Drag the expressions into the boxes to correctly complete the table, 25 points

The probability that the next 4 science fair winners will all be seventh grade students is approximately 0.316

Assuming that the probability of winning the science fair is independent of grade level, the probability of a seventh grade student winning the science fair is 12/16 or 3/4, and the probability of a student of any other grade level winning is 1/4.

If Mona placed 3 red marbles (representing seventh grade winners) and 1 green marble (representing winners from other grade levels) in the bag, then the probability of drawing a red marble (representing a seventh grade winner) is 3/4, and the probability of drawing a green marble (representing a winner from another grade level) is 1/4.

Since the events of each science fair winner are independent, we can apply the multiplication rule of probability to determine the probability that the next 4 winners will all be seventh grade students

P(4 seventh grade winners) = (3/4)^4 = 81/256 or approximately 0.316.

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According to the question, it takes Mina 80 minutes to commute to work.

Two equations are considered to be comparable when their roots and solutions line up. To create an equivalent equation, the identical quantity, symbol, or expression must always be added to or removed from both of the equation's two sides. We can also create a similar equation simply multiplying or dividing either sides of the an equation by a nonnegative value.

Let's denote the time it takes Mina to commute to work by "m" (in minutes).

The issue states that Katie's train trip requires 10 minutes or more half as much time as Mina's drive. Instead, we might write:

Katie's commute time = (1/2) * Mina's commute time + 10

We also know that it takes Katie 30 minutes to get to work, so we can write:

Katie's commute time + 30 minutes = total time to get to work

The result of putting the very first equation into to the second equation is:

[(1/2) * Mina's commute time + 10] + 30 minutes = total time to get to work

Simplifying the equation, we get:

(1/2) * Mina's commute time + 40 minutes = total time to get to work

Now we can set this equation equal to "m" to solve for Mina's commute time:

(1/2) * m + 40 = m

Subtracting (1/2) * m from both sides, we get:

40 = (1/2) * m

Multiplying both sides by 2, we get:

m = 80

Therefore, it takes Mina 80 minutes to commute to work.

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According to the question the area of the triangle with sides of 10, 9, and 6 is 45.

Area is a term used to describe the size of a two-dimensional space. It is a measure of how much surface is contained within a two-dimensional boundary. It is usually expressed in units of square meters, square kilometers, square feet, or square miles. Area can also be used to describe the size of a three-dimensional space, such as for a volume. In this case, area is expressed in units of cubic meters, cubic feet, or cubic miles. Area is an important concept in mathematics and is used to calculate the area of shapes, such as triangles and circles, as well as to solve problems related to probability, motion, and other topics. Area is also used in physics to calculate the amount of force that an object exerts on another object.

The area of a triangle can be calculated using the formula A = 1/2 * b * h, where b is the base of the triangle and h is the height of the triangle.
Using this formula, the area of a triangle with a base of 10 and a height of 9 would be A = 1/2 * 10 * 9 = 45.
Therefore, the area of the triangle with sides of 10, 9, and 6 is 45.

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The area of the triangle with sides of lengths 10, 9, and 6 is approximately 37 square units.

What is the Pythagorean theorem?

In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

To find the area of a triangle, we can use the formula:

Area = 1/2 * base * height

In this case, we need to identify the base and height of the triangle.

The longest side of the triangle is 10, which is opposite to the largest angle, so we can take that as the base. Let's call it b.

To find the height, we need to draw an altitude from the vertex opposite the base. Let's call the height h.

Now, we can use the Pythagorean theorem to find the height:

h^2 = 10^2 - (9^2 + 6^2)/2^2

h^2 = 100 - 45.25

h^2 = 54.75

h = sqrt(54.75)

h = 7.4 (rounded to one decimal place)

Now that we have the base and height, we can use the formula to find the area:

Area = 1/2 * base * height

Area = 1/2 * 10 * 7.4

Area = 37 square units (rounded to the nearest whole number)

Therefore, the area of the triangle with sides of length 10, 9, and 6 is approximately 37 square units.

