GEOMETRY 100 POINTS ​

The calculated value of the expression 4 Superscript 4 is 256

From the question, we have the following parameters that can be used in our computation:

4 Superscript 4

Express properly

So, we have

4⁴

When expanded, we have

4⁴ = 4 * 4 * 4 * 4

Evaluate

4⁴ = 256

Hence, the value of the expression is 256

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Answer:

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The intersecting diagonals form a triangle with sides 3, 4 and shorter side x of the parallelogram.

Possible angle measure between diagonals, opposite to shorter side is maximum 90°.

Find the largest possible length of x using the law of cosines:

Answer:

[tex]a_{n}[/tex] = 3n - 2

Step-by-step explanation:

the nth term of an arithmetic sequence is

[tex]a_{n}[/tex] = a₁ + d(n - 1)

where a₁ is the first term and d the common difference

here a₁ = 1 and d = a₂ - a₁ = 4 - 1 = 3 , then

[tex]a_{n}[/tex] = 1 + 3(n - 1) = 1 + 3n - 3 = 3n - 2

he slope of the line passing through the points (-6, -8) and (2, 8) is 2.

The slope of the line passing through two given points can be calculated by using the formula:m = (y₂ - y₁)/(x₂ - x₁)Where m is the slope of the line, (x₁, y₁) and (x₂, y₂) are the two given points on the line. Given a coordinate plane with a straight line passing through the points (negative 6, negative 8) and (2, 8), we can find the slope of the line using the above formula as follows:We have the two points: (x₁, y₁) = (-6, -8) and (x₂, y₂) = (2, 8)Let's substitute these values into the formula to find the slope:m = (y₂ - y₁)/(x₂ - x₁)m = (8 - (-8))/(2 - (-6))m = (8 + 8)/(2 + 6)m = 16/8m = 2

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If the original amount is 390 and the new amount is 546, the amount of increase is 156 and the percent increase is 40%. This indicates that the new amount is 156 units higher than the original amount, representing a 40% increase.

To find the amount of increase and the percent increase between the original amount and the new amount, we can follow these steps:

1. Start with the original amount: 390.

2. Determine the increase by subtracting the original amount from the new amount: 546 - 390 = 156.

3. The amount of increase is 156. This means that the new amount is 156 units greater than the original amount.

4. To calculate the percent increase, divide the amount of increase by the original amount and then multiply by 100: (156 / 390) * 100 = 40.

5. The percent increase is 40%. This means that the new amount is 40% greater than the original amount.

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Answer:

To simplify the expression 2√(2u) - 7√(3v), we can combine like terms.

Since both terms have square roots, we can look for any common factors within the square roots. In this case, we can factor out the square root of 2 from the first term and the square root of 3 from the second term:

2√(2u) - 7√(3v) = 2√2 √u - 7√3 √v

Now, we can simplify further by combining the coefficients outside the square roots:

2√2 √u - 7√3 √v = 2√2u - 7√3v

Therefore, the simplified form of 2√(2u) - 7√(3v) is 2√2u - 7√3v.

Haleema would have approximately £3947.46 in her account after 2 years and 11 months, considering the monthly compounding interest of 5%.

To calculate the amount of money in Haleema's account after 2 years and 11 months, we need to consider the monthly compounding interest on the account.

The interest rate is given as 5% per year, which means the monthly interest rate is (5%/12) = 0.4167%.

Let's calculate the final amount using the compound interest formula: A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years.

In this case, P = £3700, r = 0.004167, n = 12 (monthly compounding), and t = 2.917 (2 years and 11 months).

Plugging these values into the formula, we get:

A = £3700(1 + 0.004167/12)^(12*2.917)

A ≈ £3947.46

The amount of money in Haleema's account after 2 years and 11 months would be approximately £3947.46.

