Use the Generalized Power Rule to find the derivative of the function.
f(x) = (3x + 1)^5(3x - 1)^6

a) If the supplement of an angle is 30 degrees more than the measure of the angle, the measure of the angle is 75 degrees.

b) If the supplement of an angle is 12 degrees less than twice the measure of the angle, the measure of the angle is 64 degrees.

a) Let x be the measure of the angle. The supplement of the angle is 180-x. According to the problem, the supplement of the angle is 30 degrees more than the measure of the angle. This can be written as:

180 - x = x + 30

Solving for x, we get:

2x = 150

x = 75

b) Let x be the measure of the angle. The supplement of the angle is 180-x. According to the problem, the supplement of the angle is 12 degrees less than twice the measure of the angle. This can be written as:

180 - x = 2x - 12

Solving for x, we get:

3x = 192

x = 64

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The parametric equations for the tangent line to curve C at point P are: x = -t y = 6t - 1 z = 8t + 6

To find the parametric equations for the tangent line to curve C at point P, we need to first find the derivative of the curve at P.

Taking the derivative of each component of the curve, we get:

dx/dt = -1

dy/dt = -6t

dz/dt = 8

At point P = (0, -1, 6), t = -1.

Plugging this into the derivative, we get:

dx/dt = -1

dy/dt = 6

dz/dt = 8

So, the tangent vector to curve C at point P is < -1, 6, 8 >.

To get the parametric equations for the tangent line, we can use the point-slope form: r(t) = P + t< -1, 6, 8 > Plugging in the coordinates of point P, we get: r(t) = <0, -1, 6> + t< -1, 6, 8 > Expanding this out, we get: r(t) = <-t, 6t - 1, 8t + 6>

So, the parametric equations for the tangent line to curve C at point P are: x = -t y = 6t - 1 z = 8t + 6

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Answer:

After selling 98 cups, she made $98

Step-by-step explanation:

If she sold 98 cups, and each one went for $1, then you can set up an equation:

98(1)=98

Answer:

Ashley made $98.

Step-by-step explanation:

If 1 cup = $1, then you multiply the cups by 98 meaning you also must multiply $1 by 98 to get a total of $98.

Check the picture below.

[tex]\textit{area of a circular ring}\\\\ A=\pi (R^2 - r^2) ~~ \begin{cases} R=\stackrel{outer}{radius}\\ r=\stackrel{inner}{radius}\\[-0.5em] \hrulefill\\ R=5\\ r=3 \end{cases}\implies A=\pi (5^2-3^2)\implies A\approx 50.27~cm^2[/tex]

Answer:

It seems that the three students each calculated the volume of a sphere with a radius of 6 centimeters, but arrived at different results. Diego found the volume to be 288 cubic centimeters, Andre approximated it to be 904 cubic centimeters, and Noah calculated it to be 226 cubic centimeters. It's interesting to see the variation in their calculations.

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D. Both the height and weight distribution are symmetric about the mean.

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Therefore, the molarity of the sugar solution is 0.3704 M at 25°C. Therefore, the molality of the NaCl solution is 1.8994 mol/kg.

In mathematics, an equation is a statement that asserts the equality of two expressions. An equation typically consists of two sides separated by an equal sign (=). The expressions on either side of the equal sign may contain variables, constants, coefficients, and mathematical operations.

Here,

1. To calculate the molarity of a sugar solution, we need to first determine the number of moles of solute (glucose, C6H12O6) present in the solution. We can then divide this number of moles by the volume of the solution in liters to obtain the molarity. The number of moles of glucose in the solution can be calculated as follows:

Number of moles = mass of solute / molar mass of solute

Number of moles = 100.0 g / 180 g/mol

Number of moles = 0.5556 mol

Next, we can calculate the molarity of the solution using the following formula:

Molarity = number of moles / volume of solution (in L)

Molarity = 0.5556 mol / 1.50 L

Molarity = 0.3704 M

2. To calculate the molality of a solution, we need to know the number of moles of solute (NaCl) per kilogram of solvent (water).

First, let's calculate the number of moles of NaCl:

Number of moles = mass of NaCl / molar mass of NaCl

Number of moles = 200.0 g / 58.5 g/mol

Number of moles = 3.4188 mol

Next, we need to calculate the mass of the solvent (water) in kilograms:

Mass of solvent = 2.00 kg - 0.200 kg

Mass of solvent = 1.80 kg

Note that we subtracted the mass of the NaCl from the total mass of the solution to obtain the mass of the solvent.

