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25. Use the relationship in the diagrams below to solve for the given variable.
Justify your solution with a definition or theorem.​

The equivalent function that best shows the x-intercepts on the graph of f(x) = 4x² - 1 is option (a) or (b): Of(x) = (4x + 1)(4x - 1).

The correct answer to the given question is option a or b.

The given function is f(x) = 4x² - 1. We need to choose the equivalent function that best shows the x-intercepts on the graph.The x-intercepts are the points where the graph of a function intersects the x-axis. At the x-intercepts, the value of y is zero.

Therefore, to find the x-intercepts, we need to solve the equation f(x) = 0 for x. The function f(x) = 4x² - 1 can be factored as:(2x + 1)(2x - 1)

To find the x-intercepts, we set f(x) = 0:4x² - 1 = 0(2x + 1)(2x - 1) = 0So, either 2x + 1 = 0 or 2x - 1 = 0. Solving these equations, we get:

x = -1/2 or x = 1/2

These are the x-intercepts of the graph of f(x) = 4x² - 1.Now, let's look at the given options and determine which one shows the x-intercepts on the graph:

Option (a):

Of(x) = (4x + 1)(4x - 1)

This is the factored form of f(x) = 4x² - 1. It correctly shows the x-intercepts.

Option (b):

Of(x) = (2x + 1)(2x - 1)

This is the same as option (a) and correctly shows the x-intercepts.

Option (c): f(x) = 4(x² + 1)

This function does not have any x-intercepts. It has a minimum value of 4 at x = 0.

Option (d): f(x) = 2(x² - 1)

This function has x-intercepts at x = -1 and x = 1. It does not show the x-intercepts of the given function.

Therefore, the equivalent function that best shows the x-intercepts on the graph of f(x) = 4x² - 1 is option (a) or (b):

Of(x) = (4x + 1)(4x - 1).

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Step-by-step explanation:

To find the difference in total revenue projected by the two models over the six-year period ending at t = 5, we need to calculate the revenue for each model from t = 0 to t = 5 and subtract the results.

For R1:

R1 = 7.23 + 0.25t + 0.03t^2

Substituting t = 5:

R1(5) = 7.23 + 0.25(5) + 0.03(5^2)

R1(5) = 7.23 + 1.25 + 0.75

R1(5) = 9.23 + 0.75

R1(5) = 9.98 million dollars

For R2:

R2 = 7.23 + 0.1t + 0.01t^2

Substituting t = 5:

R2(5) = 7.23 + 0.1(5) + 0.01(5^2)

R2(5) = 7.23 + 0.5 + 0.25

R2(5) = 7.73 + 0.25

R2(5) = 7.98 million dollars

To find the difference, we subtract R2(5) from R1(5):

Difference = R1(5) - R2(5)

Difference = 9.98 - 7.98

Difference = 2 million dollars

Therefore, the model R1 projects 2 million dollars more in total revenue than R2 over the six-year period ending at t = 5.

Answer:

Third option

Step-by-step explanation:

-3(2x - 5) < 5(2 - x)

-6x + 15 < 10 - 5x <-- Third option

15 < 10 + x

5 < x

x > 5

There only appears to be one option. The solution to the inequality is x>5, not x<5.

Answer:

She willl be $1.40 under budget

Step-by-step explanation:

8% = 8/100 = 0.08

Adding this to 100% of the price of the shoes, we get 108% = 108/100 = 1.08.

We multiply the price of the shoes by this:

45*1.08 = 48.60

Subtract this from 50:

50 - 48.60 = 1.40

Answer: False

Step-by-step explanation:

Spherical geometry, on the other hand, is a type of non-Euclidean geometry that is specifically concerned with studying the properties of curved surfaces, such as spheres.

Hope this help! Have a good day!

