Calculate the frequency in hertz of electromagnetic radiation that has a wavelength of 585.0 nm. (c = 3.00 X 10⁸ m/s)

Answer:

Jika 16,5% dari suatu jumlah adalah 891, kita dapat menggunakan persamaan:

0,165x = 891

di mana x adalah jumlah aslinya. Kita ingin menyelesaikan persamaan ini untuk x.

Kita dapat memulai dengan membagi kedua sisi dengan 0,165:

x = 891 / 0,165

x = 5400

Jadi, jumlah aslinya adalah 5400.
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There are 92 elements in A but not in B.

In mathematics, a set is a well-defined collection of objects or elements. Sets are denoted by uppercase symbols, and the number of elements in a finite set is denoted as the cardinality of the set enclosed in curly braces {…}.

Empty or zero quantity:

Items not included. example:

A = {} is a null set.

Finite sets:

The number is limited. example:

A = {1,2,3,4}

Infinite set:

There are myriad elements. example:

A = {x:

x is the set of all integers}

Same sentence:

Two sets with the same members. example:

A = {1,2,5} and B = {2,5,1}:

Set A = Set B

Subset:

A set 'A' is said to be a subset of B if every element of A is also an element of B. example:

If A={1,2} and B={1,2,3,4} then A ⊆ B

Universal set:

A set that consists of all the elements of other sets that exist in the Venn diagram. example:

A={1,2}, B={2,3}, where the universal set is U = {1,2,3} 

n(A ∪ B) = n(A – B) + n(A ∩ B) + n(B – A)

Hence, There are 92 elements in A but not in B.

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300 is the correct answer.

Answer:

19/20 , 1 1/4 , 1 4/5

Step-by-step explanation:

1.8 = 1 4/5 (fraction)

1.8 converts to 18/10. This can be simplified twice, firstly by making it 9/5 since both 18 and 10 are divisible by two, but can be simplified further to 1 4/5

1.25 = 1 1/4 (fraction)

1.25 converts to 125/100. This can be simplified to 5/4 or 1 1/4

0.95 = 19/20 (fraction)

0.95 converts to 95/100. This can be simplified to 19/20

Ascending Order (smallest to largest)

smallest - 19/20

middle - 1 1/4

largest - 1 4/5

I believe this is the right answer, but haven't done fractions in a while so may want to double check to make sure

Answer:

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:

m = (y2 - y1) / (x2 - x1)

Substituting the values, we get:

m = (5 - (-10)) / (6 - 3) = 15/3 = 5

Now that we have the slope, we can use the point-slope form of a linear equation to write the equation of the line:

y - y1 = m(x - x1)

Substituting the values of m, x1, and y1, we get:

y - (-10) = 5(x - 3)

Simplifying and rearranging the equation, we get:

y + 10 = 5x - 15

y = 5x - 25

Therefore, the equation of the line passing through (3,-10) and (6,5) in slope-intercept form is y = 5x - 25.

Step-by-step explanation:

#trust me bro

Sure, let's solve this step-by-step:

First, we need to solve for x in the equation x + 1/2 = 5.

We can do this by subtracting 1/2 from both sides, giving us x = 4 1/2.

Now, we can substitute x = 4 1/2 into the equation 2*x^2 - 3x + 6 - 3/x +2/x^2.

We can simplify the equation by multiplying both sides by x^2, giving us:

2*x^2 - 3x + 6 - 3/x +2 = 10*x^2 - 3x + 6.

Now, we can combine all of the terms with x:

10*x^2 - 6x + 6 = 0.

Finally, we can solve the equation using the quadratic formula:

x = 3/5 or x = 2.

Therefore, the answer to the equation is 10*(3/5)^2 - 6(3/5) + 6 = 4.8, or 10*2^2 - 6(2) + 6 = 16.

Answer: 54, 1 2/3, 1, 0.5, 0.03, 0, -1/4, -1, -4, -103

Step-by-step explanation:

54, 1, 1 2/3, 0.5, 0.03, 0, -1/4, -1, -4, -103.

First, we order the numbers by their sign: 54, 1, 1 2/3, 0.5, 0.03, 0, -1/4, -1, -4, -103.

