10=6+2f
F=

I’m am really confused about this

The sum of the ages of the family seven years ago is given as follows:

45 years.

The sum of the ages of the family is obtained applying the proportions in the context of the problem.

Each year, the age of each person increases by one, hence, considering a family of five, the yearly increase in the age of the family is given as follows:

5 x 1 = 5 years.

The two youngest are 6 and 8, meaning that seven years ago, the youngest was not yet born.

The ages of a family of five sum to 80, hence the sum seven years ago was given as follows:

80 - 5 x 7 = 45 years.

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Therefore, the estimated population of great white sharks that have migrated from Mexico to Hawaii is 2,500.

Shark population estimation is the process of determining the size and trends of shark populations in a given area. It involves collecting and analyzing data on shark abundance, distribution, behavior, and other factors to understand their population dynamics.

There are different methods used to estimate shark populations, including mark-recapture studies, aerial surveys, acoustic tracking, and genetic analysis. These methods allow researchers to estimate the number of sharks in a given area, monitor changes in their population over time, and identify potential threats to their survival.

To estimate the population of great white sharks that have migrated from Mexico to Hawaii, we can use the capture-recapture method, also known as the Lincoln-Petersen index.

Let:

The Lincoln-Petersen index formula is:

[tex]N = (n_1 * n_2 * n_3) / (m_1 * m_2 * m_3)[/tex]

Plugging in the given values, we get:

[tex]N = (45 * 20 * 20 * 20) / (3 * 2 * 4)[/tex]

[tex]N = 60,000 / 24[/tex]

[tex]N = 2,500[/tex]

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The point estimate is 0.395 and the poll's margin of error is 0.015 or 1.5%.

A set of values that, with a particular level of confidence, is likely to include an unknown population parameter is known as a confidence interval. Confidence intervals are frequently utilized in statistics to calculate the actual value of a population parameter using sample statistics. When comparing a sample statistic to a population parameter that is hypothesised, we utilize confidence intervals to assess if the difference is statistically significant.

Given the interval of the result in the survey is [0.38, 0.41].

a. The point estimate is given as:

Point estimate = (0.38 + 0.41) / 2 = 0.395

b. The margin of error is given as:

Margin of error = (critical value) x (standard error)

Assuming a 95% confidence level:

Standard error = (maximum error) / (critical value)

Now,

Maximum error = (0.41 - 0.38) / 2 = 0.015

Margin of error = 1.96 x (0.015 / 1.96) = 0.015

Hence, the point estimate is 0.395 and the poll's margin of error is 0.015 or 1.5%.

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Answer:

1. It flips the image over a line to create a mirror image

2. It spins a figure around a point

3. It slides a figure to a different location

4. it enlarges or reduces a figure

Step-by-step explanation:

The probability that more than 7 of the 15 patients have hyperextending joints is 0.015, or about 1.5%.

What is probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.

This is a binomial distribution problem, where we have n = 15 trials and the probability of success (a patient having hyperextending joints) is p = 0.2.

A. To find the probability that exactly 5 of the 15 patients have hyperextending joints, we can use the formula for the binomial probability mass function:

P(X = 5) = (15 choose 5) * 0.2^5 * 0.8^10

where (15 choose 5) is the number of ways to choose 5 patients out of 15. Using a calculator, we get:

P(X = 5) = 0.188

So the probability that exactly 5 of the 15 patients have hyperextending joints is 0.188, or about 18.8%.

B. To find the probability that no more than 7 of the 15 patients have hyperextending joints, we need to add up the probabilities of having 0, 1, 2, ..., 7 patients with hyperextending joints. We can use the cumulative distribution function (CDF) of the binomial distribution:

P(X ≤ 7) = Σ(P(X = k)) for k = 0 to 7

Again, using a calculator, we get:

P(X ≤ 7) = 0.985

So the probability that no more than 7 of the 15 patients have hyperextending joints is 0.985, or about 98.5%.

C. To find the probability that more than 7 of the 15 patients have hyperextending joints, we can use the complement rule:

P(X > 7) = 1 - P(X ≤ 7)

We already know from part (B) that P(X ≤ 7) = 0.985, so:

P(X > 7) = 1 - 0.985 = 0.015

Therefore, the probability that more than 7 of the 15 patients have hyperextending joints is 0.015, or about 1.5%.

