Solve the following using elimination, substitution, or graphing. Tickets for The Phantom of the Opera cost $6 for children and $12 for adults. A local school group paid $252 for 36 tickets. How many of each did they purchase?

Check the picture below.

The area οf the hοt spring decreases by apprοximately 540.24 square meters.

The area οf a circle is fοund οut using the fοrmula. Area = πr²

Given that, r = d/2 = 110/2 = 55 meters

The οriginal area οf the hοt spring can be calculated as:

[tex]A = \pi r^2 = \pi(55)^2 = 9,525.69[/tex] square meters (rοunded tο twο decimal places)

After the diameter οf the hοt spring decreases by 3 meters, its new diameter is 110 - 3 = 107 meters. Therefοre, its new radius is:

r' = d'/2 = 107/2 = 53.5 meters

The new area οf the hοt spring can be calculated as:

[tex]A' = \pi r'² = \pi(53.5)² = 8,985.45[/tex] square meters (rοunded tο twο decimal places)

The decrease in the area οf the hοt spring is the difference between the οriginal area and the new area:

ΔA = A - A' = 9,525.69 - 8,985.45 = 540.24 square meters (rοunded tο twο decimal places)

Therefοre, the area οf the hοt spring decreases by apprοximately 540.24 square meters.

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Answer:

(a) To find the probability that a randomly selected individual is at least 55 years of age, we need to add up the values in the last row of the contingency table corresponding to the age group 55+ and divide it by the total number of respondents:

P(age 55+) = (515 + 617 + 567 + 560) / 2259 ≈ 0.2934

So the probability is approximately 0.2934.

(b) To find the probability that a randomly selected individual is more likely to buy a product emphasized as "Made in our country," given the individual is at least 55 years of age, we need to add up the values in the second row of the contingency table (corresponding to "Yes, more likely") for the age group 55+ and divide it by the total number of respondents in that age group:

P(more likely|age 55+) = 515 / (515 + 617 + 567 + 560) ≈ 0.2281

So the probability is approximately 0.2281.

(c) To determine whether 18-34-year-olds are more likely to buy a product emphasized as "Made in our country" than individuals in general, we need to compare the proportion of respondents who answered "Yes, more likely" in the 18-34 age group to the proportion in the entire sample. We can calculate these proportions by adding up the values in the second row of the contingency table for the 18-34 age group and for the entire sample, respectively, and dividing each by the total number of respondents in each group:

P(more likely|age 18-34) = 206 / (206 + 388 + 393 + 410) ≈ 0.1913

P(more likely|entire sample) = 1397 / (1397 + 69 + 793) ≈ 0.6251

The proportion of 18-34-year-olds who are more likely to buy a product emphasized as "Made in our country" is approximately 0.1913, while the proportion in the entire sample is approximately 0.6251. Therefore, it appears that individuals in general are more likely to buy a product emphasized as "Made in our country" than 18-34-year-olds.

Step-by-step explanation:

So you borrowed $3333.33 from the bank and $1666.67 from your uncle.

Percent is a way of expressing a number as a fraction of 100. The word "percent" comes from the Latin per centum, which means "out of a hundred". Percentages are often used to express proportions, rates, or changes in quantities. For example, if you got 80 questions correct out of 100 on a test, you got 80% correct. In other words, 80 is 80% of 100. Percentages can also be used to express changes, such as if the price of a product increases from $10 to $12, that's a 20% increase ($2 increase is 20% of the original price of $10). Percentages are represented using the % symbol, for example, 50% means 50 out of 100, or 0.5 as a decimal.

Here,

a. Let's say you borrowed x dollars from the bank. Then, you borrowed (5000 - x) dollars from your uncle. Since you borrowed from the bank at 8% interest, the interest on this loan would be 0.08x dollars.

b. Since you borrowed from your uncle at 2% interest, the interest on this loan would be 0.02(5000 - x) dollars.

c. The total interest owed to the bank is 0.08x dollars, as calculated in part (a).

d. The total interest owed is the sum of the interest owed to the bank and the interest owed to your uncle:

Total interest = 0.08x + 0.02(5000 - x)

e. If the total interest is $300, we can write the equation:

0.08x + 0.02(5000 - x) = 300

f. To solve this equation for x, we can simplify and solve for x:

0.08x + 0.02(5000 - x) = 300

0.08x + 100 - 0.02x = 300

0.06x = 200

x = 3333.33 (rounded to the nearest cent)

Therefore, the answers are:

a. You borrowed $3333.33 from the bank.

b. You owe your uncle 0.02(5000 - 3333.33) = $133.33 in interest.

c. You owe the bank 0.08(3333.33) = $266.67 in interest.

d. The total interest you owe is $133.33 + $266.67 = $400.

e. The total interest equation is 0.08x + 0.02(5000 - x) = 300.

f. You borrowed $3333.33 from the bank and $1666.67 from your uncle.

