simplify algebraic rational expressions, with numerators and denominators containing monomial bases with integer exponents, to equivalent forms.

The cardinality of set from the given vein diagram is found as 2.

Think about set A. The set A is said to be finite and so its cardinality is same to the amount of elements n if it includes precisely n items, where n  ≥  0. |A| stands for the cardinality of such a set A.

It turns out that there are two kinds of infinite sets that we need to determine between since one form is much "bigger" than the other. Particularly, one type is referred to as countable and the other as uncountable.

From the given figure

Set A = {8 , 8, 3, 6}

Compliment of Set B (elements not present in set B):

Set [tex]B^{c}[/tex] = {8, 8, 6(pink), 3(white)}

Thus,

(A∩ [tex]B^{c}[/tex] ) = {8, 8} (present in both set)

n (A∩ [tex]B^{c}[/tex] ) = 2 (cardinal number)

Thus, the cardinality of the set from the given vein diagram is found as 2.

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Answer: hope its help

Let's start by assigning variables to the unknown quantities in the problem. Let h be the height of the triangle in inches, and let b be the base of the triangle in inches.

According to the problem, the base of the triangle is 3 inches shorter than its height. This can be expressed as:

b = h - 3

The formula for the area of a triangle is:

A = (1/2)bh

We are given that the area of the triangle is 275 square inches, so we can substitute these values into the formula to get:

275 = (1/2)(h)(h-3)

Simplifying the right-hand side, we get:

275 = (1/2)(h^2 - 3h)

Multiplying both sides by 2 to eliminate the fraction, we get:

550 = h^2 - 3h

Rearranging this equation to standard quadratic form, we get:

h^2 - 3h - 550 = 0

Now we can solve for h using the quadratic formula:

h = (-b ± sqrt(b^2 - 4ac)) / (2a)

In this case, a = 1, b = -3, and c = -550, so we can substitute these values into the formula to get:

h = (-(-3) ± sqrt((-3)^2 - 4(1)(-550))) / (2(1))

Simplifying the expression inside the square root, we get:

h = (3 ± sqrt(2209)) / 2

We can ignore the negative solution since height must be positive, so we get:

h = (3 + sqrt(2209)) / 2 ≈ 29.04

Now that we know the height of the triangle is approximately 29.04 inches, we can use the equation b = h - 3 to find the length of the base:

b = 29.04 - 3 = 26.04

Therefore, the base of the triangle is approximately 26.04 inches, and the height is approximately 29.04 inches.

Step-by-step explanation:

Answer:

g(y) = √(3/2 y)

Step-by-step explanation:

To find the inverse of a function, we need to solve for x in terms of y and interchange x and y. That is, we need to write the given function f(x) = 2/3x^2 in the form y = 2/3x^2 and then solve for x in terms of y.y = 2/3x^2

Multiplying both sides by 3/2, we get:

3/2 y = x^2

Taking the square root of both sides, we get:x = ± √(3/2 y)

Note that we have two possible values of x for each value of y, because the square root can be either positive or negative. However, for a function to have an inverse, it must pass the horizontal line test, which means that each value of y can only correspond to one value of x.Therefore, we need to restrict the domain of the original function to ensure that it is one-to-one. The simplest way to do this is to take the range of the function and use it as the domain of the inverse function.The range of f(x) = 2/3x^2 is all non-negative real numbers, or [0, ∞). Therefore, we can define the inverse function g(y) as:

g(y) = ± √(3/2 y)

where we choose the positive square root to ensure that the function is one-to-one.Thus, the inverse of the function f(x) = 2/3x^2 is:

g(y) = √(3/2 y)

with domain [0, ∞).

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Answer:

Step-by-step explanation:

We can find dy/dx by differentiating y with respect to x using the quotient rule.

First, we need to rewrite y using the trigonometric identity for the tangent of the difference of two angles:

y = (cos x - sin x)/(cos x + sin x) = [(cos x - sin x)/(cos x + sin x)] * [(cos x - sin x)/(cos x - sin x)]

y = (cos^2 x - 2cos x sin x + sin^2 x)/(cos^2 x - 2sin x cos x + sin^2 x)

y = (cos 2x - sin 2x)/(cos 2x + sin 2x)

Now we can apply the quotient rule:

dy/dx = [(-sin 2x - cos 2x)(cos 2x + sin 2x) - (cos 2x - sin 2x)(-sin 2x + cos 2x)]/(cos 2x + sin 2x)^2

dy/dx = (-sin^2 2x - cos^2 2x - 2sin 2x cos 2x + sin^2 2x + cos^2 2x + 2sin 2x cos 2x)/(cos 2x + sin 2x)^2

dy/dx = 0/(cos 2x + sin 2x)^2

Therefore, dy/dx = 0.