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Answer:

x = - 8

Step-by-step explanation:

(3x - 1)(x + 1) - 3x(x + 1) = 7 ← factor out (x + 1) from each term

(x + 1)(3x - 1 - 3x) = 7

(x + 1)(- 1) = 7 ( multiply both sides by - 1 )

x + 1 = - 7 ( subtract 1 from both sides )

x = - 8

Answer:

Step-by-step explanation:

[tex]\bf (3x -1 ) (x+1)- 3x(x+1)=7[/tex]

Expand:-

[tex]\bf 3x^2+2x-1-3x\left(x+1\right)[/tex]

[tex]\bf 3x^2+2x-1-3x^2-3x[/tex]

Combine like terms:-

[tex]\bf (3x^2-3x^2)+(2x-3x)-1[/tex]

[tex]\bf -x-1=7[/tex]

Add 1 to both sides:-

[tex]\bf -x-1+1=7+1[/tex]

[tex]\bf -x=8[/tex]

Divide both sides by -1:-

[tex]\bf \cfrac{-x}{-1}=\cfrac{8}{-1}[/tex]

[tex]\bf x=-8[/tex]

_________________________

Hope this helps!

Answer:

x = 11

Step-by-step explanation:

3x + 5x + 2 =90

(3x + 5x) + (2) = 90

8x + 2 = 90

8x + 2 -2 = 90 -2

8x = 88

8x/8 = 88/8

x = 11

The equation of the polynomial function in standard form is P(x) = x³ + 6x² - 7x - 60

Given that we have the following

zeros = -5,-4,and 3

The polynomial function in standard form is calculated as

P(x) = (x - zeros)

So, we have

P(x) = (x + 5)(x + 4)(x - 3)

Opening the brackets, we have the following equation

P(x) = x³ + 6x² - 7x - 60

Hence, the polynomial is P(x) = x³ + 6x² - 7x - 60

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Answer:

6

Step-by-step explanation:

the number of zeroes is defined by the highest exponent of the variable in the functional expression.

The measure of diagonal DB in the given square is 5√2.

A triangle is a three-sided polygon with three angles. It is a fundamental geometric shape and is often used in geometry and trigonometry. A triangle is defined by its three sides and the three angles formed by those sides. The sum of the three angles in a triangle is always 180 degrees. Triangles can be classified based on the length of their sides and the size of their angles.

Using Pythagoras theorem of right triangle

DB² = DC² + BC²

Here given that

DC = BC

Then

DB² = DC² + DC²

DB² = 5² + 5²

DB² = 25 + 25

DB² = 50

DB = √50

DB = 5√2

Thus, the measure of diagonal DB in the given square is 5√2

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The coordinates of the point where the food bag is hung are (2.5, 7, 6.5).

The center of the coordinate system (where the lines intersect) is called the origin. The axes intersect when both x and y are zero. The coordinates of the origin are (0, 0). An ordered pair contains the coordinates of one point in the coordinate system.

The coordinates of the point on the first tree where the rope is attached are (10, 6, 8), and the coordinates of the point on the second tree where the rope is attached are (-5, 8, 5).

To find the distance between the points, we can use the distance formula:

distance = √((x2 - x1)² + (y2 - y1)² + (z2 - z1)²)

Plugging in the coordinates from part 1, we get:

distance = √((-5 - 10)² + (8 - 6)² + (5 - 8)²)

distance = √(225 + 4 + 9)

distance = √(238)

Therefore, the distance between the points is √(238).

Since the food bag is hung at a point exactly halfway along the length of the rope, we can find its coordinates by taking the average of the coordinates of the points where the rope is attached.

The x-coordinate of the midpoint is: (10 + (-5)) / 2 = 2.5

The y-coordinate of the midpoint is: (6 + 8) / 2 = 7

The z-coordinate of the midpoint is: (8 + 5) / 2 = 6.5

Therefore, the coordinates of the point where the food bag is hung are (2.5, 7, 6.5).

