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The various trigonometric ratios of the right angle triangle are:

cos 38 = x/15

sin 38 = y/15

cos 52 = y/15

sin 52 = x/15

The three most common trigonometric ratios are:

sin θ = opposite/hypotenuse

cos θ = adjacent/hypotenuse

tan θ = opposite/adjacent

Using trigonometric ratios, we can see that:

cos 38 = x/15

Similarly:

sin 38 = y/15

cos 52 = y/15

sin 52 = x/15

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Answer:

[tex]\textsf{A.} \quad (2+x)^x\left[\dfrac{x}{2+x}+\ln(2+x)\right][/tex]

Step-by-step explanation:

Replace f(x) with y in the given function:

[tex]y=(x+2)^x[/tex]

Take natural logs of both sides of the equation:

[tex]\ln y=\ln (x+2)^x[/tex]

[tex]\textsf{Apply the log power law to the right side of the equation:} \quad \ln a^n=n \ln a[/tex]

[tex]\ln y=x\ln (x+2)[/tex]

Differentiate using implicit differentiation.

Place d/dx in front of each term of the equation:

[tex]\dfrac{\text{d}}{\text{d}x}\ln y=\dfrac{\text{d}}{\text{d}x}x\ln (x+2)[/tex]

First, use the chain rule to differentiate terms in y only.

In practice, this means differentiate with respect to y, and place dy/dx at the end:

[tex]\dfrac{1}{y}\dfrac{\text{d}y}{\text{d}x}=\dfrac{\text{d}}{\text{d}x}x\ln (x+2)[/tex]

Now use the product rule to differentiate the terms in x (the right side of the equation).

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Product Rule for Differentiation}\\\\If $y=uv$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=u\dfrac{\text{d}v}{\text{d}x}+v\dfrac{\text{d}u}{\text{d}x}$\\\end{minipage}}[/tex]

[tex]\textsf{Let}\; u=x \implies \dfrac{\text{d}u}{\text{d}x}=1[/tex]

[tex]\textsf{Let}\; v=\ln(x+2) \implies \dfrac{\text{d}v}{\text{d}x}=\dfrac{1}{x+2}[/tex]

Therefore:

[tex]\begin{aligned}\dfrac{1}{y}\dfrac{\text{d}y}{\text{d}x}&=x\cdot \dfrac{1}{x+2}+\ln(x+2) \cdot 1\\\\\dfrac{1}{y}\dfrac{\text{d}y}{\text{d}x}&= \dfrac{x}{x+2}+\ln(x+2)\end{aligned}[/tex]

Multiply both sides of the equation by y:

[tex]\dfrac{\text{d}y}{\text{d}x}&=y\left( \dfrac{x}{x+2}+\ln(x+2)\right)[/tex]

Substitute back in the expression for y:

[tex]\dfrac{\text{d}y}{\text{d}x}&=(x+2)^x\left( \dfrac{x}{x+2}+\ln(x+2)\right)[/tex]

Therefore, the differentiated function is:

[tex]f'(x)=(x+2)^x\left[\dfrac{x}{x+2}+\ln(x+2)\right][/tex]

[tex]f'(x)=(2+x)^x\left[\dfrac{x}{2+x}+\ln(2+x)\right][/tex]

a. To find the sum of 2 (a rational number) and (-2 + √3), we add them together:

2 + (-2 + √3) = 0 + √3 = √3

The sum is √3. Since √3 is an irrational number, the sum is irrational.

b. The expression -2 + √3 cannot be rational because a rational number can be expressed as the ratio of two integers, while √3 is an irrational number. If -2 + √3 were rational, it could be expressed as a fraction a/b, where a and b are integers. However, we know that √3 is irrational, so it cannot be written as a fraction of two integers. Therefore, -2 + √3 cannot be rational.

The most suspicious set of outcomes of 30 games played between the two computers would be ABBABABABABBABABABABABABABABAA.

In a fair game where player B is supposed to win 4 out of 5 times, we would expect player B to win approximately 24 out of 30 games.  However, option C shows player B winning only 6 out of 30 games, which deviates significantly from the expected outcome.

This set of outcomes suggests a strong bias towards player A winning, as player B's wins are significantly lower than expected.

It is highly unlikely to observe such a low success rate for player B if the game was truly biased in favor of player B winning 4 out of 5 times.

Option C raises suspicions of an unfair advantage given to player A, potentially indicating some manipulation or incorrect implementation in the programming of the game.