Then we order the positive numbers in decreasing order: 54, 1 2/3, 1, 0.5, 0.03, 0.

Finally, we order the negative numbers in increasing order: -103, -4, -1, -1/4.

Putting it all together, we have: 54, 1 2/3, 1, 0.5, 0.03, 0, -1/4, -1, -4, -103.

there is a 20% chance that the point chosen at random will lie on bg.

To find the probability that a point chosen at random will be on the line segment bg, we need to consider the length of bg in relation to the length of the entire line segment ak.

Let us assume that ak is a straight line segment, and bg is a smaller segment that lies entirely within it. To find the probability, we need to divide the length of bg by the length of ak.

Let the length of bg be x and the length of ak be y. Then the probability that a point chosen at random will be on bg is:

Probability = Length of bg / Length of ak

Probability = x / y

However, we need to be careful here. If we choose a point anywhere on ak, it may not necessarily lie on bg. There are an infinite number of points on ak, but only one segment bg. Therefore, the probability we are looking for is actually the ratio of the lengths of bg to ak.

So, if we know the lengths of bg and ak, we can find the probability by dividing them. For example, if bg is 2 units long and ak is 10 units long, the probability of choosing a point on bg is:

Probability = 2 / 10

Probability = 0.2 or 20%

In this case, there is a 20% chance that the point chosen at random will lie on bg.

In conclusion, the probability of a point chosen at random on ak being on bg is directly proportional to the length of bg in relation to the length of ak. Therefore, we need to find the ratio of the lengths of the two line segments to determine the probability.

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what is the probability  a point is chosen at random on ak and then the point will be on bg. dont forget to reduce the products?

Answer:

Boy = 40%

Girl = 66.6%

Step-by-step explanation:

1) Work out probability that student is a boy who wears cap

8 boys out of 20 wears a cap so to find the probability we have to do 8 divided by 20

2) Work out probability for girls who doesn't wear spectacles

To find the probability of girls who doesn't wear spectacles we have to do 8 divided by 12

Hope this helps, have a lovely day! :)

Answer:

x = 7

Step-by-step explanation:

The two angles are equal so the opposite sides are equal.

5x-2 =33

Add two to each side.

5x-2+2 = 33+2

5x=35

Divide by 5

5x/5 =35/5

x = 7

The matches of Histogram, Bin, Descriptive Statistics, Mean, Median and Standard Deviation are C, E, D, A, F, G and B respectively.

The Match the definition are given.

Histogram - C). is a graph of the frequency distribution of a set of data

Bin - E). a group in a histogram

Descriptive Statistics - D). values calculated from a data set and used to describe some basic characteristics of the data set

Mean - A). The scatter around a central point

Median - F). the middle value of a sorted set of data

Mode - G). is the most commonly occurring value in a data set

Standard Deviation - B). is a measure of a data’s variability

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4.5y

subtract 2.9y from 7.4y, and you get 4.5y

From the discriminant of the give quadratic equation, the temperature of the machine will part after 50 minutes of operation.

The given equation that models the temperature of the machine is;

T = -0.005x² + 0.45x + 125

Let check if there's a value that exists for T = 135

Putting T = 135 in the given equation,

135 = -0.005x² + 0.45x + 125

We can simplify this to;

0.005x² - 0.45x + 10 = 0

From the general form of quadratic equation which is ax² + bx + c = 0, where a = 0.005, b = -0.45, and c = 10.

The discriminant of this quadratic equation is given by:

D = b² - 4ac

= (-0.45)² - 4(0.005)(10)

= 0.2025 - 0.2

= 0.0025

The discriminant of the equation is positive which indicates we have two roots. Therefore, the temperature of the machine part will cross 135°F at some point during the operation.

We can also find the roots of the quadratic equation using the formula:

[tex]x = (-b \± \sqrt(D)) / 2a[/tex]

Substituting the values of a, b, and D, we get:

[tex]x = (0.45 \± \sqrt(0.0025)) / 2(0.005)\\= (0.45 \± 0.05) / 0.01[/tex]

Taking the positive value, we get:

x = 50

Therefore, the temperature of the machine part will cross 135°F after 50 minutes of operation.