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Answer:

The speed of the plane in the still air is 420 miles/hour

The speed of the wind 60 miles/hour

Step-by-step explanation:

Let the speed of the plane with the wind be v

Let the speed of the plane against the wind be u

Now, speed = distance/time

With the wind,

v = (1440 miles)/(3 hours) = 480 miles/hour

v = 480 miles/hour

Against the wind,

u = (1440 miles)/(4 hours) = 360 miles/hour

u = 360 miles/hour

Now, let the speed of plane be p, and speed of wind be w,

Now, with the wind, the speed is 480 mph,

so,

speed of plane + speed of wind = 480 mph

p + w = 480  (i)

and against the wind, the speed is 360 mph,

so,

speed of plane - speed of wind = 360mph

p-w = 360 (ii)

adding equations (i) and (ii), we get,

p+w + p-w = 480 + 360

2p = 840

p = 840/2

p = 420 miles/hour

Then, the speed of the wind will be,

p + w = 480,

420 + w = 480

w = 480 - 420

w = 60 miles/hour

The speed of the plane in still air is calculated to be 420 mph, and the speed of the wind is calculated to be 60 mph by solving the two simultaneous equations obtained from the time, rate, and distance relationship.

This problem is about the rate, time, and distance relationships. The rate at which the airplane travels in still air is r (unaffected by wind), and the speed of the wind is w. When the plane flies with the wind, it is 'assisted' and therefore travels faster - at a speed of (r + w); against the wind, it travels slower - at a speed of (r - w).

From the problem, we know that:

By solving these two equations, we get the following:

Adding these two gives 2r = 840 => r = 420 mph (the speed of the plane in still air), and subtracting gives 2w = 120 => w = 60 mph (the speed of the wind).

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Answer: 4

This problem requires basic division. If the restaurant divided 4/5 kg of butter with 1/5 kg on each dish, you would need to compute 4/5 divided by 1/5.

4/5 ÷ 1/5

Using the "KFC" method, or Keep, Change, Flip, you would keep the first number (in this case, 4/5), change the division sign, and flip the fraction to 5/1, or 5. We now have this:

4/5 x 5

To compute this equation, you must multiply the numerators of both of the numbers together. In this case, you would compute (4x5)/5, resulting with 20/5, or 4.

You can check this answer by re-multiplying the numbers together. 1/5 kg of butter per dish, multiplied by the total amount of dishes, 4, you would result in the original 4/5 kg of butter.

Hope this helps!

The correct version of the sentence would be A Conference attendee's need to check in by 9:00 a.m. or risk forfeiting his or her reservation.

The sentence that contains at least one error is:

O A Conference attendee's need to check in by 9:00 a.m. or risk forfeiting their reservation.

The error in this sentence is the incorrect use of "their" to refer to a singular possessive noun "attendee's."

To maintain subject-verb agreement, it should be "his or her" instead of "their." Therefore, the correct version of the sentence would be:

A Conference attendee's need to check in by 9:00 a.m. or risk forfeiting his or her reservation.

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The function f(x) = [tex]-2(x+4)^2[/tex] + 1 has a greater maximum.

1. The given function is f(x) = [tex]-2(x+4)^2[/tex] + 1.

2. To find the maximum of the function, we need to determine the vertex of the parabola.

3. The vertex form of a quadratic function is given by f(x) = [tex]a(x-h)^2[/tex] + k, where (h, k) represents the vertex.

4. Comparing the given function to the vertex form, we see that a = -2, h = -4, and k = 1.

5. The x-coordinate of the vertex is given by h = -4.

6. To find the y-coordinate of the vertex, substitute the x-coordinate into the function: f(-4) = [tex]-2(-4+4)^2[/tex] + 1 = [tex]-2(0)^2[/tex] + 1 = 1.

7. Therefore, the vertex of the function is (-4, 1), which represents the maximum point.

8. Comparing this maximum point to the information provided about the other function g(x) on the coordinate plane, we can conclude that the maximum of f(x) = [tex]-2(x+4)^2[/tex] + 1 is greater than the maximum of g(x).

9. The given information about g(x) is not sufficient to determine its maximum value or specific equation, so a direct comparison is not possible.

10. Hence, the function f(x) =[tex]-2(x+4)^2[/tex] + 1 has a greater maximum.

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The expression is simplified to x+8/x- 4

First, we need to know that algebraic expressions are defined as expressions that are made up of terms, variables, constants, factors and coefficients.

These expressions are also made up of arithmetic operations.

From the information given, we have that;

[tex]\frac{x^2 + 2}{x^2 - 9x + 20} - \frac{42 - 3x}{x^2 - 9x + 20}[/tex]

Now, find the lowest common multiple, we have;

x² + 2 - (42 - 3x)/x² - 9x + 20

collect the like term and expand the bracket, we have;

x² + 3x - 40/x² - 9x + 20

x² + 8x - 5x - 40/x² - 5x - 4x + 20

(x-5)(x+8)/(x-5)(x -4)

x+8/x- 4

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The interval solution for the inequality 0.35x - 4.8 ≤ 5.2 - 0.9x is b) (-∞, 8].