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Answer:

B

Step-by-step explanation:

calculate the slopes m of the 2 lines using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (- 6, 5 ) and (x₂, y₂ ) = (- 2, 7 )

m = [tex]\frac{7-5}{-2-(-6)}[/tex] = [tex]\frac{2}{-2+6}[/tex] = [tex]\frac{2}{4}[/tex] = [tex]\frac{1}{2}[/tex]

repeat with

(x₁, y₁ ) = (4, 2 ) and (x₂, y₂ ) = (6, 6 )

m = [tex]\frac{6-2}{6-4}[/tex] = [tex]\frac{4}{2}[/tex] = 2

• Parallel lines have equal slopes

[tex]\frac{1}{2}[/tex] ≠ 2 , then lines are not parallel

the product of the slopes of perpendicular lines equals - 1

[tex]\frac{1}{2}[/tex] × 2 = 1 ≠ - 1 , thus lines are not perpendicular

the lines are neither perpendicular nor parallel

Either a rotation of a reflection is used to transform Shape P so that exactly one vertex is invariant.

The possible transformations on a figure are listed as follows:

An invariant vertex means that the coordinates of the vertex remain constant even after the transformation.

A translation and a dilation will always change the coordinates of every vertex, as:

For reflection and rotation, they can happen over one vertex of the shape, hence that vertex would remain constant.

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Option (C) is correct i.e., To determine the ratio of AB to DE and use SAS to conclude the triangle(s) are similar

A triangle is a polygon that has three(3) edges and three(3) vertices. It is one of the fundamental forms in geometry. Triangle DEF refers to a triangle with vertices D, E, and F. In Euclidean geometry, any three points that are not collinear identify a distinct triangle and a distinct plane at the same time.

Triangle ABC and Triangle DEF are similar if one of this property satisfy are , AAA, SAS, ASA and RHS.

It is Given that ∠B and ∠E are Similar and She determine ratio of BC to EF is 3/5.

Therefore, for similarity of two triangle she have to To determine the ratio of AB to DE and use SAS to conclude the triangles are similar.

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Complete Question:

When trying to determine whether triangles ABC and DEF are similar, Mae notices in the image that angles B and E are congruent. Next, she determines the ratio of BC to EF and obtains 3/5. What else does she need to do?

A. To determine the ratio of DE to AB and use SSS to conclude the triangles are similar

B. To determine the ratio of AB to DE and conclude the triangles are not similar

C. To determine the ratio of AB to DE and use SAS to conclude the triangles are similar

D. To determine the ratio of DE to AB and use AA to conclude the triangles are not similar.

Option (C) is correct i.e., To determine the ratio of AB to DE and use SAS to conclude the triangle(s) are similar

A triangle is a polygon that has three(3) edges and three(3) vertices. It is one of the fundamental forms in geometry. Triangle DEF refers to a triangle with vertices D, E, and F. In Euclidean geometry, any three points that are not collinear identify a distinct triangle and a distinct plane at the same time.

Triangle ABC and Triangle DEF are similar if one of this property satisfy are , AAA, SAS, ASA and RHS.

It is Given that ∠B and ∠E are Similar and She determine ratio of BC to EF is 3/5.

Therefore, for similarity of two triangle she have to To determine the ratio of AB to DE and use SAS to conclude the triangles are similar.

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Complete Question:

When trying to determine whether triangles ABC and DEF are similar, Mae notices in the image that angles B and E are congruent. Next, she determines the ratio of BC to EF and obtains 3/5. What else does she need to do?

A. To determine the ratio of DE to AB and use SSS to conclude the triangles are similar

B. To determine the ratio of AB to DE and conclude the triangles are not similar

C. To determine the ratio of AB to DE and use SAS to conclude the triangles are similar

D. To determine the ratio of DE to AB and use AA to conclude the triangles are not similar.

The single translation that maps shape A onto shape C is given as follows:

A translation happens when either a figure or a function are moved horizontally or vertically on the coordinate plane.

The four translation rules for coordinates are defined as follows:

Translations can often be composed, that is, a combination of multiple translations can result in a single translation, as is the case for this problem.

Considering the translation of 3 units left(shape B) and then 4 units right(shape C), the horizontal translation is given as follows:

-3 + 4 = 1 -> 1 unit right.

