simplify -2(2x - 5) -4x

a. The shape of the distribution of the sample mean will be approximately normal, according to the Central Limit Theorem.

b. The standard error of the mean is given by:

SE = σ / sqrt(n)

where σ is the population standard deviation (8 seconds), and n is the sample size (26). Substituting the given values, we get:

SE = 8 / sqrt(26) ≈ 1.57 seconds

Rounded to 2 decimal places, the standard error of the mean is 1.57 seconds.

c. To find the percentage of sample means that will be greater than 152 seconds, we need to calculate the z-score corresponding to a sample mean of 152 seconds:

z = (x - μ) / (σ / sqrt(n))

where x is the sample mean (152 seconds), μ is the population mean (148 seconds), σ is the population standard deviation (8 seconds), and n is the sample size (26).

Substituting the given values, we get:

z = (152 - 148) / (8 / sqrt(26)) ≈ 1.98

Using Appendix B.1, we find that the area to the right of a z-score of 1.98 is 0.0242, or 2.42%. Therefore, approximately 2.42% of the sample means will be greater than 152 seconds.

d. To find the percentage of sample means that will be greater than 144 seconds, we need to calculate the z-score corresponding to a sample mean of 144 seconds:

z = (x - μ) / (σ / sqrt(n))

where x is the sample mean (144 seconds), μ is the population mean (148 seconds), σ is the population standard deviation (8 seconds), and n is the sample size (26).

Substituting the given values, we get:

z = (144 - 148) / (8 / sqrt(26)) ≈ -1.98

Using Appendix B.1, we find that the area to the right of a z-score of -1.98 is also 0.0242, or 2.42%. Therefore, approximately 2.42% of the sample means will be less than 144 seconds.

e. To find the percentage of sample means that will be greater than 144 but less than 152 seconds, we need to find the area between the z-scores corresponding to sample means of 144 and 152 seconds.

The z-score corresponding to a sample mean of 144 seconds is:

z1 = (144 - 148) / (8 / sqrt(26)) ≈ -1.98

The z-score corresponding to a sample mean of 152 seconds is:

z2 = (152 - 148) / (8 / sqrt(26)) ≈ 1.98

Using Appendix B.1, we find that the area to the right of a z-score of -1.98 is 0.0242, and the area to the right of a z-score of 1.98 is 0.0242. Therefore, the area between these two z-scores is:

0.5 - 0.0242 - 0.0242 = 0.4516

Multiplying by 100, we get that approximately 45.16% of the sample means will be greater than 144 but less than 152 seconds.

Answer:

Step-by-step explanation:

blank 1. no

blank 2. not constant because it is a nonlinear function.

See the image linked below.

To graph system of inequalities y ≥ 5x + 2 and y > -3x - 2, draw lines for y = 5x + 2 and y = -3x - 2, then shade areas above each line. The intersection of shaded areas shows solutions to the system of inequalities.

In order to graph the system of inequalities y ≥ 5x + 2 and y > -3x - 2 you would start by graphing each inequality as if it was an equality. For the first inequality, you would graph the line y = 5x + 2 and shade the area above the line because it's y 'greater than or equal to'. For the second inequality, you draw the line y = -3x - 2 and also shade the area above because it's y 'greater than'. The intersection of the shaded areas represents the solution to the system of inequalities. Therefore, the graph would have two shaded lines, with the common shaded area above both lines representing the solutions to the system of inequalities.

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The values of all angles [tex] \angle \: GAL, \angle \: LAO, \angle CAO, \: angle \: KAC[/tex] are 71°, 90°, 90°, 90° respectively.

Given angle GAK is 19°.

It is clear that angle KAL is equal to 90°.

