uppose m professors randomly choose from n time slots to hold their final exams. If two professors pick the same time slot, we say that they are in conflict. (If three professors all pick the same time slot, that gives three pairs of professors in conflict.) What is the expected number of pairs of professors in conflict? Your answer should depend on m and n.

Answer:

x=15

Step-by-step explanation:

use the side splitter theorem

(x-6)/x = 3/5 (cross-multiply)

3x=5(x-6) distributive property

3x=5x-30

2x = 30

x = 15

Answer

Initial velocity = 32 ft/s

Initial height = 24 ft.

Explanation

The function gives us the function that models the height, h, in feet for the ball as a function of time, t, in seconds.

h(t) = -16t (t - 2) + 24

We are then asked to find the initial velocity and the initial height of the ball.

This means we find the velocity and the height of the ball at t = 0

Velocity is given as the first derivative of the height function

v = (dh/dt)

h(t) = -16t (t - 2) + 24

h(t) = -16t² + 32t + 24

v = (dh/dt) = -32t + 32

When t = 0

v = -32t + 32 = -32 (0) + 32 = 0 + 32 = 32 ft/s

Initial velocity = 32 ft/s

For the initial height, t = 0

h(t) = -16t (t - 2) + 24

h(t) = -16t² + 32t + 24

h(0) = -16(0²) + 32(0) + 24

h(0) = 0 + 0 + 24

h(0) = 24 ft

Initial height = 24 ft.

Answer:

The initial velocity of the ball is 32 feet per second.

The initial height of the ball is 24 feet.

Step-by-step explanation:

Velocity is the rate of change of distance with respect to time.

Therefore, to find the equation for velocity, differentiate the height function with respect to time.

[tex]\begin{aligned} h(t) &= -16t(t - 2) + 24\\&= -16t^2+32t+24\\\\\implies v=h'(t)&=-32t+32\end{aligned}[/tex]

The initial velocity of the ball is its velocity when t = 0.

Therefore, at time t = 0, the velocity is:

[tex]\begin{aligned}\implies v=h'(0)&=-32(0)+32\\&=32\sf\;ft\;s^{-1}\end{aligned}[/tex]

Therefore, the initial velocity of the ball is 32 feet per second.

The initial height of the ball is the height when t = 0.

Therefore, at time t = 0, the height of the ball is:

[tex]\begin{aligned} \implies h(0) &= -16(0)(0 - 2) + 24\\&= 0+24\\&=24\sf\;ft\end{aligned}[/tex]

Therefore, the initial height of the ball is 24 feet.

The T(U) is a subspace of W.

Since U is a subspace of V, the zero vector of V, 0v, is in U. Since T is linear, T(0v) = 0w, where 0w is the zero vector of W. So 0w is in T(U). Therefore, T(U) is a subspace of W.

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3BE² + 3DF²- (2AC²- AB) is the answer of the following question

The calculation is as follows

Let E and F be the feet of the perpendiculars from B and D, respectively, to AC. We can use the fact that the area of a quadrilateral is equal to half the product of the diagonals multiplied by the sine of the angle between them to find the length of the diagonal BD.

Since ABCS is a quadrilateral, we have:

Area of ABCS = (1/2) * AC * BD * sin(angle between AC and BD)

Substituting the given values, we get:

924 = (1/2) * 33 * BD * sin(angle between AC and BD)

sin(angle between AC and BD) = 924 / (16.5 * BD)

Now, consider triangles ABC and ACD. Using the Pythagorean theorem, we can write:

AB² + BC² = AC² (1)

CD²+ BC² = AC² (2)

Adding equations (1) and (2), we get:

AB² + 2BC²+ CD²= 2AC²

Substituting AC = 33 and rearranging, we get:

BC² = (2AC²- AB² - CD²) / 2

We can also write:

BE²= AB²- AE² (3)

DF² = CD²- CF²(4)

Adding equations (3) and (4), we get:

BE² + DF²= AB²+ CD² - AE²- CF²

Substituting BC²from earlier, we get:

BE²+ DF² = 2AC² - BC²- AE²- CF²

We want to find BE + DF. Squaring both sides of equation (3), we get:

BE²= AB² - AE²

AE²= AB²- BE²

Similarly, squaring both sides of equation (4), we get:

DF²= CD²- CF²

CF²= CD² - DF²

Substituting these expressions into the equation for BE²+ DF², we get:

BE²+ DF² = 2AC² - BC² - (AB² - BE²) - (CD²- DF²)

Simplifying, we get:

BE² + DF²= 2AC² - BC²- AB²- CD² + 2BE² + 2DF²

Collecting like terms, we get:

BE²+ DF²- 2BE² - 2DF²= 2AC²- BC² - AB² - CD²

Simplifying, we get:

BE²- DF² = 2AC²- BC²- AB²- CD²- 2BE² - 2DF²

Substituting the values we know, we get:

BE²- DF²= 2(33)²- BC²- AB² - CD²- 2BE²- 2DF²

Rearranging, we get:

3BE²+ 3DF² - BC²- AB²- CD²= 2(33)² - 924

Substituting BC^2 from earlier, we get:

3BE²+ 3DF² - (2AC²- AB)

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The solid has a surface area of about 401.92 cm2.