Answer:

Step-by-step explanation:

your given is not cleared repost it then post

For the computation of confidence interval for the difference in two population proportions following are negative,

p₁(cap) - p₂(cap)

Lower bound of the confidence interval

Upper bound of the confidence interval

For the computation of confidence interval,

The difference in two population proportions,

p₁ - p₂, can be negative or positive.

This implies,

The sample estimate of the difference in proportions,

p₁(cap) - p₂(cap), can also be negative or positive.

The standard error and critical value are always positive values and cannot be negative.

The lower and upper bounds of the confidence interval can be negative or positive.

Depending on the sample estimate and the margin of error.

So, both the lower and upper bounds can be negative.

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The above question is incomplete, the complete question is:

You are computing a confidence interval for the difference in 2 population proportions. which of the following could be negative?

Select all.

a. p₁

b. p₁(cap) - p₂(cap)

c. Standard error

d. Critical value

e. Lower bound of the confidence interval

f. Upper bound of the confidence interval

The likelihood that Mitsugu will have a quiz on Wednesday, Thursday, or Friday is 0.57, or 57%, based on the facts given.

To determine how probable something is to occur, use probability. Many things are difficult to forecast with absolute precision. Using it, we can only make predictions about how probable an occurrence is to happen, or its chance of happening.

Let's first examine each day's specific probabilities:

Now, all we have to do to determine the overall chance is combine the partial probabilities that were previously provided, as shown below:

0.16 + 0.21 + 0.20 = 0.57

Finally, to determine the chance as a percentage, multiply this figure by 100:

0.57 x 100 = 57%

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We can see here the values of the interior angles will be: A) m∠X = 131º, m∠Y = 16º, m∠Z = 33º.

An interior angle is an angle created between two adjacent sides of a polygon. To put it another way, it is the angle created by two polygonal sides that have a shared vertex.

Sum of interior angles of a triangle = 180°

[tex]2p + \frac{1}{4} p + \frac{1}{2} p = 180[/tex]

11p/4 = 180°

p = 720°/11

m∠X = 2p = 2 ×  720°/11 = 130.9 ≈ 131°

m∠Y = [tex]\frac{1}{4} p[/tex] = 1/4 × 720°/11 = 16.3 ≈ 16°

m∠Z = [tex]\frac{1}{2} p[/tex] = 1/2 × 720°/11 = 32.7 ≈ 33°

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For the triangle ABC, the given trigonometric ratios are -

a. sin A = 8/17

b. cos A = 15/17

c. tan A = 8/15

d. tan B = 8/15

What is trigonometric ratio?

Triangle side length ratios are known as trigonometric ratios. In trigonometry, these ratios show how the ratio of a right triangle's sides to each angle. Sine, cosine, and tangent ratios are the three fundamental trigonometric ratios.

For a right-angled triangle ABC, the hypotenuse AB is given as 17.

The base CB is given as 15 and the perpendicular AC is given as 8.

The angle C is given to be 90°.

Using the given values of the sides of the right triangle ABC, we can calculate the trigonometric ratios as follows -

a. sin A = opposite/hypotenuse = AC/AB = 8/17 (reduced fraction)

b. cos A = adjacent/hypotenuse = CB/AB = 15/17 (reduced fraction)

c. tan A = opposite/adjacent = AC/CB = 8/15 (reduced fraction)

d. tan B = opposite/adjacent = AC/CB = 8/15 (reduced fraction)

Note that since angle C is 90°, angles A and B are acute angles, so their tangent ratios are equal to each other.

Therefore, the ratios expressed as reduced fractions are -

a. sin A = 8/17

b. cos A = 15/17

c. tan A = 8/15

d. tan B = 8/15

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the sum of the series is 8/3. The series consists of reciprocals of positive integers whose only prime factors are 2s and 3s.

In other words, each term of the series can be expressed as a fraction of the form 1/n, where n is a positive integer that can be factored into only 2s and 3s. For example, the first term of the series is 1/1, the second term is 1/2, and the fourth term is 1/4.

