * 165 + 44

Simplifying and rearranging, we get:

44 = -2 * 8051 + 34 * 539 + 326

Therefore, 8051 can be factored as:

8051 = 44 * 183 + 1

= (-2 * 44) * 183 + 2

= (-2 * (-2 * 8051 + 34 * 539 + 326)) * 183 + 2

= 4 * 8051 - 2486 * 539 - 366

So, the prime factorization of 8051 is 8051 = 13 * 619.

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Answer:

i think its 8

Step-by-step explanation:

The required farmer should choose a length of 56 feet for the garden to maximize its area.

we need to maximize A with respect to x, subject to the constraint that the perimeter of the garden (the amount of fencing needed) is 200 feet. The perimeter is given by:

P = x + 2y + π*x/2

Substituting y = x/2, we get:

P = x + 2(x/2) + π*x/2
200 = (2 + π/2)*x
x = 56

Therefore, the farmer should choose a length of 56 feet for the garden to maximize its area.

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Answer:

false

Step-by-step explanation:

False. We can use multilinear regression analysis when some or all of the independent variables in the model are continuous, categorical, or a combination of both.

Answer:

3201.3ft³

Step-by-step explanation:

V=πr²h

Large container:

V=π·11²·19

V=7222.52

Small container

V=π·8²·20

V=4021.24

7222.52-4021.24=3201.28

Rounded to the nearest tenth is 3201.3

Answer: A) 2010

Step-by-step explanation:

Using the relation between velocity, distance and time, the equation that relates x and y is given by x + y - 3 = 0.

Velocity is distance divided by time, so

v = d/t

In this case , Austin wants to run 30 km at a rate of 10 km per hour, this can be represented as

10t = 30

t = 3.

The total time is 3 hours.

Looking at x as the number of hours he runs and y the number of hours he walks, along with the total time, the equation is given by

x + y = 3.

In standard form

x + y - 3 = 0.

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Let the total number of goats that McDonald had be x. After evenly dividing them between Lilly and Hawk, each of them would have received x/2 goats. However, Lilly lost 35 goats, so she was left with (x/2 - 35) goats. Similarly, Hawk lost 40 goats and was left with (x/2 - 40) goats.

When Lilly sold each goat for $80, she earned (x/2 - 35) * $80 = 80x/2 - 35*80 = 40x - 2800 dollars. When Hawk sold each goat for $60, he earned (x/2 - 40) * $60 = 60x/2 - 40*60 = 30x - 2400 dollars.

Given that Lilly earned $1100 more than Hawk, we can set up the equation:

40x - 2800 = 30x - 2400 + 1100

Solving for x, we get x = 400. Therefore, McDonald had a total of 400 goats.

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To find the area of the region bounded by the polar curves θ = π and r = 10θ, 0 ≤ θ ≤ 1.5π, we use the formula for the area enclosed by a polar curve: A = 1/2 ∫[θ1,θ2] (r(θ))^2 dθ,where θ1 and θ2 are the angles at which the curves intersect.

In this case, the curves intersect at θ = π and r = 10π, so θ1 = π and θ2 = 1.5π. We substitute r = 10θ into the formula and integrate:

A = 1/2 ∫[π,1.5π] (10θ)^2 dθ

= 1/2 ∫[π,1.5π] 100θ^2 dθ

= 50 ∫[π,1.5π] θ^2 dθ

= 50 [θ^3/3] [π,1.5π]

= 50 (1.5π)^3/3

= 562.5π^3

Therefore, the area of the region bounded by the polar curves θ = π and r = 10θ, 0 ≤ θ ≤ 1.5π is 562.5π^3 square units.

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Considering the dot plot and visual inspection, it is likely that group B has a lower mean. The reason for this is because it has a higher proportion of it's measures to the left of the dot plot than group A.

The mean of a data-set is given by the sum of all observations in the data-set divided by the cardinality of the data-set, which represents the number of observations in the data-set.

The dot plot shows the number of instances that each observation appeared in the data-set, hence we use it to identify the position of the measures.

Group B has more dots at the left of the graph, meaning that the smaller measures are more common than in group A, and thus it more than likely has a lower mean.

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After considering all the given data we conclude that yes  41 is a square modulo 1 000 000, under the condition that both congruences x₂ ≡ 41 mod 26 and x₂ ≡ 41 mod 56 have solutions.

We can apply the Chinese Remainder Theorem (CRT) to solve this problem.

