a rectangular box has a square base. if the sum of the height and the perimeter of the square base is 14 14 in, what is the maximum possible volume?

Hello and regards 24kendalllove.

The given function is y = x^2 - 2x - 1.

The domain of a quadratic function, like this one, is the set of all values of x for which the function is defined. Since a quadratic function is defined for all real values of x, the domain of this function is all real numbers.

To find the range of the quadratic function, we must first identify whether the parabola opens up or down. In this case, the coefficient of the x^2 term is positive (1), which means that the parabola opens up.

Since the parabola opens up, the vertex of the parabola will be the lowest point on the graph. To find the vertex, we use the formula x = -b / 2a, where a and b are the coefficients of the terms x^2 and x, respectively. In this case, a = 1 and b = -2, so x = -(-2) / (2 * 1) = 1. We then plug this value of x into the function to find the corresponding y value: y = (1)^2 - 2(1) - 1 = -2.

So, the vertex of the parabola is (1, -2). Since the parabola opens up, the range of the function will be all y-values greater than or equal to the y-value of the vertex. Therefore, the range of the function is y ≥ -2.

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Answer:

4.75 kg

Step-by-step explanation:

Bottles of jam given by Sakshi to her friends =7

Bottles of jam prepare by sakshi for herself =12

Total bottles of jam made =7+12=19

Weight of jam in 1 bottles =250 gm

Total weight of jam made by sakshi

=250×19

=4,750 gm

Total weight of jam =4.75 kg.

The given surface is the pyramid and the area of the pyramid is [tex]301.98978[/tex].

The formula to calculate the total surface area of a triangular pyramid is[tex]\frac{1}{2} (a*b)+\frac{3}{2}(b*s)[/tex]

Calculate the area of each triangular face. This can be done using the formula for the area of a triangle: [tex]\frac{1}{2} bh[/tex], where b is the base of the triangle and h is the height of the triangle.

In this case, the base is one of the sides of the base triangle, and the height is the slant height of the pyramid.

Add up the areas of all the triangular faces and the area of the base to get the total surface area of the pyramid.

Add the area of the square.

Therefore the  given surface is the pyramid and the area of the pyramid is [tex]301.98978[/tex].

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There is only one child in the park who does not like to play either cricket or football or soccer.

Additionally, you should ignore any typos or irrelevant parts of the question .Here's how you can solve the given problem :Given that ,In a park, 25 children play every day in the evening.

12 children like to play cricket.8 children like to play football.4 children like to play soccer.Total children who like to play either of these three sports = [tex]12 + 8 + 4 = 24.[/tex]

Therefore, the number of children who do not like to play any of these sports = Total number of children - Number of children who like to play any of these sports= [tex]25 - 24= 1[/tex].

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The value of k for the  logarithmic function f(x)=log0.5 x+k is k = log2 (1/2x).

Here we need to understand that the logarithmic function f(x)=log0.5 x+k can be written in the form f(x)=log0.5 x + log0.5 b.

The given logarithmic function is of the form f(x) = log0.5 x + k.

We want to express this function in terms of a logarithm with base 0.5 and a constant b.

Using the property of logarithms that states that the logarithm of a product is the sum of the logarithms of the factors, we can write:

f(x) = log0.5 x + log0.5 b

where b is a constant that we need to determine.

We want to find a value of b such that the expression above is equivalent to the original function f(x) = log0.5 x + k. We can do this by setting the two expressions equal to each other:

log0.5 x + log0.5 b = log0.5 x + k

b = [tex]0.5^k[/tex]

Substituting this value of b into the expression we obtained earlier gives:

f(x) = log0.5 x + log0.5 (0.5^k)

f(x) = log0.5 (x([tex]0.5^k[/tex]))

Using the property of logarithms that states that the logarithm of a power is the product of the exponent and the logarithm of the base, we can simplify this expression:

f(x) = log2 x - k

We are now given that the function f(x) T z the x-axis, which means that f(x) = 0.

Setting this equal to the expression we obtained above, we get:

log2 x - k = 0

log2 x = k

Solving for k gives:

k = log2 x

Substituting this expression for k back into the original function f(x) = log0.5 x + k, we get:

f(x) = log0.5 x + log0.5 ([tex]0.5^{(log2 x)[/tex])

f(x) = log0.5 (x ( [tex]0.5^{(log2 x)[/tex]))

f(x) = log0.5 (x ( [tex](1/2)^{log2[/tex]))

f(x) = log0.5 (x ( (1/2)))

f(x) = log0.5 (x/2)

Therefore, the value of k for the function f(x) = log0.5 x + k is k = log2 x, and the equivalent expression for the function is f(x) = log0.5 (x/2).

