Identify the domain of the function shown in the graph.

Answer:

Step-by-step explanation:

To show that a² + b² = p² + q², we can start by squaring the given equations and adding them together:

(a sin θ - b cos θ)² + (a cos θ - b sin θ)² = p² + q²

Expanding the squared terms:

(a² sin² θ - 2ab sin θ cos θ + b² cos² θ) + (a² cos² θ - 2ab sin θ cos θ + b² sin² θ) = p² + q²

Combining like terms:

a² sin² θ + a² cos² θ + b² sin² θ + b² cos² θ - 2ab sin θ cos θ - 2ab sin θ cos θ = p² + q²

Using the trigonometric identity sin² θ + cos² θ = 1:

a² + b² - 2ab sin θ cos θ - 2ab sin θ cos θ = p² + q²

Rearranging and factoring out a common factor of 2ab:

(a² + b² - 2ab sin θ cos θ - 2ab sin θ cos θ) = p² + q²

2ab (1 - sin θ cos θ - sin θ cos θ) = p² + q²

2ab (1 - 2 sin θ cos θ) = p² + q²

Using the trigonometric identity 2 sin θ cos θ = sin 2θ:

2ab (1 - sin 2θ) = p² + q²

2ab - 2ab sin 2θ = p² + q²

2ab(1 - sin 2θ) = p² + q²

Since 1 - sin 2θ = cos 2θ:

2ab cos 2θ = p² + q²

Dividing both sides by 2ab:

cos 2θ = (p² + q²) / (2ab)

Using the double angle formula for cosine: cos 2θ = 2cos² θ - 1:

2cos² θ - 1 = (p² + q²) / (2ab)

Rearranging and simplifying:

2cos² θ = (p² + q²) / (2ab) + 1

2cos² θ = (p² + q² + 2ab) / (2ab)

Dividing both sides by 2:

cos² θ = (p² + q² + 2ab) / (4ab)

Using the Pythagorean identity sin² θ + cos² θ = 1, we have:

sin² θ = 1 - cos² θ

sin² θ = 1 - (p² + q² + 2ab) / (4ab)

Rearranging and simplifying:

4ab sin² θ = 4ab - p² - q² - 2ab

4ab sin² θ = 2ab - p² - q²

Multiplying both sides by a² + b²:

4ab (a² + b²) sin² θ = (2ab - p² - q²) (a² + b²)

Expanding both sides:

4a³b sin² θ + 4ab³ sin² θ = 2a³b - p²a² - q²a² - 2ab

Step-by-step explanation:

Perimeter= 12ft

Length= 4ft

Width= ?

Bedroom window = rectangle

Rectangle= l×b

= 4× ?

12÷ 4=3

Answer:

Step-by-step explanation:

When conducting hypothesis testing, the p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true.In this case, if the p-value is 0.15 and the significance level (alpha) is set at 0.01, we can interpret the results as follows:Since the p-value (0.15) is greater than the significance level (0.01), we fail to reject the null hypothesis. There is not enough evidence to conclude that the observed results are statistically significant at the 0.01 level. However, it's important to note that the decision of whether to reject or fail to reject the null hypothesis is based on the chosen significance level, and it's possible that the result may be considered statistically significant at a higher significance level.In summary, the conclusion would be that there is not enough evidence to reject the null hypothesis at the 0.01 significance level.

Answer:

Step-by-step explanation:

Answer:

The function value.of cos (-305.3)° is 0.5921

Explanation:

To find the value of cos (-305.3)°, we can use the periodicity property of the cosine function. Since cos (x) is a periodic function with a period of 360 degrees, we can add or subtract multiples of 360 degrees from the angle without changing the value of the cosine.

Step 1:  Convert -305.3° to an equivalent angle within period one (0 to 360 degrees).

-305.3° + 360° = 54.7°

Step 2: Calculate the cosine of the equivalent angle.

cos (54.7°)

Step 3: Use a scientific calculator or trigonometric table to find the cosine value.

cos (54.7°) ≈ 0.5921

Therefore, the value of cos(-305.3)° is approximately 0.5921

The polynomial -8m²n - 7m²n can be factored using the common factor -m²n. The complete factored form of the polynomial is (-m²n) (8 + 7a).

