If y= cos x - sin x /cos x + sin x then dy /dx is :​

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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Answer: The poster should have width 17.50 CM

Step-by-step explanation:

Given that boasted I have a total area 245 cm square area of a poster is 200 and 45 20 m square. And it is in the rectangle format. So we know that the poster is always in the rectangle format and the area of rectangular area is equal to L N T W. So 245 will be equal to Ln tW. From this. We need to find L. So L is equal to 245 divided by W. Now consider the diplomatic representation of the poster. So it has mentioned that there is a margin around the edges of six cm at the top. This is 6cm and four cm at the sites. All the four sides 3 sides are four cm. So from this we need to find land and the doctor posted area that is printed area. The first wine printed with www. Z. Quilter. Now let this be total birth will be W. And this will be four and this will be four. Therefore posted with will be we need to find this part alone. So W -4 -4 will give the this part with. So W -4 -4. So post printed with PW will be equal to W -8. Similarly printed lunch will be equal to The total length is already we have found 245 by W. And we need to find this part length. So we have to subtract six and four from the total length so that that will give them this part length, So -6 -4. So printed length will be equal there 245 Divided by W -10. And we know that formula for area of a rectangle. Urz D is equal to L W. Now substitute the printed with and printed length in the area formula. We have to find the printed area. So Printed area Zeke Walter W -8 into 245 Divided by W -10. To simplify this, we get 325 minus 10. W -100 and 30,960 W. to the power of -1. Now fine D A by D. T. Not D D. This is D. W. So this is equal to differentiation of constant alma zero, this is minus 10 plus 900 and 60 W. to the power of -2 and equate this to be equal to zero. We out to find the maximum width so D A by D W is equal to zero, therefore minus 10 1960 W. To the power of -2 will be equal to zero. To simplify this, we get them maximum with value the W is equal term I wrote off 196. Therefore the value of W. Z quilter plus or minus 14. We will neglect the negative values since we cannot be negative. So one we assume the positive values. So what will be equal to 14 and length will be equal to 245 divided by 14, So which will be equal to 17.50 centimeters. And they have concluded that At W is equal to 14 cm and lunch will be equal to 17.50 cm needed to print the largest area. I hope you found the answer to school. Thank you.

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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Answer:

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

For the first question, the equation for the line is y = -x + 11. This comes from the fact that the slope for a line perpendicular to the line x = 5 is -1. From there, we can use the point (5,6) to calculate the y-intercept, which is 11.

For the second question, the equation for the line is y = 7x. This comes from the fact that the slope for a line parallel to the line 7x - y = -7 is 7. Since the point (0,0) is already on the line, the equation is already solved.

For the third question, the equation for the line is x = 8. This comes from the fact that the slope for a line with an undefined slope is 0. Since the point (8,2) is already on the line, the equation is already solved.

Answer:

x = 2

y = -3/4

Step-by-step explanation:

1. Substitute y=1/8x -1 in −5x+4y=−13

-5x+4(1/8x -1) = -13

2. Solve for x

-5x + 4/8x - 4 = -13

-9/2x - 4 = -13

-9/2x = -9

x = 2

3. Now that you know x = 2, plug it into y=1/8x - 1 to find what y is.

y= 1/8(2) - 1

y= 2/8 - 1

y= -3/4

Answer:

E.  x² + (y - 3)² = 16

Step-by-step explanation:

The equation of a circle in the standard (x, y) coordinate plane with center (h, k) and radius r is given by:

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

To find the equation of the circle with a radius of 4 units and the same center as the circle determined by x² + y² - 6y + 4 = 0, we need to first write the equation of the second circle in the standard form.