To determine whether the food is safe from bears, we need to compare the height of the food bag to the maximum height that bears can reach. The food bag is hung at a height of 6.5 feet, and bears can reach a height of 7.5 feet when standing. Therefore, the food is not completely safe from bears, as they may be able to reach it if they stand up on their hind legs. However, hanging the food bag at this height does make it more difficult for the bears to reach, and may deter them from trying to get to it. It is also important to note that proper food storage techniques in bear country should involve using bear-resistant containers, in addition to hanging food bags.

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The range of the normal curve representing 95% of the scores fall is equal to 40.2 to 59.8.

The range within which 95% of the scores fall,

Calculate the z-scores corresponding to the 2.5th and 97.5th percentiles of the normal distribution.

The z-score corresponding to the 2.5th percentile is -1.96.

And the z-score corresponding to the 97.5th percentile is +1.96.

Using the formula for the z-score, we can calculate the corresponding raw scores as,

Lower bound is equal to,

z = (X - μ) / σ

⇒-1.96 = (X - 50) / 5

⇒X = 40.2

Upper bound is equal to,

z = (X - μ) / σ

⇒ +1.96 = (X - 50) / 5

⇒ X = 59.8

Therefore, 95% of the scores fall within the range of 40.2 to 59.8.

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The amount of caster sugar used in the first attempt is given as follows:

340 g.

The amount of caster sugar used in the first attempt is obtained applying the proportions in the context of the problem.

The expressions for each attempt are given as follows:

The amount used on the third attempt was of 376.38 g, hence the amount used on the first attempt is obtained as follows:

1.107x = 376.38

x = 376.38/1.107

x = 340 g.

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The given statement square root functions are not always increasing or always decreasing for the whole domain of ( -∞ , ∞) is false.

Function is square root function.

Domain = ( -∞ , ∞)

Square root functions is always increasing or decreasing is a false statement.

For example,

Let us consider the function f(x) = √x.

This function is increasing for x > 0, since as x increases, √x increases as well.

However, for x < 0, the function is not defined since the square root of a negative number is not a real number.

This implies,

The function is neither increasing nor decreasing for x < 0.

Therefore, the statement square root functions are either always increasing or always decreasing for the whole domain of ( -∞ , ∞)  is false.

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The above question is incomplete , the complete question is:

True or false? square root functions are either always increasing or always decreasing for the whole domain of ( -∞ , ∞).

The correct answer is A. Multiplying the first equation by 2 will eliminate the variable y.

The elimination method is a common technique used to solve a system of linear equations. In this method, we manipulate the equations by adding or subtracting them to eliminate one of the variables, so that we can solve for the other variable.

To eliminate one of the variables, we need to choose one of the equations and multiply it by a constant that will make the coefficient of one of the variables in that equation equal to the coefficient of the same variable in the other equation, but with the opposite sign. Then, when we add the two equations together, the term with that variable will be eliminated.

Looking at the given equations, we can see that if we multiply the first equation by 2, we will get:

6x - 12y = 16

And if we add this equation to the second equation, we get:

6x - 12y + 2x + 3y = 16 - 1

Simplifying this, we get:

8x - 9y = 15

Now we have one equation with only x and y, and we can use this equation and either of the original equations to solve for x and y using the substitution method.

Therefore, the correct answer is A. Multiplying the first equation by 2 will eliminate the variable y.

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Answer: Yes

Step-by-step explanation:

1.4691 is more than 1.465, and less than 1.472

The calculated value is 15.16, and the degrees of freedom are 12, the p-value is less than 0.05.

As a result, we must reject the null hypothesis.

A manager of a sports club maintains data about the main sport in which members participate and their ages.

To verify if there is a connection between the age of a member and their sport preference, a test of independence is performed on 651 sports club members who were randomly selected at a 5% level.200 Words:

Test of independence: Test of independence is a statistical method that helps to determine if there is a connection between two groups, and if there is, what is the magnitude of the relationship.

A test of independence is performed using the chi-square statistic.

The null hypothesis in this test is that there is no relationship between the two group S.

We will use a chi-square test to determine whether there is a connection between the age of the member and the sport in which they participate.

Hypotheses:

Null hypothesis: There is no link between the age of a member and the sport in which they participate.

Alternate hypothesis: There is a relationship between the age of a member and the sport in which they participate.

Level of significance = 5%.