So,

[tex] \angle KAL = {90}^{o} [/tex]

Now,

[tex] \angle \: KAL = \angle KAG+ \angle \: GAL[/tex]

So,

[tex] \angle \: GAL = {90}^{o} - {19}^{o} \\ = {71}^{o} [/tex]

Now,

[tex] \angle \: KAO = 180°[/tex]

Now

[tex] \angle \: KAL + \angle \: LAO = {180}^{o} \\ \angle \: LAO = {180}^{o} - {90}^{o} \\ \angle \: LAO = {90}^{o} [/tex]

Again,

[tex] \angle \: CAO = {180}^{o} - \angle \: LAO \\ \angle \: CAO = {180}^{o} - {90}^{o} = {90}^{o} [/tex]

Similarly,

[tex] \angle \: KAC = \angle \: KAO - \angle \: CAO \\ \angle \: KAC = {180}^{o} - {90}^{o} \\ = {90}^{o} [/tex]

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The probability of people who run red lights is

=0.465

Probability is the measure of the likelihood or chance that an event will occur.

Probability = Possible outcome of an event ÷ Total outcome

Total number of people asked if they had run a red light = number of people that responded 'yes' + number of people that responded 'no'

Total outcome = 348+400

Total outcome = 748

Possible outcome = number of people that responded yes = 394

The probability that if a person is chosen at random, they have run a red light in the last year will be

[tex]=\frac{348}{748}\\\\=0.465[/tex]

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Teri can drive approximately 27.47 miles per gallon of gas.

To find how many miles per gallon Teri can drive, we need to divide the total distance she can travel on a full tank of gas by the amount of gas she needs to fill the tank. This gives us the average number of miles she can travel on one gallon of gas.

Teri's car can hold 17.4 gallons of gas.

Teri can drive 478.5 miles on a full tank of gas.

Mathematically, we can represent this as:

Miles per gallon = Total distance traveled ÷ Amount of gas used

Plugging in the values we have:

Miles per gallon = [tex]\frac{478.4 miles}{17.4 gallons}[/tex]

Performing the division:

Miles per gallon = 27.47126437

Rounding to two decimal places, we get:

Miles per gallon ≈ 27.47

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The function for the graph of the parabola given in the picture attached will be Option (H). f(x) = x² - 6x -1

What is the equation of the parabola?

The equation of a parabola depends on its vertex and focus. There are two possible orientations for a parabola, depending on whether it opens upward or downward (in the case of a vertical axis of symmetry) or leftward or rightward (in the case of a horizontal axis of symmetry).

Equation of a parabola in vertex form:

Equation of a parabola in vertex form is given by,

         y = a(x - h)² + k

         Here, (h, k) is the vertex of the parabola

Given in the graph,

Vertex of the parabola → (0, 2)

Parabola passes through the origin (4, 8).

The equation of the graph with vertex (0, 2) will be,

y = a(x - 0)² + 2

Since, the parabola passes through (4, 8),

8 = a(4 - 0)² + 2

8 = 16a + 2

a = 6/16

Substitute the value of 'a' in the equation of the parabola,

y = a(x - 0)² + 2

y = 6/16(x - 0)² + 2

y = 2/8(x - 0)² + 2

y = x² - 6x + 9 - 9

y = x² - 6x -1

Hence, the function representing the graph will be f(x) = x² - 6x -1

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It seems that the problem solved incorrectly is the one represented by letter A, which is trying to find the missing angle in a triangle. The correct answer is 74.1°, which can be obtained by subtracting the sum of the other two given angles (83.2° and 22.7°) from 180°.

The problem that Anna solved incorrectly is the last one: 32 + x = 108. The correct answer should be x = 76, not x = 108.

In this math problem, Anna seems to be solving for unknown angles in several triangle problems. An important principle here is that the sum of the angles in a triangle always adds up to 180 degrees. Looking at the problems she solved:

So, the problem that was solved incorrectly is the last one: 32 + x = 108.

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Answer:

Step-by-step explanation:

First you have to find out how many red were in the original 27 marbles.

If there were 7 Black and 4 Yellow - this is 11 so 27 - 11 = 16 RED.