We can complete the square for y to rewrite this equation in standard form. To do this move the constant to the right side of the equation:

[tex]\implies x^2 + y^2 - 6y + 4 = 0[/tex]

[tex]\implies x^2 + y^2 - 6y = -4[/tex]

Add the square of half the coefficient of the term in y to both sides of the equation:

[tex]\implies x^2 + y^2 - 6y +\left(\dfrac{-6}{2}\right)^2= -4+\left(\dfrac{-6}{2}\right)^2[/tex]

[tex]\implies x^2 + y^2 - 6y +9= -4+9[/tex]

[tex]\implies x^2 + y^2 - 6y +9=5[/tex]

Factor the perfect square trinomial in y:

[tex]\implies x^2+(y-3)^2=5[/tex]

[tex]\implies (x-0)^2 + (y-3)^2=5[/tex]

So the center of this circle is (0, 3) and its radius is √5 units.

Since the new circle has the same center, its center is also (0, 3).

We know its radius is 4 units, so we can write the equation of the new circle as:

[tex]\implies (x - 0)^2 + (y - 3)^2 = 4^2[/tex]

[tex]\implies x^2 + (y - 3)^2 = 16[/tex]

Therefore, the equation of the circle in the standard (x, y) coordinate plane with a radius of 4 units and the same center as the circle determined by x² + y² - 6y + 4 = 0 is x² + (y - 3)² = 16.

To find:-

Answer:-

The given equation of the circle is ,

[tex]\implies x^2+y^2-6y + 4 = 0 \\[/tex]

Firstly complete the square for y in LHS of the equation as ,

[tex]\implies x^2 + y^2 -2(3)y + 4 = 0 \\[/tex]

Add and subtract 3² ,

[tex]\implies x^2 +\{ y^2 - 2(3)(y) + 3^2 \} -3^2 + 4 = 0 \\[/tex]

The term inside the curly brackets is in the form of a²-2ab+b² , which is the whole square of "a-b" . So we may rewrite it as ,

[tex]\implies x^2 + (y-3)^2 -9 + 4 = 0 \\[/tex]

[tex]\implies x^2 + (y-3)^2 - 5 = 0 \\[/tex]

[tex]\implies x^2 + (y-3)^2 = 5\\[/tex]

can be further rewritten as,

[tex]\implies (x-0)^2 + (y-3)^2 = \sqrt5^2\\[/tex]

now recall the standard equation of circle which is ,

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

where,

So on comparing to the standard form, we have;

[tex]\implies \rm{Centre} = (0,3)\\[/tex]

Now we are given that the radius of second circle is 4units . On substituting the respective values, again in the standard equation of circle, we get;

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

[tex]\implies (x-0)^2 + (y-3)^2 = 4^2 \\[/tex]

[tex]\implies \underline{\underline{\red{ x^2 + (y-3)^2 = 16}}}\\[/tex]

and we are done!

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

Answer:

The unit circle has many different angles that each have a corresponding point on the circle. The coordinates of each point give us a way to find the tangent of each angle. The tangent of an angle is equal to the y-coordinate divided by the x-coordinate.

The inequality is X > 0 and a word sentence represent the graph is  X  the graph of a number line with an open circle on zero and an arrow pointing to the right.

The inequality X > 0 represents the graph of a number line with an open circle on zero to left and an arrow pointing to the right. This means that any value of X that is greater than zero is a valid solution for the inequality.

In other words, X can be any positive number, such as 1, 2, 3, and so on. However, X cannot be zero or any negative number, as those values do not satisfy the inequality. Therefore, the word sentence that represents this inequality is "X is greater than zero."

This means that X must be a positive number, and it can be any value that is greater than zero.

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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:

D

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 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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there is an 85% chance that the two cards drawn at random from an ordinary deck of 52 cards will not be sixes.

The probability of drawing a card from an ordinary deck without replacement can be determined using the concept of conditional probability. Conditional probability is the probability of an event occurring, assuming that another event has already occurred.

In order to calculate the probability that the two cards drawn are not sixes, we can use the formula:

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

Where A and B represent two independent events, P(A) is the probability of event A occurring, and P(B|A) is the conditional probability of event B occurring given that event A has already occurred.