The value of k for the function f(x) = log0.5 x + k is k = log2 (1/2x).

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The point (1, -5) is not a solution to the given system of inequalities.

In order to graph the solution for the given system of inequalities on a coordinate plane, we would use an online graphing calculator to plot the given system of inequalities and then check the point of intersection;

y ≤ 3x + 2          .....equation 1.

y > -2x - 3         .....equation 2.

Based on the graph (see attachment), we can logically deduce that the solution to the given system of inequalities is the shaded region below the solid and dashed line, and the point of intersection of the lines on the graph representing each, which is given by the ordered pair (-1, -2).

Next, we would use the point (1, -5) to test the system of inequalities mathematically:

y ≤ 3x + 2    

-5 ≤ 3(1) + 2

-5 ≤ 5  (True).

y > -2x - 3

-5 > -2(1) - 3

-5 > -5 (False)

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Answer:

the answer is that the lines overlap each other

Step-by-step explanation:

If you put both linear questions into desmos both linear equation have the same slope and x intercept so they would be shown to overlap on the graphing calculator.

Answer:c. They Overlap

Step-by-step explanation: This is because when we change both terms from standard to slope intercept, they are both y=-3/2x -2, causing them to have the same solution, which means they overlap, or they have infinite solutions

We can use the dot product formula to find the angle between two vectors:

cos(theta) = (U dot V) / (|U| * |V|)

where U dot V is the dot product of U and V, and |U| and |V| are the magnitudes of U and V, respectively.

First, let's calculate U dot V:

U dot V = (4 * -6) + (-6 * -4) = -24 - (-24) = 0

Next, let's calculate the magnitudes of U and V:

|U| = sqrt(4^2 + (-6)^2) = sqrt(52)

|V| = sqrt((-6)^2 + (-4)^2) = sqrt(52)

Now we can substitute these values into the formula for cos(theta):

cos(theta) = 0 / (sqrt(52) * sqrt(52)) = 0

Since cos(theta) = 0, this means that the angle between U and V is 90 degrees or π/2 radians.

Answer:

  C.  25

Step-by-step explanation:

You want to know the first quartile as shown in the given box plot.

A box plot has vertical lines at (left to right) ...

The left end of the "box" is the first quartile.

The first quartile of the dataset represented by this box plot is 25.

At the Digital animation studio, character designers outnumber interns by 320.

Let's break down the information given in the circle graph and find out how many more character designers there are than interns.

1. Determine the number of employees for each job type by multiplying the percentage by the total number of employees (1600):
- Interns: 5% * 1600 = 80 employees
- Set Shading: 10% * 1600 = 160 employees
- Character Shading: 10% * 1600 = 160 employees
- Story Artist: 20% * 1600 = 320 employees
- Character Design: 25% * 1600 = 400 employees
- Animator: 30% * 1600 = 480 employees

2. To find how many more character designers there are than interns, subtract the number of interns from the number of character designers:
- Character Design - Interns = 400 - 80 = 320 employees

In conclusion, there are 320 more character designers than interns at the digital animation company.

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55% percentage  of the cupcakes had sprinkles.

A percentage is a number, οr ratiο, that expresses a fractiοn οut οf 100. Percentages are easy tο recοgnize because they are always fοllοwed by a percent sign (%). A percentage is an alternative way tο represent a fractiοn οut οf 100 instead οf using a traditiοnal fractiοn fοrmat. Fοr example, we can write ½ οr 50/100 tο represent a half fractiοn, using percentage we can alsο write 50%.

Add the total no. of cupcakes = 11 + 9 = 20 cupcakes

Out of 20 cupcakes only 11 cupcakes had sprinkles = 11/20

To find percentage multiply be 100 percent = 11/20 × 100

                                                                          = 55%

Thus, 55% percentage  of the cupcakes had sprinkles.

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The linear scale factor of the enlargement is 6.7 (580 cm³/87 cm³). To the nearest hundredth, the linear scale factor of the enlargement is 6.70.
The surface area scale factor of the enlargement is 33.2 (290 cm³/8.7 cm³). To the nearest hundredth, the surface area scale factor of the enlargement is 33.17.