Removing 3 Black - changes the the can to 24 marbles.

4 Black - 4 Yellow and 16 Red.

So probability the random pick is red after the removal of 3 Black is

16 out of 24.       [tex]\frac{16}{24}[/tex]  

This reduces to 2/3.

Choice (K)

Answer:

Step-by-step explanation:

There are 27 marbles which include 7 black marbles

                                                            4 yellow marbles

Therefore, the number of red marbles=27-(7+4) marbles

                                                              =16 red marbles

When 3 black marbles are removed from the can, the total number of marbles become 24, which includes the 4 black marbles,4 yellow marbles and 16 red marbles.

So, the probability of the picking of a random red marble from the can after removing the black marbles = 16÷24

This when reduced gives the result 2÷3.

Hence, the answer is option K which means the probability that it was red is 2/3.

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Amplitude of attached graph is 800 and here amplitude explains the maximum change in volume as compare to average volume.

Let us consider two point on the attached graph.

Lowest point as ( 0.5, 1000 )

Highest point as ( 2.5 , 2600 )

In a periodic function the volume of air in a person's lungs over time,

The amplitude is the distance between the maximum value of the function and its average value.

It is represented by y-coordinate.

Amplitude  = ( 2600 - 1000 ) / 2

                  = 1600 / 2

                   = 800

In the context of the volume of air in a person's lungs,

The amplitude represents the maximum change in the volume of air from the average volume.

This is an important measure of lung function.

As it indicates how much air a person can take in and expel from their lungs with each breath.

The higher the amplitude, the greater the lung capacity and respiratory function of the person.

Therefore, the amplitude in the graph is equal to 800 and it represents the maximum variation in the volume of air from the average volume.

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The above question is incomplete, the complete question is:

The volume of air in a persons lungs can be molded with a periodic function the graph below represents the volume of air ,in mL, in a persons long overtime measured in seconds, t. What is the amplitude and what does it represent in this context?

Graph is attached.

The tower is approximately 95 meters(m) tall.

What is a meter?

A meter is the basic unit of length in the International System of Units (SI). It is defined as the distance light travels in a vacuum in 1/299,792,458 of a second.

What kind of unit is meter?

The meter is a unit of measurement for length or distance. It is commonly used in science, engineering, and daily life to measure the size or distance between objects.

According to the given information

Let's call the height of the tower h. We can use similar triangles to set up a proportion:

3 / 1.55 = h / 49.25

Simplifying this expression gives:

h = (3 * 49.25) / 1.55

h = 95.48

Rounding this answer to the nearest meter gives:

h ≈ *95 meters*.

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The linear relationship between the number of bean stalks, n, and the yield, y, is y = -0.5n + 40. This equation can be used to predict the yield of beans for any number of bean stalks planted by the farmer.

To find the linear relationship between the number of bean stalks, n, and the yield, y, we can use the two data points provided by the farmer. We know that when 30 stalks are planted, each plant yields 25 oz of beans, and when 32 stalks are planted, each plant yields 24 oz of beans.

First, we need to find the slope, m, of the line. We can use the formula:

m = (y2 - y1) / (x2 - x1)

where x1 = 30, y1 = 25, x2 = 32, and y2 = 24.

m = (24 - 25) / (32 - 30) = -0.5

Next, we need to find the y-intercept, b, of the line. We can use the point-slope form of the equation:

y - y1 = m(x - x1)

where x1 = 30 and y1 = 25.

y - 25 = -0.5(x - 30)
y - 25 = -0.5x + 15
y = -0.5x + 40

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Answer:

Step-by-step explanation:

3

x

−

4

y

+

15

=

0

.

y

=

3

4

x

−

1

2

Correct answers 5.4

Distance from point (2,3) to the line 3x+4y+9=0

=>r=

3

2

+4

2

∣3(2)+4(3)+9∣

=>r=

9+16

∣16+12+9∣

=>r=

5

27

=>r=5.4

By SAS congruence, triangles ΔVTW and ΔUTX are congruent.

