Calculate the rate of change for each function over the interval [1, 4], then tell which function is growing faster.
f (x)=3x+1
g(x)=3^x+1

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 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?

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.

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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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:

20 kilograms.

Step-by-step explanation:

First, we need to find the cost per kilogram.

We can do this by dividing 374 by 34.

That gives us 11. So, our cost per kilogram is $11.

Next, to find how many kilograms we can get with $220 dollars, we need to divide 220 by 11.

That leaves us with a final answer of 20 kilograms.

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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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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the length of the missing side is approximately 4.73 units. Rounded to two decimal places, the answer is 4.73.

To solve this problem, we can use the Law of Sines, which states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and angles in the triangle.

Let's label the missing side as b, and use sin(A) = opposite/hypotenuse to find the length of the side opposite angle A:

sin(A) = opposite/hypotenuse

sin(33.7°) = b/8

b = 8 × sin(33.7°)

b ≈ 4.726

Therefore, the length of the missing side is approximately 4.73 units. Rounded to two decimal places, the answer is 4.73.

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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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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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After descending [tex]21.5[/tex] feet, the scientist's new position in the submarine would be  [tex]30.8[/tex] feet below sea level.

The scientist was initially in a submarine, located 52.3 feet below sea level. This means that her position was  [tex]52.3[/tex]feet lower than the surface of the sea.

Over the next ten minutes, the scientist descended further into the ocean by [tex]21.5[/tex] feet. This means that she traveled down an additional [tex]21.5[/tex]feet from her initial position.

To calculate her new position below sea level, we can subtract the distance she traveled downward (21.5 feet) from her initial position (52.3 feet below sea level).

[tex]52.3 feet - 21.5 feet = 30.8 feet[/tex]

Therefore, after descending 21.5 feet, the scientist's new position in the submarine would be 30.8 feet below sea level.

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a. Hypotheses are: [tex]\rm H_{0}:p=0.20,H_{a}:p > 0.20[/tex]

b. Z-statistics will be 2.47 and p-value = 0.0068

c. you should proceed with plans to develop and market the upgrade.

d. Confidence interval for population proportion will be (0.23, 0.45)

e. The populations are different because they are frοm different regiοns sο prefrences οf peοple may different.

Probability is a mathematical concept that measures the likelihood or chance of an event occurring. It is a number between 0 and 1, where 0 indicates that the event is impossible and 1 indicates that the event is certain.

a. Hypotheses are:

[tex]\rm H_{0}:p=0.20,H_{a}:p > 0.20[/tex]

b) Here we have following information:

[tex]\rm n=50, \hat{p}=\frac{17}{50}=0.34[/tex]

Standard deviation of the proportion is:

[tex]\rm \sigma=\sqrt{\frac{p\left ( 1-p \right )}{n}}=\sqrt{\frac{0.20\left ( 1-0.20 \right )}{50}}=0.0566[/tex]

Test statistics will be:

[tex]$ \rm z=\frac{\hat{p}-p}{\sigma}=\frac{0.34-0.20}{0.0566}=2.47[/tex]

Alternative hypothesis shows that the test is right tailed so p-value of the test is

[tex]\rm p-value = P(z > 2.47) = 1 - P(z \leq 2.47)=1-0.9932=0.0068[/tex]

c. Since p-value of the test is less than 0.05 so we reject the null hypothesis at 0.05 level of significance. So based on this sample, you should proceed with plans to develop and market the upgrade.

d. For 90% confidence interval [tex]\alpha[/tex] = 0.10 so critical value of z will be[tex]\rm z_{c}=z_{\alpha/2}=1.645[/tex]

Confidence interval for population proportion will be

[tex]\rm \hat{p}\pm z_{c}\sqrt{\frac{\hat{p}\left ( 1-\hat{p} \right )}{n}}=0.34\pm 1.645\sqrt{\frac{0.34\left (1-0.34 \right )}{50}}=0.34\pm 0.11 =(0.23, 0.45)[/tex]

e. The populations are different because they are frοm different regiοns sο prefrences οf peοple may different.

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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

Answer:

C) Stock Market.

Brainly wants me to put at least 20 characters and since I only provided the answer choice and was too lazy to do an explanation I am putting this here.

Answer:

C) Stock Market.

Step-by-step explanation:

The term stock market refers to several exchanges in which shares of publicly held companies are bought and sold.

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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To fold and tape a net into a solid shape, cut out the net, fold it along the lines, tape the edges together, and flatten the edges

The net cannot be fold and tape as it is given as a picture, however the general steps to fold and tape a net into a solid shape:

When the above stapes are followed, some of the nets that can be gotten from the given shape are triangular pyramids and triangles

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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]

According to the assumption the translated triangle would have vertices at (4,0), (6,2), and (5,-1).

Given,

To translate a triangle 3 units to the right and 2 units down, we would take each point of the original triangle and add 3 to the x-coordinate and subtract 2 from the y-coordinate.

For example, if the original triangle has vertices at (1,2), (3,4), and (2,1), the translated triangle would have vertices at:

(1+3, 2-2) = (4,0)

(3+3, 4-2) = (6,2)

(2+3, 1-2) = (5,-1)

So, the translated triangle would have vertices at (4,0), (6,2), and (5,-1).

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There are 9 parts to the trail, and each part is the same length.  and hence each part of the trail is 0.2 kilometers long.