Answer:

Step-by-step explanation:

Answer:

y = π/3

Step-by-step explanation:

To find the value of y, we can use the trigonometric identity for the sum of angles:

sin(x + y) = sin x * cos y + cos x * sin y

Comparing this with the given equation:

sin(x + y) = 1/2(sin x) + √3/2(cos x)

We can equate the corresponding terms:

sin x * cos y = 1/2(sin x) ----(1)

cos x * sin y = √3/2(cos x) ----(2)

From equation (1), we can see that cos y = 1/2.

From equation (2), we can see that sin y = √3/2.

To determine the values of y, we can use the trigonometric values of cosine and sine in the first quadrant of the unit circle.

In the first quadrant, cos y is positive, so cos y = 1/2 corresponds to y = π/3 (60 degrees).

Similarly, sin y is positive, so sin y = √3/2 corresponds to y = π/3 (60 degrees).

Therefore, the value of y is y = π/3 (or 60 degrees).

Answer:

--------------------------

Angle G is central angle and angle E is inscribed angle, both with same endpoints.

According to the inscribed angle theorem the inscribed angle is half of the central angle.

Hence the central angle G measures:

Answer:

1. x = 18

2. x = 31

Step-by-step explanation:

1. Angles in a quadrilateral sum to 360°.

360 -(60 +130 +70) = 360 -260 = 100

So, x +82 = 100. x = 100 -82 = 18

2. Angles in a triangle sum to 180°. 180 -(29 +75) = 180 -104 = 76.

So, -17 +3x = 76. 3x = 93. x = 93/3 = 31

Answer:

p =1 0

Step-by-step explanation:

x³(x - y)⁹ - y³(x - y)⁹ = (x - y)⁹[x³ - y³]

= (x - y)⁹(x - y)(x² + y² + xy)

= (x - y)¹⁰(x² + y² + xy)

where p = 10 and q = 1

Answer:

p = 10

Step-by-step explanation:

Given expression:

[tex]x^3(x-y)^9 - y^3(x-y)^9[/tex]

Factor out the common term (x - y)⁹:

[tex](x-y)^9(x^3- y^3)[/tex]

[tex]\boxed{\begin{minipage}{5cm}\underline{Difference of two cubes}\\\\$x^3-y^3=(x-y)(x^2+y^2+xy)$\\\end{minipage}}[/tex]

Rewrite the second parentheses as the difference of two cubes.

[tex](x-y)^9(x-y)(x^2+y^2+xy)[/tex]

[tex]\textsf{Apply\:the\:exponent\:rule:} \quad a^b\cdot \:a^c=a^{b+c}[/tex]

[tex](x-y)^{9+1}(x^2+y^2+xy)[/tex]

[tex](x-y)^{10}(x^2+y^2+xy)[/tex]

Comparing the rewritten original expression with the given expression:

[tex](x-y)^{10}(x^2+y^2+xy)=(x-y)^p(x^2+y^2+qxy)[/tex]

We can see that [tex](x-y)^p[/tex] corresponds to [tex](x-y)^{10}[/tex] in the given expression.

Therefore, we can conclude that p = 10.

Given the information in the diagram, the theorem that best justifies why lines j and k must be parallel include the following: D. converse alternate exterior angles theorem.

In Mathematics and Geometry, parallel lines are two (2) lines that are always the same (equal) distance apart and never meet or intersect.

In Mathematics and Geometry, the alternate exterior angle theorem states that when two (2) parallel lines are cut through by a transversal, the alternate exterior angles that are formed lie outside the two (2) parallel lines, are located on opposite sides of the transversal, and are congruent angles.

Since the alternate exterior angles are congruent, we can logically deduce the following based on the converse alternate exterior angles theorem;

93° ≅ 93° (lines j and k are parallel lines).

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Complete Question:

Given the information in the diagram, which theorem best justifies why lines j and k must be parallel?

alternate interior angles theorem

alternate exterior angles theorem

converse alternate interior angles theorem

converse alternate exterior angles theorem

Using basic inference rules, we can establish the validity of the argument: p ⟹ ¬q, q V r, p V u, ¬r ├ u.