SAS congruence (Side-Angle-Side) is a criterion for the congruence of two triangles in Euclidean geometry. Two triangles are said to be congruent by SAS if they have two pairs of corresponding sides that are congruent, and the included angle between those sides is also congruent.

Formally, the SAS congruence theorem states that if two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of another triangle, then the triangles are congruent. In symbols, if triangle ABC is congruent to triangle DEF by SAS, we write:

AB = DE

AC = DF

∠A = ∠D

In the triangle ΔVTW and ΔUTX

TX/TW=TU/TV (given)

∠VTW=∠UTX(Vertically opposite angle)

By SAS congruence, ΔVTW≈ΔUTX

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a. the total number of different passcodes possible if the digits can be repeated is 10⁶

b. the total number of different passcodes possible is 151,200.

The possibility or chance of an event occurring is measured by probability. It is stated as a number between 0 and 1, where 0 denotes an impossibility (i.e., an event that won't happen) and 1 denotes a certainty (i.e., an event that will happen) (i.e., an event that will always occur).

a. Since there are 10 digits (0-9) that can be used for each digit in the passcode, there are 10 options for each of the six digits.

Therefore, the total number of different passcodes possible if the digits can be repeated is 10⁶, which is equal to 1,000,000.

b. If the digits cannot be repeated, there are 10 options for the first digit, 9 options for the second digit, 8 options for the third digit, and so on.

Therefore, the total number of different passcodes possible is 10x9x8x7x6x5, which is equal to 151,200.

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The equation will be y = 2.4x. Thus, 2nd option is correct.

Equations are mathematical statements with two algebraic expressions on either side of an equals (=) sign. It illustrates the equality between the expressions written on the left and right sides. To determine the value of a variable representing an unknown quantity, equations can be solved. A statement is not an equation if there is no "equal to" symbol in it. It will be regarded as an expression.

We can use a proportion to model the relationship between the number of cans and the total weight. Since we have three cans, we can represent the weight of each can as x, and the total weight as 2.4 pounds:

3x = 2.4

Solving for x, we get:

x = 0.8

Therefore, each can weighs 0.8 pounds.

Hence the equation will be y = 2.4x. Thus, 2nd option is correct.

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Emery bought 3 cans of beans that had a total weight of 2.4 pounds. If each can of beans weighed the same amount, which model correctly illustrates the relationship? Check all that apply.

y = 0.8x

y = 2.4x

y = 3x

y = x/2.4

Equation: [tex]y=k\sqrt{z}[/tex].

Value of the constant of proportionality (k): 6.

So if y varies directly as the square root of z, there's a constant coefficient (k) multiplying the square root of z. Why? Because taking the square root of z will reduce it's value, then a number (k) must be multiplying it to make it match the value of y.

So:

[tex]y=k\sqrt{z}[/tex]

[tex]y = 36;\\ \\z = 36;\\ \\(36)=k\sqrt{(36)}\\ \\[/tex]

[tex]\frac{36}{\sqrt{(36)}} =\frac{k\sqrt{(36)}\\ \\}{\sqrt{(36)}} \\ \\\frac{36}{6} =k\\ \\k=6[/tex]

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[tex]\blue{\huge {\mathrm{SOLVING \; EQUATIONS}}}[/tex]

[tex]{===========================================}[/tex]

[tex]{\underline{\huge \mathbb{Q} {\large \mathrm {UESTION : }}}}[/tex]

[tex]{===========================================}[/tex]

[tex] {\underline{\huge \mathbb{A} {\large \mathrm {NSWER : }}}} [/tex]

[tex]\qquad\qquad\begin{aligned}\bold{y = k\sqrt{z}}\\\\\bold{\:k = 6\:\:\:}\end{aligned}[/tex]