Considering the translation of 7 units down(shape B) and then 3 units down(shape C), the vertical translation is given as follows:

-7 - 3 = 10 units down.

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Approximately 342,857 empty cans would weigh 200 pounds.

A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0. A proportion is an equation in which two ratios are set equal to each other. For example, if there is 1 boy and 3 girls you could write the ratio as 1 : 3 (for every one boy there are 3 girls).

To solve this problem, we need to use proportional reasoning. We know that six empty cans weigh 3.5 ounces, so we can set up a proportion to find out how many cans are needed to weigh 200 pounds:

6 cans : 3.5 ounces = x cans : 200 pounds

To solve for x, we need to cross-multiply and simplify:

6 cans * 200 pounds = 3.5 ounces * x cans

1200 pounds = 3.5 ounces * x cans

x cans = 1200 pounds / 3.5 ounces

x cans = 342857.14 cans (rounded to two decimal places)

Therefore, approximately 342,857 empty cans would weigh 200 pounds.

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Answer:

x = - 2 , 5

Step-by-step explanation:

to find the roots let y = 0 , that is

x² - 3x - 10 = 0 ← in standard form

(x + 2)(x - 5) = 0 ← in factored form

equate each factor to zero and solve for x

x + 2 = 0 ⇒ x = - 2

x - 5 = 0 ⇒ x = 5

Therefore, the values of a and b are:

[tex]a = 2[/tex]

[tex]b = 9[/tex]

[tex]So, the functionf(x)=2x+9.[/tex]

In programming and mathematics, a function is a set of instructions or a block of code that performs a specific task or calculation. A function takes in one or more input values, performs a series of operations on them, and then returns a single output value. Functions are used to modularize code, making it easier to read, understand, and maintain. They can be defined and called within a program to perform repetitive tasks, manipulate data, or solve complex problems. Functions can also be used to create reusable code that can be called from multiple parts of a program or shared between different programs.

To find the values of a and b, we need to use the given information to create a system of two equations.

Let's start by using the first two rows of the table:

[tex]f(0) = a(0) + b = 7[/tex]

[tex]f(1) = a(1) + b = 9[/tex]

Simplifying these equations, we get:

[tex]b = 7\\a + b = 9[/tex]

Substituting b = 7 into the second equation, we get:

[tex]a + 7 = 9[/tex]

Solving for a, we get:

[tex]a = 2[/tex]

Now we can use this value of a to find b. Let's use the third and fourth rows of the table:

[tex]f(2) = a(2) + b = 11\\f(3) = a(3) + b = 13[/tex]

Substituting a = 2 into these equations, we get:

[tex]2 + b = 11\\3 + b = 13[/tex]

Solving for b, we get:

[tex]b = 9[/tex]

Therefore, the values of a and b are:

[tex]a = 2\\b = 9[/tex]

[tex]So, the functionf(x)=2x+9.[/tex]

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The most reliable results would come from randomly selecting 50 batteries throughout the day and testing to determine if they function.

A voltage reading, a pulse or AC impedance measurement of internal resistance, coulomb counting, and electrochemical impedance spectroscopy are some of the test techniques available (EIS).

The red probe should be connected to the positive terminal of the battery, and the black probe should be connected to the negative terminal. Take the multimeter's reading. The battery is still safe to use if the reading for a 9V battery is greater than 7V.

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The value of x is equal to: B. 12.

The measure of <BCD, in degrees is 143 degrees.

In Geometry, the vertical angles theorem is also referred to as vertically opposite angles theorem and it states that two (2) opposite vertical angles that are formed whenever two (2) lines intersect each other are always congruent, which simply means being equal to each other.

By applying the vertical angles theorem to the geometric figure, we have the following:

37 = 3x + 1

3x = 37 - 1

3x = 36

x = 36/3

x = 12.

For the measure of ∠BCD, in degrees, we have:

∠BCD + (3x + 1) = 180

∠BCD + (3(12) + 1) = 180

∠BCD + 37 = 180

∠BCD = 143°.

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The answer is (D) (7, 3), (9, 7) and (8, 3) when A triangle with vertices (2, 3), (4, 7), and (3, 3) is translated a distance of 5 units to the right.

To translate a two-dimensional figure such as a triangle, we move all its points a certain distance in a certain direction. In this case, we need to move the triangle 5 units to the right. This means we need to add 5 to the x-coordinate of each vertex, while leaving the y-coordinate unchanged.

In this case, we need to add 5 units to the x-coordinates of each vertex of the triangle.