Surface area is the combined area of all the faces of a three-dimensional object. It is the area that is visible when looking at the outside of the object. It is often used to calculate the amount of material needed for a certain project or product. It is also used to calculate the cost and energy required to heat or cool a certain space. Surface area can be calculated using geometry and calculus, or it can be measured directly.

1. Linear scale factor: The linear scale factor of the enlargement is the ratio of the volume of the larger jar to the volume of the smaller jar. The volume of the larger jar is 0.58 L which is equivalent to 580 cm³. The volume of the smaller jar is 87 cm³. Therefore, the linear scale factor of the enlargement is 6.7 (580 cm³/87 cm³). To the nearest hundredth, the linear scale factor of the enlargement is 6.70.

2. Surface area scale factor: The surface area scale factor of the enlargement is the ratio of the surface area of the larger jar to the surface area of the smaller jar. The surface area of a jar depends on its radius. Since the radius of the larger jar is larger than the radius of the smaller jar, the surface area of the larger jar is larger than the surface area of the smaller jar.

Therefore, the surface area scale factor of the enlargement is greater than 1. To calculate this factor, we can use the formula for the surface area of a cylinder: A = 2πrh, where r is the radius and h is the height. The height of both jars is the same, so we can calculate the surface area scale factor by dividing the radius of the larger jar by the radius of the smaller jar. The radius of the larger jar is 0.29 L, which is equivalent to 290 cm³. The radius of the smaller jar is 8.7 cm³.

Therefore, the surface area scale factor of the enlargement is 33.2 (290 cm³/8.7 cm³). To the nearest hundredth, the surface area scale factor of the enlargement is 33.17.
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Complete questions as follows-
Use the following information to answer the next two questions Raj Jars Ltd. Sells different types of similar jars. One of their jars has a volume of 87 cm³ and another has a volume of 0.58 L. 1. What is the linear scale factor of the enlargement to the nearest hundredth? Remember 1L = 1000 cm³ 2. What is the surface area scale factor of the enlargement to the nearest hundredth? Remember 1L = 1000 cm³

In this case, the IQR should be used because the data in both Sunny Town and Desert Landing is not normally distributed and has different shapes.

What is outliers ?

Outliers are pieces of information that dramatically deviate from the rest of the information in a dataset. These could be values that are atypically high or low or values which vary significantly from the data's central trend. Outliers can appear for a number of causes, including measurement or record errors, inherent data variability, or unusual events. It is frequently required to recognize and effectively handle outliers because of their potential to significantly affect statistical analyses.

given,

IQR, as it can provide a better measurement of the variability of the middle 50% of the data and is less impacted by outliers.

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[tex]\huge\bold{Solution }[/tex]

[tex]\large\mathfrak{given}[/tex]

[tex] \large \: \mathfrak{ {formula}}[/tex]

Let height be : x

[tex]\sf\Rightarrow{ \: 968 = 2\pi \: r(r \: + h) } \: [/tex]

[tex]\sf\Rightarrow{968 = 2 \times \frac{22}{7} \times 7 \times (7 + x) }[/tex]

[tex]\sf\Rightarrow{ 968 = 2 \times 22 \times(7 + x)}[/tex]

[tex]\sf\Rightarrow{ 968 = 44(7 + x)}[/tex]

[tex]\sf\Rightarrow{ \frac{968}{44} = 7 + x}[/tex]

[tex]\sf\Rightarrow{22 = 7 + x }[/tex]

[tex]\sf\Rightarrow{22 - 7 = x }[/tex]

[tex]\sf\Rightarrow{x = 15 }[/tex]

[tex]\bf\Rightarrow{ x = 15}[/tex]

[tex]\sf\Rightarrow{ \underline{{ height \: = 15}}} \\ \\ \sf \: option \: A is \: correct \: [/tex]

[tex] {\underline{ \rule {200pt}{6pt}}}[/tex]

Answer:

(a) To calculate Mr. James's total wage for a basic week, we can use the following formula:

The total wage for the basic week = Basic rate x Number of hours worked

Given that Mr. James works 40 hours a week at a rate of $16 per hour, his total wage for a basic week would be:

Total wage for basic week = $16 x 40 = $640

Therefore, Mr. James's total wage for a basic week is $640.