1. We are given the following premises:

  - p ⟹ ¬q (Premise 1)

  - q V r (Premise 2)

  - p V u (Premise 3)

  - ¬r (Premise 4)

2. To prove the conclusion, u, we need to use the premises and apply inference rules.

3. From Premise 4 (¬r) and the Disjunctive Syllogism rule, we can deduce ¬q: (¬r, q V r) ⟹ ¬q.

4. From Premise 1 (p ⟹ ¬q) and Modus Ponens, we can conclude ¬p: (p ⟹ ¬q, ¬q) ⟹ ¬p.

5. From Premise 3 (p V u) and Disjunctive Syllogism, we obtain ¬p V u.

6. Using Disjunctive Syllogism with ¬p V u and ¬p, we can derive u: (¬p V u, ¬p) ⟹ u.

7. From Premise 2 (q V r) and Disjunctive Syllogism, we have q.

8. Finally, using Modus Tollens with q and ¬q, we can deduce ¬p: (q, p ⟹ ¬q) ⟹ ¬p.

9. Therefore, combining ¬p and u, we can conclude the desired result: ¬p ∧ u.

10. Since ¬p ∧ u is logically equivalent to u, we have established the validity of the argument: p ⟹ ¬q, q V r, p V u, ¬r ├ u.

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The area of the obtuse triangle is 34 square feet with a base of 10 ft and height of 6.8 ft.

To find the area of the obtuse triangle, we can use the formula A = (1/2) * base * height. Let's denote the unknown part of the base as x.

In the given triangle, we have the width of the obtuse angle triangle as 10 ft, the height (perpendicular) as 6.8 ft, and the unknown part of the base as x.

Using the formula, we can calculate the area as:

A = (1/2) * (10 + x) * 6.8

Simplifying this expression, we get:

A = 3.4(10 + x)

Now, we need to determine the value of x. From the given information, we know that the width of the obtuse angle triangle is 10 ft. This means the sum of the two parts of the base is 10 ft. Therefore, we can write the equation:

x + 10 = 10

Solving for x, we find:

x = 0

Since x = 0, it means that one part of the base has a length of 0 ft. Therefore, the entire base is formed by the width of the obtuse angle triangle, which is 10 ft.

Now, substituting this value of x back into the area formula, we have:

A = 3.4(10 + 0)

A = 3.4 * 10

A = 34 square feet

Hence, the area of the obtuse triangle is 34 square feet.

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a) The summation (Ex) of the given data set, we add up all the values

Ex = 77

b) n = 5

c) x = 15.4

a) To compute the summation (Ex) of the given data set, we add up all the values:

19 + 10 + 15 + 17 + 16 = 77

b) The sample size (n) is the total number of data points in the set. In this case, there are 5 data points, so:

n = 5

c) To calculate the sample mean (x), we divide the summation (Ex) by the sample size (n):

x = Ex / n

x = 77 / 5

x = 15.4

Therefore, the answers are:

a) Ex = 77

b) n = 5

c) x = 15.4

The summation is 77, the sample size is 5, and the sample mean is 15.4.

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Answer:

47 cents or $0.47

Step-by-step explanation:

1 dime = 10 cents (or $0.1)

1 quarter = 25 cents or ($0.25)

12 cents + 1 dime + 1 quarter = 12 + 10 + 25 = 47 cents

Step-by-step explanation:

In a randomized block design blocked by gender, treatments should be assigned randomly within each gender block. The correct assignment maintains a distribution of one treatment for each gender. Looking at the given options, only one meets this criterion:

OA: (1f, 2f), B: (1m, 2m). C: (3f, 3m). D: (4f, 4m)

Each treatment group A, B, C, and D contains one male and one female, making the distribution of treatments blocked by gender.

The probability that the dial stops on yellow is 2/5 or 0.4 (or 40%).

Therefore, there is a 40% chance that the spinner will stop on yellow.