To find the sum of the series, we can first list out the terms and their corresponding values:

1/1 = 1

1/2 = 0.5

1/3 = 0.333...

1/4 = 0.25

1/6 = 0.166...

1/8 = 0.125

1/9 = 0.111...

1/12 = 0.083...

and so on.

We can see that the terms of the series decrease in value as n increases, so we can use this fact to estimate the sum of the series. For example, we can take the sum of the first few terms to get an idea of how large the sum might be:

1 + 0.5 + 0.333... + 0.25 = 2.083...

We can see that the sum is greater than 2, but less than 3. To get a more accurate estimate, we can add a few more terms:

2.083... + 0.166... + 0.125 + 0.111... = 2.486...

We can continue adding terms in this way to get a more and more accurate estimate of the sum. However, it is not easy to find a closed-form expression for the sum of the series.

Alternatively, we can use a formula for the sum of a geometric series to find the sum of the series. A geometric series is a series of the form a + ar + ar^2 + ... + ar^n, where a is the first term and r is the common ratio between terms. In our series, the first term is 1 and the common ratio is 1/2 or 1/3, depending on whether n is even or odd. Therefore, we can split the series into two separate geometric series:

1 + 1/2 + 1/8 + 1/32 + ... = 1/(1 - 1/2) = 2

1/3 + 1/12 + 1/48 + 1/192 + ... = (1/3)/(1 - 1/2) = 2/3

The sum of the two geometric series is the sum of the original series:

2 + 2/3 = 8/3

Therefore, the sum of the series is 8/3.

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When space is spanned by the two functions of linear transformation from into itself with respect to the basis we need to apply T to each basis vector vi to get the column vectors T(vi) = [T(vi)]B.

where [T(vi)]B is the coordinate vector of T(vi) with respect to the basis B. Arrange the column vectors [T(v1)]B, [T(v2)]B, ..., [T(vn)]B into a matrix. This matrix is the matrix of T with respect to the basis B.

In this case, you have two functions that span a vector space, so you need to specify the basis B. Once you have chosen the basis, you can apply the above steps to find the matrix of the linear transformation.

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The function rule for this graph is  y = -1/2(x) + 2.

In Mathematics, the point-slope form of a straight line can be calculated by using the following mathematical expression:

y - y₁ = m(x - x₁) or [tex]y - y_1 = \frac{(y_2- y_1)}{(x_2 - x_1)}(x - x_1)[/tex]

Where:

At data point (0, 2), a linear equation in slope-intercept form for this line can be calculated by using the point-slope form as follows:

[tex]y - y_1 = \frac{(y_2- y_1)}{(x_2 - x_1)}(x - x_1)\\\\y - 2 = \frac{(0- 2)}{(4 -0)}(x -0)[/tex]

y - 2 = -1/2(x)

y = -1/2(x) + 2.

In this context, we can reasonably infer and logically deduce that an equation of the line that represents this graph in slope-intercept form is y = -1/2(x) + 2.

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Answer:

Step-by-step explanation:

t+t+t= 3t

3t = 12

12/3=t

4=t

Step-by-step explanation:

that height splits GK (32) into 2 parts :

8 and 32-8 = 24

then we use the geometric mean theorem for right-angled triangles

height = sqrt(p×q)

with p and q being the parts of the Hypotenuse.

so,

height = sqrt(8×24) = sqrt(192)

and now we can use Pythagoras

c² = a² + b²

with c being the Hypotenuse (the side opposite of the 90° angle), a and b are the legs,

to get HK.

HK² = height² + 24² = 192 + 576 = 768

HK = sqrt(768)

In linear equation, 65.625% of the pupils travel by bus.

What is  linear equation?

A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept. The variables in the previous sentence, y and x, are referred to as a "linear equation with two variables" at times.

A) 1800 * 0.55 * 0.3 = 297 Girls.

B)  1800 * 0.45 * 0.7 = 567 boys

C)  Girl

      297/864 * 100%  = 34.375%

  boy -

       567 ÷ (297 + 567 ) * 100%  = 65.625%

         864 = 297 + 567

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The properties of a probability distribution for a discrete random variable ensure that the probabilities assigned to each possible value of the variable are consistent with the axioms of probability and allow for meaningful inference and prediction.

The properties of a probability distribution for a discrete random variable are.

The probability of each possible value of the random variable must be non-negative.

The sum of the probabilities of all possible values must equal 1.

The probability of any event A is the sum of the probabilities of the values in the sample space that correspond to A.

