21.7.3 Quiz: Intersecting Lines and Proofs
Which pairs of angles in the figure below are vertical angles?
Check all that apply.
R
P
T
D
B

Step-by-step explanation:

a probability is always the ratio of

desired cases / totally possible cases.

so, in this case it is the

area of the triangle / area of the circle.

as everything of the triangle is also a part of the circle.

and so, that fraction of the area of the whole circle that is the area of the triangle in refutation to the area of the whole circle is the probability that a random point inside the circle would be also inside the triangle.

the area of a right-angled triangle is

leg1 × leg2 / 2

in our case

12 × 12 / 2 = 72 units²

the area of a circle is

pi × r²

in our case that is

pi × 12² = 144pi units²

the requested probability is

P = 72 / 144pi = 1/2pi = 0.159154943... ≈ 0.16

These are general considerations, and specific experiences may vary depending on geographical location, cultural practices, and individual preferences.

On days when the temperature was less than 58 degrees, it indicates that the weather was relatively cool. This could imply various conditions and experiences depending on the context and location. In general, some possible scenarios on such days may include:

Cooler outdoor activities: People might engage in activities such as hiking, jogging, or outdoor sports that are more enjoyable in cooler temperatures.

Layered clothing: Individuals may choose to wear warmer clothing, including jackets, sweaters, or scarves, to stay comfortable in the cooler weather.

Indoor activities: Cooler temperatures may encourage people to spend more time indoors, engaging in activities such as reading, watching movies, or pursuing hobbies.

Increased energy consumption: Cold weather often leads to an increased need for heating systems, resulting in higher energy consumption to maintain indoor comfort.

Changes in vegetation: Cooler temperatures can affect plant growth and flowering patterns. Certain plants may thrive in cooler conditions, while others may enter a dormant phase.

Changes in animal behavior: Some animals may adapt to cooler temperatures by seeking shelter or adjusting their activities and migration patterns.

Possible health effects: Cooler temperatures may impact individuals with certain health conditions, such as respiratory issues or joint pain, requiring them to take appropriate measures to stay comfortable.

These are general considerations, and specific experiences may vary depending on geographical location, cultural practices, and individual preferences.

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Reason:

The order of ABCD and EFGH is important. This is because the letters pair up based on their position.

D and H pair up because they're the last letters of ABCD and EFGH respectively. Similar polygons have congruent corresponding angles.

Take note of how the angles are marked to indicate which angles pair up.

D = H

4x = 100

x = 100/4

x = 25

Answer:

25 is the answer by matching the equial sides

Step-by-step explanation:

100°/4=X

25=X

Answer:

[tex]\frac{2.5}{1}[/tex]

Step-by-step explanation:

To express it as a fraction, divide 250 by 100 first to get 2.5. Then put that over 1.

Hope this helps!

The statement that correctly answers the question "A square prism has a base length of 5 m, and a square pyramid of the same height also has a base length of 5 m. Are the volumes the same?" is "No, because only the bases have the same area, not every cross-section at every level parallel to the bases."

Explanation: A square prism is a three-dimensional shape that has two square bases that are parallel to each other, and every side is a rectangle. In contrast, a square pyramid is a three-dimensional figure that has a square base and triangular faces that meet at a point called an apex or vertex. The height of a square pyramid is the distance from the base to the apex.

Therefore, the volume of a square prism can be calculated by multiplying the area of the base by the height, whereas the volume of a square pyramid can be determined by multiplying the area of the base by one-third of the height.

Thus, even though the base length is 5 m in both cases, the cross-sectional areas at every level parallel to the bases in a square pyramid are not the same. This implies that the answer is No, because only the bases have the same area, not every cross-section at every level parallel to the bases.

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To find the value of x, we can set the two angle measures equal to each other and solve for x.

Given:

mZA = (4x - 2)°

mZB = (6x - 20)°

Setting them equal to each other:

4x - 2 = 6x - 20

Now, we can solve for x:

4x - 6x = -20 + 2

-2x = -18

Dividing both sides by -2:

x = -18 / -2

x = 9

Therefore, the value of x is 9.

Answer:

The answer is 9.