(b) To calculate Mr. James's wage for a week in which he worked 47 hours, we need to consider his basic and overtime rates. Since his overtime rate is $4 per hour more than his basic rate, his overtime rate would be:

Overtime rate = Basic rate + $4

Overtime rate = $16 + $4 = $20 per hour

Now, we can calculate his wage for the week as follows:

The wage for the week = (Number of basic hours x Basic rate) + (Number of overtime hours x Overtime rate)

Wage for the week = (40 x $16) + (7 x $20)

Wage for the week = $640 + $140

Wage for the week = $780

Therefore, Mr. James's wage for a week in which he worked 47 hours is $780.

(c) To find the number of hours Mr. James worked during one week if he was paid a wage of $860, we can use the same formula as in part (b) but rearrange it to solve for the number of hours:

Number of hours worked = (Wage for the week - (Number of basic hours x Basic rate)) / (Overtime rate - Basic rate)

Number of hours worked = ($860 - (40 x $16)) / ($20 - $16)

Number of hours worked = ($860 - $640) / $4

Number of hours worked = $220 / $4

Number of hours worked = 55

Therefore, Mr. James worked 55 hours during the week, for which he was paid $860.

Answer:

A) i think 64,000

B) 66,800 or 66,900

C)53.75 maybe sorry if im incorrect for all

Step-by-step explanation:

A) James works a basic week for 40 hours at the rate of $1,600 an hour.

therefore, total wage of James for a basic week will be = $1,600 × 40 = $64,000

B) Now he worked for 47 hours

his basic week work is 40 hours

No of hours he worked extra = 47 - 40 = 7 hours

Now his wage for 7 hours overtime at the rate of $400 will be = $400 ×7 = $2,800

and his wage for basic week for 40 hours is $64,000 as we have calculated above.

therefore, his total wage for a week in which he worked for 47 hours = $64,000 + $2,800 = $66,800

C) Now he worked for one week and was paid a wage of $86,000

we will calculate the number of hours he worked in a week by unitary method

For 40 hours work he used to get $64,000 Therefore for 1 hour work he will get $1600

Now,

$1600 is earned by working for 1 hour

$86,000 is earned by working for = 53.75

He worked for 53.75 hours to get $86,000

Again sorry if incorrect, i gave it all that i could. If it was correct then good but if wrong im am super sorry anyways, Have a great weekend/day!

Answer:

  I = 12my -p

Step-by-step explanation:

You want to express the interest I on a mortgage of principal p that has a monthly payment of m for y years.

The number of monthly payments in y years is 12y.

The value of those monthly payments is (12y)(m).

The interest paid is the difference between the value of payments and the principal amount of the loan:

  I = 12my -p

The formula for calculating the angle of the related regression line in degrees is:

angle = (180/π) * arctan(k)

The angle of the related regression line (model) in degrees can be calculated using the coefficient involving a column of the independent variable (k) and the arctangent function.

Arctang of the coefficient (k) gives the angle in radians between the horizontal axis and the regression line. To convert this angle to degrees, we can multiply the angle in radians by 180/π.

Therefore, the formula for calculating the angle of the related regression line in degrees is:

angle = (180/π) * arctan(k)

where arctan(k) is the arc tangent of the coefficient associated with a column of the independent variable.

This formula gives the angle of the regression line to the horizontal axis.

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To check whether the point (2, √5) lies on the circle centered at the origin with radius 3, we can use the distance formula for a point (x, y) on the circle:

d = √((x - 0)^2 + (y - 0)^2)

Since the center of the circle is at the origin, the x-coordinate is 0 and the y-coordinate is 0. The radius is given as 3. So, substituting these values in the above formula, we get:

3 = √((2 - 0)^2 + (√5 - 0)^2)

Simplifying the right side of the equation:

3 = √(4 + 5)

3 = √9

3 = 3

Since both sides of the equation are equal, the point (2, √5) lies on the circle centered at the origin with radius 3.

The geometric sequence for [tex]a_{1} = -8[/tex], r = 1/2, and n=9 is -1/32.

The geometric sequence formula has the general form [tex]a_{n} = a_{1} * r^{n-1}[/tex], where r denotes the common ratio, [tex]a_{1}[/tex] denotes the first term, and n denotes the term's position in the series.