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Answer:

Step-by-step explanation:

I’m not good at explaining but basically if you do 20n - 15n it would be 5n

After computing the composition of f(g(x)) and g(f(x)), it is clear that f(x) and g(x) are not inverse functions of each other.

To determine whether f(x) and g(x) are inverse functions of each other, we need to check whether the composition of the two functions f(g(x)) and g(f(x)) result in the identity function, which is equal to x.

First, let's find f(g(x)):

f(g(x)) = f(5x - 5/4) (substituting g(x) into f(x))

f(g(x)) = 4/5(5x - 5/4) + 1 (substituting the expression for f(x))

f(g(x)) = 4x - 1 + 1

f(g(x)) = 4x

Now let's find g(f(x)):

g(f(x)) = g(4/5x + 1) (substituting f(x) into g(x))

g(f(x)) = 5(4/5x + 1) - 5/4 (substituting the expression for g(x))

g(f(x)) = 4x + 5 - 5/4

g(f(x)) = 4x + 20/4 - 5/4

g(f(x)) = 4x + 15/4

Since f(g(x)) = 4x and g(f(x)) = 4x + 15/4, we can see that the two compositions are not equal to x, which means that f(x) and g(x) are not inverse functions of each other.

Therefore, we can conclude that f(x) and g(x) are not inverse functions of each other.

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The volume of the cone with radius as 9 inches and height as 11 inches is 933 cubic inches.

The formula for the volume of a cone is the product of the following multiplication, given as V = π×r²×h/3.

In this formula, π is the pie or 22/7, r is the radius, and h is the height.

The radius of the base of cone, r = 9 inches

The height of the cone, h = 11 inches

π = 22/7

The volume of the cone = πr²h/3 cubic units.

= 22/7 x 9 x 9 x 11/3

= 22/7 x 81 x 3.667

= 933 cubic inches

= 933 in³

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The interval estimate for the population mean, μ, is 65 ± 3.38 mph, or 61.62 to 68.38 mph.

The z-value for a 96.6% confidence level is 2.58. This value is used to determine the margin of error for a sample mean in a population. The formula for margin of error is ME = z*σ/√n, where σ is the population standard deviation, n is the sample size, and z is the z-value for the desired confidence level. In this case, ME = 2.58*22/√121 = 3.38 mph.Therefore, the interval estimate for the population mean, μ, is 65 ± 3.38 mph, or 61.62 to 68.38 mph.

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Answer:

To subtract −5x² − 9x − 5 from 4x² − 5x + 3, we need to subtract the coefficients of each term with the same degree.

(4x² - 5x + 3) - (-5x² - 9x - 5)

4x² - 5x + 3 + 5x² + 9x + 5

9x² + 4x + 8

To subtract -5x² - 9x - 5 from 4x² - 5x + 3, you subtract each corresponding term, which results in 9x² + 4x + 8.

If you would like to subtract -5x² - 9x - 5 from 4x² - 5x + 3, you need to subtract every term separately. So, the steps will look like the following:

So, 4x² - 5x + 3 - (-5x² - 9x - 5) = 9x² + 4x + 8.

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Answer:

Step-by-step explanation:

We can solve the equation 4x+2 = 12 for x using the change of base formula by isolating x on one side of the equation:

4x + 2 = 12

Subtracting 2 from both sides:

4x = 10

Dividing both sides by 4:

x = 2.5

We do not need to use the change of base formula to solve this equation as it is a simple linear equation that can be solved by basic algebraic manipulations. Therefore, none of the given options are correct.

Check the picture below.

so the cars are 276 miles apart, let's say the southbound car is going "r" mph fast, that means the northbound car is going "r + 12" mph fast as you see in the picture, so at some time they meet, hmmm wait, we know they're 3 hours apart from passing each other, so that means by the time they meet, 3 hours have passed, so by the time that happens, the northbound car has been traveling for 3 hours and the southbound car has been traveling for 3 hours as well.