*Please read and understand my solution. Don't just rely on my direct answer*

[tex]{===========================================}[/tex]

[tex] {\underline{\huge \mathbb{S} {\large \mathrm {OLUTION : }}}} [/tex]

From the given information, we are given that:

Since y varies directly as the square root of z, we can write this as an equation:

where:

Substitute the given values into the equation and solve for k:

[tex]\qquad\qquad\begin{aligned}\sf \red{y} &=\sf k\sqrt{\blue{z}}\\\sf \red{36} &=\sf k\sqrt{\blue{36}}\\\sf \dfrac{\red{36}}{\sqrt{\blue{36}}} &=\sf \dfrac{k \cancel{\sqrt{\blue{36}}}}{ \cancel{\sqrt{\blue{36}}}}\\\sf \dfrac{\red{36}}{\blue{6}}&=\sf k\\\sf 6& =\sf k\\\sf \bold{\:k}& = \bold{6}\:\end{aligned}[/tex]

[tex]{===========================================}[/tex]

Equations are mathematical statements that show that two quantities are equal. They typically contain an equal sign (=) and one or more variables. Equations are used to express relationships between quantities and to solve problems in various fields of science, engineering, and mathematics.

Example equations include:

To solve an equation, we want to determine the value of the variable(s) that make the equation true.

There are different techniques to solve equations, but some common steps include:

1. Simplify both sides of the equation by combining like terms and using the order of operations if necessary.

2. Isolate the variable on one side of the equation by undoing any operations that were performed on it.

3. Check the solution by plugging it back into the original equation to see if it makes the equation true.

[tex]{===========================================}[/tex]

[tex]- \large\sf\copyright \: \large\tt{AriesLaveau}\large\qquad\qquad\qquad\tt 04/01/2023[/tex]

The volume of a cube with volume 27/64 cubic units is 3/4.

All edge lengths of the cube are equal. Then the volume of the cube is 3/4 units.

It deals with the size of geometry, region, and density of the different forms both 2D and 3D.

The volume of the cube is 27/64 cubic units.

We know the formula of the volume is given as

Volume = a³

[tex]a^3 = \sqrt[3]{\dfrac{27}{64} }[/tex]

a = 3/4

where a is the edge length of the cube.

The edge length of the cube is 3/4 units.

Therefore, the volume of a cube with a volume of 27/64 cubic units is 3/4.

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The complete question is: What is the side of a cube with volume 27/64 cubic units?

Answer:

A) The histogram is bimodal.

Answer:

To create a frequency distribution, we first need to determine the range of the data. The smallest measurement is 3.4 inches and the largest is 5.4 inches, so the range is 5.4 - 3.4 = 2 inches. To create 6 classes with a width of 0.4 inches, we divide the range by 0.4 and round up to the nearest integer:

Number of classes = (range / class width) rounded up = 2 / 0.4 = 5

So we will use 5 classes with a width of 0.4 inches each. The classes and their corresponding frequency counts are:

Class 1: 3.4 - 3.8 | Frequency: 3

Class 2: 3.9 - 4.3 | Frequency: 8

Class 3: 4.4 - 4.8 | Frequency: 6

Class 4: 4.9 - 5.3 | Frequency: 7

Class 5: 5.4 - 5.8 | Frequency: 1

To create a histogram, we can plot the frequency counts on the y-axis and the class intervals on the x-axis. The shape of the histogram can give us information about the distribution of the data. In this case, the histogram is bimodal, meaning there are two peaks in the data. This suggests that the data may be composed of two separate subpopulations.

The exact area of the inscribed circle is 12π square units.

A circle is a geometric shape consisting of all the points in a plane that are equidistant from a given point called the center of the circle. The distance between the center and any point on the circle is called the radius of the circle.

The radius of the inscribed circle is also the distance from the center of the hexagon to each of its sides. To find this distance, we can divide the hexagon into six equilateral triangles, each with side length 2 units. The height of each triangle is √3 times the side length, or 2√3 units. The distance from the center of the hexagon to each side is equal to this height, or 2√3 units.