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By looking at the vertex of the graph, we can see that the fourth graph is the correct option.

Here we want to see which one of the given graphs represents the given absolute value function.

Remember that for the absolute value function:

f(x) = |x - a| + b

Has a vertex at the point (a, b) and opens up.

Then in this particular case, with the function f(x)=∣x+1∣−3, the vertex will be at the point (-1, -3), so we just need to identify which one of the given graphs has that vertex, we can see that the correct option is the fourth option.

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Answer: x = -3, y = -4, or (-3, -4)

Step-by-step explanation:

To solve the system of equations, we can substitute the first equation into the second equation, replacing y with 3x + 5:

x + (3x + 5) = -7

Simplifying:

4x + 5 = -7

Subtracting 5 from both sides:

4x = -12

Dividing by 4:

x = -3

Now that we know x = -3, we can substitute that value into either of the original equations to find y:

y = 3(-3) + 5 = -4

Therefore, the solution to the system of equations is x = -3, y = -4, or (-3, -4).

Between 3.26 on the low end and 3.30 on the high end is where UGPAS's middle 50% lies.

Provided that the parabola's vertex is at the origin and that it is symmetric about the y-axis. So, depending on whether the parabola expands upward or downward, the equation can take the form x2 = 4ay or x2 = -4ay.

Because we are interested in the top 5%, the region to the right of the z-score is 0.05. n,... As a result, we can apply the following z-score formula:

z = (x - μ) / σ

x = z * σ + μ

Substituting the values we have, we get:

x = 1.645 * 0.15 + 3.25 = 3.66

Therefore, the z-scores corresponding to the 25th and 75th percentiles are:

z1 = -0.675

z2 = 0.675

Using the same formula as before, we can find the UGPAs corresponding to these z-scares:

x1 = -0.675 * 0.15 + 3.25 = 3.26

x2 = 0.675 * 0.15 + 3.25 = 3.30

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Answer:

To shift the graph of f(x) = |x| to have a domain of [-3, 6], we need to move the left endpoint from -6 to -3 and the right endpoint from 3 to 6.

A translation to the right by 3 units will move the left endpoint of the graph of f(x) to -3, but it will also shift the right endpoint to 6 + 3 = 9, which is outside the desired domain.

A translation to the left by 3 units will move the right endpoint of the graph of f(x) to 3 - 3 = 0, which is outside the desired domain.

A translation upward or downward will not change the domain of the graph, so options B and D can be eliminated.

Therefore, the correct answer is C g(x) = x - 3. This translation will move the left endpoint to -3 and the right endpoint to 6, which is exactly the desired domain.

OR is the perpendicular bisector and R is the vertex of a pair of congruent angles in the diagram.

What is perpendicular bisector and a pair of congruent angles ?

A perpendicular bisector is a geometric tool that is commonly used in mathematics and geometry. It is used to divide a line segment into two equal parts and is constructed by first finding the midpoint of the line segment. The perpendicular bisector is then constructed by drawing a line perpendicular to the line segment at its midpoint.

In geometry, two angles are said to be congruent if they have the same measure. This means that they have the same size and shape, even if they are oriented differently. In other words, if you were to place one angle on top of the other, they would perfectly overlap.

Explanation of the true statements:

In the given diagram ,

PR=RN and OR is perpendicular .

Hence, OR is the perpendicular bisector because a perpendicular bisector is a line that passes through the midpoint of a line segment and intersects it at a right angle, dividing the segment into two equal parts.

The vertex of a pair of congruent angles is the point where the two angle bisectors intersect.
Hence, R is the vertex of a pair of congruent angles in the diagram.

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The probability of choosing a letter from n to q would be = 2/5

Probability is defined as the expression that is used to represent the possibility of an outcome of an event which can be solved with the formula = chosen events/ total outcomes.

The number of cards in the box = 10

The various cards are labelled as follows= j, k, l, m, n, o, p, q, r, and s.

The number of cards from n to q = 4

Therefore the probability that a number from n to q will be chosen = 4/10 = 2/5.

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The inverse of the coefficient matrix is A^-1 = [-2 2]. The solution to the system of equations is x = -1 and y = 1/5.