Since we know they're 276 miles apart, let's say by the time they meet the southbound car has covered "d" miles, so the northbound car has covered the slack then, that is "276 - d" miles.

[tex]{\Large \begin{array}{llll} \underset{distance}{d}=\underset{rate}{r} \stackrel{time}{t} \end{array}} \\\\[-0.35em] ~\dotfill\\\\ \begin{array}{lcccl} &\stackrel{miles}{distance}&\stackrel{mph}{rate}&\stackrel{hours}{time}\\ \cline{2-4}&\\ Southbound&d&r&3\\ Northbound&276-d&r+12&3 \end{array}\hspace{5em} \begin{cases} d=(r)(3)\\\\ 276-d=(r+12)(3) \end{cases} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{\textit{using the 1st equation}}{d=3r}\hspace{5em}\stackrel{\textit{substituting on the 2nd equation}}{276-(3r)~~ = ~~3(r+12)} \\\\\\ 276-3r=3r+36\implies 240-3r=3r\implies 240=6r \\\\\\ \cfrac{240}{6}=r\implies \stackrel{ Southbound~car }{40=r}\hspace{5em}\underset{ Northbound~car }{\stackrel{ 40~~ + ~~12 }{\text{\LARGE 52}}}[/tex]

Starting with 110mg of a radioactive substance, after 21 hours, 55mg remains. Using the exponential decay model, after 31 hours,  39.3mg of the substance will remain.

We can use the formula for exponential decay, which is:

N(t) = N0 * e^(-λt)

where:

N0 is the initial amount of the substance

N(t) is the amount remaining after time t

λ is the decay constant

To solve for λ, we can use the fact that after 21 hours, 55 mg remains:

55 = 110 * e^(-λ*21)

Simplifying:

0.5 = e^(-λ*21)

Taking the natural logarithm of both sides:

ln(0.5) = -21λ

Solving for λ:

λ = ln(0.5) / (-21) ≈ 0.033

Now we can use this value of λ to find the amount remaining after 31 hours:

N(31) = 110 * e^(-0.033*31) ≈ 39.3 mg

Therefore, approximately 39.3 milligrams of the substance will remain after 31 hours.

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The answer of the given question based on the finding the area of rectangle is  QRST is approximately 116.56 square units.

Area is a measure of the size or extent of a two-dimensional surface, like a shape or a region on a flat surface. It is typically measured in square units, like square meters (m²), square feet (ft²), or square centimeters (cm²).

To calculate the area of a shape, you need to know its dimensions and shape. The formula for finding the area of a shape depends on its shape. Area is an important concept in mathematics and geometry, and it is used in many real-life applications, such as measuring the size of land, determining the amount of material needed for a project, and calculating the volume of a container or room.

To find the area of the rectangle, we can use the distance formula to find the lengths of its sides. Then, we can use the formula for the area of a rectangle, which is the product of its length and width.

Length QR = √[(6 - (-4))² + (3 - (-3))²] = √[10² + 6²] = √(136)

Length ST = √[(9 - (-1))² + (-2 - (-8))²] = √[10² + 6²] = √(136)

Width QS = √[(6 - (-1))² + (3 - (-8))²] = √[7² + 11²] = √(170)

Width RT = √[(-4 - 9)² + (-3 - (-2))²] = √[13² + 1] = √(170)

Therefore, the area of the rectangle is:

Area = Length x Width = √(136) x √(170) = 116.56 (rounded to two decimal places)

So, the area of the rectangle QRST is approximately 116.56 square units.

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1. Ben's mistake in solving the area of circle is, he has taken the value of diameter at the place of radius in formula.