Finally, we can calculate the molality of the solution using the following formula:

Molality = number of moles of solute / mass of solvent (in kg)

Molality = 3.4188 mol / 1.80 kg

Molality = 1.8994 mol/kg

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To find the critical points of f(x) = x - 18x² + 96x, we need to find the values of x where f'(x) = 0.

f'(x) = 1 - 36x + 96

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

-36x + 97 = 0

x = 97/36

So the critical point is (97/36, f(97/36)).

To use the Second Derivative Test, we need to find f''(x):

f''(x) = -36

At the critical point x = 97/36, f''(97/36) = -36 < 0.

Since f''(97/36) is negative, the Second Derivative Test tells us that the critical point corresponds to a local maximum.

Therefore, the critical point (97/36, f(97/36)) is a local maximum.
To find the critical points of the function f(x) = x - 18x² + 96x, we first need to find its first derivative, f'(x), and then set it to zero to find the critical points.

1. Find the first derivative, f'(x):
f'(x) = d/dx (x - 18x² + 96x) = 1 - 36x + 96

2. Set f'(x) to zero and solve for x:
0 = 1 - 36x + 96
36x = 95
x = 95/36

Now, let's use the Second Derivative Test to determine if this critical point corresponds to a local minimum or maximum.

3. Find the second derivative, f''(x):
f''(x) = d/dx (1 - 36x + 96) = -36

4. Evaluate f''(x) at the critical point x = 95/36:
f''(95/36) = -36

Since f''(95/36) is negative, the Second Derivative Test tells us that the critical point x = 95/36 corresponds to a local maximum.

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The length of the diagonal of the house is 23.43 meters based on the dimensions of the John's guest house.

The diagonal of the house will be calculated using Pythagoras theorem. We know that diagonal will be hypotenuse and length and width will be base and perpendicular.

Diagonal = ✓length² + width²

Keep the values in formula

Diagonal = ✓18² + 15²

Taking square

Diagonal = ✓324 + 225

Adding the values

Diagonal = ✓549

Taking square root

Diagonal = 23.43 meters

Hence, the diagonal of the house is 23.43 meters.

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The equation expressing y, as the total number of volumes in the library, in terms of x, the number of mouths after march 2004 is y = 400x + 60,000.

The equation expressing y, as the total number of volumes in the library, in terms of x, the number of mouths after march 2004 is determined as follows;

Using general linear equation;

y = mx + b

where:

The initial volume as at March = 60,000

the rate = 400 volumes / month

The equation is determined as;

y = 400x + 60,000

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The correct statement regarding the solution to the system of equations is given as follows:

There is one unique solution (−5, 9).

The slope-intercept representation of a linear function is given by the equation shown as follows:

y = mx + b

The coefficients m and b have the meaning presented as follows:

For the first line, the points are given as follows:

(-10, 10) and (0,8).

Hence the equation is:

y = -0.2x + 8.

For the second line, the points are given as follows:

(-8, 12) and (0,4).

Hence the equation is:

y = -x + 4.

Then the x-coordinate of the solution is obtained as follows:

-0.2x + 8 = -x + 4

0.8x = -4

x = -5.

The y-coordinate is given as follows:

y = -(-5) + 4 = 9.

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Therefore, the equation of the line that passes through the points (-3, 6) and (-5, 9) in slope-intercept form is:

y = -3/2 x + 3/2

Step-by-step explanation:

To find the equation of the line that passes through the points (-3, 6) and (-5, 9) in slope-intercept form (y = mx + b), we need to first find the slope of the line using the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) = (-3, 6) and (x2, y2) = (-5, 9).

m = (9 - 6) / (-5 - (-3))

m = 3 / (-2)

m = -3/2

Now that we have the slope, we can use either of the two given points and the slope to find the y-intercept (b) of the line:

y = mx + b

6 = (-3/2)(-3) + b

6 = 9/2 + b

b = 6 - 9/2

b = 3/2

It will take approximately 60.9 days for 650 mg of chromium-51 to decay to 200 mg.

What is Equation ?