To solve the system of equations:

2x + 2y = 1

2x - 3y = 0

We can write this system in matrix form as:

[2 2] [x] [1]

[2 -3] [y] = [0]

The coefficient matrix is:

[2 2]

[2 -3]

To find the inverse of the coefficient matrix, we can use the following formula:

A^-1 = (1/|A|) adj(A)

where |A| is the determinant of A and adj(A) is the adjugate of A.

The determinant of the coefficient matrix is:

|A| = (2)(-3) - (2)(2) = -10

The adjugate of the coefficient matrix is:

adj(A) = [-3 2]

[-2 2]

Therefore, the inverse of the coefficient matrix is:

A^-1 = (1/-10) [-3 2]

[-2 2]

Multiplying both sides of the matrix equation by A^-1, we get:

[x] 1 [-3 2] [1]

[y] = -10 [-2 2] [0]

Simplifying the right-hand side, we get:

[x] [-1]

[y] = [1/5]

Therefore, the solution to the system of equations is:

x = -1

y = 1/5

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_____The given question is incomplete, the complete question is given below:

solve the systems of equations by finding the inverse of the coefficient matrix. a. 2x+2y=1 2x-3y-0

The probability that a randomly chosen card that is less than 7 is the 5 of diamonds is 5%.

There are four suits in a standard deck of 52 cards: diamonds, clubs, hearts, and spades. Each suit has 13 cards, with ranks ranging from 2 (low) to 10, jack, queen, king, and ace (high).

If a randomly chosen card from the deck is less than 7, there are only two possibilities: it is either a 2, 3, 4, 5, or 6 of any suit, or it is the 5 of diamonds.

There are 20 cards that are less than 7 in the deck (4 cards of each of the 5 ranks). Out of these 20 cards, only one is the 5 of diamonds.

Therefore, the probability that a randomly chosen card that is less than 7 is the 5 of diamonds is:

1/20 = 0.05 = 5%

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The value of local maximum and local minimum for the function f(x) = x^2/(x -1 ) is equal to f(0) = 0 at x = 0 and f(2) = 4 at x = 4 respectively.

Local maximum and minimum values of the function

f(x) = x^2 / (x - 1),

Use both the first and second derivative tests.

First, let's find the critical points of the function,

By setting its derivative equal to zero and solving for x,

f'(x) = [2x(x - 1) - x^2] / (x - 1)^2

⇒ [2x(x - 1) - x^2] / (x - 1)^2 = 0

Simplifying this expression, we get,

x(x - 2) = 0

This gives us two critical points,

x = 0 and x = 2.

These critical points correspond to local maxima, local minima, or neither.

Use the second derivative test,

f''(x) = [2(x - 1)^2 - 2x(x - 1) + 2x^2] / (x - 1)^3

At x = 0, we have,

f''(0) = 2 / (-1)^3

       = -2

Since the second derivative is negative at x = 0, this critical point corresponds to a local maximum.

f(0) = 0^2/ (0 -1 )

      = 0

At x = 2, we have,

f''(2) = 2 / 1^3

       = 2

Since the second derivative is positive at x = 2, this critical point corresponds to a local minimum.

f(2) = 2^2/ (2 - 1)

     = 4

Therefore, at x = 0, the local maximum value is f(0) = 0, and at x = 2, the local minimum value is f(2) = 4.

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The median wait time for Speed Slide is 2 minutes longer than the median wait time for Wave Machine and the IQR for both rides is 6 minutes.

A box plot, also known as a box and whisker plot, is a graphical representation of a data set that shows the distribution of the data along a number line.

In the box plots show a random sample of wait times for two rides at a water park is shown in figure.

If we compare the wait times in the box plots,

then for, Speed Slide: Median = 11

IQR= Q1 - Q3           (Calculation formula of IQR)

      = 12 - 6

IQR = 6 minutes

for wave slide: median: 9

IQR= Q1 - Q3           (Calculation formula of IQR)

      = 11 - 9

IQR = 2 minutes

The median wait time for Speed Slide is 2 minutes longer than the median wait time for Wave Machine and the IQR for both rides is 6 minutes.