2. Luna's mistake is, he did the incorrect squaring of radius 4.5.

The formula makes use of the diameter. The units are squared at all times. When the diameter is known, the area can be calculated by first determining the radius. The radius of the circle is the area. You can use circumference to determine area.

area of circle A =  πr²

where, π = 3.14 and r is the radius of circle

r = diameter /2 = 9/2 = 4.5 cm

A = πr²

A = 3.14*(4.5)²

A = 3.14*20.25

A = 63.585 cm² (correct area)

Thus,

1. Ben's mistake in solving the area of circle is, he has taken the value of diameter at the place of radius in formula.

2. Luna's mistake is, he did the incorrect squaring of radius 4.5.

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Answer:

Step-by-step explanation:

[tex]\sqrt{87x} =\sqrt{29\times3\times x}[/tex]

When [tex]x=21[/tex] we get:

    [tex]\sqrt{87x} =\sqrt{29\times3\times 21}[/tex]

              [tex]=\sqrt{29\times3\times3\times7}[/tex]

              [tex]=3\sqrt{29\times7}[/tex]

               [tex]=3\sqrt{203}[/tex]

So when [tex]x=21[/tex] we can simplify [tex]\sqrt{87x}[/tex].

a) The plot that represents a dot plot of the data is plot (B).

b) The answer is (C)

c) The answer is (B)

A normal distribution is a continuous probability distribution that has a symmetric bell-shaped curve, with the mean, median, and mode all being equal.

(a) The plot that represents a dot plot of the data is plot (B).

(b) The configuration of the points does not suggest that the volumes are from a population with a normal distribution. The answer is (C) - The dot plot does not approximate a "bell" form, hence the population does not seem to have a normal distribution.

(c) There are no outliers in the data. The answer is (B) - No, there does not appear to be any outliers.

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The answers are:

1). B. The plot that depicts a data dot plot is plot (B).

2). C. Because the dotplot does not match a "bell" shape, the population does not appear to have a normal distribution.

3). B. No, there does not appear to be any outlie

A normal distribution is a kind of continuous distribution of probability in which the majority of data points cluster in the centre of the range, while the remainder taper off symmetrically towards either extreme. The mean of the distribution is also known as the centre of the range.

Because of its flared form, a normal distribution resembles a bell curve graphically. The exact shape can vary depending on the population's value distribution. The population is the total number of data elements in the distribution.

a). The plot depicts a data dot plot and is called plot. (B).

b). Because the dot plot does not resemble a "bell" form.(C).

c). The data does not contain any anomalies. (B).

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The Complete question is,

a). question a is attached below.

b). Does the configuration of the points appear to suggest that the volumes are from a population with a normal distribution?

A. Yes, the population appears to have a normal distribution because the dotplot resembles a "bell shape

B. No, the population does not appear to have a normal distribution because the frequencies of the volume decrease from left to right.

C. No, the population does not appear to have a normal distribution because the dotplot does not resemble a "bell" shape

D. Yes, the population appears to have a normal distribution because the frequencies of the volume increase from left to right.

c). Are there any outliers?

A. Yes, the volumes of 0 oz and 200 oz appear to be outliers because they are far away from the other temperatures

B. No, there does not appear to be any outliers °

C. Yes, the volume of 50 oz appears to be an outlier because it is far away from the other volumes

D. Yes, the volume of 70 oz appears to be an outlier because many sodas had this as their volume

The area of the circle is 63.585 sq. cm.

An object's area is how much space it takes up in two dimensions. It is the measurement of the quantity of unit squares that completely cover the surface of a closed figure. The square unit, which is frequently expressed as square inches, square feet, etc., is the accepted unit of area. Each shape's area may be calculated by counting how many unit squares will fit inside of it.

Given that the diameter of the circle is 9 cm

The radius = 9/2 = 4.5 cm

The area of the circle is:

A = πr²

Substitute the values:

A = (3.14)(4.5)²

A = 63.585 sq. cm.

Hence, the area of the circle is 63.585 sq. cm.