These properties are satisfied because the probabilities of each possible value of a discrete random variable are defined in such a way that they are non-negative and sum to 1. Additionally, any event A can be expressed as a collection of possible values of the random variable, and the probability of A is then computed as the sum of the probabilities of those values.

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Answer:

x = 21

Step-by-step explanation:

Given the angle measurements of two angles, we can deduce that this is a 30-60-90 triangle. The formula for side lengths of this type of triangle is given by s → s√3 → 2s with s being the shortest side (the side opposite the 30° angle).

We are looking for the side length represented by s[tex]\sqrt{3}[/tex]. We can see that the hypotenuse is represented by 2s, so...

2s = 24

s = 12

The side length of the shortest side is 12. Therefore, we can say that

x = s  [tex]\sqrt{3}[/tex]

x = 12 * [tex]\sqrt{3}[/tex]

x = 20.785

The question asks for our answer to the nearest whole number, so

x = 21

We cannot determine the actual value of the average rate of change without knowing the function f(x) or having a graph of the function.

The visual representation of mathematical functions or data points on a Cartesian coordinate system is an x-y axis graphic. The vertical or dependent variable is represented by the y-axis, while the horizontal or independent variable is represented by the x-axis. The difference between the change in output values and the change in input values is known as the average rate of change of a function over a period.

Let's assume that the function is denoted by f(x). Then, the average rate of change on the interval (a, b) can be calculated as

average rate of change = (f(b) - f(a)) / (b - a)

Using this formula, we can calculate the average rate of change on the given intervals as follows:

For the interval (-3, -2):

average rate of change = [tex]\frac{[f(-2) - f(-3)]}{[-2 - (-3)]}[/tex]

For the interval (1, 3):

average rate of change = [tex]\frac{(f(3) - f(1))}{(3 - 1)}[/tex]

For the interval (-1, 1):

average rate of change = [tex]\frac{(f(1) - f(-1))}{ (1 - (-1))}[/tex]

Note that we cannot determine the actual value of the average rate of change without knowing the function f(x) or having a graph of the function. If you provide the function or the graph, I can help you find the actual values of the average rate of change on these intervals.

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To satisfy the inequality x less-than StartFraction 7 over 12 EndFraction.

Inequalities are called as the mathematical expressions in which both sides are nonequal. Unlike to equations, we compare two values in inequality. Less than (or less than or equal to), greater than (or greater than or equal to), or not equal to signs can be used in place of the equal sign in between.

The inequality is 1/4 + x < 5/6 in order to solve this inequality we need to isolate the value of x, that is our variable of interest. This is shown bellow:

1/4 + x < 5/6

x < 5/6 - 1/4

LMC is used to subtract the fractions we have as follows:

x < (2*5 - 3*1)/12

x < (10 - 3)/12

x< 7/12

The inequality must be satisfied for x to be smaller than 7/12.

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Answer:  x < 7/12

Step-by-step explanation:

Answer:

0.048 in %

Step-by-step explanation:

firstly: remove the decimal point

= 48/1000

secondly : Simplify

48/1000*100

=48/10

=4.8%

Answer:

We need to find dz/dt given:

z = cos(x + 8y), x = 7t^5, y = 5/t

Using the chain rule, we can find dz/dt by taking the derivative of z with respect to x and y, and then multiplying by the derivatives of x and y with respect to t:

dz/dt = dz/dx * dx/dt + dz/dy * dy/dt

First, let's find dz/dx and dz/dy:

dz/dx = -sin(x + 8y)

dz/dy = -8sin(x + 8y)

Now, let's find dx/dt and dy/dt:

dx/dt = 35t^4

dy/dt = -5/t^2

Substituting these values, we get:

dz/dt = (-sin(x + 8y)) * (35t^4) + (-8sin(x + 8y)) * (-5/t^2)

Simplifying this expression, we get:

dz/dt = -35t^4sin(x + 8y) + 40sin(x + 8y)/t^2

Substituting x and y, we get:

dz/dt = -35t^4sin(7t^5 + 40/t) + 40sin(7t^5 + 40/t)/t^2

Therefore, dz/dt is given by -35t^4sin(7t^5 + 40/t) + 40sin(7t^5 + 40/t)/t^2.

The graph that shows the electricity usage on a record-breaking summer day is Sacramento, California is a function.

The domain is 24 hours of a day.