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2 gallons of blue paint are used for the walls, which cover 700 square feet.

What is surface area?

The space occupied by a two-dimensional flat surface is called the area. It is measured in square units. The area occupied by a three-dimensional object by its outer surface is called the surface area.

Let's call the surface area of the trim "T" and the surface area of the walls "W". We know that the ratio of T to W is 1:10, which means that:

T = (1/11) * W

We also know that 2 gallons of blue paint are used for the walls. Let's call the amount of white paint needed for the trim "P" (in pints).

We can use the fact that the total surface area of the room is equal to the surface area of the walls plus the surface area of the trim:

W + T = total surface area

Since T = (1/11) * W, we can substitute and simplify:

W + (1/11) * W = total surface area

(12/11) * W = total surface area

Now we can use the fact that 2 gallons of blue paint are used for the walls to find the surface area of the walls:

2 gallons = 16 pints

2 gallons = W / 350 (since 1 gallon covers 350 square feet)

W = 700 square feet

Now we can use the formula above to find the total surface area of the room:

total surface area = (12/11) * W

total surface area = (12/11) * 700

total surface area = 763.64 square feet

We know that the blue paint covers the walls, so we don't need to worry about that. We only need to find the amount of white paint needed for the trim. Let's call the amount of white paint needed per square foot of trim "p" (in pints). Then the total amount of white paint needed is:

P = p * T

We know that the ratio of the surface area between the trim and the walls is 1:10, so we can use that to find the surface area of the trim:

T = (1/11) * W

T = (1/11) * 700

T = 63.64 square feet

Now we just need to find the amount of white paint needed per square foot of trim. Since the trim is white, we don't need to worry about coverage, so we just need to find the surface area of the trim in square pints:

P = p * T

P = p * 63.64

Finally, we know that 1 gallon of paint is equal to 8 pints, so we can convert the total amount of white paint needed from pints to gallons:

P = p * 63.64

P / 8 = gallons of white paint needed

Putting it all together, we get:

2 gallons of blue paint are used for the walls, which cover 700 square feet.

The total surface area of the room is (12/11) * 700 = 763.64 square feet.

The surface area of the trim is (1/11) * 700 = 63.64 square feet.

The total amount of white paint needed is P = p * 63.64.

The amount of white paint needed in gallons is P / 8.

We don't know the value of p, so we can't solve for P directly. However, we do know that the ratio of the surface area between the trim and the walls is 1:10. This means that the surface area of the trim is 1/11 of the total surface area of the room.

Therefore, we can solve for p as follows:

T = (1/11) * W

63.64 = (1/11) * 700

p = P / T

p = P / 63.64

Hence, 2 gallons of blue paint are used for the walls, which cover 700 square feet.

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The equation that represents the value of the collection after 5 years is:

Value of collection after 5 years = 190 x (1 + 0.06)^5

Explanation:

To calculate the value of the collection after 5 years, we need to use the compound interest formula. This formula is represented as A = P x (1 + r)^n, where P is the principal amount (initial value of the collection), r is the rate of interest (in this case, 6%), and n is the number of years (in this case, 5).

Therefore, the equation for the value of the collection after 5 years is:

Value of collection after 5 years = 190 x (1 + 0.06)^5

This can also be written as:

Value of collection after 5 years = 190 x 1.31 (1.31 is the result of (1 + 0.06)^5)

Therefore, the value of the collection after 5 years is $246.90.

Answer: 254.26

Step-by-step explanation:

a.(1) When both r and s are odd integers, the quadratic polynomial obtained by multiplying out (x - r)(x - s) will have a coefficient of 1 for x^2 term, and both the coefficient of x term and constant term will be odd integers.

(2) When both r and s are even integers, the polynomial obtained by multiplying out (x - r)(x - s) will also have a coefficient of 1 for x^2 term, but the coefficient of x term and constant term will be even integers.

(3) When one of r and s is even and the other is odd, the polynomial obtained by multiplying out (x - r)(x - s) will have a coefficient of 1 for x^2 term, the coefficient of x term will be an odd integer, while the constant term will be an even integer.

b. x^2 - 1253x + 255 cannot be written as a product of two polynomials with integer coefficients.

a. When both r and s are odd integers, the product (x − r)(x − s) will have a coefficient of 1 for x^2 term, and both the coefficient of x term and constant term will be odd integers. This is because the sum of two odd integers and the product of two odd integers is also an odd integer.