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The complete question is:

The perimeter of ABCDEFGH is 8s, with s = 3, so the perimeter of ABCDEFGH is 8x3 = 24 units.

The perimeter of a regular octagon can be calculated using the formula P = 8s, where s is the length of each side. To solve for the perimeter of ABCDEFGH, we need to first calculate the length of each side.We can use the distance formula to calculate the length of AH. The distance formula is [tex]d = √((x2-x1)^2 + (y2-y1)^2).[/tex] In this case, x1 = 4, y1 = 5, x2 = 7, and y2 = 5. Plugging these values into the formula, we get [tex]d = √((7-4)^2 + (5-5)^2) = √(3^2 + 0) = √9 = 3[/tex] units.Since each side of a regular octagon is equal, we can use the length of AH, 3 units, to calculate the perimeter of the octagon. The perimeter of ABCDEFGH is 8s, with s = 3, so the perimeter of ABCDEFGH is 8x3 = 24 units.

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The amount of paint needed in cubic centimeters to apply a coat of paint 0.03 cm thick to a hemispherical dome with a diameter of 30 meters is 1.413737 × 10¹⁰ cm³.

To estimate the amount of paint needed to apply a coat of paint 0.03 cm thick to a hemispherical dome with a diameter of 30 meters, we will use linear approximation.

The volume of a hemisphere with a diameter of 30 meters is given by:

V = (2/3)πr³Where r = 30/2 = 15 meters

We need to find the amount of paint required for a thickness of 0.03 cm.

We convert this to meters as:

0.03 cm = 0.0003 m

The linear approximation formula is given by:

ΔV ≈ dV/dx × Δx

where Δx = 0.0003 is the change in thickness, and

dV/dx is the rate of change of the volume with respect to thickness x.

We have:

V = (2/3)πr³ = (2/3)π(15)³ = 14137.2 m³

dV/dx = 4πr² = 1800π m²

Now, we can estimate the change in volume:

ΔV ≈ dV/dx × Δx = 1800π × 0.0003 = 0.1696 m³

The total amount of paint needed is:

V + ΔV = 14137.2 + 0.1696 = 14137.37 m³

We can convert this to cubic centimeters as:

1 m³ = 10⁶ cm³

So, the amount of paint needed in cubic centimeters is:

V = 14137.37 * 10⁶ cm³ = 1.413737 × 10¹⁰ cm³

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The probability that a randomly selected box of cold medicine will cost more than $13 is 0.4542 or 45.42%.


Given the mean of the normally distributed cold medicine = $12.75 and the standard deviation = $2.15 and the random variable x, the probability of a randomly selected box of cold medicine costing more than $13 needs to be found.
Now, as we are given mean and standard deviation, we can standardize the normal distribution and then use the Z table or calculator to find the probability.
The formula for standardizing the normally distributed curve:
Z = (X - μ) / σ
where
Z = Standardized score
X = Score value
μ = Mean
σ = Standard deviation
Here, we have to find the probability of a randomly selected box of cold medicine will cost more than $13.
So, the formula becomes:
Z = (X - μ) / σ = ($13 - $12.75) / $2.15 = 0.116
Using the Z-table, the area to the left of Z = 0.116 is 0.5458
Thus, the probability that a randomly selected box of cold medicine will cost more than $13 is:
1 - 0.5458 = 0.4542 or 45.42%
Therefore, the probability of a randomly selected box of cold medicine costing more than $13 is 0.4542 or 45.42%.

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Part a: Model for m∠A : cos A = 200/d.

Part b: If d = 300 m,  m∠A = 48.19°.

Part c: If d = 550 m, m∠A =  68.67° .

The trigonometric functions sine, cosine, tangent, and its inverses make up these trigonometric ratios.

Given an angle measurement, so every function represents a distinct ratio in either a right triangle. Trigonometric ratios are the ratios of a right triangle's sides to one of the triangle's non-right angles. Remember that a right triangle has one right angle that is always 90 degrees in length.