Therefore, the radius of the inscribed circle is 2√3 units. The area of the circle is given by the formula A = πr², where r is the radius. Substituting in the value of the radius, we get:

A = π(2√3)² = 12π

So the exact area of the inscribed circle is 12π square units.

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By iterating through different alpha values, you can identify the optimal alpha for this data set that results in the smallest MSE.

a. To construct a time series plot, you would typically use software like Excel or programming languages like Python or R.

However, I can describe the general trend based on the given data points.

Week 1: 7.35

Week 2: 7.40

Week 3: 7.55

Week 4: 7.56

Week 5: 7.60

Week 6: 7.52

Week 7: 7.52

Week 8: 7.70

Week 9: 7.62

Week 10: 7.55

Based on the data, there seems to be a slight overall upward trend in the Commodity Futures Index over the ten weeks, but there are also small fluctuations up and down throughout the period.

b. To find the best exponential smoothing coefficient (alpha) using the trial and error method, you would typically use software or programming languages to calculate the Mean Squared Error (MSE) for each alpha value.

Choose an initial value for alpha (e.g., 0.1).

Apply the exponential smoothing formula to the data series: St = α * Xt + (1 - α) * St-1, where St is the smoothed value at time t, α is the smoothing coefficient, Xt is the observed value at time t, and St-1 is the smoothed value at time t-1.

Calculate the forecast errors (Et) as the difference between the observed value and the smoothed value for each time period: Et = Xt - St-1.

Compute the Mean Squared Error (MSE) by taking the average of the squared forecast errors.

Repeat steps 2-4 with different values of alpha (e.g., 0.2, 0.3, 0.4, etc.) until you find the alpha value that minimizes the MSE.

By iterating through different alpha values, you can identify the optimal alpha for this data set that results in the smallest MSE.

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Assume Z is a regular, standard random variable. The answer is P(Z 1.2), which is written as P(Z 1.2) = (1.2) =. 8849.

An variables is just a sum that can change according to the fundamental math concept. The generic letters x, y, and z are often employed in mathematical expressions and equations. Thus a variable is a sign for a numeric value that is unknown.

Every feature, number, or amount that can be gauged or quantified is referred to as a variable. A statistic may also be referred to as a data item. Examples of variables include age, sex, company revenue and costs, birth country, capital expenditures, class grades, eye colour, and vehicle kind.

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The probability of Z being greater than 1.2 is 0.11507, or approximately 11.51% (rounded to two decimal places).

Probability is a mathematical concept that expresses the possibility or chance of an event happening.

A number between 0 and 1, with 0 signifying impossibility and 1 signifying certainty, is used to symbolize it.

The likelihood of an event occurring increases as the probability approaches 1.

To find the probability P(Z > 1.2) where Z follows a standard normal distribution, we need to use the standard normal distribution table or a calculator with a normal distribution function.

Using a standard normal distribution table, we look for the probability value that corresponds to a standard normal deviate of 1.2.

The table gives us the area under the standard normal curve to the left of 1.2, which is 0.88493. Therefore, the probability of Z being greater than 1.2 is:

P(Z > 1.2) = 1 - P(Z ≤ 1.2)

= 1 - 0.88493

= 0.11507

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I believe you meant to write the expression as follows: 71 = -8x + 7y + 7z + 5. This equation represents a relation between the values of x, y, and z in a rhombus where 71 is the sum of the diagonals of the rhombus.

How to calculate value of x,y and z for the given rhombus?

To solve for x, y, and z, we need to use additional information about the rhombus. Without any further information, we cannot determine unique values for x, y, and z.

However, we can make some observations based on the given equation:

The coefficient of x is negative (-8), which means that increasing x would decrease the value of the left-hand side of the equation. Therefore, we can conclude that x must be positive to satisfy the equation.