To find the geometric sequence, we can use the formula,

[tex]a_{n} = a_{1} * r^{n-1}[/tex]

where [tex]a_{n}[/tex] is the geometric sequence of n term

           [tex]a_{1}[/tex] = first term

           n = term number

In given problem [tex]a_{1}[/tex] = -8, r = 1/2, n=9

Put all values in the above equation,

[tex]a_{n} = (-8) * (\frac{1}{2})^{8-1} \\a_{n} = (-8) * (\frac{1}{2})^{7} \\a_{n} = (-8) * \frac{1}{2^{7} } \\a_{n} = (-8) *\frac{1}{128} \\a_{n} = \frac{-1}{32}[/tex]

Therefore the [tex]9^{th}[/tex] term of the geometric sequence is -1/32.

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The mean of the fitted exponential distribution is 66.67.

To find the mean of the fitted exponential distribution, we first need to estimate the parameter lambda using maximum likelihood estimation.

The probability density function of the exponential distribution is given by

f(x; lambda) = lambda * exp(-lambda * x)

where x is the claim size and lambda is the parameter to be estimated.

The likelihood function for the observed data is the product of the individual probabilities of each claim

L(lambda) = lambda^n * exp(-lambda * sum(x_i))

where n is the number of observed claims and x_i is the i-th claim size.

The log-likelihood function is given by:

ln L(lambda) = n * ln(lambda) - lambda * sum(x_i)

To estimate the parameter lambda, we need to maximize the log-likelihood function with respect to lambda:

d/d(lambda) ln L(lambda) = n/lambda - sum(x_i) = 0

Solving for lambda, we get

lambda = n / sum(x_i)

Substituting the observed values, we get

lambda = 6 / (20 + 45 + 50 + 80 + 100 + 2*100) = 0.015

Therefore, the estimated parameter of the fitted exponential distribution is lambda = 0.015.

The mean of the exponential distribution is given by

E(X) = 1/lambda

Substituting the estimated value of lambda, we get

E(X) = 1/0.015 = 66.67

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After converting 5²/53³  in simplest form we get 25/148,877 as the correct answer.

To simplify the expression 5²/53³, we first evaluate the exponents of 5 and 53.

5² means 5 multiplied by itself, or 5 × 5, which equals 25.

53³ means 53 multiplied by itself three times, or 53 × 53 × 53. This can be calculated using a calculator or by multiplying the numbers out by hand. The result is 148,877.

So, we can rewrite the expression 5²/53³ as: 25/148,877

This expression cannot be simplified any further because 25 and 148,877 have no common factors other than 1.

Therefore, the expression in its simplest form is 25/148,877.

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Answer:

B

Step-by-step explanation:

you would fill in x for 0 and 3 times 0 is 0

then it would be zero plus 1 which would be !

Answer:

B. 1

Explanation:
3(0)+1

0+1=1

Answer:

[tex]≈7.07 \: {ft}^{2} [/tex]

Step-by-step explanation:

Given:

d = 3 ft

Find: A (area) - ?

First, we can find the length of the radius, since we know the diameter:

r = 0,5× d

r = 0,5 × 3 = 1,5 ft

[tex]a = \pi {r}^{2} = \pi \times( {1.5})^{2} = 2.25\pi ≈7.07 \: {ft}^{2} [/tex]

The area of the circular tablecloth is approximately 7 square feet.

To find the area of a circle, you use the formula A = πr^2, where A is the area and r is the radius of the circle. The diameter of the tablecloth is 3 feet, so the radius is 1.5 feet. Plugging this into the formula, we get A = π(1.5)^2 ≈ 7.07 square feet. However, the question asks for the measurement closest to the area of the tablecloth in feet, so we can round the answer to 7 square feet.

Therefore, the area of the tablecloth is approximately 7 square feet.

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Answer:

9(2x + 1) < 9x - 18

9(2x + 1) < 9(x - 2)

2x + 1 < x - 2

x < -3

So -4 is in the solution set.

The function  f(x) = [tex]\frac{3}{x^2+2}[/tex] is even.

What is function?

A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.

Here the given function is f(x) = [tex]\frac{3}{x^2+2}[/tex]

Now put x = -x then

=> f(-x) = [tex]\frac{3}{(-x)^2+2}[/tex]

=> f(-x) = [tex]\frac{3}{x^2+2}[/tex]

=> f(-x)=f(x)

Here the given f(-x)=f(x) then the function is even.