To find the complete factored form of the polynomial -8m²n - 7m²n, we can factor out common terms from both the terms. The common factor in the terms -8m²n and -7m²n is -m²n. We can write the polynomial as:

-8m²n - 7m²n = (-m²n) (8 + 7a)

Therefore, the complete factored form of the polynomial -8m²n - 7m²n is (-m²n) (8 + 7a). This expression represents the original polynomial in a multiplied form. We can expand this expression using distributive law to verify that it is equivalent to the original polynomial.

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We can substitute this value into the function f(x) = 1/(x + 1):

1) f(g(0)) = 1/3

2) g(f(0)) = 4.

To find the value of f(g(0)), we need to first evaluate g(0) and then substitute that result into f(x).

Given the function g(x) = 2x + 2, we can find g(0) by substituting 0 for x:

g(0) = 2(0) + 2 = 0 + 2 = 2

Now that we know g(0) = 2, we can substitute this value into the function f(x) = 1/(x + 1):

f(g(0)) = f(2)

Substituting 2 for x in f(x):

f(2) = 1/(2 + 1) = 1/3

Therefore, f(g(0)) = 1/3.

Now, let's find the value of g(f(0)).

Given the function f(x) = 1/(x + 1), we need to evaluate f(0) and then substitute that result into g(x).

f(0) = 1/(0 + 1) = 1/1 = 1

Now, using the function g(x) = 2x + 2, we substitute f(0) = 1 into g(x):

g(f(0)) = g(1)

Substituting 1 for x in g(x):

g(1) = 2(1) + 2 = 2 + 2 = 4

Therefore, g(f(0)) = 4.

In summary:

f(g(0)) = 1/3

g(f(0)) = 4

Thus, fg(0) = 1/3 and g(f(0)) = 4.

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Answer:

x < -3 or x > 2

Step-by-step explanation:

x² + x - 6 > 0

Convert the inequality to an equation.

x² + x - 6 = 0

Factor using the AC method and get:

(x - 2) (x + 3) = 0

If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.

x - 2 = 0

x = 2

x + 3 = 0

x = -3

So, the solution is x < -3 or x > 2

Answer:

3(8 + 2) = 3(8) + 3(2) = 3(10) = 30

Answer:

The equivalent equation is -9i-48

Step-by-step explanation:

i = √-1, i² = -1

Therefore (-√-9 + √-4) - (2√576 +√-64)

=(-√(9*(-1)) + √(4*(-1))) - (2 *24 + √(64 *(-1)))

= (-3i + 2i) - (2 * 24 + 8i)

= -i - 48 -8i

= -9i - 48

The expression that represent the relationship is 7 / 2 x - 15 / 2 = 4.

An equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign =.

Therefore, let's find the equation that represents the relationship as follows:

Three consecutive numbers, four times the third decreased by on-half, the second is 4.

Therefore,

Let

x = first number

second number = x + 1

third number  = x + 2

Therefore,

4(x + 2)  - 1 / 2(x + 1) = 4

4x + 8 - 1 / 2x - 1 / 2 = 4

4x - 1 / 2x + 8 - 1 / 2 = 4

7 / 2 x - 15 / 2 = 4

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1) The expression that describes this scenario where Edmond decided to save €10,000 from his first year of work for an emergency fund, earning 6% per year and compounded annually is that the fund will grow to become equal to [tex]10,000 (1.06)^t[/tex] in t years.

2) After ten years of investing at 6%, compounded annually, the amount (future value) in the fund will be €17,908.

3) Given that the annual rate is 6%, the monthly rate will be 0.5%.

4) The exponent for the exponential function will be 120 with the 10 years in the scenario.

5) The complete expression that describes the scenario over 10 years is [tex]10,000(1.005)^{120}[/tex].

6) After ten years of investing at 6% APR compounded monthly, the amount (future value) in the fund will be €18,193.97.

7) There is more money with monthly compounding than annual compounding because the monthly interest is added and compounded more regularly, in this case, twelve times yearly than once a year.

Thus, more regular compounding increases the future value than periodic (annual, monthly, semiannual) compounding.

The future values can be determined using an online finance calculator that compounds interest per period as follows:

We can also derive the future values using the FV formula given as [tex]PV(1+r)^t[/tex].