The new coordinates of the triangle's vertices would be:

(2+5, 3) = (7, 3)

(4+5, 7) = (9, 7)

(3+5, 3) = (8, 3)

Therefore, the answer is (D) (7, 3), (9, 7) and (8, 3).

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The complete question is :

A triangle with vertices (2, 3), (4, 7), and (3, 3) is translated a distance of 5 units to the right. What are the new coordinates?

A.(-3, 3), (-1, 7), (-2, 3)

B.(2, -2), (4, 2), (3, -2)

C.(2, 8), (4, 12), (3, 8)

D.(7, 3), (9, 7) and (8, 3)

Answer:

x = 16

Step-by-step explanation:

∠1 + ∠2 = 90

42 + 3x = 90

3x = 90 - 42

x = 48/3 = 16

Answer:

  (a)  5.62 m/s²

  (b)  22.5 m/s

Step-by-step explanation:

You want the magnitude of the acceleration of a crate on a frictionless plane inclined at 35° from the horizontal. And you want the speed after 4 seconds.

The acceleration will be the component of the acceleration due to gravity that is in the direction down the plane.

  (9.8 m/s²)·sin(35°) ≈ 5.62 m/s²

The speed of the crate will be the product of the acceleration and time:

  v = at = (5.6 m/s²)(4 s) = 22.5 m/s

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Let the number of 50 cents coins in the container be "x".

We know that the total number of coins in the container is 600, so we can write:

[tex]x + 220 = 600[/tex]

Solving for x, we get:

[tex]x = 600 - 220[/tex]

[tex]x = 380[/tex]

Therefore, there are 380 50 cents pieces in the container.

Hope you can understand :)

Answer:

(0.3)^x has the slowest growth rate.

Step-by-step explanation:

Out of the given functions, the function with the slowest growth rate is (0.3)^x.

To see why, we can compare the growth rates of each function as x increases.

For 5(e)^x, the base e is approximately equal to 2.718, which means that the function grows very quickly as x increases.

For 6(1.03)^x, the base is slightly greater than 1, which means that the function grows at a moderate rate as x increases.

For 2(3)^x, the base is greater than 1, which means that the function grows more quickly than (1.03)^x, but less quickly than (e)^x.

For (0.3)^x, the base is less than 1, which means that the function decreases as x increases. Therefore, this function has the slowest growth rate out of the given functions.

So, (0.3)^x has the slowest growth rate.

Options A and B are the most likely to occur, but they still have a small probability of occurring. The other options are much less likely to occur.

what is probability?

Probability is a branch of mathematics that deals with the study of random events or outcomes.

Since there are 20 cubes in the bag, each cube has a probability of 1/20 of being drawn on any given draw, assuming that each cube is equally likely to be drawn.

A) The probability of drawing a blue cube on any given draw is 3/20. Drawing a blue cube 15 times in 100 draws would have a probability of (3/20)^15 * (17/20)^85 * 100C15, which is an extremely small probability. However, since the question asks what we would expect to occur, we might still select this option.

B) The probability of drawing a red cube on any given draw is 8/20, or 2/5. Drawing more than two times as many red cubes as blue cubes in 100 draws would mean drawing at least 10 red cubes and at most 5 blue cubes. This is equivalent to drawing at least 10 red cubes in 90 draws, which has a probability of approximately (2/5)^10 * (3/5)^90 * 100C90. This is also a small probability, but again, we might still select this option.

C) The probability of drawing a red cube on any given draw is 8/20, or 2/5. Drawing a red cube 40 times in 100 draws would have a probability of (2/5)^40 * (3/5)^60 * 100C40, which is also a small probability.

D) The probability of drawing a green cube on any given draw is 9/20. Drawing a green cube 45 times in 100 draws would have a probability of (9/20)^45 * (11/20)^55 * 100C45, which is again a small probability.

E) Drawing 50 blue cubes in 100 draws is equivalent to drawing a blue cube on every draw. This would have a probability of (3/20)^100, which is an extremely small probability.

Therefore, options A and B are the most likely to occur, but they still have a small probability of occurring. The other options are much less likely to occur.

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Therefore, we have proven that cos (0) - sin (0) = √2 sin (0), given that sin (0) + cos (0) = √2 cos (0).

Trigonometry is a branch of mathematics that deals with the study of the relationships between the angles and sides of triangles. It explores the properties and functions of angles, as well as the relationships between them in various geometrical shapes.

Given by the question.