When both r and s are even integers, the product (x − r)(x − s) will also have a coefficient of 1 for x^2 term, but the coefficient of x term and constant term will be even integers. This is because the sum of two even integers and the product of two even integers is also an even integer.

When one of r and s is even and the other is odd, the product (x − r)(x − s) will have a coefficient of 1 for x^2 term, and the coefficient of x term will be an odd integer, while the constant term will be an even integer. This is because the sum of an odd and even integer is an odd integer, and the product of an odd and even integer is an even integer.

b. If x^2 - 1253x + 255 can be written as a product of two polynomials with integer coefficients, then we can write it as (x - r)(x - s) where r and s are integers. From part (a), we know that both r and s cannot be odd integers since the coefficient of x term would be odd, but 1253 is an odd integer. Similarly, both r and s cannot be even integers since the constant term would be even, but 255 is an odd integer. Therefore, one of r and s must be odd and the other must be even. However, the difference between an odd integer and an even integer is always odd, so the coefficient of x term in the product (x - r)(x - s) would be odd, which is not equal to the coefficient of x term in x^2 - 1253x + 255. Hence, x^2 - 1253x + 255 cannot be written as a product of two polynomials with integer coefficients.

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The  angles can be named with one letter  are 2.

Angles are a type of geometric shape which are formed when two straight lines intersect at a point. They are measured in degrees, usually between 0 and 360. Angles can be classified as acute, right, obtuse, straight and reflex, and the sum of any three angles in a triangle will always equal 180 degrees. Angles can be used in various ways, from providing stability in construction work to helping to identify other shapes. They can also be used to measure the size and shape of objects and to calculate distances and areas on maps. Angles are an important part of mathematics, geometry and trigonometry, and are used in a variety of applications in science and engineering.

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Answer:

The answer to your problem is, B, 0.5

Step-by-step explanation:

We are given the following table below;

              X             Y               Z             Total

A             32           10             28              70

B              6             5              25              36

C             18            15              7                40

Total       56           30            60              146

As we know that the conditional probability formula of P(A/B) is given by:

P(A/B) =  [tex]\frac{P(AnB)}{P(b)}[/tex]

P(C/Y) = [tex]\frac{P(CnY)}{P(Y)}[/tex]

P ( Y ) = [tex]\frac{30}{146}[/tex] and P(CnY) = [tex]\frac{15}{146}[/tex] [ because of the third column shown ]

Thus, the answer is, B. 0.5

Feel free to ask any questions down below \/ !

Therefore, the distance between the 90 degree angle and the hypotenuse is approximately 0.829 units.

A triangle is a two-dimensional geometric shape that is formed by three straight line segments that connect to form three angles. It is one of the most basic shapes in geometry and has a wide range of applications in mathematics, science, engineering, and everyday life. Triangles can be classified by the length of their sides (equilateral, isosceles, or scalene) and by the size of their angles (acute, right, or obtuse). The study of triangles is an important part of geometry, and their properties and relationships are used in many areas of mathematics and science.

Here,

1. To find HF, we can use the angle bisector theorem, which states that if a line bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the adjacent sides. Let's denote the length of HF as x. Then, by the angle bisector theorem, we have:

JF/FH = JG/HG

Substituting the given values, we get:

15/x = 18/21

Simplifying and solving for x, we get:

x = 15 * 21 / 18

x = 17.5

Therefore, HF is 17.5 cm.