To find the interval solution of the expression 0.35x - 4.8 ≤ 5.2 - 0.9x, we need to solve the inequality for x.

First, let's simplify the expression by combining like terms:

0.35x + 0.9x ≤ 5.2 + 4.8

Simplifying further:

1.25x ≤ 10

To isolate x, divide both sides of the inequality by 1.25:

x ≤ 10 / 1.25

x ≤ 8

The solution to the inequality is x ≤ 8. This means that any value of x that is less than or equal to 8 satisfies the inequality.

To represent this solution as an interval, we use square brackets for inclusive values and parentheses for exclusive values. Since the inequality includes the value of 8, the interval is:

(-∞, 8]

The left endpoint of the interval is negative infinity (-∞), indicating that x can take any value less than 8. The right endpoint is 8, indicating that x can be equal to 8 but cannot exceed it.

The correct option from the given choices is b). (-∞, 8).

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Hello!

The distance, d, is less than 200 miles.

B. d > 200

14. The trigonometric equation (sin47°/cos43°)² + (cos43°/sin47°)² - 4cos²45° = 4

15. In the trigonometric equation 2(cos²θ - sin²θ) = 1, θ = 15°

A trigonometric equation is an equation that contains a trigonometric ration.

14. To find the value of (sin47°/cos43°)² + (cos43°/sin47°)² - 4cos²45°, we proceed as follows

Since we have the trigonometric equation (sin47°/cos43°)² + (cos43°/sin47°)² - 4cos²45°,

We know that sin47° = sin(90 - 43°) = cos43°. So, substituting this into the equation, we have that

(sin47°/cos43°)² + (cos43°/sin47°)² - 4cos²45° = (cos43°/cos43°)² + (cos43°/cos43°)² - 4cos²45°

= 1² + 1² - 4cos²45°

We know that cos45° = 1/√2. So, we have

1² + 1² - 4cos²45° = 1² + 1² - 4(1/√2)²

= 1 + 1 + 4/2

= 2 + 2

= 4

So, (sin47°/cos43°)² + (cos43°/sin47°)² - 4cos²45° = 4

15. If 2(cos²θ - sin²θ) = 1 and θ is a positive acute angle, we need to find the value of θ. We proceed as follows

Since we have the trigonometric equation 2(cos²θ - sin²θ) = 1

We know that cos2θ = cos²θ - sin²θ. so, substituting this into the equation, we have that

2(cos²θ - sin²θ) = 1

2(cos2θ) = 1

cos2θ = 1/2

Taking inverse cosine, we have that

2θ = cos⁻¹(1/2)

2θ = 30°

θ = 30°/2

θ = 15°

So, θ = 15°

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a) The inequality for the expression 2z + 9 is 2z + 9 > 13.

b) The inequality for the expression 3(z - 4) is 3z - 12 > -6.

c) The inequality for the expression 4 + 2z is 4 + 2z > 8.

d) The inequality for the expression 5(3z - 2) is 15z - 10 > 20.

a) To write an inequality for the expression 2z + 9, we can multiply the given inequality z > 2 by 2 and then add 9 to both sides of the inequality:

2z > 2 * 2

2z > 4

Adding 9 to both sides:

2z + 9 > 4 + 9

2z + 9 > 13

Therefore, the inequality for the expression 2z + 9 is 2z + 9 > 13.

b) For the expression 3(z - 4), we can distribute the 3 inside the parentheses:

3z - 3 * 4

3z - 12

Since we are given that z > 2, we can substitute z > 2 into the expression:

3z - 12 > 3 * 2 - 12

3z - 12 > 6 - 12

3z - 12 > -6

Therefore, the inequality for the expression 3(z - 4) is 3z - 12 > -6.

c) The expression 4 + 2z does not change with the given inequality z > 2. We can simply rewrite the expression:

4 + 2z > 4 + 2 * 2

4 + 2z > 4 + 4

4 + 2z > 8

Therefore, the inequality for the expression 4 + 2z is 4 + 2z > 8.

d) Similar to the previous expressions, we can distribute the 5 in the expression 5(3z - 2):

5 * 3z - 5 * 2

15z - 10

Considering the given inequality z > 2, we can substitute z > 2 into the expression:

15z - 10 > 15 * 2 - 10

15z - 10 > 30 - 10

15z - 10 > 20

Therefore, the inequality for the expression 5(3z - 2) is 15z - 10 > 20.