Firstly, we have to evaluate the solutions of x² ≡ 41 mod 26 and x² ≡ 41 mod 56.

For x² ≡ 41 mod 26, we clearly see that x² ≡ 15 mod 26 is a solution since 15² = 225 ≡ 41 mod 26.

For x² ≡ 41 mod 56, we can apply the fact that x² ≡ a mod p has solutions if and only if [tex]a^{(P-1)} /2[/tex] ≡ 1 mod p (Euler's criterion).

Since p = 56 = 7 × 8, we have:

[tex]a^{(p-1)} /2[/tex] = a²¹ ≡ (a⁷)³ ≡ (-1)³ ≡ -1 mod p

Hence, x² ≡ 41 mod 56 has no solutions.

Now we can apply CRT to find the solutions of x² ≡ 41 mod (26 × 56) = 1456.

Since gcd(26,56) = 2, we have:

26 × u + 56 × v = gcd(26,56) = 2

Evaluating  this equation gives us u = -13 and v = 6.

So, the solutions of x² ≡ 41 mod (26 × 56) are:

x ≡ (15 × 56 × 6 - (-13) × 26 × (-1)) mod (26 × 56) = 937 or

x ≡ (-15 × 56 × 6 - (-13) × (-26) × (-1)) mod (26 × 56) = 519.

Hence, there are two solutions for x modulo one million: 519 and 481.

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The bearing of the ship from boston is approximately s36.

to solve this problem, we need to use vector addition to find the displacement of the ship from boston to its current position. we can start by breaking down the ship's motion into two parts: the first hour of motion at 10 nm/hr on a bearing of s80°e, and the next 2 hours of motion at 20 nm/hr on a bearing of s45°w (which is equivalent to n45°e).

for the first hour of motion, we can find the ship's initial displacement as follows:

distance = speed × time         = 10 nm/hr × 1 hr

        = 10 nm

using trigonometry, we can find the horizontal and vertical components of this displacement:

horizontal distance = 10 nm × cos(80°)                     = 1.68 nm (rounded to two decimal places)

vertical distance = 10 nm × sin(80°)

                  = 9.92 nm (rounded to two decimal places)

, the ship's initial displacement from boston is 1.68 nm to the east and 9.92 nm to the south.

for the next 2 hours of motion, we can find the ship's additional displacement as follows:

distance = speed × time         = 20 nm/hr × 2 hr

        = 40 nm

using trigonometry again, we can find the horizontal and vertical components of this displacement:

horizontal distance = 40 nm × cos(45°)

                    = 28.28 nm (rounded to two decimal places)

vertical distance = 40 nm × sin(45°)                   = 28.28 nm (rounded to two decimal places)

, the ship's additional displacement is 28.28 nm to the northeast.

to find the ship's total displacement, we can add the initial and additional displacements using vector addition:

horizontal displacement = 1.68 nm - 28.28 nm

                       = -26.60 nm (rounded to two decimal places)

vertical displacement = 9.92 nm + 28.28 nm                      = 38.20 nm (rounded to two decimal places)

the negative sign for the horizontal displacement indicates that the ship is west of boston. we can find the bearing of the ship from boston using trigonometry:

tan(θ) = horizontal displacement / vertical displacement

θ = arctan(horizontal displacement / vertical displacement)

θ = arctan(-26.60 nm / 38.20 nm)

θ ≈ -36.6° (rounded to one decimal place)

however, we need to adjust this   angle    by adding 180° since the ship is now in the southern hemisphere.

θ = -36.6° + 180°θ ≈ 143.4° (rounded to one decimal place) 6°w (or n36.6°e).

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The y-intercept of the function f(x) is 1.

The y-intercept of a function is the point where the graph of the function intersects the y-axis. It represents the value of the function when x=0. To find the y-intercept of a function, we can substitute x=0 into the function and evaluate it.

In the case of the function [tex]f(x) = 2x^2 - 2x + 1[/tex], when x=0, we have:

[tex]f(0) = 2(0)^2 - 2(0) + 1 = 1[/tex]

Therefore, the y-intercept of the function f(x) is 1. This means that the graph of the function intersects the y-axis at the point (0, 1).

Knowing the y-intercept is important when graphing the function, as it provides a reference point for drawing the graph. Additionally, the y-intercept can provide information about the behavior of the function as x approaches infinity or negative infinity.

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The absolute maximum value is 3√3, and the absolute minimum value is 3 on the set d.