2. Let's denote the length of the hypotenuse as h and the length of the leg opposite the 18-unit perpendicular as a. We can then use the Pythagorean theorem to write:

h² = a²  + 18²

We are told that the hypotenuse is divided into segments of length x and 6 units, so we can write:

h = x + 6

Substituting this expression into the first equation, we get:

(x + 6)² = a² + 18²

We are also told that the leg adjacent to the angle opposite the 4-unit segment is divided into segments of length 4 and (a - 4), so we can write:

a = 4 + (a - 4)

Simplifying this equation, we get:

a = a

Now we can substitute this expression for a into the previous equation and solve for x:

(x + 6)² = (4 + (a - 4))² + 18²

Expanding and simplifying, we get:

x² + 12x - 36 = 0

Using the quadratic formula, we get:

x = (-12 ± √(12² - 4(1)(-36))) / (2(1))

x = (-12 ± √(288)) / 2

x = -6 ± 6√(2)

Since the length of a segment cannot be negative, we take the positive root:

x = -6 + 6sqrt(2)

x ≈ 1.46

Therefore, the value of x is approximately 1.46 units.

3. Let's denote the length of the hypotenuse as h and the length of the leg adjacent to the angle opposite the 9-unit perpendicular as b. We can then use the Pythagorean theorem to write:

h² = b² + 9²

We are told that the hypotenuse is divided into segments of length x and 6 units, so we can write:

h = x + 6

Substituting this expression into the first equation, we get:

(x + 6)² = b² + 9²

Expanding and simplifying, we get:

x² + 12x - b² = 27

We also know that the length of the leg opposite the 9-unit perpendicular is:

a = √(h² - 9²)

= √((x + 6)² - 9²)

= √(x² + 12x + 27)

Now we can use the fact that the tangent of the angle opposite the 9-unit perpendicular is equal to the ratio of the lengths of the opposite and adjacent sides:

tan(θ) = a / b

Substituting the expressions for a and b, we get:

tan(θ) = √(x² + 12x + 27) / (x + 6)

We also know that the tangent of the angle theta is equal to the ratio of the length of the opposite side to the length of the adjacent side:

tan(θ) = 9 / b

Substituting the expression for b, we get:

tan(θ) = 9 / √(h² - 9²)

Substituting the expression for h, we get:

tan(θ) = 9 / √((x + 6)² - 9²)

Since the tangent function is the same for equal angles, we can set these two expressions for the tangent equal to each other:

√(x² + 12x + 27) / (x + 6) = 9 / √((x + 6)² - 9²)

Squaring both sides, we get:

(x² + 12x + 27) / (x + 6)² = 81 / ((x + 6)² - 81)

Cross-multiplying and simplifying, we get:

x⁴ + 36x³ + 297x² - 1458x - 2916 = 0

Using a numerical method such as the Newton-Raphson method or the bisection method, we can find the approximate solution to this equation:

x ≈ 9.449

Therefore, the value of x is approximately 9.449 units.

4. Let's denote the length of the hypotenuse as h and the length of the leg adjacent to the angle opposite the distance we want to find as b. We can use the Pythagorean theorem to write:

h² = b² + d²

We are told that the hypotenuse is divided into segments of length 9 and 4 units, so we can write:

h = 9 + 4 = 13

Substituting this expression into the first equation, we get:

13² = b² + d²

Simplifying and solving for d, we get:

d = √(13² - b²)

Now, we need to find the value of b. We know that the hypotenuse is divided into segments of length 9 and 4 units, so we can use similar triangles to write:

b / 4 = 9 / 13

Simplifying and solving for b, we get:

b = 36 / 13

Substituting this expression for b into the equation we found earlier for d, we get:

d = √(13² - (36/13)²)

Simplifying and finding a common denominator, we get:

d =√ (169*13 - 36²) / 13²

Simplifying further, we get:

d = √(169169 - 3636) / 169

Calculating this expression, we get:

d ≈ 0.829

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The function f(x,y) has only minimum value at (1,0) is -1 and maximum value does not exist.

The given function is f(x,y)=x²+y²-2x

First find the partial derivative with respect to x and y

f'(x)=2x-2

f'(y)=2y

f'(x)=0=f'(y)

2x-2=0

x=1

and y=0

Now we will cheak maxima and minima at (1,0)

f''(x,y)=2 and f"(x,y)=2 and f"(x,y)=0( derivative of first order of x with respect to y)

We know that

rt-s²≥0 and r positive then f is minimum and  r negative maximum

r=2 , t=2 and s=0

rt-s²≥0 and r is positive so f(x,y) is minimum at (1,0)

f(1,0)=1-2=-1

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