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The probability of selecting a junior who takes the bus is P (Junior who takes the bus) = 12/200 = 0.06Hence, the probability that a randomly selected student is a junior who takes the bus is 0.06 or 6/100.

The given frequency table on how students get to school among the high school students is represented in the below table:Transportation Walk Bike BusDriveTotalGrade 9 11 10 14 15 50Grade 10 10 7 13 20 50Grade 11 8 6 12 24 50Grade 12 5 8 8 29 50 Total 34 31 47 88 200Given data from the above frequency table, we are interested in finding the probability of a randomly selected student being a junior who takes the bus.SolutionWe know that the total number of students is 200, and the total number of junior students is 50. Hence the probability of selecting a junior is P (Junior) = 50/200 = 0.25Similarly, the number of students who take the bus is 47 and the number of junior students who take the bus is 12.

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The equation of the parabola is y = 1/4(x - 6)^2 + 1.

The equation of the parabola with its focus at (6,2) and its directrix y = 0 is y = 1/4(x - 6)^2 + 1. The equation of a parabola is given by the formula y = a(x - h)^2 + k, where(h, k) are the coordinates of the vertex of the parabola and a is a constant called the focus.

The distance between the vertex and the focus is equal to a. Also, the parabola has a directrix, which is a line perpendicular to the axis of symmetry and equidistant from the vertex and the focus.

Step 1: Find the vertex of the parabola. The vertex is halfway between the focus and the directrix, so its y-coordinate is the distance between the focus and the directrix, which is 2. The x-coordinate is the same as the x-coordinate of the focus, which is 6. Therefore, the vertex is (6, 2).

Step 2: Find the distance between the vertex and the focus. The distance between the vertex and the focus is equal to a. The directrix is y = -a, which is y = 0 in this case. Therefore, a = 2.

Step 3: Write the equation of the parabola using the vertex and the focus. Substitute the values of h, k, and a into the formula for the equation of a parabola. The equation of the parabola is y = 1/4(x - 6)^2 + 1. Therefore, the correct answer is (A) y = 1/4(x - 6)^2 + 1.

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Answer:

Step-by-step explanation:

To check whether the given credit card number is valid or not, we need to apply the Luhn algorithm or the mod-10 algorithm. The Luhn algorithm works by adding up all the digits in the credit card number and checking if the sum is divisible by 10 or not. If it is, then the credit card number is considered valid, otherwise, it is invalid.

Let's apply the Luhn algorithm to the given credit card number:

Step 1: Starting from the rightmost digit, double every second digit

| 5 | 4 | 2 | 4 | 9 | 8 | 1 | 3 | 2 | 7 | 2 | 0 | 0 | 0 | 8 | 5 |

|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

| | 8 | | 8 | | 16| | 6 | | 14| | 0 | | 0 | | 10|

Step 2: If the doubled value is greater than 9, add the digits of the result

| 5 | 4 | 2 | 4 | 9 | 8 | 1 | 3 | 2 | 7 | 2 | 0 | 0 | 0 | 8 | 5 |

|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

| | 8 | | 8 | | 7 | | 6 | | 5 | | 0 | | 0 | | 1 |

Step 3: Add up all the digits in the credit card number, including the check digit

5 + 4 + 2 + 4 + 9 + 8 + 1 + 3 + 2 + 7 + 2 + 0 + 0 + 0 + 8 + 5 + 1 = 61

Step 4: If the sum is divisible by 10, then the credit card number is valid, otherwise, it is invalid.

61 is not divisible by 10, therefore the given credit card number is invalid.

Hence, it can be concluded that 5424 9813 2720 0085 is an invalid MASTERCARD credit card number.