Further investigation would be warranted to determine the cause of this biased outcome and to ensure fair play between the two computers in future games.

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Answer:

To factor the polynomial y = x³ + 2x² - 5x - 6 by dividing by (x + 1), we can use synthetic division:

-1  |   1   2   -5   -6

        |_______-1___-1___6

        | 1   1   -6   0

The result of the synthetic division is: x² + x - 6.

Therefore, we can write the polynomial as:

y = (x + 1)(x² + x - 6)

To factor it completely, we can factor the quadratic expression x² + x - 6:

y = (x + 1)(x + 3)(x - 2)

Therefore, the polynomial function y = x³ + 2x² - 5x - 6 can be factored completely as y = (x + 1)(x + 3)(x - 2).

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To find the probability of randomly selecting a Spade or a face card given it's black, we need to determine the number of favorable outcomes and the total number of possible outcomes. Let's break it down step by step:

Step 1: Determine the number of favorable outcomes:

First, let's determine the number of black cards in a deck of 52 cards. In a standard deck, there are 26 black cards, which include 13 Spades (black suits) and 13 Clubs (black suits). Therefore, we have 26 favorable outcomes.

Next, we need to determine the number of face cards that are black. In each suit, there are three face cards: Jack, Queen, and King. So, there are a total of 12 face cards in the deck, out of which 6 are black (2 black face cards in each black suit). Therefore, we have 6 additional favorable outcomes.

Step 2: Determine the total number of possible outcomes:

In a standard deck of 52 cards, there are 52 possible outcomes.

Step 3: Calculate the probability:

The probability is the ratio of favorable outcomes to the total number of possible outcomes.

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

Probability = (26 + 6) / 52

Probability = 32 / 52

Probability = 8 / 13

So, the probability of randomly selecting a Spade or a face card given it's black is 8/13.

Please note that this calculation assumes that the selection is random and the deck of cards is well-shuffled.

\(2 \cdot m_A = m_B\) is true about the mass of Ball A vs. Ball B.

To determine the relationship between the masses of Ball A and Ball B given that Ball A has double the acceleration of Ball B, let's analyze the situation.

We know that the net force acting on both balls is [tex]50 \ N[/tex] . According to Newton's second law of motion, the net force acting on an object is directly proportional to its acceleration and inversely proportional to its mass.

Let's denote the mass of Ball A as [tex]\(m_A\)[/tex] and the mass of Ball B as [tex]\(m_B\)[/tex]. We also know that the acceleration of Ball A is twice the acceleration of Ball B.

Using Newton's second law, we can express the relationship between the forces, masses, and accelerations as:

[tex]\[F = m \cdot a\][/tex]

For Ball A:

[tex]\[50 = m_A \cdot (2a)\][/tex]

Simplifying:

[tex]\[50 = 2m_A \cdot a\][/tex]

For Ball B:

[tex]\[50 = m_B \cdot a\][/tex]

Comparing the two equations, we can see that the mass of Ball A,[tex]\(m_A\)[/tex], must be twice the mass of Ball B, [tex]\(m_B\)[/tex], for both balls to experience the same net force with Ball A having double the acceleration of Ball B.

Therefore, the correct statement about the mass of Ball A vs. Ball B is:

[tex]\[2 \cdot m_A = m_B\][/tex]

Hence, [tex]\(2 \cdot m_A = m_B\)[/tex] must be true.

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Check the picture below.

Answer:

3

Step-by-step explanation:

since 1 is one and 2 is 2

Hello !

1 + 2 = 3

Have a good day !

Answer: The box of pencils has a total of 12 pencils inside the box.

Step-by-step explanation:

Each pencil weighs 3/10 or 0.3 oz.

Since the total weight of the box is 0.4 ounces, subtract that with 4 oz- the total weight of the box. You then get 3.6 ounces worth of pencils.

To find how many pencils that are in the box, you will need to divide 3.6 with the amount 0.3.

3.6/0.3 = 12

Therefore, the box of pencils has a total of 12 pencils inside the box.

1. The velocity of the object is 3 m/s.

2. The object is moving in positive direction.

3. The total distance travelled is 5√10.

1. The velocity of the object is

= displacement / time

= 12/ 4

= 3 m/s

2. The object is moving in positive direction.

3. The total distance travelled

= √225+ 25

= √250

= 5√10

4. The displacement is

= 1/2 x base x height

= 1/2 x 5 x 15

= 37.5 m

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Saving for a down payment was worth it for Isabel. It reduces the total amount paid for the car by lowering the loan amount, resulting in less interest paid over time.