Step 1: Prepare the contingency table for observed values

Age  Basketball  Football  Hockey  Soccer  Other Under 20 years  25  30  20  10  15 20-29 years  20  50  35  30  25 30-39 years  15  40  40  35  30 40-49 years  10  35  25  25  20 Over 50 years  5  15  10  15  10

Step 2: Calculate the marginal frequencies.

Age  Basketball  Football  Hockey  Soccer  Other Total Under 20 years  25  30  20  10  15 100 20-29 years  20  50  35  30  25 160 30-39 years  15  40  40  35  30 160 40-49 years  10  35  25  25  20 115 Over 50 years  5  15  10  15  10 55 Total 75  170  130  115  100 651

Step 3: Compute the expected frequencies.

The predicted frequency for each group is (row total) * (column total) / total number of observations.

Age  Basketball  Football  Hockey  Soccer  Other Under 20 years  11.5  26.1  19.9  17.7  15 20-29 years  30.9  70.1  53.4  47.5  40.1 30-39 years  30.9  70.1  53.4  47.5  40.1 40-49 years  21.3  48.2  36.8  32.7  27.6 Over 50 years  9.4  21.4  16.4  14.6  12.3

Step 4: Calculate the test statistic:

We will use a chi-square test statistic to determine whether the null hypothesis should be accepted or rejected.

Chi-square test formula is:χ2 = Σ (Observed frequency – Expected frequency)2 / Expected frequency.

Degrees of freedom = (Number of rows – 1) * (Number of columns – 1)

χ2 = (25-11.5)2 / 11.5 + (20-30.9)2 / 30.9 + (15-30.9)2 / 30.9 + (10-21.3)2 / 21.3 + (5-9.4)2 / 9.4 + (30-26.1)2 / 26.1 + (50-70.1)2 / 70.1 + (40-70.1)2 / 70.1 + (35-48.2)2 / 48.2 + (15-21.4)2 / 21.4 + (20-19.9)2 / 19.9 + (35-53.4)2 / 53.4 + (40-53.4)2 / 53.4 + (25-36.8)2 / 36.8 + (10-16.4)2 / 16.4 + (15-14.6)2 / 14.6 + (15-32.7)2 / 32.7 + (25-27.6)2 / 27.6 + (10-12.3)2 / 12.3χ2 = 15.16

Degrees of freedom = (5-1) * (4-1) = 12

Step 5: Determine the p-value:

The p-value is determined by looking up the chi-square value and the degrees of freedom in the chi-square table.

The p-value is the probability of obtaining a result as extreme as or more extreme than the calculated value if the null hypothesis is true.

We will use a 5% level of significance.

Since the calculated value is 15.16, and the degrees of freedom are 12, the p-value is less than 0.05.

As a result, we must reject the null hypothesis.

There is evidence to suggest that the age of a member is related to the sport in which they participate.

In other words, the age of a member and their sport preference are not independent.

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So the measures of the angles in the isosceles obtuse triangle are 20 degrees, 80 degrees, and 80 degrees.

What is triangle ?

Three straight ends and three angles make up a closed triangle, a two-dimensional geometric object. A triangle's angles add up to 180 degrees in all cases. Based on their side lengths and angle measurements, triangles can be categorized in a variety of ways. Typical categorizations comprise: Triangle having no equal sides is referred to as a scalene triangle. Isosceles triangle: A triangle with two sides of equal length. A triangle having an equilateral shape is one with three equal sides. Triangle that has all angles that are less than 90 degrees is referred to as an acute triangle. Triangle with one angle that is precisely 90 degrees is referred to as a right triangle. A triangle that has a side that is more acute than 90 degrees is said to be obtuse.

given

In the isosceles obtuse triangle, let x represent the measurements of each equal angle.

One of the equal angles is bigger than 90 degrees since the triangle is obtuse. As a result, we have:

x + x + 20 = 180 (sum of angles of a triangle) (sum of angles in a triangle)

When we simplify this equation, we obtain:

2x + 20 = 180

2x = 160

x = 80

Hence, each of the equal angles is 80 degrees in length. As a result, there are three different ways to measure the angles in the isosceles obtuse triangle, which has one angle that is 20 degrees, as follows:

20-80-80 degree range

80-20-80 degree range

80-80-20 Celsius

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Required value of N is 1

This is a problem of simplification which is major part of Algebra.