In the given query:

Base  = 200 m.

Hypotenuse = d m.

Part a: Model for m∠A :

cos A = Base/ Hypotenuse

cos A = 200/d.

Thus, Model for m∠A : cos A = 200/d.

Part b: If d = 300 m,  m∠A = ?

cos A = 200/d.

cos A = 200/300

cos A = 2/3

m∠A =  48.19°

Part c: If d = 550 m, m∠A = ?

cos A = 200/d.

cos A = 200/550

cos A = 4/11

m∠A = 68.67°

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Meg has 12 different options to choose from when it comes to buying a new pair of skates based on combinations.

Meg has three options to choose from - roller skates, figure skates, and hockey skates, and four options for the color of her skates - black, white, blue, and silver. To find the total number of different combinations, we have to multiply the number of options for each category.

Total number of combinations of skates that Meg can choose from is:

3 (types of skates) x 4 (colors of skates) = 12

Meg has 12 different options to choose from when it comes to buying a new pair of skates.

It is important to note that this calculation assumes that Meg can choose any combination of skate type and color. If certain types of skates only come in certain colors, or if certain colors are only available for certain types of skates, then the number of possible combinations will be less than 12. Additionally, other factors such as size and brand of the skates may also affect the total number of options available to Meg.

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It will take 5.71 hours for Pam and Erin to be 14 miles apart.

If Pam and Erin start at the same point and rollerblade in different directions, then the total distance between them at any given time can be represented by the Pythagorean Theorem. The Pythagorean Theorem states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse.

Let x represent the number of hours it takes for Pam and Erin to be 14 miles apart. Then, the distance traveled by Pam will be 2x, and the distance traveled by Erin will be 4x.

Using the Pythagorean Theorem, the equation that represents the distance between them can be written as 2x^2 + 4x^2 = 14^2.

Solving this equation, we get 6x^2 = 196, which can be written as x^2 = 32.75. Taking the square root of both sides, we get x = 5.71.

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The pair of supplementary angles in the given situation is:

136 + 44 = 180°

135 + 45 = 180°

154 + 26 = 180°

A supplementary angle is an angle that sums to 180 degrees.

For example, the 130° and 50° angles are complementary because the sum of 130° and 50° is 180°.

Similarly, the sum of the supplementary angles is 90 degrees.

An apex angle is an angle opposite at the intersection of two straight lines, and an adjacent angle is two angles next to each other.

So, the pair of supplementary angles would be:

136 + 44 = 180°

135 + 45 = 180°

154 + 26 = 180°

Therefore, the pair of supplementary angles in the given situation is:

136 + 44 = 180°

135 + 45 = 180°

154 + 26 = 180°

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Answer:

Given :

By selling an article for rupees 144, a man loses 17

of his outlay. It is sold for rupees 189.

To do :

We have to find the gain or loss percent.

Solution :

Let the cost price of article be Rs. x

.

This implies,

SP =CP −Loss

=x−x7

=6x7

Therefore,

6x7=144

x=144×76

x=Rs. 168

If the article is sold for Rs. 189, then,

SP =Rs. 189

Profit =

SP − CP

=189−168

=Rs. 21

Gain %=GainCP×100%

=21168×100%

=12.5%

The gain percent is 12.5%

.

Step-by-step explanation:

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Two airplanes are flying in the air at the same height. airplane a is flying east at 453 mi/h and airplane b is flying north at 508 mi/h. if they are both heading to the same airport, located 7 miles east of airplane a and 8 miles north of airplane b, at the rate at which the distance between the two airplanes is changing is approximately 473 mi/h.

What is the distance between the airplanes?

We may use the Pythagorean theorem to find the distance between the two airplanes. Let’s use A to represent the position of Airplane A and B to represent the position of Airplane B.