The coefficients of y and z are positive (7 and 7, respectively), which means that increasing y and/or z would increase the value of the left-hand side of the equation. Therefore, we can conclude that y and z must be non-negative to satisfy the equation.

With these observations in mind, we can come up with multiple solutions for x, y, and z that satisfy the given equation. Here are a few examples:

If we let x = 0, y = 11, and z = 9, then the equation is satisfied:

71 = -8(0) + 7(11) + 7(9) + 5

71 = 77

If we let x = 2, y = 10, and z = 8, then the equation is also satisfied:

71 = -8(2) + 7(10) + 7(8) + 5

71 = 71

Therefore, we cannot determine a unique solution for x, y, and z based on the given equation alone.

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Answer: Sampling is very often used in our daily life. For example, while purchasing fruits from a shop, we usually examine a few to assess the quality. A doctor examines a few drops of blood as a sample and draws a conclusion about the blood constitution of the whole body.

Prahar would need approximately [tex]2.0625[/tex] cups of sugar to make apple pies that serve [tex]22[/tex]  people. We can round this up to [tex]2 1/8[/tex] cups of sugar for practical purposes.

To determine the amount of one of the ingredients needed to make an apple pie that serves [tex]22[/tex] people, we can set up a proportion comparing the number of servings:

Number of servings of the original recipe: [tex]8[/tex]

Number of servings needed: [tex]22[/tex]

Let's choose sugar as the ingredient to calculate:

Original amount of sugar for 8 servings:  [tex]3/4[/tex] cup

Unknown amount of sugar for  [tex]22[/tex] servings: x

We can set up the proportion as follows:

[tex]8/22 = 3/4x[/tex]

To solve for x, we can cross-multiply:

[tex]8x = 22 \times 3/4[/tex]

[tex]8x = 16.5[/tex]

[tex]x = 16.5/8[/tex]

[tex]x = 2.0625[/tex]

Therefore, Prahar would need approximately [tex]2.0625[/tex] cups of sugar to make apple pies that serve [tex]22[/tex]  people. We can round this up to [tex]2 1/8[/tex] cups of sugar for practical purposes.

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Answer:

Step-by-step explanation:

3

Therefore, to the nearest tenth of a minute, it will be about 9.4 minutes until there are 110 grams of the element remaining.

"Initial amount" usually refers to the starting quantity or value of something, such as money, an investment, or a substance. It is the amount or quantity that exists at the beginning of a particular time period or situation. For example, if you invest $1,000 in a savings account, the initial amount is $1,000. Similarly, if you are given 10 liters of water, the initial amount of water is 10 liters.

We can use the exponential decay formula to solve this problem:

[tex]N(t) = N₀ * e^{(-kt)}[/tex]

where N(t) is the amount of the element remaining at time t, N₀ is the initial amount of the element, k is the decay constant (which is equal to ln (1 - 0.128) = -0.142), and e is the base of the natural logarithm.

We can plug in the given values and solve for the time t when the remaining amount N(t) is equal to 110 grams:

[tex]110 = 540 * e^{(-0.142t)}[/tex]

Dividing both sides by 540, we get:

[tex]0.2037 = e^{(-0.142t)}[/tex]

Taking the natural logarithm of both sides, we get:

[tex]ln(0.2037) = -0.142t[/tex]

Solving for t, we get:

t = ln (0.2037) / (-0.142) ≈ 9.4 minutes

Therefore, to the nearest tenth of a minute, it will be about 9.4 minutes until there are 110 grams of the element remaining.

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Statement-Reason Proof:

If the circle has center O, then:

1.) BD = CA [Definition of Diameters of a Circle]

2.) Angle BAD = Angle CDA [Angle of Semicircle are Right Angles (90°)]

3.) AD = AD [Reflexive Property]

4.) Triangle DCA ≅ Triangle ABD [RHS Congruency Rule]