Step-by-step explanation:

We need to use the fact that the sum of the angles in a triangle is 180 degrees. Let A, B, and C be the three angles in the triangle. Then we have:

mZA + mZB + mZC = 180°

Substituting the given values, we get:

(4x - 2)° + (6x - 20)° + mZC = 180°

Simplifying the left side, we get:

10x - 22 + mZC = 180°

Next, we use the fact that angles opposite congruent sides of a triangle are congruent. Since we know that segment AC and segment BC are congruent, we have:

mZA = mZB

Substituting the given values and simplifying, we get

4x - 2 = 6x - 20

Solving for x, we get:

x = 9

Therefore, the value of x is 9.

Answer: E

Step-by-step explanation:

Answer:

  -3

Step-by-step explanation:

You want the multiplier of the second equation that would result in eliminating the x-terms when the first equation is multiplied by 7 and added to the multiplied second equation.

The desired multiplier will have the effect of making the coefficient of x be zero when the multiplications and addition are carried out. If k is that multiplier, the resulting x-term will be ...

 7(-3x) +k(-7x) = 0

  -21x -7kx = 0 . . . . . . simplify

  3 +k = 0 . . . . . . . . . divide by -7x

  k = -3 . . . . . . . . . . subtract 3

The multiplier of the second equation should be -3.

__

Additional comments

Carrying out the suggested multiplication and addition, we have ...

  7(-3x -7y) -3(-7x +10y) = 7(-56) -3(1)

  -49y -30y = -395

  y = -395/-79 = 5

The solution is (x, y) = (7, 5).

In general, the multipliers will be the reverse of the coefficients of the variable, with one of them negated. Here the coefficients of x are {-3, -7}. When these are reversed, you have {-7, -3}. When the first is negated, the multipliers of the two equations are {7, -3}. That is, the second equation should be multiplied by -3, as we found above.

Note that if you subtract the multiplied equations instead of adding, you can use the reversed coefficients without negating one of them. The choice of where the minus sign appears (multiplication or subtraction) will depend on your comfort level with minus signs.

The number of minus signs in this system can be reduced by multiplying the first equation by -1 to get 3x +7y = 56.

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Answer:

Step-by-step explanation:

x=0.5, y=12.

x=3, y=2.

x=4, y=1.5.

x=6, y=1.

Answer:

21.6

Step-by-step explanation:

Tan 27= 11

y

y×tan27=11

y=21.6

The answer is:

y = 21.6

Work/explanation:

We are asked to use SOH-CAH-TOA. But what does it mean?

SOH stands for Sine = Opposite ÷ Hypotenuse

CAH stands for Cosine = Adjacent ÷ Hypotenuse

TOA stands for Tangent = Opposite ÷ Adjacent

Since we do not have the hypotenuse, we will use the TOA ratio:

[tex]\sf{Tangent=\dfrac{Opposite}{Adjacent}}[/tex]

The opposite is 11, and the adjacent is y:

[tex]\sf{\tan27=\dfrac{11}{y}}[/tex]

Take the tangent of 27 & approximate it:

[tex]\sf{0.5095=11\div y}[/tex]

Multiply each side by y

[tex]\sf{0.5095y=11}[/tex]

Divide each side by 0.5095

[tex]\sf{y=21.6}[/tex]

Answer:

Step-by-step explanation:

a) To solve the equation tan θ = 1/√3, we can find the angle whose tangent is 1/√3 by taking the inverse tangent (arctan) of 1/√3.

θ = arctan(1/√3)

θ ≈ 30.0°

Therefore, the angle in standard position that satisfies tan θ = 1/√3 is approximately 30.0°.

b) To solve the equation 2cos θ = √3, we can isolate the cosine term by dividing both sides of the equation by 2.

cos θ = √3 / 2

Now, we can find the angle whose cosine is √3/2 by taking the inverse cosine (arccos) of √3/2.

θ = arccos(√3/2)

θ ≈ 30.0°

Therefore, the angle in standard position that satisfies 2cos θ = √3 is approximately 30.0°.

Step-by-step explanation:

1 job divided by the sum of the rates = 3 hours

1 / ( 1/5 + 1/x )  = 3  

x = 7.5 hrs    for assistant alone

Answer:A

Step-by-step explanation:the graph shows the values that are greater than 2

The square root of 184 is approximately 13.5647. It is a non-repeating, non-terminating decimal.

The square root is obtained by finding the number that, when multiplied by itself, equals 184. In this case, 13.5647 multiplied by itself is approximately equal to 184. To explain the answer further, the square root is a mathematical operation that determines the value which, when multiplied by itself, gives the original number.