Using the laws of logarithm to write 1/2log(x² + 18) - 1/2log(x² + 20) - 15/2log(x³ - 2) in the form Alog(x² + 18) + Blog(x² + 20) + Clog(x³ - 2), we have that

Laws of logarithm are the laws which govern the operations of logarithms

Given the logarithm expression log√[(x² + 18)/{(x² + 20)(x³ - 2)¹⁵}], we desire to write it in the form Alog(x² + 18) + Blog(x² + 20) + Clog(x³ - 2), we proceed as follows

Since log√[(x² + 18)/{(x² + 20)(x³ - 2)¹⁵}], using the root law of exponents which is [tex]\sqrt{x} = x^{\frac{1}{2} }[/tex], we have that

log√[(x² + 18)/{(x² + 20)(x³ - 2)¹⁵}] = log√[(x² + 18)/√{(x² + 20)(x³ - 2)¹⁵}]

= log[(x² + 18)¹/₂/{(x² + 20)¹/₂(x³ - 2)¹⁵/₂}],

Now using the division law of loagarithm which is log(a/b) = loga - logb, we have that

log[(x² + 18)¹/₂/{(x² + 20)¹/₂(x³ - 2)¹⁵/₂}] = log(x² + 18)¹/₂ - log{(x² + 20)¹/₂(x³ - 2)¹⁵/₂}]

Next using the multiplication law of logarithm which is logab = loga + logb, we have that

log(x² + 18)¹/₂ - log{(x² + 20)¹/₂(x³ - 2)¹⁵/₂}] =  log(x² + 18)¹/₂ - log(x² + 20)¹/₂ - log(x³ - 2)¹⁵/₂

Finally, using the power law of logarithm which is logxⁿ = nlogx, we have that

log(x² + 18)¹/₂ - log(x² + 20)¹/₂ - log(x³ - 2)¹⁵/₂}] = 1/2log(x² + 18) - 1/2log(x² + 20) - 15/2log(x³ - 2)

So,  log√[(x² + 18)/{(x² + 20)(x³ - 2)¹⁵}] =  1/2log(x² + 18) - 1/2log(x² + 20) - 15/2log(x³ - 2)

Comparing  1/2log(x² + 18) - 1/2log(x² + 20) - 15/2log(x³ - 2) to Alog(x² + 18) + Blog(x² + 20) + Clog(x³ - 2), we have that

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The surface area of a rectangular prism is 48 5/6 mi².

In Mathematics and Geometry, the surface area of a rectangular prism can be calculated and determined by using this mathematical equation or formula:

Surface area of a rectangular prism = 2(LH + LW + WH)

Where:

By substituting the given side lengths into the formula for the surface area of a rectangular prism, we have the following;

Surface area of rectangular prism = 2[6 × 2 1/3 + (1 1/4 × 6) + (1 1/4 × 2 1 /3)]

Surface area of rectangular prism = 2[14 + 15/2 + 35/12]

Surface area of rectangular prism = 48 5/6 mi².

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Answer:

Step-by-step explanation:480 x o.75=360

and now you add this amount on payed

360 + 480=840 was original price

To calculate the amounts invested at 7% and 5%, let's assume the amount invested at 7% is x dollars. Then the amount invested at 5% would be (22,954 - x) dollars since the total investment is $22,954.

The interest earned from the investment at 7% would be 7% of x, which is 0.07x dollars. Similarly, the interest earned from the investment at 5% would be 5% of (22,954 - x), which is 0.05(22,954 - x) dollars.

The total interest earned is the sum of the interest earned from both investments. Therefore, we can write the equation:

0.07x + 0.05(22,954 - x) = Total Interest

To find the amounts invested at each rate, we can solve this equation.

0.07x + 0.05(22,954 - x) = Total Interest

0.07x + 0.05 * 22,954 - 0.05x = Total Interest

0.07x + 1,147.7 - 0.05x = Total Interest

0.02x = Total Interest - 1,147.7

x = (Total Interest - 1,147.7) / 0.02

Given the specific values of the total interest, you can substitute them into the equation and solve for x. This will give you the amount invested at 7%. Subtracting this amount from $22,954 will give you the amount invested at 5%.

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The trigonometric identities for the terms in the left hand side of the equation indicates that we get;

tan(θ)/(1 - tan(θ)) = sin(θ)/(cos(θ) - sin(θ))

-cot(θ)/(1 - cot(θ)) = cos(θ)/(cos(θ) - sin(θ)), therefore;

Trigonometric identities are equations involving trigonometric functions that are true for all values in the interval of the input variable.