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Adjacent angles are a pair of angles that have a similar vertex and side and add up to 180 degrees. Contrarily, complementary angles are two angles that together measure 90 degrees and have a number of mathematical uses.

There are numerous angles created when two lines intersect. When two of these angles are referred to as neighbouring, it signifies that they have a similar vertex and side. A linear pair of angles is any two angles that are next to one another on one side of the intersection.

The sum of two linear angles is 180 degrees. As a result, it is simple to determine the measure of another angle if you know the size of one. Geometry relies on linear pairings of angles to solve problems involving angles and lines, hence they are crucial.

Complementary angles, on the other hand, are two angles that sum up to 90 degrees. Although they are not need to be, they can be adjacent angles. When two angles are parallel to one another, the sum of their respective measures is 90 degrees.

Trigonometry and geometry are two areas of mathematics where complementary angles are helpful. In issues involving right triangles, where one of the angles is always 90 degrees, they are frequently used.

In conclusion, neighbouring angles are a pair of parallel angles that have the same vertex and side and add up to 180 degrees. Contrarily, complementary angles are two angles that together measure 90 degrees and have a number of mathematical uses.

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By answering the presented question, we may conclude that (1/7) × 23 parallelograms inches x (1/6) inches Equals 23/42 inches to the power of 2

In Euclidean geometry, a parallelogram is a simple quadrilateral with two sets of parallel sides. A parallelogram is a kind of quadrilateral in which both sets of opposite sides are parallel and equal. Parallelograms are classified into four types, three of which are unique. The four distinct shapes are parallelograms, squares, rectangles, and rhombuses. A quadrilateral is a parallelogram when it has two sets of parallel sides. The opposing sides and angles of a parallelogram are both the same length. The internal angles on the same side of the horizontal line are also angles. The total number of internal angles is 360.

Let's start with the formula for parallelogram area:

Base x Height = Area

We know that the parallelogram's height is 1/6 inch and its area is 23/42 inch to the power of 2. So, by plugging these values into the formula, we get:

23/42 inches multiplied by 2 Equals base x 1/6 inch

6 x 23/42 inches multiplied by 2 = base

(6/42) x 23 inches multiplied by 2 = base

base = (1/7) x 23 inches

Base x Height = Area

(1/7) × 23 inches x (1/6) inches Equals 23/42 inches to the power of 2

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Answer:

To find the circumcenter of a triangle, we need to find the point where the perpendicular bisectors of the sides of the triangle intersect.

Let's start by finding the equation of the line passing through the midpoint of the segment AB and perpendicular to AB. The midpoint of AB is ((2+8)/2, (4+2)/2) = (5,3), and the slope of AB is (2-4)/(8-2) = -1/3. The slope of a line perpendicular to AB is the negative reciprocal of -3, which is 3. So, the equation of the line passing through (5,3) and perpendicular to AB is y - 3 = 3(x - 5), or y = 3x - 12.

Similarly, we can find the equation of the line passing through the midpoint of BC and perpendicular to BC. The midpoint of BC is ((8+4)/2, (2-2)/2) = (6,0), and the slope of BC is (-2-2)/(4-8) = 1/2. The slope of a line perpendicular to BC is the negative reciprocal of 1/2, which is -2. So, the equation of the line passing through (6,0) and perpendicular to BC is y - 0 = -2(x - 6), or y = -2x + 12.

Finally, we can find the equation of the line passing through the midpoint of AC and perpendicular to AC. The midpoint of AC is ((2+4)/2, (4-2)/2) = (3,1), and the slope of AC is (-2-4)/(4-2) = -3/2. The slope of a line perpendicular to AC is the negative reciprocal of -3/2, which is 2/3. So, the equation of the line passing through (3,1) and perpendicular to AC is y - 1 = (2/3)(x - 3), or y = (2/3)x - 1/3.

Now, we need to find the intersection point of these three lines, which will give us the circumcenter of the triangle. To do this, we can solve the system of equations:

y = 3x - 12

y = -2x + 12

y = (2/3)x - 1/3

Substituting the first equation into the second equation, we get:

3x - 12 = -2x + 12

5x = 24

x = 24/5

Substituting x = 24/5 into the first equation, we get:

y = 3(24/5) - 12 = 36/5

So, the circumcenter of the triangle is the point (24/5, 36/5)