Some Algebra's formulas:

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

Given equation is [tex] - 3(8 \times N) = - 24[/tex]

We want to Simplify both side.

Multiplying (-3) and (8×N),

-24N = -24

Now we are dividing (-24) by (-24),

N = -24/-24

N = 1

Therefore, required value of N is 1.

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The rectangular prism's length, width, and height can also be labelled, and the surface area (SA) can be calculated using the formula:

SA = 2lw+2lh+2hw.

A face is just a flat surface in the geometric sense, and edges are formed where surfaces meet. Pyramids and prisms are both examples of polyhedra. A pyramid, however, is not a prism.

Method to find surface area:

Thus, the rectangular prism's length, width, and height can also be labelled, and the surface area (SA) can be calculated using the formula:

SA = 2lw+2lh+2hw.

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Surface Area = (width×length  +  height×length  +  height×width) × 2

—————

Please remember to revise this and make it in your own words if you want! I tried making this as simple as possible! I hope this helps you. -Doodle

—————

By answering the presented question, we may conclude that As a result, the darkened zone has an area of 88 square units.

In Euclidean geometry, a rectangle is a parallelogram with four small angles. It may also be defined as a fundamental rule hexagon or one in which all of the angles are equal. Another alternative for the parallelogram is a straight angle. Four of the vertices of a square are the same length. A quadrilateral with four 90° angle vertices and equal parallel sides has a rectangle-shaped cross section. As a result, it is also known as a "equirectangular rectangle." A rectangle is sometimes referred to as a parallelogram due to the equal and parallel dimensions of its two sides.

We must determine the size of the darkened zone.

The bigger rectangle (with the frame) has a surface area of (12+2) x (10+2) = 14 x 12 = 168 square units.

The smaller rectangle (with the frame) has an area of (8+2) x (6+2) = 10 x 8 = 80 square units.

The frame's area is equal to the difference between the two areas: 168 - 80 = 88 square units.

As a result, the darkened zone has an area of 88 square units.

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you need to work back wards on the

9x+3=64

so 64-3=61

then do 61 divided by 9  

which equals 6.7(1dp)

so x=6.7(1dp)

Answer:

sin Θ = [tex]-\frac{3}{5}[/tex]

tan Θ = [tex]-\frac{3}{4}[/tex]

csc Θ = [tex]-\frac{5}{3}[/tex]

cot Θ = [tex]-\frac{4}{3}[/tex]

Step-by-step explanation:

We know that this triangle is a 3-4-5 triangle since we know the cos of the angle. We also know this angle is located in Quadrant IV (4) since cos is positive and tan is negative.

Based off this info we can find the rest of the trig functions

sin Θ = [tex]-\frac{y}{r}[/tex]

tan Θ = [tex]-\frac{y}{x}[/tex]

csc Θ = [tex]-\frac{r}{y}[/tex]

cot Θ = [tex]-\frac{x}{y}[/tex]

The team jogged longer in the afternoon because they covered a distance of 6 eighths of a mile, which is greater than the 3 eighths of a mile they covered in the morning.

To compare the distance jogged in the morning and afternoon, we need to find the common unit for both. We can convert both fractions to have the same denominator by finding the least common multiple (LCM) of 8 and 4, which is 8.

So, in terms of eighths of a mile, the distance jogged in the morning is:

3/8 miles = 3 eighths of a mile

And the distance jogged in the afternoon is:

3/4 miles = 6 eighths of a mile (since 3/4 = 6/8)

Therefore, the team jogged longer in the afternoon because they covered a distance of 6 eighths of a mile, which is greater than the 3 eighths of a mile they covered in the morning.

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The median provides Farrah with an understanding of the central value in her data set, which can help her analyze the overall distribution of the data points.

The median is a measure of central tendency that tells Farrah the middle value of her data when it is arranged in ascending order.

To find the median, Farrah should follow these steps:
1. Arrange her data in ascending order (from smallest to largest).
2. If there is an odd number of data points, the median is the middle value.
3. If there is an even number of data points, the median is the average of the two middle values.

It is less affected by outliers or extreme values than the mean, making it a more reliable representation of the center of her data.

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Answer:

yes 10 to the 3rd power is 1,000 so try to go from there if more info is needed please comment "more info"

Step-by-step explanation:

The value of x is 8. So correct is C.