An equation is a mathematical statement that shows that two expressions are equal. It typically contains variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

The decay of a radioactive substance can be modeled by the following equation:

N(t) = N₀ * [tex](1/2)^{(t/T) }[/tex]

where:

N(t) is the amount of the substance remaining after time t

N₀ is the initial amount of the substance

T is the half-life of the substance

We can use this equation to find how long it will take for 650 mg of chromium-51 to decay to 200 mg.

Let's first find the decay constant (λ) for chromium-51:

λ = ㏒(2) ÷ T = ㏒(2) ÷ 28 = 0.0248 (rounded to 4 decimal places)

Now we can use the equation:

N(t) = N₀ * [tex]e^{(-λ*t)}[/tex]

We know that N₀ = 650 mg and N(t) = 200 mg, so we can solve for t:

200 = 650 * [tex]e^{(-0.0248*t)}[/tex]

Dividing both sides by 650:

0.3077 =   [tex]e^{(-0.0248*t)}[/tex]

Taking the natural logarithm of both sides:

㏒(0.3077) = -0.0248*t

Solving for t:

t = ㏒(0.3077) : (-0.0248) ≈ 60.9 days

Therefore, it will take approximately 60.9 days for 650 mg of chromium-51 to decay to 200 mg.

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In this case, the mean absolute deviation of 8.5 indicates that, on average, the data points deviate from the mean by about 8.5 units.

Mean absolute deviation (MAD) is a statistical measure that represents the average distance between each data point and the mean of the data set. It is calculated by finding the absolute value of the difference between each data point and the mean, and then taking the average of these absolute differences. MAD is a useful measure of the variability or spread of a data set, and is often used as an alternative to the more common measure of standard deviation. Like standard deviation, MAD gives an indication of how spread out the data is, but unlike standard deviation, MAD is less sensitive to extreme values or outliers.

Here,

To find the mean absolute deviation of the data, we first need to calculate the mean (average) of the data:

Mean = (46 + 54 + 43 + 57 + 50 + 62 + 78 + 42) / 8

Mean = 52

The mean of the data is 52.

Next, we need to calculate the absolute deviation of each data point from the mean. The absolute deviation is simply the absolute value of the difference between each data point and the mean:

|46 - 52| = 6

|54 - 52| = 2

|43 - 52| = 9

|57 - 52| = 5

|50 - 52| = 2

|62 - 52| = 10

|78 - 52| = 26

|42 - 52| = 10

Now, we can calculate the mean absolute deviation by taking the average of the absolute deviations:

Mean Absolute Deviation = (6 + 2 + 9 + 5 + 2 + 10 + 26 + 10) / 8

Mean Absolute Deviation = 8.5

The mean absolute deviation of the data is 8.5.

Interpretation: The mean absolute deviation represents the average distance between each data point and the mean of the data. In this case, the mean absolute deviation of 8.5 indicates that, on average, the data points deviate from the mean by about 8.5 units. This means that the data points are relatively spread out, with some points being much higher or lower than the mean.

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Andre's claim that the greatest product possible in the game is 987x65 is incorrect.

While 987x65 is a large product, it is not the greatest possible product in the game. In fact, a larger product can be obtained by multiplying the two largest numbers available in the game. Without knowing the specific rules of the game, it is impossible to determine the exact greatest product, but it is certain that 987x65 is not it.

To elaborate, the game likely has certain constraints or rules that limit the numbers that can be multiplied. It may be possible to combine multiple numbers to create a larger product, or to find a different pair of numbers that yield a larger product. Therefore, without knowing the specifics of the game's rules, it is impossible to determine the greatest possible product.

The actual greatest product possible in the game will depend on the specific rules and constraints that are in place. It may be possible to combine multiple numbers to create an even larger product, or to find a different pair of numbers that yield a larger product. Without knowing the specifics of the game's rules, it is impossible to determine the greatest possible product.

In mathematics, the concept of maximum or greatest products is important and is studied in various areas such as algebra, number theory, and calculus. In real-world applications, maximum products play a critical role in determining profits, yields, and returns on investments in economics and finance. In engineering, the concept of maximum product is used in optimization problems, where the goal is to maximize or minimize a certain function or output.

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If ABC is similar to XYZ and YZX, then the corresponding angles of those triangles are same, and the corresponding sides are proportional. because of this triangle ABC is a special type of triangle known as a "similar triangle."