The following equation may be used to determine a rectangle's perimeter:

Perimeter is equal to 2 x (length + breadth).

where "length" denotes the rectangle's length and "width" denotes the rectangle's width. In order to account for both sides of the rectangle, we add the length and breadth together and multiply the result by 2, which is the distance around the outside of the rectangle.

The distance ran by Brurt is:

1.8 km ÷ 9 = 0.2 km

Hence, each part of the trail is 0.2 kilometers long.

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The random variable x is normally distributed with mean u=89 and standard deviation o=4. The indicated probability, ​P(x<8​7)= 0.3085.

In statistics, standard deviation is a measure of the amount of variation or dispersion in a set of data. It is a commonly used measure of the spread of a distribution and is calculated as the square root of the variance.

A higher standard deviation indicates that the data points are more spread out from the mean, while a lower standard deviation indicates that the data points are more tightly clustered around the mean.

Standard deviation is an important concept in statistics and is used in many statistical analyses, such as hypothesis testing and confidence interval estimation. It is also used in finance and economics to measure the risk associated with investments and to assess the variability of economic data.

To solve this problem, we need to standardize the value using z-score formula.

z = (x - mu) / sigma

Here, x = 87, mu = 89, and sigma = 4.

z = (87 - 89) / 4 = -0.5

We can look up the corresponding area under the standard normal distribution curve using a table or calculator. The area to the left of z = -0.5 is 0.3085.

Therefore, P(x < 87) = P(z < -0.5) = 0.3085.

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The equation of a circle given by the graph is (x-3)²+(y+2)²=29,

hence option b is correct.

A circle is a closed curve that extends outward from a set point known as the center, with each point on the curve being equally spaced from the center. A circle with an (h, k) center and a radius of r has the equation:

(x-h)² + (y-k)²= r²

This is the equation's standard form. Hence, we can quickly determine the equation of a circle if we know its radius and center coordinates.

We know that the equation of the circle is -

[tex](x-h)^2+(y-k)^2=r^2[/tex]

where (h,k) is the center of the circle and r is the radius,

given that,

center of the circle is = (3,-2)

and the radius is=√29

Hence the equation of a circle is

[tex](x-3)^2+(y+2)^2=(\sqrt29)^2\\\\(x-3)^2+(y+2)^2=29[/tex]

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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 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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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

To prove: Area of triangle MAP = 1/2 area of trapezium MNOP

Given: MN//OP and A is the midpoint of NO

Proof:

Let h be the height of trapezium MNOP.

The area of trapezium MNOP is given by:

Area of trapezium MNOP = (1/2) × (MN + OP) × h

Since MN//OP, we have MN = OP.

Therefore, Area of trapezium MNOP = (1/2) × 2MN × h = MN × h

Now, consider triangle MAP.

The base of triangle MAP is MA, which is half of NO.

Therefore, the base of triangle MAP is (1/2) × NO.

The height of triangle MAP is h, which is the same as the height of trapezium MNOP.

Therefore, the area of triangle MAP is given by:

Area of triangle MAP = (1/2) × (base) × (height) = (1/2) × (1/2NO) × h

Substituting NO = 2OA, we get:

Area of triangle MAP = (1/2) × (1/2 × 2OA) × h = (1/2) × OA × h

Since A is the midpoint of NO, we have OA = 1/2NO.

Therefore, Area of triangle MAP = (1/2) × (1/2NO) × h = (1/2) × (base of trapezium MNOP) × h

Hence, Area of triangle MAP = 1/2 Area of trapezium MNOP.

Therefore, the given statement is proved.

Answer: Given that MNOP is a trapezium with MN parallel to OP and A is the midpoint of NO.

To prove that the area of triangle MAP is half the area of trapezium MNOP, we need to use the following theorem:

Theorem: If a line segment joins the midpoint of one side of a triangle to a vertex opposite to that side, then the segment divides the triangle into two equal areas.

Proof:

Since A is the midpoint of NO, we can draw a line segment AP from vertex P to point A on NO as shown in the figure below:

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      M _______N

       |\      |

       |  \    |

       |    \  |

       |      \|

      O--------P

            A

We can see that triangle MAP is formed by the line segment AP and side MP of the trapezium.

Also, we can see that triangle MAN is congruent to triangle NOP because they are both right triangles with corresponding sides parallel. Therefore, they have the same area.

Since MNOP is a trapezium, the area of trapezium is given by:

Area of trapezium MNOP = (1/2)(MN+OP) × height

Since MN is parallel to OP, the height of the trapezium is the same as the height of triangle MAN and NOP.

Therefore, the area of trapezium MNOP can be written as:

Area of trapezium MNOP = (1/2)(MN+OP) × height

= (1/2)(MA+AP+OP) × height

= (1/2)(MA+MP) × height (Since AP = NO/2 = height)

So, we have proved that:

Area of trapezium MNOP = (1/2)(MA+MP) × height

Using the theorem, we know that AP divides triangle MAP into two equal areas. Therefore,

Area of triangle MAP = (1/2) × Area of triangle MAP

Substituting the above in the expression for the area of trapezium, we get:

Area of trapezium MNOP = Area of triangle MAP + (1/2) × Area of triangle MAP

= (3/2) × Area of triangle MAP

Therefore, we have:

Area of triangle MAP = (1/2) × Area of trapezium MNOP

Thus, we have proved that the area of triangle MAP is half the area of trapezium MNOP.

Step-by-step explanation:

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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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.