In mathematics, a function is a unique arrangement of the inputs (also referred to as the domain) and their outputs (sometimes referred to as the codomain), where each input has exactly one output and the output can be linked to its input.

To find the absolute maximum and minimum values of the function f(x, y) = xy² on the set d = {(x, y) | x ≥ 0, y ≥ 0, x² y² ≤ 3}, we can employ the method of Lagrange multipliers. This method allows us to optimize a function subject to certain constraints.

Let's define the function g(x, y) = x² y² - 3, which represents the constraint x² y² ≤ 3. We can now set up the following equations:

1. ∇f = λ∇g

2. x² y² = 3 (constraint equation)

Here, ∇f represents the gradient of f(x, y), and ∇g represents the gradient of g(x, y). λ is the Lagrange multiplier.

First, let's calculate the partial derivatives of f(x, y) and g(x, y):

∇f = (∂f/∂x, ∂f/∂y) = (y², 2xy)

∇g = (∂g/∂x, ∂g/∂y) = (2xy², 2x²y)

Setting up the equations:

1. y² = λ * 2xy²

2. 2xy = λ * 2x²y

3. x² y² = 3 (constraint equation)

From equation 1, we can deduce two possibilities:

  a) y² = 0 (which implies y = 0)

  b) λ = 1/2x

For case a) y = 0, substituting it into equation 3 gives us x² * 0² = 3, which is not possible since x² * 0 = 0 ≠ 3. Therefore, case a) is not valid.

Now let's consider case b) λ = 1/2x. Substituting this into equation 2, we get:

2xy = (1/2x) * 2x²y

2xy = xy

Cancelling out the common factors of xy, we have x = 1.

Substituting x = 1 into equation 3, we find:

1 * y² = 3

y² = 3

y = √3

Thus, we have the critical point (1, √3) that satisfies the constraints.

Next, we need to check the boundaries of the feasible region, which is defined by x ≥ 0, y ≥ 0, and x² y² ≤ 3.

When x = 0, the constraint equation becomes 0 * y² = 3, which is not valid.

When y = 0, the constraint equation becomes x² * 0² = 3, which is not valid.

Now, let's consider the boundary when x² y² = 3:

When x = √3 and y = √3, the constraint equation is satisfied.

In summary, we have the following critical points and boundary points:

- Critical Point: (1, √3)

- Boundary Point: (√3, √3)

Finally, we need to evaluate the function f(x, y) = xy² at these points to find the absolute maximum and minimum values.

For the critical point (1, √3):

f(1, √3) = 1 * (√3)² = 1 * 3 = 3

For the boundary point

(√3, √3):

f(√3, √3) = √3 * (√3) = √3 * 3 = 3√3

Therefore, the absolute maximum value is 3√3, and the absolute minimum value is 3 on the set d.

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To find the extremum of the function f(x,y) = 2y^2-9x^2 subject to the constraint 3xy = 27x, we can use the method of Lagrange multipliers.

Let g(x,y) = 3xy - 27x be the constraint function. We want to find the critical points of the function f(x,y) subject to the constraint g(x,y) = 0, so we set up the following system of equations:

∇f(x,y) = λ∇g(x,y)

g(x,y) = 0

where λ is the Lagrange multiplier.

Taking the partial derivatives of f(x,y) with respect to x and y, we get:

∂f/∂x = -18x

∂f/∂y = 4y

Taking the partial derivatives of g(x,y) with respect to x and y, we get:

∂g/∂x = 3y - 27

∂g/∂y = 3x

Setting ∇f(x,y) = λ∇g(x,y), we get the following system of equations:

-18x = λ(3y - 27)

4y = λ(3x)

Multiplying the first equation by 4 and the second equation by -6, we get:

-72x = λ(12y - 108)

-24y = λ(-18x)

Simplifying these equations, we get:

4x = λ(y - 9)

y = 3λx/2

Substituting y = 3λx/2 into the first equation, we get:

4x = λ(3λx/2 - 9)

8x = λ^2x - 18λ

x(λ^2 - 8) = 18λ

If x = 0, then y = 0, which is not a critical point since f(0,0) = 0. Therefore, we can divide both sides by x to get:

λ^2 - 8 = 18/ x

If λ^2 - 8 < 0, then there are no critical points since the equation above has no real solutions. Therefore, we assume λ^2 - 8 ≥ 0, which gives:

λ = ±√(8 + 18/x)