The number of megawatts used at 8 am is 1, 200 megawatts.

The time with the most electricity used was 4 pm to 6 pm and least used was 4 am.

f ( 12 ) would be 1, 900 megawatts.

Usage is increasing from 4 am to 5 pm and decreasing from 5 pm to 4 am.

The graph is a function because each point on the graph represents a distinct megawatt usage. The domain would be 24 hours of a day as this graph of electricity usage shows the usage per day.

The megawatts used at 8 am is:

= 1, 300 - ( 200 / 2 )

= 1, 200 megawatts

From 4 am to 5 pm, we see that electricity usage is increasing as people are getting ready for work and going to work, but from 5 pm to 4 am, electricity usage decreases.

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Therefore, the slope of the line that passes through (-2, 8) and (4, 6) is -1/3.

What is the slope?

In mathematics, the slope is a measure of the steepness of a line.

The solution provided involves three steps to find the slope of the line that passes through two points: (-2, 8) and (4, 6).

Step 1 involves finding the change in y-coordinates, which is the difference between the y-coordinate of the second point and the y-coordinate of the first point. In this case, the second point has a y-coordinate of 6 and the first point has a y-coordinate of 8.

Therefore, the change in y-coordinates is 6 - 8 = -2.

Step 2 involves finding the change in x-coordinates, which is the difference between the x-coordinate of the second point and the x-coordinate of the first point. In this case, the second point has an x-coordinate of 4 and the first point has an x-coordinate of -2.

Therefore, the change in x-coordinates is 4 - (-2) = 6.

Step 3 involves dividing the change in y-coordinates by the change in x-coordinates to find the slope of the line. In this case, the change in y-coordinates is -2 and the change in x-coordinates is 6, so the slope is -2/6 or -1/3.

Since all the steps are correct and properly executed, there is no error.

Therefore, the slope of the line that passes through (-2, 8) and (4, 6) is -1/3.

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Answer:

Step-by-step explanation:

We'Re looking at a normal distribution here- let's start by drawing it out to the mean me- is 315 grams standard. Deviation is 16 point. We want to know the weight corresponding to each of these events and we can use either the appendix which i assume is a z, school table or excel so the first 1. We want the highest 20 percent up here somewhere. This is what we would call the 80 percent. It separates the bottom 80 percent from the highest 20 percent. So how do we work the well? We need to start by getting the z score for it. So how do we get that, while in excel you're going to use the norm inverse function which looks like this? So it's calls norm and then in here you just put in x, where x is your percentage and that will spit out the z score and i'm using this rather than a table, because it will give me a more exact value. So we're going to do that, and so here the percent is the 80, so it's not .8 and that spits out the z score of nort .8416 to 4 decimal places. But i'm always going to this exact score, because we now have to turn it from a z score to a piece of real data. Z score is a measure of how many standard deviations away from the mean a value is so we're. Looking for the value not .84 standard deviations, above the mean or if we write it like this x, is equal to z, sigma plus mu. So here we take our exact zedscore, because we can still just use excels, multiply it by 16 and add it on to the mean and we'll get our value of 328.447328 .47 grams to 2 decimal places for part b. We want to be middle 60 percent. Now we need to cut off points and within here where, in this interval we have 60 percent of values, which means we have 40 percent of values, not in here. So we can label our 3 sections. These 2 add up to 40 percent, so they have to be 20 percent. Each are called nor .2 and not .2, and then this middle bit here is nor .6 for the total of 100 percent point. So we need these cut off points when you, the z, scores first, and because these are equal distances away from a mine they're going to have the same z score. Just 1 is going to be positive or negative, so z is going to be equal to plus and minus. Let'S look at the lower 1. This is the 20 percent here so into this excel command. We put 20 percent nor .2 and out of it we get the z score of minus, not .8416. So it's actually very similar to the top question, because the top question asked you for the 80 percent be 80 percent. Is the upper cut off point here? 20? Is below cutoff point, so we already have the up 1, let's just calculate below 1, so we've got to be minus, nor .8416 multiplied by 16, but the standard deviation and on to the mean- and we get 301.53 and the upper cut off point is from part A so that's the middle 60 percent makes more space a part c. We want to be highest 80 percent. So now what we want is the cut off between the lowest 20 and the highest 80, which we've just got from part c part b. It'S this lower 1, here, 301.53 grams. That'S an easy! 1! Now we want the lowest 15 percent, so the lowest 15 percent is the 15 percent. So we go to our exylgamant put in the 15 for percent, so that would be no .15 and it's a z score of minus 1.036 keeping the exact value put into this formula. We multiply our z by 16, as on 315, to get 298.42 grams.