1) Principal investment = €10,000

Compound interest rate = 6% = 0.06 (6/100)

Compound factor = 1.06 (1 + 0.06)

Compounding period = annually

Investment period = t years

Expression: €10,000 x [tex]1.06^t[/tex]

= 10,000(1.06)^t

= [tex]10,000(1.06)^t[/tex]

2) After ten years, the amount in the account will be equal to [tex]10,000(1.06)^{10}[/tex]

= 10,000 x 1.7908

= 17,908

3) Annual rate of interest = 6%

Monthly rate = 0.5% (6%/12)

4) Exponent = 120 (10 years x 12)

5) Exponential function:

[tex]10,000(1.005)^{120}[/tex]

6) Future value after 10 years:

[tex]10,000(1.005)^{120}[/tex]

= 10,000 x 1.8193967

= €18,193.97

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Answer:

OK, here is your answer

Step-by-step explanation:

AI-generated answer

We are given three points, p(-2, 6), q(-8, 4), and r(1, 2), and we are asked to find the y-value of p'. Since we are not given any information about p', we can assume that it is the reflection of p across the x-axis.To find the reflection of p across the x-axis, we need to negate the y-coordinate of p. Therefore, the y-value of p' is equal to the negative of the y-coordinate of p.

So, the y-value of p' is -6.

Hence, the coordinates of p' are (-2, -6).

For quadrilateral DEFG to be shown to be a parallelogram, we need to demonstrate that: a. EF║GD; b. DE = FG = EF = GD; e. DE║FG; f. DE = FG

A parallelogram is a four-sided polygon with opposite sides that are parallel and equal in length.

To demonstrate that quadrilateral DEFG is a parallelogram, the following conditions need to be satisfied:

a. EF║GD (Opposite sides are parallel)

b. DE = FG = EF = GD (Opposite sides are congruent)

e. DE║FG (Opposite sides are parallel)

f. DE = FG (Opposite sides are congruent)

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The height of the tree is approximately 87.13 feet.

To find the height of the tree, we can use trigonometric principles based on the given information.

Let's proceed with the following steps:

Draw a diagram to visualize the problem.

Visualize a right triangle where the horizontal line represents the hillside, the vertical line represents the height of the tree, and the hypotenuse represents the line from the tip of the shadow to the top of the tree. Label the given angles and lengths accordingly.

Identify the relevant trigonometric ratios.

In this case, we can use the tangent ratio (opposite/adjacent) to relate the angles and sides of the triangle.

Determine the length of the adjacent side.

The adjacent side represents the portion of the shadow cast by the tree down the hillside.

From the given information, the length of the shadow is 200ft.

Calculate the height of the tree.

To calculate the height, we can use the tangent of the angle of inclination (b) to find the ratio of the height to the adjacent side:

tan(b) = height/adjacent

tan(25) = height/200

Rearranging the equation, we have:

height = 200 [tex]\times[/tex] tan(25)

Using a calculator, we find:

height ≈ 87.13ft.

By applying trigonometric principles and using the tangent ratio, we were able to determine the height of the tree based on the given angles and length of the shadow.

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The bench costs $375, and the garden table costs $295. Together, they amount to $670, with the table being $80 cheaper than the bench.

Let the cost of the bench be x. Then the cost of the garden table is x-80. The sum of the costs of both items is $670. So we have the equation: x + (x-80) = $670.

Simplifying this, we get 2x - 80 = $670  + 80 2x = $750  x = $375. So the cost of the bench is $375. The garden table costs $375 - $80 = $295. A garden table and a bench cost $670 combined.

The garden table costs $80 less than the bench. To find the cost of the bench, we can use algebraic equations. Let the cost of the bench be x. Then, the cost of the garden table is x - 80.

The sum of both costs is $670. Using this information, we can form an equation: x + (x - 80) = $670. Simplifying, we get 2x - 80 = $670 + 80. Solving for x, we get x = $375.

Therefore, the cost of the bench is $375, and the cost of the garden table is $295.