Let d be the distance between the airplanes. Then, using the Pythagorean Theorem, we have:

d² = (8 miles)² + (7 miles)² d² = 64 + 49d² = 113d = sqrt(113) miles

Since Airplane A is flying straight to the airport, its rate of approach to the airport is its speed, 453 mi/h.

Since Airplane B is flying straight to the airport, its rate of approach to the airport is its speed, 508 mi/h.

Let d be the distance between the airplanes at some point in time t. We need to find the rate at which the distance is changing, or the derivative of d with respect to t. We may use the Pythagorean theorem to find the distance between the two airplanes.

Let A represent the position of Airplane A and B represent the position of Airplane B.Let d be the distance between the airplanes. Then, using the Pythagorean Theorem, we have:

d² = (8 miles)² + (7 miles)² d² = 64 + 49d² = 113d = sqrt(113) miles

At some time t, let A(t) represent the position of Airplane A, and let B(t) represent the position of Airplane B. We have that:

A(t) = 453t B(t) = 508t

Therefore, the distance between the airplanes is given by:

d(t)² = (453t)² + (508t)²d(t)² = 205,609t² + 258,064t²d(t)² = 463,673t²

We take the derivative of both sides with respect to t, noting that d²/dt² = 2dd/dt:

2d(t)d'(t) = 927,346t

Then, dividing both sides by 2d(t), we have:

[tex]d'(t) = 927,346t/(2d(t))d'(t) = 927,346t/(2sqrt(113)) miles/h[/tex]

Using t = 1 hour (since we are asked for the rate at which the distance between the airplanes is changing), we have:

[tex]d'(1) = 927,346/(2sqrt(113))d'(1) ≈ 473 miles/h[/tex]

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Therefore, the result of (u*v)v is a vector with components (80, -16).

In mathematics, a vector is a mathematical object that has both magnitude and direction. Vectors are used to represent physical quantities such as force, velocity, and acceleration, as well as abstract mathematical quantities such as points, lines, and planes. A vector is usually represented geometrically as an arrow in a coordinate system, where the length of the arrow represents the magnitude of the vector, and the direction of the arrow represents the direction of the vector. Vectors can be added, subtracted, and multiplied by scalars, and these operations follow specific rules that make them a powerful tool for solving problems in mathematics, physics, engineering, and other fields. Vectors can have any number of dimensions, but in two-dimensional space, a vector can be represented as an ordered pair of numbers (x, y), where x and y represent the horizontal and vertical components of the vector, respectively. In three-dimensional space, a vector can be represented as an ordered triplet of numbers (x, y, z), and so on for higher dimensions.

Here,

To find (u*v)v, we first need to find the dot product of u and v, which is defined as:

u · v = (4, 4) · (-5, 1)

= (4 × -5) + (4 × 1)

= -20 + 4

= -16

Next, we need to multiply the dot product of u and v by v:

(u*v)v = (-16)(-5, 1)

= (80, -16)

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Answer:  Option A, [tex]\textsf{6a}^3\textsf{ + 22a}^2\textsf{ + 14a - 10}[/tex]

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Given:  [tex]\textsf{3a + 5 and 2a}^2\textsf{ + 4a - 2}[/tex]

Find:  [tex]\textsf{The product of the two given equations}[/tex]

Solution: The first step toward solving this problem would be to distribute the 3a and 5 to each of the values in the second equation.

After doing so, we can simplify each of the expressions until we have one equation.  This can be done by both some simple algebra and combining of like terms.

Therefore, the correct answer to this question is Option A, [tex]\textsf{6a}^3\textsf{ + 22a}^2\textsf{ + 14a - 10}[/tex].

In conclusion there are 33 ways to distribute seven identical balls into three distinct boxes so that each box contains at least one ball.

How to solve?

This is a combinatorics problem known as distributing identical objects into distinct boxes with the condition that each box has at least one object.