The weight that corresponds to this event are approximately 344.03 grams and 375.97 grams.

To find the weight that corresponds to each event, we need to use the standard normal distribution, which has a mean of 0 and a standard deviation of 1. We can convert the given mean and standard deviation to z-scores using the formula:

z = (x - μ) / σ

where x is the weight we want to find, μ is the mean (360 grams), and σ is the standard deviation (9 grams).

Then, we can use a standard normal distribution table or calculator to find the probability of each event, and convert it back to a weight using the inverse of the z-score formula:

x = μ + z * σ

where z is the z-score that corresponds to the desired probability.

Event 1: The weight is less than 345 grams.

z = (345 - 360) / 9 = -1.67

Using a standard normal distribution table or calculator, we find that the probability of a z-score less than -1.67 is approximately 0.0475.

x = 360 + (-1.67) * 9 = 344.03 grams

Therefore, the weight that corresponds to this event is approximately 344.03 grams.

Event 2: The weight is between 355 and 365 grams.

First, we need to find the z-scores that correspond to the two boundaries:

z1 = (355 - 360) / 9 = -0.56

z2 = (365 - 360) / 9 = 0.56

Using a standard normal distribution table or calculator, we find that the probability of a z-score less than -0.56 is approximately 0.2123, and the probability of a z-score less than 0.56 is approximately 0.7123. Therefore, the probability of a z-score between -0.56 and 0.56 is:

0.7123 - 0.2123 = 0.5

x1 = 360 + (-0.56) * 9 = 355.16 grams

x2 = 360 + (0.56) * 9 = 364.84 grams

Therefore, the weight that corresponds to this event is any weight between 355.16 and 364.84 grams.

Event 3: The weight is greater than 375 grams.

z = (375 - 360) / 9 = 1.67

Using a standard normal distribution table or calculator, we find that the probability of a z-score greater than 1.67 is approximately 0.0475.

x = 360 + (1.67) * 9 = 375.97 grams

Therefore, the weight that corresponds to this event is approximately 375.97 grams.

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Answer:

this is when you want to draw a sketch of a building

If △ABC is congruent to △DEF, then it must also be a right triangle. Thus, the two triangles have congruent sides and are congruent.

A key idea in geometry known as the Pythagorean theorem explains the relationship between the sides of a right triangle. The square of the hypotenuse, or side opposite the right angle, is said to be equal to the sum of the squares of the other two sides. It can be expressed mathematically as: a² + b² = c²

given

By defining a right triangle, DEF, with sides a, b, and x, we can demonstrate the opposite of the Pythagorean theorem. Then, we'll demonstrate that if ABC is a triangle with sides a, b, and c where a2 + b2 = c2, it is congruent to DEF and is thus a right triangle because a2 + b2 = c2.

By the Pythagorean theorem, because △DEF is a right triangle, a² + b² = x².

When a2 + b2 = c2 and a2 + b2 = x2, c2 equals x2.

If △ABC is congruent to △DEF, then it must also be a right triangle.Thus, the two triangles have congruent sides and are congruent.

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If △ABC is congruent to △DEF, then it must also be a right triangle. Thus, the two triangles have congruent sides and are congruent.

A key idea in geometry known as the Pythagorean theorem explains the relationship between the sides of a right triangle. The square of the hypotenuse, or side opposite the right angle, is said to be equal to the sum of the squares of the other two sides. It can be expressed mathematically as: a² + b² = c²

By defining a right triangle, DEF, with sides a, b, and x, we can demonstrate the opposite of the Pythagorean theorem. Then, we'll demonstrate that if ABC is a triangle with sides a, b, and c where [tex]a^2 + b^2 = c^2[/tex], it is congruent to DEF and is thus a right triangle because a2 + b2 = c2.

By the Pythagorean theorem, because △DEF is a right triangle, a² + b² = x².

When[tex]a^2 + b^2 = c^2[/tex] and [tex]a^2 + b^2 = x^2[/tex], [tex]c^2[/tex] equals [tex]x^2[/tex].

If △ABC is congruent to △DEF, then it must also be a right triangle. Thus, the two triangles have congruent sides and are congruent.