In the case of 184, the square root is an irrational number, meaning it cannot be expressed as a fraction or a terminating decimal. The approximate value of 13.5647 is derived using numerical methods or a calculator. This value represents the principal square root of 184, and it is positive since the square of a negative number would yield a positive result.

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Answer:

The square root of 184 is expressed as √184 in the radical form and as (184)½ or (184)0.5 in the exponent form. The square root of 184 rounded up to 5 decimal places is 13.56466. It is the positive solution of the equation x2 = 184. We can express the square root of 184 in its lowest radical form as 2 √46.

Square Root of 184: 13.564659966250536

Square Root of 184 in exponential form: (184)½ or (184)0.5

Square Root of 184 in radical form: √184 or 2 √46

Answer:

If Tonia eats 1/8 of the cake, then the fraction of the cake left is:

1 - 1/8 = 7/8

If Trinny eats 1/6 of the cake, then the fraction of the cake left is:

1 - 1/6 = 5/6

Since Tonia and Trinny have identical cakes, the amount of cake left is the same for both of them. Therefore, the amount of cake left is:

(7/8 + 5/6) / 2 = 41/48

So there is 41/48 of the cake left.

Answer:

Relative frequency of selecting a 2 = 8/50 = 0.16 = 16%

Step-by-step explanation:

You are selecting a random number between 1 and 5, and you perform this task 50 times.

Out of these 50 times, the outcome "2" appears 8 times.

Therefore the relative frequency of selecting the number 2 is:

f(2) = 8/50 = 0.16 which is 16%

Step-by-step explanation:

x = width

Raul    x   then    6x   then  finally   3x

Jen     x   then 5 x

5x   versus 3x

5x/3x = 1  2/3 wider

The antiderivative of ∫[[tex]x^\frac{1}{11}[/tex] - 7sin(x)]dx is:

[tex](11/12)x^\frac{12}{11} - 7cos(x) + C[/tex]

The indefinite integral, denoted as ∫f(x)dx, represents the antiderivative of a function f(x). It involves finding a function whose derivative is equal to the given function f(x).

Let's evaluate the indefinite integral of [tex]\int [x^\frac{1}{11} - 7sin(x)]dx[/tex]

To find the antiderivative of [tex]x^\frac{1}{11}[/tex], we add 1 to the exponent and divide by the new exponent:

[tex]\int x^\frac{1}{11} dx = (11/12)x^\frac{12}{11} + C_1[/tex], where C₁ is the constant of integration.

∫7sin(x)dx:

To find the antiderivative of 7sin(x), we use the trigonometric identity that the antiderivative of sin(x) is -cos(x):

∫7sin(x)dx = -7cos(x) + C₂, where C₂ is another constant of integration.

Combining the two results, the indefinite integral of [tex]\int[x^\frac{1}{11} - 7sin(x)]dx[/tex] is:

[tex](11/12)x^\frac{12}{11} - 7cos(x) + C[/tex],

where C = C₁ + C₂ represents the constant of integration.

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The interval for the graphed function that has a local minimum of 0 is [-2, 0].

To determine the interval for the graphed function that has a local minimum of 0, we need to examine the behavior of the graph around the x-values where it crosses the x-axis.

We know that the graph crosses the x-axis at -2.5, 0, and 3. Since the local minimum occurs at 0, we need to find the interval where the graph is decreasing to the left of 0 and increasing to the right of 0.

From the given information, we can determine the following:

The graph is decreasing between -2.5 and 0 because it crosses the x-axis at -2.5 and 0, and the minimum value is at (-1.56, -6).

The graph is increasing between 0 and 3 because it crosses the x-axis at 3, and the maximum value is at (1.2, 2.9).

The correct option is [–2, 0].

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Question

On a coordinate plane, a curved line with minimum values of (negative 1.56, negative 6) and (3, 0), and a maximum value of (1.2, 2.9), crosses the x-axis at (negative 2.5, 0), (0, 0), and (3, 0), and crosses the y-axis at (0, 0). Which interval for the graphed function has a local minimum of 0? [–3, –2] [–2, 0] [1, 2] [2, 4]

Answer:

the answer is y=mx+c

Step-by-step explanation:

where the answer is the coefficient of the gradient which is x

a. The  slope of the curve at the point (4,16)   is approximately 1.165, and at the point (16,4)  is approximately -0.496.

b. The   curve has a horizontal tangent line at the points(0,0) and (3,27).

c. The   curve has a vertical tangent lineat the points (0,0) and (65/2, (65/2)³).

a. To find the   slope of the curve given by the equation x³ + y³ - 65xy = 0 at the points (4,16) and (16,4),we can differentiate the equation implicitly with respect to x and solve for dy/dx.