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Answer:

The coreect answer o

is 14

i) It has been proven that CD/GH​ =  AC/FG by Corresponding Sides of Similar Triangles

ii)  △DCB ∼△HGE by the AA criterion of similarity

iii) △DCA ∼△HGF by the AA criterion of similarity

In △ABC and △FEG, we are told that they are similar and as such we can denote as:

△ABC ∼ △FEG

Thus, we can say that:

∠ACB = ∠EGF (Corresponding angles of similar triangles)

We are told that, CD and GH are bisectors of ∠ACB and ∠EGH respectively and as such we can say that:

∠ACB = 2∠ACD = 2∠BCD

∠EGF = 2∠FGH = 2∠HGE

Thus, by substitution, we can say that:

∠ACD = ∠FGH and ∠DCB = ∠HGE ...................(1)

Similarly:

∠A = ∠F and ∠B = ∠E ...............(2)

With respect to equations 1 and 2 we can establish that:

△ACD ∼ △FGH by the AA criterion of similarity

Similarly, △DCA ∼△HGF

Thus:

CD/GH​ =  AC/FG (Corresponding Sides of Similar Triangles)

Similarly, In △DCB and △HGE, we can also establish that

△DCB ∼△HGE by the AA criterion of similarity

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Answer:

- 72 - 48i

Step-by-step explanation:

note that i² = - 1

(6i)(- 2i)(- 6 - 4i)

= - 12i²(- 6 - 4i)

= - 12(- 1)(- 6 - 4i)

= 12(- 6 - 4i) ← distribute parenthesis by 12

= - 72 - 48i

Answer:

Step-by-step explanation: a = o/t

a = 18/tan(42)

a = 19.99

9 1/2 years on Venus is equivalent to approximately 2137.65 days on Earth.

To find how many days on Earth are equivalent to 9 1/2 years on Venus, we need to use the ratio of 224.7 days on Earth to 1 year on Venus. First, we convert 9 1/2 years to a decimal form, which is 9.5 years. Then, we multiply 9.5 years by the conversion ratio: 9.5 years * 224.7 days/year = 2137.65 days on Earth. Therefore, 9 1/2 years on Venus is equivalent to approximately 2137.65 days on Earth in decimal form.

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A. The initial value of the linear formula is $40,500

The growth rate of the function is $950

B. The value of the car in 2020 is $69,000

C. The year in which the car is worth $61,400 is 2012.

.A. V = 40,500 + 950n

Where,

V = value in dollars

n = number of years since 1990

The initial value of the linear formula is $40,500

The growth rate of the function is $950

B. The value of the car in 2020;

When n = 30

V = 40,500 + 950n

= 40,500 + 950(30)

= 40,500 + 28,500

= $69,000

C. The year in which the car is worth $61,400

V = 40,500 + 950n

61,400 = 40,500 + 950n

subtract 40,500 from both sides

61,400 - 40,500 = 950n

20,900 = 950n

divide both sides by 950

n = 22 years

Therefore, 22 years after 1990 is 2012

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There are 555 different ways for David to choose his fish according to the given combinations.

David to choose his fish, we need to consider all the possible combinations of fish he can select.

Let's break down the calculations for each combination:

Guppy fish and an angel fish:

David can choose one guppy fish from 7 options and one angel fish from 5 options.

This gives us a total of 7 × 5 = 35 different combinations.

Tetra fish and an angel fish:

David can choose one tetra fish from 13 options and one angel fish from 5 options.

This gives us a total of 13 × 5 = 65 different combinations.

Guppy fish, tetra fish, and an angel fish:

David can choose one guppy fish from 7 options, one tetra fish from 13 options, and one angel fish from 5 options.

This gives us a total of 7 × 13 × 5 = 455 different combinations.

To find the total number of different ways for David to choose his fish, we sum up the combinations from each category:

35 + 65 + 455 = 555

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The simplified expression for ( x - y ) ( x - y ) ( x - y ) is x³ - x²y - xy² + y³.

To simplify ( x - y ) ( x - y ) ( x - y ), we can expand the expression using the distributive property of multiplication. We can also simplify the expression by combining like terms.

In using the distributive property of multiplication, we multiply each term in the first set of parentheses by each term in the second set of parentheses. This gives us:( x - y ) ( x - y ) ( x - y ) = ( x(x - y) - y(x - y) ) ( x - y )= ( x² - xy - xy + y² ) ( x - y )= ( x² - 2xy + y² ) ( x - y )

Now we can further simplify the expression by combining like terms. We can use the difference of squares formula to simplify the second set of parentheses.( x² - 2xy + y² ) ( x - y ) = ( x - y ) ( x - y )²= ( x - y ) ( x² - 2xy + y² )= x³ - x²y - 2xy² + xy² - xy² + y³= x³ - x²y - xy² + y³

Therefore, the simplified expression is x³ - x²y - xy² + y³.