The probability of drawing the first card that is not a six is:

P(A) = 48/52 = 0.9231

The probability of drawing the second card that is not a six, given that the first card drawn was not a six, is:

P(B|A) = 47/51 = 0.9216

Therefore, the probability of drawing two cards at random from an ordinary deck of 52 cards and having neither of them be a six is:

P(A and B) = P(A) x P(B|A) = 0.9231 x 0.9216 = 0.8503 or approximately 85%.

This means that there is an 85% chance that the two cards drawn at random from an ordinary deck of 52 cards will not be sixes.

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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:

1. x = 11

Step-by-step explanation:

1. x + 7 = 18

move 7 to right then change the sign
        x = 18 - 7
        x = 11
2. x - 13 = 15

move -13 to right then change the sign
        x = 15 + 13
        x = 28
3. 8x = 64
   8      8
divided by 8 both side
        x = 8
4. 5x-13-12=0 it this the correct given?
add same variable
   5x = 13 + 12
   5x = 25
   5      5

divided by 5 both side
         x = 5

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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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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Answer: $27.98

Step-by-step explanation:

22.00 × .2= 4.40

22 + 4.40 = 26.40

26.40 × .06 = 1.584

26.40 + 1.584 = 27.984

Round to the nearest hundred so the total paid by the customer would be 27.98

Answer:

23.8 cm

Step-by-step explanation:

17 * 140% = 17 * 1.4 = 23.8 cm

The image of the house would be 23.8 cm tall on the screen.

To calculate the height of the image of the house on the screen, we can use the given zoom factor of 140%.

The zoom factor of 140% means that the images on the screen are 140% as long and 140% as wide compared to when they are printed on a sheet of paper.

To calculate the height of the image on the screen, we need to multiply the printed height by the zoom factor (140% or 1.4).

Height on the screen = Printed height * Zoom factor

Height on the screen = 17 cm * 1.4

Height on the screen = 23.8 cm

Therefore, the image of the house would be 23.8 cm tall on the screen.

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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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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:

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)

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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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:

Step-by-step explanation:

t+t+t= 3t

3t = 12

12/3=t

4=t

(a) The interval (-∞, ∞).

(b) The interval (-∞, ∞).

(c) The interval (-∞, ∞).

(d) The interval (-π/2, π/2) \ {0}.

(a) The longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution is the interval where the coefficient function, 3t, is continuous and bounded. Since 3t is a continuous and bounded function for all t in the interval (-∞, ∞), the given initial value problem is certain to have a unique twice-differentiable solution for all t in (-∞, ∞).

(b) The longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution is the interval where the coefficient functions, t(t - 4), 3t, and 4, are continuous and bounded. Since t(t - 4), 3t, and 4 are continuous and bounded functions for all t in the interval (-∞, ∞), the given initial value problem is certain to have a unique twice-differentiable solution for all t in (-∞, ∞).

(c) The longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution is the interval where the coefficient functions, cost and In|t|, are continuous and bounded. Since cost and In|t| are continuous and bounded functions for all t in the interval (-∞, ∞), the given initial value problem is certain to have a unique twice-differentiable solution for all t in (-∞, ∞).

(d) The longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution is the interval where the coefficient functions, x - 2, 1, and (x - 2)tanx, are continuous and bounded. Since x - 2, 1, and (x - 2)tanx are continuous and bounded functions for all x in the interval (-π/2, π/2) \ {0} , the given initial value problem is certain to have a unique twice-differentiable solution for all x in (-π/2, π/2) \ {0}.

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The given question is incomplete, the complete question is:

determine the longest interval in which the given initial value problem is certain to have a unique twice- differentiable solution. Do not attempt to find the solution. (a) ty" + 3y = 1, y(1) = 1, y'(1) = 2   (b)   t(t – 4)y" + 3ty' + 4y = 2, y(3) = 0, y'(3) = -1 (c)   y" + (cost)y' + 3( In |t|) y = 0, y(2) = 3, y'(2) = 1 (d) (x - 2)y"+y' +(x - 2)(tan x) y = 0, y(3) = 1, y'(3) = 2