Differentiating the equation with respect to x, we have  -

3x² + 3y²(dy/dx) - 65y - 65x(dy/dx) = 0

To find the slope at a specific point, substitute the x and y coordinates into the equation and solve for dy/dx.

For the point (4,16)  -

3(4)² + 3(16)²(dy/dx) - 65(16) - 65(4)(dy/dx) = 0

48 + 768(dy/dx) - 1040 - 260(dy/dx) = 0

508(dy/dx) = 592

(dy/dx) = 592/508

(dy/dx) ≈ 1.165

For the point (16,4)  -

3(16)² + 3(4)²(dy/dx) - 65(4) - 65(16)(dy/dx) = 0

768 + 48(dy/dx) - 260 - 1040(dy/dx) = 0

(-992)(dy/dx) = 492

(dy/dx) = 492/(-992)

(dy/dx) ≈ -0.496

Thus, the slope of the   curve at the point (4,16) isapproximately 1.165, and at the point (16,4) is approximately -0.496.

b. To find the point where the curve has   a horizontal tangent line, we need to find the x-coordinate(s)where dy/dx equals zero.

This means   the slope is zero and the tangent line is horizontal.

From the derivative we obtained earlier  -

3x² + 3y²(dy/dx) - 65y - 65x(dy/dx) = 0

Setting dy/dx equal to zero  -

3x² - 65y = 0

Substituting y = x³/65 into the equation  -

3x² - 65(x³/65) = 0

3x² - x³ = 0

Factoring out an x²  -

x²(3 - x) = 0

This equation has two solutions  -  x = 0 and x = 3.

hence, the curve has a horizontal   tangent line at the points(0,0) and (3,27).

c. To find the point where the curve has a vertical tangent line, we need to find the x-coordinate(s)   where the derivative is undefinedor approaches infinity.

From the derivative  -

3x² + 3y²(dy/dx) - 65y - 65x(dy/dx) = 0

To find the vertical tangent line, dy/dx should be undefined or infinite. This occurs when the denominator of dy/dx is zero.

Setting the denominator equal to zero:  -

65x = 65y

x = y

Substituting this condition back into the original equation  -

x³ + x³ - 65x² = 0

2x³ - 65x² = 0

x²(2x - 65) = 0

This equation has two solutions  - x = 0 and x = 65/2.

Therefore, the curve has a vertical tangent line   at the points (0,0)

and(65/2, (65/2)³).

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The graph of the function and its inverse is added as an attachment

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

f(x) = 3 + 2ln(x)

Express as an equation

So, we have

y = 3 + 2ln(x)

Swap x and  y in the above equation

x = 3 + 2ln(y)

Next, we have

2ln(y) = x - 3

Divide by 2

ln(y) = (x - 3)/2

Take the exponent of both sides

[tex]y = e^{\frac{x - 3}{2}}[/tex]

Next, we plot the graphs

The graph of the functions is added as an attachment

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Answer:

y = -3

Step-by-step explanation:

The midline of a sinusoidal function is the horizontal center line about which the function oscillates periodically.

The midline is positioned halfway between the maximum (peaks) and minimum (troughs) values of the graph. It serves as a baseline that helps visualize the oscillations of the function.

To find the equation of the midline, we need to determine the average y-value between the maximum and minimum y-values.

In this case, the maximum y-value is -1, and the minimum y-value is -5. To find the equation of the midline, sum the maximum and minimum y-values, and divide by 2:

[tex]y=\dfrac{-1 + (-5)}{2} = \dfrac{-6}{2}=-3[/tex]

Therefore, the equation of the midline for the graphed sinusoidal function is y = -3.

Answer:

Given the information you provided, we can model cellular phone usage over time with an exponential growth model. An exponential growth model follows the equation:

`y = a * b^(x - h) + k`

where:

- `y` is the quantity you're interested in (cell phone usage),

- `a` is the initial quantity (34 million in 1995),

- `b` is the growth factor (1.22, representing 22% growth per year),

- `x` is the time (the year),

- `h` is the time at which the initial quantity `a` is given (1995), and

- `k` is the vertical shift of the graph (0 in this case, as we're assuming growth starts from the initial quantity).