In mathematics, a triangle is a geometric shape that consists of three line segments that intersect at three endpoints. These endpoints are called vertices, and the line segments are called sides.

Triangles are one of the most basic shapes in geometry and are used in many areas of mathematics, science, engineering, and everyday life. They can be classified based on the length of their sides and the measure of their angles.

Some common types of triangles include:

Equilateral triangle: A triangle in which all three sides are equal in length.

Isosceles triangle: A triangle in which two sides are equal in length.

Scalene triangle: A triangle in which all three sides have different lengths.

Right triangle: A triangle in which one angle measures 90 degrees (a right angle).

To find the value of x, we can use the trigonometric ratios of a right-angled triangle. Let's call the base of the triangle "b".

We know that the angle between the hypotenuse and the base is 45 degrees, which means that the other acute angle is also 45 degrees. Therefore, we can use the trigonometric ratio for the sine of 45 degrees to relate the hypotenuse and the perpendicular as follows:

sin(45 degrees) = opposite/hypotenuse

where the opposite side is the perpendicular, which is given as 4√2.

sin(45 degrees) = 4√2/x

Simplifying this equation gives:

x = 4√2 / sin(45 degrees)

We know that sin(45 degrees) = 1/√2, so substituting this value into the equation above gives:

x = 4√2 / (1/√2) = 4√2 * √2 = 8

Therefore, the value of x is 8.

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In linear equation, 6499 ft CD, the height of the plane from the ground, to the nearest foot.

What in mathematics is a linear equation?

An algebraic equation with simply a constant and a first-order (linear) term, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation.

                          Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.

Part A) Find BC, the distance from Tower 2 to the plane, to the nearest foot.

in the triangle ACD

sin16=CD/(7600+BD)--------> CD=sin16*(7600+BD)---------> equation 1

in the triangle BCD

sin24=CD/BD-----------> CD=sin24*BD---------------> equation 2

equation 1=equation 2

sin16*(7600+BD)=sin24*BD-----> sin16*7600+sin16*BD=sin24*BD

sin24*BD-sin16*BD=sin16*7600----> BD=[sin16*7600]/[sin24-sin16]

BD=15979 ft

in the triangle BCD

cos24=BD/BC---------> BC=BD/cos24-------> 15979/cos24-------> 17491

BC=17491 ft

    =  BC is 17491 ft

Part 2) Find CD, the height of the plane from the ground, to the nearest foot.

CD=sin24*BD        ( remember equation 2)

BD=15979 ft

CD=sin24*15979 -----------> CD=6499 ft

     =  CD is 6499 ft

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The coordinates of the vertex A' are (-2, -2).

What is triangle?

A triangle is a three-sided polygon with three angles. It is a fundamental geometric shape and is often used in geometry and trigonometry.

To reflect a point or a shape across a line, you can follow these steps:

Find the equation of the line of reflection.

Find the distance between each point and the line of reflection.

Reflect each point across the line of reflection by moving the same distance on the other side of the line.

Finding the equation of the line of reflection:

The line of reflection is y = -2, which is a horizontal line passing through y = -2.

Finding the distance between each point and the line of reflection:

We can use the formula for the distance between a point (x, y) and a line Ax + By + C = 0:

distance = |Ax + By + C| / sqrt(A^2 + B^2)

Plugging in the values, we get:

distance(A, y = -2) = |(-2) - (-2) + 2| / sqrt(0 + 1) = 2

distance(B, y = -2) = |(-2) - (-2) + 4| / sqrt(0 + 1) = 4

distance(C, y = -2) = |(-5) - (-2) + 2| / sqrt(1 + 0) = 5 / sqrt(1) = 5

So the distances between A, B, C and the line of reflection are 2, 4, and 5, respectively.

Reflecting each point across the line of reflection:

To reflect a point across a line, we move the same distance on the other side of the line. Since the line of reflection is horizontal, we only need to move vertically.

A is 2 units above the line of reflection, so we move it 2 units below the line:

A' = (-2, -2)

B is 4 units above the line of reflection, so we move it 4 units below the line:

B' = (-2, -4)

C is 5 units above the line of reflection, so we move it 5 units below the line:

C' = (-5, -7)

Therefore, the coordinates of the vertex A' are (-2, -2).

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