There are three such cases: when the first box is empty, when the second box is empty, and when the third box is empty. Each of these cases can be represented by a single bar that is placed at the beginning or end of the line of stars. For example, the case where the first box is empty can be represented as:

|****|

There are three ways to place this single bar, so we need to subtract three from our previous count of 36 to get the final answer:

36 - 3 = 33

Therefore, there are 33 ways to distribute seven identical balls into three distinct boxes so that each box contains at least one ball.

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If we know that f(7) = 0, we can say that:

x - 7 is a factor of the function f(x).

This is because when we substitute x = 7 into the function, we get:

f(7) = 7^3 - 7(7)^2 + 25(7) - 175 = 0

This means that (x - 7) is one of the factors of the function.

Now we can use polynomial division or synthetic division to find the other factors. If we divide the function f(x) by (x - 7), we obtain:

           x^2 - 4x + 25

Therefore, we can say that:

f(x) = (x - 7)(x^2 - 4x + 25)

The quadratic term, x^2 - 4x + 25, has no real roots because the discriminant is negative. Therefore, we can write the factorization of f(x) in terms of complex numbers as:

f(x) = (x - 7)(x - 2 + 5i)(x - 2 - 5i)

Thus, the complete factorization of f(x) is:

f(x) = (x - 7)(x - 2 + 5i)(x - 2 - 5i)

Si nous savons que f(7) = 0 alors,

x - 7 est un facteur de la fonction f(x).

on substitue x = 7 dans la fonction

f(7) = 7^3 - 7(7)^2 + 25(7) - 175 = 0

donc  (x - 7) est l’un des facteurs de la fonction.

Maintenant, on utilise division polynomiale ou la division synthétique pour avoir les autres facteurs. Si on divise la fonction f(x) par (x - 7), on obtient:

x^2 - 4x + 25

Du coup peut dire que :

f(x) = (x - 7)(x^2 - 4x + 25)

Le terme quadratique, x^2 - 4x + 25, n’a pas de racines réelles car le discriminant est négatif. Par conséquent, nous pouvons écrire la factorisation de f(x) en termes de nombres complexes comme:

f(x) = (x - 7)(x - 2 + 5i)(x - 2 - 5i)

la factorisation complète de f(x) est:

f(x) = (x - 7)(x - 2 + 5i)(x - 2 - 5i)

Using the trigonometric Identities, [tex]tan(A - B) =\frac{-5+12\sqrt{13} }{12+5\sqrt{13} }[/tex]

Trigonometric Identities are the equalities that involve trigonometry functions and holds true for all the values of variables given in the equation.

Given

[tex]sin(A) =\dfrac{5}{13}[/tex]

[tex]\dfrac{\pi }{2} < A < \pi[/tex]

Using the trigonometric identity

[tex]sin^2A+cos^2A=1[/tex]

[tex]cosA =\sqrt{1-sin^2A}[/tex]

[tex]cosA =\sqrt{1-(\dfrac{5}{13})^2 }[/tex]

[tex]cosA =-\dfrac{12}{13}[/tex]

[tex]tanA=\dfrac{sinA}{cosA}[/tex]

[tex]tanA =\dfrac{\frac{5}{13} }{\frac{-12}{13} }[/tex]

[tex]tanA =-\dfrac{5}{12}[/tex]

[tex]tan(A-B) =\dfrac{tanA-tanB}{1+tanAtanB}[/tex]

[tex]=\dfrac{-\frac{5}{12}-(-\sqrt{13}}{1+(-\frac{5}{12})(-\sqrt{3}) }[/tex]

[tex]=\dfrac{5-12\sqrt{13}}{-12-5\sqrt{3} }[/tex]

[tex]=\dfrac{-5+12\sqrt{13} }{12+5\sqrt{13} }[/tex]

Option D is correct.

Hence, [tex]tan(A - B) =\dfrac{-5+12\sqrt{13} }{12+5\sqrt{13} }[/tex]

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