We have:

sin (0) + cos (0) = √2 cos (0)

Simplifying the left-hand side:

sin (0) + cos (0) = 1

Substituting back into the original equation:

1 = √2 cos (0)

Dividing both sides by √2:

1/√2 = cos (0)

Now we need to prove that:

cos (0) - sin (0) = √2 sin (0)

Substituting our value for cos (0):

1/√2 - sin (0) = √2 sin (0)

Multiplying both sides by √2:

√2/2 - √2 sin (0) = 2 sin (0)

Adding √2 sin (0) to both sides:

√2/2 = 3 sin (0)

Dividing both sides by 3:

sin (0) = √2/6

Substituting back into the equation we were trying to prove:

cos (0) - sin (0) = cos (0) - √2/6

We can now use our earlier equation to substitute for cos (0):

cos (0) - √2/6 = √2/2 - √2/6

Simplifying:

cos (0) - sin (0) = √2 sin (0)

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Answer:

From the information given, we can draw the following diagram:

   S ------- P

       / \       / \

      /   \ QR /   \

     /     \ / /     \

    /       Q       \

   R-----------------P

           MN

Here, SP || QR, so we have ∠QSP = ∠PQR and ∠SPQ = ∠QRP.

Since m∠QRS = 120, we have m∠QRP = 360 - m∠QRS = 360 - 120 = 240.

Since m∠QPS = 135 and ∠QSP = ∠PQR, we have m∠PQR = 360 - m∠QPS = 360 - 135 = 225.

Let x = MN. Since MN is the median of the trapezoid, we have MP = NR = x.

Now, consider the triangles SPQ and RPQ. We have:

tan(∠SPQ) = x/12 (using the tangent ratio in triangle SPQ)

tan(∠RPQ) = x/12 (using the tangent ratio in triangle RPQ)

Since ∠SPQ = ∠RPQ (as they are corresponding angles), we have:

x/12 = x/12

x = 12

Therefore, MN = x = 12.

Answer:

18

Step-by-step explanation:

To determine the number of days until they both work out at the gym on the same day again, Find the lowest common multiples of 6 days and 9 days.

Ahmed (6-days)=6,12,18,24

Brother(9-days)=9,18,24

Therefore , the solution of the given problem of angle comes out to be Judy is therefore looking at her red balloon at a height of about 81.06°.

In Euclidean space, the top and bottom of the wall divide the two circular faces that constitute the sides of a tilt. When two beams collide, they can form a junction point. Angle is another outcome of two entities interacting. They mirror dihedral forms the most. A two-dimensional curve can be created by arranging two line beams in various configurations at their ends.

Here,

We must use trigonometry to determine the run's duration. Call the run's duration "y" for now. We can use the tangent function since we are aware that the angle of height is 75°:

=> tan(75°) = opposite / adjacent

The slope's 800-foot height is on the opposing side. We want to determine the run's length, which is on the opposite side:

=> tan(75°) = 800 / y

We can multiply both sides by y and then split both sides by tan(75°) to find the answer to y:

=> y = 224.12 feet / tan(75°) = 800

The course therefore measures about 224.12 feet in length.

We must once more turn to trigonometry to determine the angle of height.

=> tan(θ) = opposite / adjacent

=> tan(θ) = 2240 / 350

We can use the inverse tangent (or arctan) of both sides to find :

=> 81.06° = arctan(2240 / 350).

Little Judy is therefore looking at her red balloon at a height of about 81.06°.

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Answer:

it cant be if the term 9p and 9c is together but if it 9p and 9c both are in product form then your answer is -p+c

Since two-way table shows the distribution of high school students The correct option is C: No, events A and B are not independent because P(A) ≠ P(A|B).

To determine whether events A and B are independent, we need to compare the probability of event A occurring without any knowledge of event B (i.e., P(A)) to the probability of event A occurring given that event B has occurred (i.e., P(A|B)).

From the table, we see that P(A) = 8/32 = 1/4, since there are 8 juniors out of a total of 32 students. Also, P(B) = 8/32 = 1/4, since there are 8 students who prefer earbuds out of a total of 32 students.

From the table, we see that the probability of both events A and B occurring is P(A and B) = 4/32 = 1/8, since there are 4 juniors who prefer earbuds out of a total of 32 students.

Now, to calculate P(A|B), we use the formula:

P(A|B) = P(A and B) / P(B)

Substituting the values we have, we get:

P(A|B) = (1/8) / (1/4) = 1/2

Since P(A) = 1/4 and P(A|B) = 1/2, we can see that P(A) ≠ P(A|B). Therefore, events A and B are not independent.

Therefore, the answer is No, P(A) ≠ P(A|B).

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