When evaluating whether saving for a down payment on a car was worth it for Isabel, we need to consider the difference in the total amount paid for the car in two scenarios: with a down payment and without a down payment. Let's analyze the two scenarios to make a judgment.

Scenario 1: With a down payment

In this scenario, Isabel saved up and made a down payment on the car. As a result, her monthly loan payment fits within her budget. By making a down payment, she reduces the principal amount borrowed and, subsequently, the interest charged on the loan.

The benefits of saving for a down payment include:

Lower loan amount: By making a down payment, Isabel reduces the total amount borrowed from the lender.

Reduced interest: With a lower loan amount, the interest charged on the loan is also reduced.

Shorter loan term: A down payment can help Isabel secure a shorter loan term, reducing the overall interest paid over time.

Scenario 2: Without a down payment

In this scenario, Isabel does not make a down payment and finances the entire purchase price of the car. As a result, her monthly loan payment may be higher, as she is borrowing a larger sum and potentially paying more interest over the loan term.

The drawbacks of not saving for a down payment include:

Higher loan amount: Without a down payment, the entire purchase price is financed, leading to a higher loan amount.

Increased interest: With a higher loan amount, more interest may accumulate over the loan term, resulting in a higher total cost of the car.

By comparing the two scenarios, we can assess the impact on the total amount paid for the car. Generally, having a down payment reduces the loan amount, decreases interest charges, and may result in a shorter loan term. This, in turn, can lead to substantial savings in the long run.

To determine the exact difference in the total amount paid for the car in each scenario, we would need specific details such as the purchase price of the car, the interest rate, and the loan term. With these figures, we could calculate the total cost of the car, including interest, in both scenarios and compare the results. Based on this analysis, we can then determine if saving for a down payment was worth it for Isabel.

In conclusion, saving for a down payment on a car can offer financial benefits such as lower loan amounts, reduced interest charges, and potentially shorter loan terms. Considering the potential cost savings and improved financial stability, it is generally advisable for Isabel to save for a down payment, making it worth the effort and consideration.

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Question

The down payment makes the monthly loan payment fit into Isabel's budget. Besides this benefit, do you think saving for a down payment on the car was worth it for Isabel? Use each monthly payment with and without a down payment) to find the difference of the total amount paid for the car in each scenario to justify your answer.

Answer:

[tex]\huge\boxed{\sf x = 12}[/tex]

Step-by-step explanation:

Let the smaller number be x and greater number be y.

y = 7 + 2x ----------------(1)

x + y = 43 ----------------(2)

Put Eq. (1) in Eq. (2)

x + 7 + 2x = 43

x + 2x + 7 = 43

3x + 7 = 43

3x = 43 - 7

3x = 36

x = 36/3

Put x = 12 in Eq. (1)

y = 7 + 2(12)

y = 7 + 24

[tex]\rule[225]{225}{2}[/tex]

The box plot which correctly shows displays the given data is the box plot where the whisker ranges from 1 to 6 and the box ranges from 2.5 to 4 with a line at 3.

Given a data set,

0, 1, 2, 3, 3, 3, 3, 3, 3, 3, 5, 5, 6

Box shows the interquartile range.

Starting line of the box is first quartile, middle line is the median and the ending line is the third quartile.

There are 13 data points.

Here median = middle point = 7th point = 3

So the middle line of the box is at 3.

Median divides data plot in to two : 0, 1, 2, 3, 3, 3 and 3, 3, 3, 5, 5, 6.

First quartile is the median of the data points 0, 1, 2, 3, 3, 3.

First quartile, Q1 = average of middle two elements = (2 + 3) / 2 = 2.5

Third quartile is the median of the data points 3, 3, 3, 5, 5, 6.

Third quartile, Q3 = (3 + 5) / 2 = 4

IQR = Q3 - Q1 = 1.5

So the box ranges from 2.5 to 4 with a line at 3.

Now, we have to find the extension of whiskers, that is minimum and maximum value.

Lower whisker = Q1 - 1.5 (IQR) = 0.25

No values less than 0.25 is on the line.

Upper whisker = Q3 + 1.5 (IQR) = 6.25

No values greater than 6.25 will be on the line.

From the data, 0 belong as an outlier outside the whisker.

So whisker ranges from 1 to 6.