Substituting λ into y = 3λx/2, we get:

y = ±√(2x(8 + 18/x))/2

We want to find the extremum of f(x,y) = 2y^2-9x^2, so we evaluate this function at the critical points:

f(x,y) = 2y^2-9x^2 = 2(2x(8 + 18/x))/4 - 9x^2 = (4x^2 + 36) / x - 9x^2

Taking the derivative of f(x,y) with respect to x, we get:

f'(x,y) = (8x - 36)/x^2 - 18

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

8x - 36 = 18x^2

18x^2 - 8x + 36 = 0

Solving for x, we get:

x = (2 ± √13)/9

Substituting x into y = ±√(2x(8 + 18/x))/2, we get:

y = ±(4 ± √13)√2/3

Therefore, the critical points are (x,y) = x = (2 ± √13)/9, y = ±(4 ± √13)√2/3

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Therefore, The approximate normal probability that more than 39 accounts receivable will contain errors is 2.28%.

The problem involves calculating the probability of finding errors in a sample of accounts receivable. We know that the true proportion of accounts receivable with errors is 0.20. The sample size is 225 accounts receivable. We want to find the probability of finding more than 39 accounts with errors. We can use the normal distribution formula to calculate this probability. By converting the problem to a standard normal distribution, we can use a z-score table to find the probability. The probability is approximately 0.0228, or 2.28%. This means that there is a 2.28% chance of finding more than 39 accounts with errors in the sample.

Therefore, The approximate normal probability that more than 39 accounts receivable will contain errors is 2.28%.

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Answer:

Linear

Step-by-step explanation:

A linear graph is a straight line.

To find the radius of convergence, we use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series converges absolutely. If the limit is greater than 1, then the series diverges. If the limit is equal to 1, then the test is inconclusive.

In this case, we have the series (-1)^(n-1)/n5^n. Taking the absolute value of the ratio of consecutive terms, we get |((-1)^n)/(n+1)(5^(n+1))) / ((-1)^(n-1)/n5^n)| = 1/(5(n+1)). Taking the limit as n approaches infinity, we get 1/5. Since the limit is less than 1, the series converges absolutely.

The radius of convergence is equal to the reciprocal of the limit we just found, which is 5. Therefore, the series converges for all x values between -5 and 5.

To find the interval of convergence, we need to test the endpoints. When x=5, the series becomes (-1)^(n-1)/(5n), which is an alternating series. The alternating series test tells us that the series converges if the absolute value of the terms decreases and approaches zero. In this case, the terms are decreasing in absolute value but do not approach zero, so the series diverges at x=5.

When x=-5, the series becomes (-1)^(n-1)/(-5n), which is also an alternating series. The same reasoning as above tells us that the series converges at x=-5.

Therefore, the interval of convergence is [-5,5).

The radius of convergence of the series (-1)^(n-1)/n5^n is 5, and the interval of convergence is [-5,5). To find the radius of convergence, we used the ratio test and found that the limit of the absolute value of the ratio of consecutive terms is 1/5. To find the interval of convergence, we tested the endpoints and found that the series converges at x=-5 and diverges at x=5.

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The interval of convergence is [-5,5). We apply the ratio test to determine the radius of convergence. The ratio test asserts that the series converges absolutely if the limit of the absolute value of the ratio of consecutive terms is smaller than 1.

The series diverges if the limit is bigger than 1. The test is not convincing if the limit is equal to 1.The series in question is (-1)(n-1)/n5n. The result is |((-1)n)/(n+1)(5(n+1))] / ((-1)(n-1)/n5n)| = 1/(5(n+1) when we take the absolute value of the ratio of successive words. When we take the limit as n gets closer to infinity, we get 1/5. Since 1, the limit, the series completely converges.

The radius of convergence is equal to the reciprocal of the limit we just found, which is 5. Therefore, the series converges for all x values between -5 and 5.

To find the interval of convergence, we need to test the endpoints. When x=5, the series becomes (-1)[tex]^(n-1)/(5n)[/tex], which is an alternating series. The alternating series test tells us that the series converges if the absolute value of the terms decreases and approaches zero. In this case, the terms are decreasing in absolute value but do not approach zero, so the series diverges at x=5.

When x=-5, the series becomes (-1)[tex]^(n-1)/(-5n),[/tex]which is also an alternating series. The same reasoning as above tells us that the series converges at x=-5.

Therefore, the interval of convergence is [-5,5).