To find the probability that the spinner stops on yellow, we need to determine the number of favorable outcomes (yellow) and the total number of possible outcomes.

The spinner has 10 equally sized slices, with 4 yellow, 4 red, and 2 blue.

The number of favorable outcomes (yellow) is 4 because there are 4 yellow slices.

The total number of possible outcomes is 10 because there are 10 slices in total.

Therefore, the probability of the spinner stopping on yellow can be calculated as:

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

Probability = 4 / 10

Simplifying this fraction, we get:

Probability = 2 / 5

So, the probability that the dial stops on yellow is 2/5 or 0.4 (or 40%).

Therefore, there is a 40% chance that the spinner will stop on yellow.

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The sides of the rectangle that have the same length as the circumference of the circles are the sides parallel to the bases of the cylinder. The sides of the rectangle that have the same length as the height of the cylinder are the sides perpendicular to the bases. While the model provides a simplified representation, practical calculations might have slight differences due to real-world factors.

In a cylinder, the two circular bases are connected by a curved surface, forming a three-dimensional shape. The rectangular shape that wraps around the curved surface of the cylinder is called the lateral surface or the lateral area.

To determine which sides of the rectangle have the same length as the circumference of the circles, we need to understand the geometry of a cylinder. The circumference of a circle is calculated using the formula:

Circumference = 2πr,

where r is the radius of the circle. In a cylinder, the bases are identical circles, so the circumference of each base is equal. Therefore, the sides of the rectangle that are parallel to the bases have the same length as the circumference of the circles.

Now, let's consider the height of the cylinder. The height is the distance between the two bases and is perpendicular to the bases. In the rectangular representation of the cylinder, the sides that are perpendicular to the bases represent the height. Hence, the sides of the rectangle that are perpendicular to the bases have the same length as the height of the cylinder.

When comparing the model (rectangular representation) with practical calculations, we may notice some differences. The model provides a simplified representation of the cylinder, assuming that the lateral surface is perfectly wrapped around the curved surface. However, in practical calculations, there might be slight variations due to factors like material thickness, manufacturing processes, or measuring precision. These variations can result in minor deviations between the model and the practical calculations.

It's important to consider that the model is an approximation and serves as a visual aid to understand the basic properties of the cylinder. In real-life applications or engineering calculations, precise measurements and considerations of tolerances are crucial.

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Answer:

D

Step-by-step explanation:

the area (A) of a triangle is calculated as

A = [tex]\frac{1}{2}[/tex] bh ( b is the base and h the perpendicular height )

here b = 16 and h = 8 , then

A = [tex]\frac{1}{2}[/tex] × 16 × 8 = 8 × 8 = 64 ft²

Answer:

B.     [tex] 9^{\frac{1}{8}x} [/tex]

Step-by-step explanation:

[tex] \sqrt[4]{9}^{\frac{1}{2}x} = [/tex]

[tex] = ({9}^{\frac{1}{4}})^{\frac{1}{2}x} [/tex]

[tex] = 9^{\frac{1}{4} \times \frac{1}{2}x} [/tex]

[tex] = 9^{\frac{1}{8}x} [/tex]

118 be the median of a positively skewed distribution with a mean of 122. Option D.

To determine which of the given values could be the median of a positively skewed distribution with a mean of 122, we need to consider the relationship between the mean, median, and skewness of a distribution.

In a positively skewed distribution, the tail of the distribution is stretched towards higher values, meaning that there are more extreme values on the right side. Consequently, the median, which represents the value that divides the distribution into two equal halves, will typically be less than the mean in a positively skewed distribution.

Let's examine the given values in relation to the mean:

A. 122: This value could be the median if the distribution is perfectly symmetrical, but since the distribution is positively skewed, the median is expected to be less than the mean. Thus, 122 is less likely to be the median.

B. 126: This value is higher than the mean, and since the distribution is positively skewed, it is unlikely to be the median. The median is expected to be lower than the mean.

C. 130: Similar to option B, this value is higher than the mean and is unlikely to be the median. The median is expected to be lower than the mean.