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Answer:

Math Quotient Verification

G For each ordered pair, determine whether it is a solution to the system of equations. 9x+2y=-5 2x-3y=-8 (x, y) (1, -7) (0, -4) (5,6) (-1,2) Is it a solution? Yes No X 5

To check if an ordered pair is a solution to a system of equations, we substitute the values of x and y into both equations and see if both equations are satisfied.

Let's check each ordered pair one by one:

(1, -7):

9x + 2y = -5 becomes 9(1) + 2(-7) = -5, which is false.

2x - 3y = -8 becomes 2(1) - 3(-7) = -8, which is true.

Therefore, (1, -7) is not a solution to the system of equations.

(0, -4):

9x + 2y = -5 becomes 9(0) + 2(-4) = -8, which is false.

2x - 3y = -8 becomes 2(0) - 3(-4) = 12, which is false.

Therefore, (0, -4) is not a solution to the system of equations.

(5, 6):

9x + 2y = -5 becomes 9(5) + 2(6) = 41, which is false.

2x - 3y = -8 becomes 2(5) - 3(6) = -8, which is true.

Therefore, (5, 6) is not a solution to the system of equations.

(-1, 2):

9x + 2y = -5 becomes 9(-1) + 2(2) = -11, which is false.

2x - 3y = -8 becomes 2(-1) - 3(2) = -8, which is true.

Therefore, (-1, 2) is not a solution to the system of equations.

Therefore, the answer is "No" for all the ordered pairs given in the problem.

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Use the vector law of addition to get the resulting vector is [tex](7i+2j-k)[/tex] and magnitude of resulting vector is [tex]3\sqrt{6}[/tex].

What is the resulting vector and its magnitude?

Vector addition can be defined as the sum of two or more vectors of corresponding components.

It is given that,

[tex]\vec{r}= < 4, -2 > \\\vec{s}= < 3,4,-1 >[/tex]

Given vectors can also be written as,

[tex]\vec{r}=4i-2j\\\vec{s}=3i+4j-k[/tex]

Add above vectors as follows:

[tex]\vec{r}+\vec{s}=(4i-2j)+(3i+4j-k)\\\\=(7i+2j-k)[/tex]

Therefore,

[tex]\vec{r}+\vec{s}= < 7,2,-1 >[/tex]

Show the resulting vector as follows:

Now calculate the magnitude of the vector [tex](\vec{r}+\vec{s})[/tex]

[tex]|\vec{r}+\vec{s}|=\sqrt{(7)^2+(2)^2+(-1)^2}\\=\sqrt{49+4+1}\\=\sqrt{54}\\=3\sqrt{6}[/tex]

Hence the resulting vector is [tex](7i+2j-k)[/tex] and magnitude of resulting vector is [tex]3\sqrt{6}[/tex].

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Therefore, the scale factor of the dilation of the shape is 2.

A scale factor is a ratio that describes the proportional relationship between two similar figures. It represents how much larger or smaller one figure is compared to the other, and it is calculated by dividing a corresponding measurement (such as side length, perimeter, or area) of the larger figure by the corresponding measurement of the smaller figure. Scale factor is used in mathematics, particularly in geometry and measurement, to describe the transformation of one figure into another through dilation or resizing. It is represented by a number or a ratio, such as 2:1 or 1/2, which indicates how many times larger or smaller the new figure is compared to the original.

Here,

In the given picture, it appears that the distance from the center of dilation (the origin) to the pre-image (the original figure) is 4 units, and the distance from the center of dilation to the image (the transformed figure) is 8 units. The scale factor of the dilation is equal to the ratio of the distance from the center of dilation to the image and the distance from the center of dilation to the pre-image.

So, the scale factor of the dilation is:

8 units ÷ 4 units = 2

Therefore, the scale factor of the dilation is 2.

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The rental cost for each movie and each video game is $3.5 and $5.5 respectively.

Let

cost of each movie = x

Cost of each video game = y

5x + 3y = 34

2x + 12y = 73

Multiply (1) by 4

20x + 12y = 136

2x + 12y = 73

subtract the equations to eliminate y

18x = 63

divide both sides by 18

x = 63/18

x = 3.5

Substitute x = 3.5 into (1)

5x + 3y = 34

5(3.5) + 3y = 34

17.5 + 3y = 34

3y = 34 - 17.5

3y = 16.5

y = 16.5/3

y = 5.5

Therefore, $3.5 and $5.5 is the rental cost of each movie and video game respectively.

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