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Answer:

length A is 5 ft while width of B is 4

[Top Question] Answer: A. |-6| + |3|

Step-by-step explanation:

       First, we realize they are both on the same y-axis as they both have y-values of 4. This means that we can "ignore" the y-value and focus on the x-value. To find the distance between these two points, we will find the absolute value of the x-values and add them together. This gives us answer option A;

                   A. |-6| + |3|

[Bottom Question] Answer: 36 [tex]\frac{3}{4}[/tex] units²

Step-by-step explanation:

       To find the area of a parallelogram, we multiply the base by the height. I will change 8 [tex]\frac{3}{4}[/tex] into 8.75 and 4 [tex]\frac{1}{5}[/tex] into 4.2 to make multiplying with a calculator easier.

A = bh

A = (8.75 units)(4.2 units)

A = 36.75 units² = 36 [tex]\frac{3}{4}[/tex] units²

Answer:

it's A

Step-by-step explanation:

if you go to the fair they also tell you the price for one ticket then you have to add all that up to see how much money you need for that much people that you bring so it would make more since

Answer:

[tex]\huge\boxed{\sf Solution\ Set: \{x | x > 8/3\}}[/tex]

Step-by-step explanation:

4x - 7 < 7x - 15

-7 < 7x - 4x - 15

-7 < 3x - 15

-7 + 15 < 3x

8 < 3x

8/3 < x

[tex]\rule[225]{225}{2}[/tex]

sinθ = t/o

cosθ = m/o

tanθ = t/m

cscθ = o/t

secθ = o/m

cotθ = m/t

The model best fits the given situation is exponential.

Option D is the correct answer.

We have,

The given situation describes a situation where the height of a tree is increasing at a constant rate (2 feet) per year.

This type of growth is typically modeled by an exponential function, where the variable (in this case, the number of years) is in the exponent.

In an exponential model, the quantity being measured (in this case, the height of the tree) grows or decays at an exponential rate over time.

Thus,

The model best fits the given situation is exponential.

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Answer:

C, 36 3/4

Step-by-step explanation:

Multiply 8 3/4 *4 1/5

The Probability of selecting a jelly-filled donut followed by a lemon-filled donut is approximately 0.087, or 8.7%.

The probability of selecting a jelly-filled donut followed by a lemon-filled donut, we need to determine the total number of possible outcomes and the number of favorable outcomes.

In this case, we have 24 donuts in the box, and we are selecting two donuts without replacement. Let's calculate the probability step by step:

Step 1: Calculate the probability of selecting a jelly-filled donut.

Out of the 24 donuts, the number of jelly-filled donuts is not specified. Let's assume there are 8 jelly-filled donuts in the box. Therefore, the probability of selecting a jelly-filled donut on the first pick is 8/24.

Step 2: Calculate the probability of selecting a lemon-filled donut.

After the first donut is selected and eaten, there are 23 donuts remaining in the box. Let's assume there are 6 lemon-filled donuts among the remaining donuts. Therefore, the probability of selecting a lemon-filled donut on the second pick is 6/23.

Step 3: Multiply the probabilities.

To calculate the probability of both events occurring, we multiply the probabilities from Step 1 and Step 2:

(8/24) * (6/23) = 48/552 = 4/46 ≈ 0.087

Therefore, the probability of selecting a jelly-filled donut followed by a lemon-filled donut is approximately 0.087, or 8.7%.

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1. The graph of y=-2x² + 8  in the domain x ∈ [-3, 3] is attached

2. The name is a quadratic graph

3. The range is y ≤ 8

4, The equation of the axes of symmetry is x = 0

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

y =-2x² + 8

And the domain is given as x ∈ [-3, 3]

The above function is a quadratic function that has the following features

a = -2, b = 8 and c = 0

Next, we plot the graph using a graphing tool by taking note of the above features

The graph of the function is added as an attachment

Recall that

y =-2x² + 8

The above has a degree of 2

So, the name of the graph is a quadratic graph

From the graph, the maximum is 8

This means that the highest output value is 8

So, we have the range to be y ≤ 8

This is the line that divides the graph into equal parts

From the graph, the equation of the axes of symmetry is x = 0

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The breakeven point is 890 units and Molander Corporation's margin of safety is $3,600 in dollars and 14.42% as a percentage of its sales.

To find the breakeven point:

Breakeven Point = Fixed Expenses / Contribution Margin per unit

Contribution Margin per unit = Selling Price per unit - Variable Expense per unit

= $24 - $14

= $10

Breakeven Point = $8,900 / $10

= 890 units

The breakeven point is 890 units.