In a similar triangle, the angles of the triangle are equal, however the sides can be exclusive lengths. but, the ratios of the corresponding aspects are usually the same. This property is beneficial in lots of regions of mathematics and physics, which includes trigonometry and the study of geometric shapes.

inside the case of triangle ABC, the fact that it's far much like each XYZ and YZX tells us that its angles are same to those of these triangles, and its sides are proportional to the ones of these triangles. This property may be used to solve many issues regarding triangles and other geometric shape.

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Answer: 10, 911,000,000

Step-by-step explanation:

10,911,000,000 < 11,000,000,000

10,911,000,000 > 10,910,099,999

In one month, a total of 447 people stayed at the hotel.

In one month, a hotel had 382 adults and 65 children staying as guests.

To find out the total number of people who stayed at the hotel, we simply need to add the number of adults and children together.

In one month, a total of 447 people (382 adults and 65 children) stayed at the hotel.

Overall, this problem is a simple example of addition in action. By adding the number of adults and children together, we can determine the total number of people who stayed in the hotel.

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Step-by-step explanation:

180° + 42° + x = 360°

222° + x = 360°

x = 138°

Answer:

138

Step-by-step explanation:

Since a circle is 360 degrees and it is split in half, each half would equal 180 degrees so you would subtract 42 from 180

Answer:

The formula for the volume of a cone is ⅓ r2h cubic units, where r is the radius of the circular base and h is the height of the cone.The volume of any sphere is 2/3rd of the volume of any cylinder with equivalent radius and height equal to the diameter.The formula for the volume of a sphere is 4⁄3πr³. For a cylinder, the formula is πr²h. A cone is ⅓ the volume of a cylinder, or 1⁄3πr²h

Step-by-step explanation:

The formula for volume is: Volume = length x width x height

Answer:

Step-by-step explanation:

Volume of a sphere:  4/3 π r³

4/3 (3.14) (2.5)³ =

4/3 (3.14) (15.625) = 65.42 units³

Volume of a cylinder = π r² h

(3.14) (2.5)² (4)

(3.14) (6.25)(4) = 78.5 units²

Volume of a Cone = 1/3 π r² h

(1/3)(3.14)(2.5)²(4) =

(1/3)(3.14)(6.25)(4) = 26.17 units²

A data set has many very large data values, then the mean will be larger than the median. The correct option is a.

In a dataset with large values, the mean is influenced by outliers and extreme values, whereas the median is not. The median is the value that divides the data into two equal halves, while the mean is calculated by summing up all values and dividing by the total number of values.

Thus, in a skewed dataset with large values, the mean tends to be pulled towards the larger values, resulting in a larger mean than the median. This effect is more pronounced in datasets with a high variance. Therefore, option (a) is the correct answer.

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The new total bill including the tip of 15% is $59.01.

To find the amount of the tip, we can multiply the total bill by the percentage as a decimal:

tip = 0.15 * $51.31 = $7.70

To find the new total including the tip, we can add the tip to the original bill:

new total = $51.31 + $7.70 = $59.01

So the new total, including a 15% tip, will be $59.01.

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The derivatives of the given functions are:

A) h'(x) = (1/2)√xen(x)[2+In(x)]

B) h'(x) = (1/x) - V2V

C) h'(x) = (2/x) + 2

D) h'(x) = 0

A) To find the derivative of h(x) = √xen(x), we use the product rule of differentiation. Let u = √x and v = en(x).

Then, h(x) = uv, and h'(x) = u'v + uv'.

We have u' = (1/2)x^(-1/2) and v' = en(x)(1/x).

Substituting the values, we get h'(x) = (1/2)√xen(x)[2+In(x)].

B) To find the derivative of h(x) = In(x) + V2V, we use the sum rule of differentiation.

Using the properties of logarithms, we rewrite the function as h(x) = In(x) + (1/2)ln(x).

Taking the derivative, we get h'(x) = (1/x) - V2V.

C) To find the derivative of h(x) = V2 In(x) + 2, we use the sum rule of differentiation.

Taking the derivative, we get h'(x) = (2/x) + 2.

D) To find the derivative of h(x) = In(30) + 2ln(x), we use the sum rule of differentiation.

Taking the derivative, we get h'(x) = 0, since the derivative of a constant is always zero.