Summary: To simplify ( x - y ) ( x - y ) ( x - y ), we expand the expression using the distributive property of multiplication. We simplify the expression by combining like terms. We use the difference of squares formula to simplify the second set of parentheses. Finally, we simplify the expression further by combining like terms. The simplified expression is x³ - x²y - xy² + y³.

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Given statement solution is :-The decision to renegotiate the egg prices should be based on the calculated probability (P) obtained from the z-score calculation.

To determine whether the diner should renegotiate their egg prices with their vendor, we need to assess the likelihood that the sample mean price will be higher than their current price of $4.75 per dozen. We can use the concept of the sampling distribution of the sample mean to make this assessment.

Given that the population standard deviation is known ($0.75) and the sample size is small (n = 5), we can assume the sampling distribution of the sample mean will be approximately normally distributed. The mean of the sampling distribution will be equal to the population mean ($5 per dozen), and the standard deviation of the sampling distribution (also known as the standard error) can be calculated by dividing the population standard deviation by the square root of the sample size.

Standard error (SE) = population standard deviation / √(sample size)

SE = 0.75 / √(5)

SE ≈ 0.3354

Now, we can calculate the probability that the sample mean price will be higher than $4.75 by standardizing the sample mean using the z-score formula:

z = (sample mean - population mean) / standard error

z = ($4.75 - $5) / 0.3354

z ≈ -0.445

Using a standard normal distribution table or calculator, we can find the probability corresponding to a z-score of -0.445. Let's assume this probability is P.

If P is sufficiently small (typically less than 0.05), it indicates that the sample mean being higher than $4.75 is unlikely to occur by chance. In such a case, it would suggest that the diner should consider renegotiating their egg prices with their vendor.

However, if P is relatively large (greater than or equal to 0.05), it suggests that the sample mean being higher than $4.75 is reasonably likely to occur by chance. In this scenario, there may not be sufficient evidence to justify renegotiating the egg prices based solely on the sample mean.

Therefore, the decision to renegotiate the egg prices should be based on the calculated probability (P) obtained from the z-score calculation.

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Answer:

2 and 12

Step-by-step explanation:

we have 2 number cubes (dice) and try to roll specific numbers as the sum of both cubes.

4 can be achieved by rolling

1 3

2 2

3 1

so, 3 out of 36 possible combinations.

11 can be achieved by

5 6

6 5

so, 2 out of 36 possible combinations.

so, no, they do not have the save probabilty distribution, and would not have equal bars.

2 can be achieved by

1 1

so, 1 out of 36 possible combinations.

12 can be achieved by

6 6

so, 1 out of 36 possible combinations.

yes, they have the same probability distribution and would have equal bars.

6 can be achieved by

1 5

2 4

3 3

4 2

5 1

so, 5 out of 36 possible combinations.

9 can be achieved by

3 6

4 5

5 4

6 3

so, 4 out of 36 possible combinations.

so, no, they do not have the save probabilty distribution, and would not have equal bars.

5 can be achieved by

1 4

2 3

3 2

4 1

so, 4 out of 36 possible outcomes.

10 can be achieved by

4 6

5 5

6 4

so, 3 out of 36 possible outcomes.

so, no, they do not have the save probabilty distribution, and would not have equal bars.

Answer:

1. Regular pay amount:

$17.40 × 37.25 = $648.15

2. Overtime pay: $0.00

3. Gross pay: $648.15

The midpoint formula and distance formula can be used to find the coordinates of the midpoints of line segments RR' and SS' and the lengths of these segments. Then an equation involving these lengths can be set up and solved for m which is equal to 5.

Given a trapezoid PQRS and its image P'Q'R'S' after a reflection across Line W Y, the line of reflection.

We are also given that line segment R R prime has a midpoint at point Z and that line segment S S prime has a midpoint at point X.

We need to find the value of m. We can use the midpoint formula to find the coordinates of the midpoints Z and X.

Using the coordinates of P, Q, R, S, and the line of reflection WY, we can find the coordinates of the reflected image P', Q', R', and S'.

Now we can find the length of line segments RR' and SS' using the distance formula. Then we can set up an equation involving the length of line segments RR' and SS' and solve for m. Thus, m=5.

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