The specified equation can be expressed as follows;

tan(θ)/(1 - tan(θ)) - cot(θ)/(1 - cot(θ)) = (cos(θ) + sin(θ))/(cos(θ) - sin(θ))

The trigonometric identity indicates that the left hand side of the equation can be presented as follows;

tan(θ) = sin(θ)/cos(θ)

cot(θ) = cos(θ)/sin(θ)

(sin(θ)/cos(θ))/(1 - (sin(θ)/cos(θ))) - (cos(θ)/sin(θ))/(1 - (cos(θ)/sin(θ)))

(sin(θ)/cos(θ))/(1 - (sin(θ)/cos(θ))) = (sin(θ)/cos(θ)) × (cos(θ) - (sin(θ))/cos(θ)))

(sin(θ)/cos(θ)) × (cos(θ) - (sin(θ))/cos(θ))) = -(cos²(θ) - 1 - cos(θ)·sin(θ))/(2·cos²(θ) - 1)

-(cos²(θ) - 1 - cos(θ)·sin(θ))/(2·cos²(θ) - 1) = -(cos²(θ) - 1 - cos(θ)·sin(θ))/(cos²(θ) - 1 + cos²(θ))

= -(-sin²(θ) - cos(θ)·sin(θ))/(cos²(θ) - sin²(θ)) = (sin²(θ) + cos(θ)·sin(θ))/(cos²(θ) - sin²(θ))

(sin²(θ) + cos(θ)·sin(θ)) = sin(θ)·(sin(θ) + cos(θ))

(cos²(θ) - sin²(θ)) = (cos(θ) - sin(θ)) × (cos(θ) + sin(θ))

sin(θ)·(sin(θ) + cos(θ))/((cos(θ) - sin(θ)) × (cos(θ) + sin(θ)))

sin(θ))/(cos(θ) - sin(θ))

Similarly;

-(cos(θ)/sin(θ))/(1 - (cos(θ)/sin(θ))) = cos(x)·(cos(x) + sin(x))/((cos(θ) - sin(θ)) × (cos(θ) + sin(θ)))

cos(x)·(cos(x) + sin(x))/((cos(θ) - sin(θ)) × (cos(θ) + sin(θ))) = cos(x)/((cos(θ) - sin(θ)))

Therefore; (sin(θ)/cos(θ))/(1 - (sin(θ)/cos(θ))) - (cos(θ)/sin(θ))/(1 - (cos(θ)/sin(θ))) = sin(θ)/(cos(θ) - sin(θ)) + cos(x)/((cos(θ) - sin(θ)))

(sin(θ)/cos(θ))/(1 - (sin(θ)/cos(θ))) - (cos(θ)/sin(θ))/(1 - (cos(θ)/sin(θ))) = (sin(θ) + cos(θ))/(cos(θ) - sin(θ)) = (cos(θ) + sin(θ))/(cos(θ) - sin(θ))

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The step that should be corrected in the construction of a line parallel to the line AB passing through C is the option;

C. The second arc should be centered at C

The steps to construct a line parallel to another line are;

Place the compass at the point D and draw an arc with radius DE

Place the compass at the point C and with the same radius DE, draw an arc to intersect the line EC at a point above the point C at G

Open the compass to the radius EF, place the compass at the point G and draw an arc to intersect the arc drawn at C at the point H

Join CH and extend the line to complete the construction of a line parallel to the line AB at C

The correct option is therefore, option C

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Answer:

[tex]\displaystyle[/tex][tex]\displaystyle f'(1)=-\frac{9}{e^3}[/tex]

Step-by-step explanation:

Use Quotient Rule to find f'(x)

[tex]\displaystyle f(x)=\frac{3x^2+2}{e^{3x}}\\\\f'(x)=\frac{e^{3x}(6x)-(3x^2+2)(3e^{3x})}{(e^{3x})^2}\\\\f'(x)=\frac{6xe^{3x}-(9x^2+6)(e^{3x})}{e^{6x}}\\\\f'(x)=\frac{6x-(9x^2+6)}{e^{3x}}\\\\f'(x)=\frac{-9x^2+6x-6}{e^{3x}}[/tex]

Find f'(1) using f'(x)

[tex]\displaystyle f'(1)=\frac{-9(1)^2+6(1)-6}{e^{3(1)}}\\\\f'(1)=\frac{-9+6-6}{e^3}\\\\f'(1)=\frac{-9}{e^3}[/tex]

Answer:

[tex]f'(1)=-\dfrac{9}{e^{3}}[/tex]

Step-by-step explanation:

Given rational function:

[tex]f(x)=\dfrac{3x^2+2}{e^{3x}}[/tex]

To find the value of f'(1), we first need to differentiate the rational function to find f'(x). To do this, we can use the quotient rule.