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The boat must move 785.82 m closer to the cliff  for the angle of depression to become 19°.

We need to find how much closer to the cliff the boat must move for the angle of depression to change from 7° to 19°.

Calculate the distance from the boat to the base of the cliff at 7° angle of depression.
Using the tangent function, we have:
tan(angle) = height/distance
tan(7°) = 150m/distance
distance = 150m/tan(7°)

distance=1221.49

Calculate the distance from the boat to the base of the cliff at 19° angle of depression.
Using the tangent function, we have:
tan(angle) = height/distance
tan(19°) = 150m/distance
distance = 150m/tan(19°)

distance=435.6665

Calculate the difference between the two distances to find out how much closer the boat must move.
difference = distance at 7° angle of depression - distance at 19° angle of depression

Plugging in the values from Steps 1 and 2, we get:
difference = (150m/tan(7°)) - (150m/tan(19°))

difference=1221.49-435.6665

difference=785.8235

After calculating, we find that the boat must move approximately 785.82 meters closer to the cliff for the angle of depression to change from 7° to 19°.

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9. Let's start by defining the variable.

Let's say the number we're trying to find is "x".

Now, let's translate the problem into an equation:

2(x - 4) = x + 6

We can simplify this equation by distributing the 2:

2x - 8 = x + 6

Next, let's isolate the variable on one side of the equation by subtracting x from both sides:

x - 8 = 6

Finally, we can solve for x by adding 8 to both sides:

x = 14

Therefore, the number we're looking for is 14.

10. Let's start by defining the variable.

Let's say the number we're trying to find is "x".

Now, let's translate the problem into an equation:

-(x) - 5 = 3(x + (-7))

We can simplify the right side of the equation by first simplifying the expression inside the parentheses:

-(x) - 5 = 3(x - 7)

Next, we can distribute the 3:

-(x) - 5 = 3x - 21

Let's isolate the variable on one side of the equation by adding x to both sides:

-(x) + x - 5 = 3x + x - 21

Simplifying the left side of the equation:

-5 = 4x - 21

Now, we can isolate the variable by adding 21 to both sides:

-5 + 21 = 4x

Simplifying the left side of the equation:

16 = 4x

Finally, we can solve for x by dividing both sides by 4:

x = 4

Therefore, the number we're looking for is 4.

The proportion of this word problem is 6 : 1 where Neil Armstrong weighed approximately 27.5 pounds on the moon.

The proportion of a word problem represents the relationship between two or more quantities. In this case, the proportion can be set up as:

Weight on Earth : Weight on Moon = 6 : 1

Using the information provided in the problem, we know that Neil Armstrong weighed 165 pounds on Earth. We can use this information to find his weight on the moon by setting up a proportion:

165 : x = 6 : 1

where x represents his weight on the moon. To solve for x, we can cross-multiply and simplify:

165 * 1 = 6 * x

x = 165/6

x ≈ 27.5

Therefore, Neil Armstrong weighed approximately 27.5 pounds on the moon.

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Answer:

x=32°

Step-by-step explanation:

3x-2=2x+30

x-2=30

x=32°

The most likely factor to push the price of a company's stock higher is an increase in demand for the company's stock. Option A.

A stock's price will often increase when there is greater demand than there is supply. Various things, such as good news about the company's financial performance, new product releases or innovations, positive analyst reports, or general market trends, can contribute to this increased demand.

The company's financial performance may benefit from a rise in product demand, but if investors do not view it as a key growth engine for the business, it may not necessarily transfer into a higher stock price.

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Answer:

Step-by-step explanation:

[tex]x^{2}[/tex]=a²+b²

x²=6²+2.5²

x²=36+6.25

x²=42.25

√x=√42.25

x=6.5

Answer:

6.5

Step-by-step explanation:

6^2 + 2.5^2 = c^2

•36 + 6.25 = c^2

•42.25 = c^2

• find square root of 42.25 = 6.5

• 6.5 = C

Hi! To solve for the unknown value in the equation "x + 61 + 42 = degrees", follow these steps:

1. Combine the known values (61 and 42) by adding them together: 61 + 42 = 103.
2. Rewrite the equation with the combined values: x + 103 = degrees.
3. To isolate the unknown value (x), subtract 103 from both sides of the equation: x = degrees - 103.

So, the unknown value x can be found by subtracting 103 from the given degrees value.

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