So, our specific model becomes:

`y = 34 * 1.22^(x - 1995)`

To find the cellular usage in 2003, we plug 2003 in for x:

`y = 34 * 1.22^(2003 - 1995)`

Calculating this out will give us the cellular usage in 2003.

Let's calculate this:

`y = 34 * 1.22^(2003 - 1995)`

So,

`y = 34 * 1.22^8`

Calculating the above expression gives us:

`y ≈ 97.97` million.

So, the cellular phone usage in 2003, according to this model, is approximately 98 million.

The value of y in the figure is

35.134 degrees

The value of y is worked using SOH CAH TOA

Sin = opposite / hypotenuse - SOH

Cos = adjacent / hypotenuse - CAH

Tan = opposite / adjacent - TOA

The figure shows a right angle triangle of

opposite = 19

adjacent = 27

The angle is calculated using tan, TOA let the angle be y

tan y= Opposite / Adjacent

tan y = 19 / 27

y = arc tan (19/27)

y = 35.134 degrees

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Answer:

25

Step-by-step explanation:

5+5+5+5+5=25

Answer:25

Step-by-step explanation:

Answer:

thousandths place value

Step-by-step explanation:

Answer:

C. 335

Step-by-step explanation:

We will need a system of equations to determine how many senior citizen tickets were sold, where

First equation:

The sum of the revenues earned from the adult and senior citizen tickets equals $8579.00:

(price of adult tickets * quantity) + (price of senior citizen tickets * quantity) = total revenue earned

Since adult tickets cost $22/adult, senior citizen tickets cost $13/senior citizen, and the total revenue earned is $8579.00, our first equation is given by:

22A + 13S = 8579

Second Equation:

The sum of the quantities of adult and senior citizen tickets equals the total number of ticket sold:

quantity of adult tickets + quantity of senior citizen tickets = total quantity of tickets sold

Since the theater sold 527 tickets in total, our second equation is given by:

A + S = 527

Method to Solve:  Elimination:

We can solve for S by first eliminating.  To eliminate A, we'll first need to multiply the second equation by -22:

Multiplying -22 by A + S = 527

-22(A + S = 527)

-22A - 22S = -11594

Now we can add this equation to the first equation to find S, the number of senior citizen tickets:

Adding 22A + 13S = 8579 to -22A - 22S = -11594:

    22A + 13S = 8579

+

   -22A - 22S = -11594

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

(22A - 22A) + (13S - 22S) = (8579 - 11594)

(-9S = -3015) / -9

S = 335

Thus, 335 senior citizen tickets (answer C.)

Optional:  Find A (the number of adult tickets sold) to check the validity of our answers:

We can find A by plugging 335 for S in any of the two equations in our system.  Let's use the second one:

Plugging in 335 for S in A + S = 527:

(A + 335 = 527) - 335

A = 192

Thus, 192 adult tickets were sold.

Checking the validity of our answers:

Now we can check that our two answers for S and A are correct by plugging in 335 for S and 192 for A in both of the equations in our system and seeing if we get the same answer on both sides:

Plugging in 335 for S and 192 for A in 22A + 13S = 8579:

22(192) + 13(335) = 8579

4224 + 4355 = 8579

8579 = 8579

Plugging in 335 for S and 192 for A in A + S = 527:

192 + 335 = 527

527 = 527

Thus, our answers for S and A are correct.

The expression g(x) = f(x) + represents the relationship between the two functions expression for g(x) in terms of f(x).

To express the function g(x) in terms of f(x), we need to understand the relationship between the two functions.

The given expression g(x) = f(x) + indicates that the function g(x) is obtained by adding a certain value or expression to the function f(x). expression for g(x) in terms of f(x).

In general, if we have the function g(x) = f(x) + c, where c is a constant value, then g(x) can be expressed in terms of f(x) as:

g(x) = f(x) + c

In this case, g(x) is obtained by adding the constant value c to the corresponding values of f(x).

It's important to note that without additional information about the specific relationship between f(x) and g(x), such as a functional equation or given values, we cannot provide a more precise expression for g(x) in terms of f(x).

Therefore, the expression g(x) = f(x) + represents the relationship between the two functions expression for g(x) in terms of f(x).

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Answer: 3

Step-by-step explanation: just 3

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