D. 118: This value is lower than the mean, which is consistent with a positively skewed distribution. In such a distribution, the median is expected to be less than the mean, so 118 is a plausible value for the median.

In summary, among the given options, (118) is the most likely value to be the median of a positively skewed distribution with a mean of 122. So Option D is correct.

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Note the complete question is

If the mean of a positively skewed distribution is 122, which of these values could be the median of the distribution?

A. 122

B. 126

C. 130

D. 118

Answer: 10 and -4

Step-by-step explanation: 10 + - 4 = 6 and 10 x -4 = -40

Answer:

(3, -2) is the correct choice.

Answer: D: The cost of one hand towel is $8 and the cost of one bath towel is $5.

Step-by-step explanation:

Let's assume the cost of one hand towel is 'x' dollars and the cost of one bath towel is 'y' dollars.

For the first hotel, Mr. Miller ordered 27 hand towels and 48 bath towels, resulting in a bill of $540. This can be expressed as the equation:

27x + 48y = 540 ...(equation 1)

For the second hotel, Mr. Miller ordered 50 hand towels and 24 bath towels, resulting in a bill of $416. This can be expressed as the equation:

50x + 24y = 416 ...(equation 2)

To solve this system of equations, we can use any suitable method such as substitution or elimination. Let's use the elimination method:

Multiplying equation 1 by 2 and equation 2 by 3, we get:

54x + 96y = 1080 ...(equation 3)

150x + 72y = 1248 ...(equation 4)

Now, subtracting equation 4 from equation 3, we have:

(54x + 96y) - (150x + 72y) = 1080 - 1248

-96x + 24y = -168

Dividing both sides of the equation by -24, we get:

4x - y = 7 ...(equation 5)

Now, we have a system of equations:

4x - y = 7 ...(equation 5)

50x + 24y = 416 ...(equation 2)

Solving this system of equations, we find that x = 8 and y = 5.

Therefore, the cost of one hand towel is $8 and the cost of one bath towel is $5.

So, the correct answer is option D: The cost of one hand towel is $8 and the cost of one bath towel is $5.

Answer:

Step-by-step explanation:

Total cost of hand towels for first hotel = 27 * $5 = $135

Total cost of bath towels for first hotel = 48 * $8 = $384

Total cost of hand towels for second hotel = 50 * $5 = $250

Total cost of bath towels for second hotel = 24 * $8 = $192

Total cost of all hand towels = $135 + $250 = $385

Total cost of all bath towels = $384 + $192 = $576

Total number of hand towels = 27 + 50 = 77

Total number of bath towels = 48 + 24 = 72

Average cost of one hand towel = $385 / 77 = $5

Average cost of one bath towel = $576 / 72 = $8

The resulting matrix after the rows are interchanged is given as follows:

[tex]\left[\begin{array}{cccc}2&9&4&5\\8&-2&1&7\\1&4&-4&9\end{array}\right][/tex]

The matrix for this problem is defined as follows:

[tex]\left[\begin{array}{cccc}8&-2&1&7\\2&9&4&5\\1&4&-4&9\end{array}\right][/tex]

The row 1 is given as follows:

[8 -2 1 7].

The row 2 is given as follows:

[2 9 4 5].

Interchanging the rows means that the elements of the row 1 in the matrix is exchanged with the elements of row 2, hence the resulting matrix is given as follows:

[tex]\left[\begin{array}{cccc}2&9&4&5\\8&-2&1&7\\1&4&-4&9\end{array}\right][/tex]

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Answer:

-27+15 (N-1)

Step-by-step explanation:

-27, -12, 3, 18

Take the second term and subtract the first term to find the common difference

-12 - (-27)

-12+27 = 15

The common difference is +15

We are adding 15 each time

The formula for an arithmetic sequence is

an = a1+d(n-1)

an = -27 +15(n-1)

1. Using a calculator, we find that cos 10 ≈ 0.98.

2. Using a calculator, we find that sin 30 ≈ 0.50.

3. Using a calculator, we find that sin 20 ≈ 0.34.

4. Using a calculator, we find that tan 25 ≈ 0.47.

5. Using a calculator, we find that tan 48.5 ≈ 1.14.

Using a calculator to evaluate the given trigonometric functions, rounded to the nearest hundredth, we have:

Cos 10:

Using a calculator, we find that cos 10 ≈ 0.98.