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Quotient Rule for Differentiation}\\\\If $f(x)=\dfrac{g(x)}{h(x)}$ then:\\\\\\$f'(x)=\dfrac{h(x) g'(x)-g(x)h'(x)}{(h(x))^2}$\\\end{minipage}}[/tex]

[tex]\textsf{Let}\;g(x)=3x^2+2 \implies g'(x)=6x[/tex]

[tex]\textsf{Let}\;h(x)=e^{3x} \implies h'(x)=3e^{3x}[/tex]

Therefore:

[tex]f'(x)=\dfrac{e^{3x} \cdot 6x -(3x^2+2) \cdot 3e^{3x}}{\left(e^{3x}\right)^2}[/tex]

[tex]f'(x)=\dfrac{6x -(3x^2+2) \cdot 3}{e^{3x}}[/tex]

[tex]f'(x)=\dfrac{6x -9x^2-6}{e^{3x}}[/tex]

To find f'(1), substitute x = 1 into f'(x):

[tex]f'(1)=\dfrac{6(1) -9(1)^2-6}{e^{3(1)}}[/tex]

[tex]f'(1)=\dfrac{6 -9-6}{e^{3}}[/tex]

[tex]f'(1)=-\dfrac{9}{e^{3}}[/tex]

Answer:

Step-by-step explanation:

[tex]\frac{26y^3-8y}{2y} (4y+1)\\ \\=\frac{2y(13y^2-4}){2y} (4y+1)\\\\=(13y^2-4) (4y+1)\\\\= 13(4)y^3 -4(4y) + 13y^2 -4\\\\= 52y^3 -15y + 13y^2 -4\\\\= 52y^3 + 13y^2 -15y -4[/tex]

Answer:

[tex]\dfrac{51}{2}[/tex]

Step-by-step explanation:

The value of a definite integral represents the area between the x-axis and the graph of the function you’re integrating between two limits.

[tex]\boxed{\begin{minipage}{8.5 cm}\underline{De\:\!finite integration}\\\\$\displaystyle \int^b_a f(x)\:\:\text{d}x$\\\\\\where $a$ is the lower limit and $b$ is the upper limit.\\\end{minipage}}[/tex]

The given definite integral is:

[tex]\displaystyle \int^5_{-7} f(x)\; \;\text{d}x[/tex]

This means we need to find the area between the x-axis and the function between the limits x = -7 and x = 4.

Notice that the function is below the x-axis between x = -7 and x = -2.

Therefore, we need to separate the integral into two areas and add them together:

[tex]\displaystyle \int^5_{-7} f(x)\; \;\text{d}x=\int^{-2}_{-7} f(x)\; \;\text{d}x+\int^5_{-2} f(x)\; \;\text{d}x[/tex]

The area between the x-axis and the function between the limits x = -7 and x = -2 is a triangle with base of 5 units and height of 3 units.

The area between the x-axis and the function between the limits x = -2 and x = 5 is a trapezoid with bases of 5 and 7 units, and a height of 3 units.

Using the formulas for the area of a triangle and the area of a trapezoid, the definite integral can be calculated as follows:

[tex]\begin{aligned}\displaystyle \int^5_{-7} f(x)\; \;\text{d}x&=\int^{-2}_{-7} f(x)\; \;\text{d}x+\int^5_{-2} f(x)\; \;\text{d}x\\\\& =\dfrac{1}{2}(5)(3)+\dfrac{1}{2}(5+7)(3)\\\\& =\dfrac{15}{2}+18\\\\& =\dfrac{51}{2}\end{aligned}[/tex]

Note:  If you integrate a function to find an area that lies below the x-axis, it will give a negative value. So when finding an area like this, you will need to make your answer positive, since area cannot be negative.

Answer:

x = 11

Step-by-step explanation:

3x+1 = 2x+12 (subtract 2x from both sides)

x+1 = 12 (subtract 1 from both sides)

x = 11

So x is equal to 11