To calculate the margin of safety:

Margin of Safety = Actual Sales - Breakeven Point

Actual Sales = Unit Sales per month

= 1,040 units

Margin of Safety = 1,040 units - 890 units

= 150 units

Margin of Safety in dollars = Margin of Safety × Selling Price per unit

= 150 units × $24

= $3,600

Margin of Safety as a percentage of sales = (Margin of Safety / Actual Sales) × 100

= (150 units / 1,040 units) * 100

= 14.42% (rounded to 2 decimal places)

Therefore, Molander Corporation's margin of safety is $3,600 in dollars and 14.42% as a percentage of its sales.

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Answer:

x = 31°

Step-by-step explanation:

the sum of the 3 angles in a triangle = 180° , that is

x + 54° + 95° = 180°

x + 149° = 180° ( subtract 149° from both sides )

x = 31°

The experimental group consists of the volunteers who receive the bottle of solution with ingredients intended to minimize wrinkles among women aged 40 to 45.

In the given situation, the experimental group refers to the group of participants who receive the bottle of solution that is specifically formulated to minimize wrinkles among women.

The research team is experimenting to test the effectiveness of the product in reducing wrinkles. To do this, they divided the participants into two groups. One group is given a bottle of the solution that contains ingredients intended to lessen wrinkles, while the other group is given a similar bottle of the solution but without any ingredients targeted at reducing wrinkles.

The purpose of the experiment is to compare the outcomes between these two groups and determine whether the solution with the specific wrinkle-reducing ingredients has any effect on minimizing wrinkles among women aged 40 to 45.

The experimental group, in this case, is the group of volunteers who receive the bottle of solution with the ingredients intended to lessen wrinkles. They are the group that is exposed to the independent variable, which in this case is the wrinkle-reducing solution. The research team will monitor and evaluate the effects of the solution on this group to determine its effectiveness in minimizing wrinkles.

Meanwhile, the control group is the group of participants who receive the bottle of solution without any ingredients aimed at reducing wrinkles. The control group serves as a baseline for comparison, allowing the research team to assess the specific impact of the wrinkle-reducing ingredients in the experimental group.

Therefore, in this scenario, the experimental group consists of the volunteers who receive the bottle of solution with ingredients intended to minimize wrinkles among women aged 40 to 45.

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The value of ∂x/∂s is  4s^2 * t^6.

To find ∂x/∂s, we need to differentiate x with respect to s while treating t as a constant. Let's begin by substituting the given expressions for x, y, and z into the function f(x, y, z):

f(x, y, z) = (x^2)(y^3) + z^3

Substituting x, y, and z:

f(s, t) = ((s^2)(t^3))^2 * (s*t)^3 + ((s^2)*t)^3

Expanding the expression:

f(s, t) = (s^4 * t^6) * (s^3 * t^3) + (s^6 * t^3)

Now, let's differentiate x with respect to s:

∂x/∂s = ∂/∂s [(s^2)(t^3)]^2

Using the chain rule, we can differentiate each term separately:

∂x/∂s = 2 * (s^2 * t^3)^1 * ∂/∂s [(s^2)(t^3)]

Differentiating (s^2)(t^3) with respect to s:

∂/∂s [(s^2)(t^3)] = 2s * t^3

Substituting back into the expression for ∂x/∂s:

∂x/∂s = 2 * (s^2 * t^3)^1 * 2s * t^3

Simplifying:

∂x/∂s = 4s^2 * t^6

Therefore, ∂x/∂s = 4s^2 * t^6.

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Answer:

3³-8*3 = 3

Step-by-step explanation:

Use order of operations:

P -> Parentheses

E -> Exponents

M -> Multiplication (left to right)

D -> Division (left to right)

A -> Addition (left to right)

S -> Subtraction (left to right)

3³=27 is done first because of the exponent. Next, we would do 8*3=24 because that has multiplication. Finally, we do 27-24=3 because of subtracting being the last step in PEMDAS.

Therefore, the final answer is 3

Answer:

7

Step-by-step explanation:

x=people who wear glasses

y=people who don't

3y=x

x+y=28

3y+y=28

4y=28

y=7

Answer:

[tex]\huge\boxed{\sf 7 \ students}[/tex]

Step-by-step explanation:

Total students = 28

There are total 4 parts in the class.

1 part does not wear glasses.

3 parts do wear glasses.

We will have to divide total students by 4.

= 28 / 4

7 students do not wear glasses.

[tex]\rule[225]{225}{2}[/tex]