The radius of convergence of the series (-1)^(n-1)/n[tex]5^n[/tex] is 5, and the interval of convergence is [-5,5). To find the radius of convergence, we used the ratio test and found that the limit of the absolute value of the ratio of consecutive terms is 1/5. To find the interval of convergence, we tested the endpoints and found that the series converges at x=-5 and diverges at x=5.

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If you randomly draw one card from the deck, there is about a 26.92% chance that you will get either a Jack or a Spade.

Since there is only one Jack of Spades, we have one favorable outcome for drawing a Jack. Additionally, there are 13 Spades in the deck, including the Jack of Spades. Therefore, the number of favorable outcomes for drawing a Spade is 13.

Total number of favorable outcomes = Number of Jacks + Number of Spades

= 1 + 13

= 14

Total number of possible outcomes

In a deck of 52 cards, each card is unique. Therefore, the total number of possible outcomes is equal to the total number of cards in the deck, which is 52.

Now that we have determined the number of favorable outcomes and the total number of possible outcomes, we can calculate the probability using the following formula:

Probability = Number of favorable outcomes / Total number of possible outcomes

Substituting the values we found:

Probability = 14 / 52

Simplifying the fraction:

Probability = 7 / 26

So, the probability of drawing a Jack or a Spade from a standard deck of 52 cards is 7/26, or approximately 0.2692, which can also be expressed as 26.92%.

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Answer:

C

Step-by-step explanation:

(k - 1)(6k + 5)

each term in the second factor is multiplied by each term in the first factor , that is

k(6k + 5) - 1(6k + 5) ← distribute parenthesis

= 6k² + 5k - 6k - 5 ← collect like terms

= 6k² - k - 5

Answer; X=14

Step-by-step explanation:

14 - 6 = 8 and 8+4 =12

CYA BESTIE!

S is a function from the set of positive integers to the set of positive integers, defined as the sum of the positive divisors of a given integer. The questions to be answered are whether S is one-to-one, onto, or a one-to-one correspondence.

To determine if S is one-to-one, we need to check whether different inputs to the function produce different outputs. In other words, if S(a) = S(b) for some positive integers a and b, does it follow that a = b? To prove that S is not one-to-one, we can provide a counterexample. For example, S(6) = 1 + 2 + 3 + 6 = 12, and S(28) = 1 + 2 + 4 + 7 + 14 + 28 = 56, but 6 ≠ 28. Therefore, S is not one-to-one.

To determine if S is onto, we need to check whether every positive integer is in the range of the function. In other words, for every positive integer y, is there some positive integer x such that S(x) = y? To prove that S is not onto, we can provide a counterexample. For example, there is no positive integer x such that S(x) = 2. Therefore, S is not onto.

A function is a one-to-one correspondence if it is both one-to-one and onto. Since S is not one-to-one and not onto, it is not a one-to-one correspondence.

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The area of the trapezoid is 25. 8 ft²

We can see from information given that the shape is a trapezoid.

Hence, the formula for calculating the area of a trapezoid is expressed as;

A = a + b/2 h

Such that the parameters of the given equation are;

Substitute the value, we have that;

Area = 2.7 + 5.9)/2 × 6

add the values, we have;

Area = 8. 6/2 ×6

Divide the values, we have;

Area = 25. 8 ft²

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Answer: =133.3%

Step-by-step explanation:

Formula for percent change:

[tex]\frac{difference of 2 numbers}{original} *100[/tex]                 >substitute

[tex]\frac{\frac{7}{8} -\frac{3}{8} }{\frac{3}{8} } *100[/tex]                                         >subtract top

[tex]=\frac{\frac{4}{8} }{\frac{3}{8} }*100[/tex]                                         >simplify/reduce top

[tex]=\frac{\frac{1}{2} }{\frac{3}{8} }*100[/tex]                                         >Divide fractions(Keep the first, Change

                                                           the sign, Flip the 2nd fraction

= [tex]\frac{1}{2} *\frac{8}{3} *100\\[/tex]                                     >Reduce fractions and multiply

= [tex]\frac{4}{3} *100[/tex]

=133.3%

The solution is: the value of m in the equation below when j = 24 and

n = 3 is: m=4

Here, we have,

given that,

the equation is:

j=2mn,

and, when j = 24 and n = 3.

now, we have to find the value of m in the equation,

Let j = 24 and n=3

24 = 2*m*3

Simplify

so, we have,

24 = 6*m

Divide each side by 6

we get,

24/6 = 6m/6

4=m

Hence, The solution is: the value of m in the equation below when j = 24 and  n = 3 is: m=4

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