Sin 30:

Using a calculator, we find that sin 30 ≈ 0.50.

Sin 20:

Using a calculator, we find that sin 20 ≈ 0.34.

Tan 25:

Using a calculator, we find that tan 25 ≈ 0.47.

Tan 48.5:

Using a calculator, we find that tan 48.5 ≈ 1.14.

These values represent the approximate decimal values of the trigonometric functions at the given angles, rounded to the nearest hundredth.

Just a reminder, when using a calculator, make sure it is set to the correct angle mode (degrees or radians) as per the given problem.

It's important to note that these values are approximate since they are rounded to the nearest hundredth. If you need more precise values, you can use a calculator that allows for a greater number of decimal places or use trigonometric tables.

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Answer: -24

Step-by-step explanation:

To evaluate the expression, I guess we need to break it down into steps:

Expression: -3[-4(3-10)-12] / -2(-1)

Step1: Simplify the innermost parentheses Inside the square brackets: 3 - 10 = -7 Expression becomes: -3[-4(-7) - 12] / -2(-1)

Step2: Simplify the multiplication in square brackets: -4 * (-7) = 28. Expression becomes: -3[28 - 12] / -2(-1)

Step3: Simplify the subtraction inside the square brackets: 28 - 12 = 16. Expression becomes: -3[16] / -2(-1)

Step4: Simplify the multiplication outside the square brackets: -3 * 16 = -48. Expression becomes: -48 / -2(-1)

Step5: Simplify the multiplication inside the denominator: -2 * (-1) = 2 Expression becomes: -48 / 2

Step 6: Perform the division -48 divided by 2 is equal to -24

Therefore, the value of the expression -3[-4(3-10)-12] / -2(-1) is -24.

The steps used in the solution of the equation -6 = -2/3(x + 12) + 1/3x are based on the principles of the Distributive Property, combining like terms, addition property of equality, symmetric property, subtraction property of equality, and multiplication property of equality.

Let's analyze the steps of the solution for the given equation -6 = -2/3(x + 12) + 1/3x:

Step 1: Distributive Property

The equation begins with the Distributive Property, which states that you can distribute a factor to each term inside parentheses. In this case, we distribute -2/3 to (x + 12), resulting in -2/3 * x and -2/3 * 12.

Step 2: Simplification

We simplify the expression -2/3 * 12 to -8, as multiplying -2/3 by 12 gives us -24, and simplifying the fraction -24/3 yields -8.

Step 3: Combine Like Terms

We combine the like terms -2/3x and -8. The equation becomes -2/3x - 8 + 1/3x.

Step 4: Combine Like Terms

We combine the like terms -2/3x and 1/3x by adding their coefficients. The sum of -2/3x and 1/3x is -1/3x.

Step 5: Addition Property of Equality

We add -1/3x to both sides of the equation to isolate the constant term. The equation becomes -6 - 1/3x = -1/3x.

Step 6: Symmetric Property

Since the equation has a form of -1/3x = -6 - 1/3x, we can rearrange the terms using the Symmetric Property.

Step 7: Addition Property of Equality

We add 1/3x to both sides of the equation to isolate the constant term. The equation becomes -6 = 0.

Step 8: Subtraction Property of Equality

We subtract 0 from both sides of the equation to simplify it further. The equation remains -6 = 0.

Step 9: Multiplication Property of Equality

We multiply both sides of the equation by any non-zero number to check for consistency. In this case, there is no need for multiplication as the equation is already in its simplified form.

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