A three-dimensional shape's surface area is the sum of all of its faces. The surface area of a shape can be calculated by finding the area of each face and combining them.

The surface areas of the cylinder and the hemisphere must be added to determine the solid's surface area.

The cylinder's surface area is equal to 2rh + 2r2, where r is the cylinder's radius and h is its height.

The hemisphere's surface area is equal to 2r2, where r is the hemisphere's radius.

As the cylinder's and hemisphere's radiuses are equal, we can sum their two surface areas as follows:

The solid's surface area is equal to 2rh plus 2r2 plus 2r2.

= 2πrh + 4πr²

Solid's surface area is equal to 2(4)(8) + 4(4)(2).

= 64π + 64π

= 128π

≈ 401.92 cm²

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Answer:

32.1m

Step-by-step explanation:

[tex]2\pi r*(\frac{x}{360})=[/tex] arc length of sector of a circle

2 radii + arc length of a sector of a circle= perimeter of the sector

x=90 degrees

r=9

substitute values to find the arc length

[tex]2(9)*\pi *(\frac{90}{360} )= arc length\\arclength=14.1\\[/tex][tex]14.1+2(9)=32.1[/tex]

The number of positive integers with exactly four decimal digits, that is, positive integers between 1000 and 9999 inclusive, are not divisible by either 5 or 7 is 4680.

Step by step explanation:
The number of positive integers with exactly four decimal digits between 1000 and 9999 inclusive can be obtained as follows:

Total number of four decimal digits = 9999 − 1000 + 1 = 9000
Numbers that are multiples of 5 are obtained by starting with 1000 and adding 5, 10, 15, 20, ..., 1995, that is, 5k, where k = 1, 2, 3, ..., 399.

Therefore, the number of positive integers with exactly four decimal digits that are multiples of 5 is 399.

Numbers that are multiples of 7 are obtained by starting with 1001 and adding 7, 14, 21, 28, ..., 1428, that is, 7m, where m = 1, 2, 3, ..., 204.

Therefore, the number of positive integers with exactly four decimal digits that are multiples of 7 is 204.

Note that some numbers in the interval [1000, 9999] are divisible by both 5 and 7. Since 5 and 7 are relatively prime, the product of any number of the form 5k by a number of the form 7m is a multiple of 5 × 7 = 35.

The numbers of the form 35n in the interval [1000, 9999] are

1035, 1070, 1105, 1140, ..., 9945, 9980.
We can check that there are 285 numbers of this form.

To find the number of positive integers with exactly four decimal digits that are not divisible by either 5 or 7, we will subtract the number of multiples of 5 and 7 and add the number of multiples of 35.

Therefore, the number of positive integers with exactly four decimal digits, that is, positive integers between 1000 and 9999 inclusive, are not divisible by either 5 or 7 is

9000 - 399 - 204 + 285 = 4680.

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Yes, there is a chance that the bottled water you are drinking is just tap water packaged in a bottle. According to a study by the Natural Resources Defense Council, 25% of bottled water is just tap water that has been packaged in a bottle. [Scientific American, July 2003]

Bottled water is drinking water that has been packaged in bottles or containers for sale. Bottled water can come from a variety of sources, such as municipal supplies, springs, wells, and other sources. In some cases, bottled water is treated or purified to remove impurities and contaminants. However, not all bottled water is purified or treated, and some may be just tap water that has been packaged in a bottle.

Purified water is water that has been treated to remove impurities and contaminants. Purified water can come from any source, including municipal supplies, wells, and other sources. Purification methods may include filtration, reverse osmosis, distillation, or other methods. Purified water is commonly used in medical and industrial applications, as well as in homes for drinking and cooking.

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The sample size is 12 with an error calculation of 5 if 97% of the confidence interval is around a sample mean of 31. 3.

Percent confidence = 97

Sample mean =  31. 3

Standard deviation = 7. 6

Sample size = 40

Error calculation = 5

The formula for a confidence interval using the t-distribution is:

CI = x ± tα/2 * (s/√n)

CI = 31.3 ± 2.423 * (7.6/√40)

CI = 31.3 ± 3.940

CI = (27.36, 35.24)

The population mean with an error of 5,

n = (Zα/2 * σ / E)^2

Assuming a 95% confidence level, the sample size is,

n = (1.96 * 7.6 / 5)^2

n = 11.69

Therefore we can conclude that the sample size is 12 with an error calculation of 5.

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The inequality that shows the remaining distance Rina must climb to reach the peak is 0 < d < P

Let's assume that the distance from Rina's current position to the peak of the mountain is "P" feet and the distance from her current position to the base camp is "B" feet.

Then, the remaining distance, "d", that she must climb to reach the peak can be calculated as follows

d = P - B

Since Rina has not yet reached the base camp, the distance she has traveled so far is less than the distance to the base camp, which means:

B > 0

Substituting this inequality in the equation for "d", we get

d = P - B < P

Therefore, the inequality that shows the remaining distance Rina must climb to reach the peak is:

0 < d < P

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The width of the rectangle whose length of one side is 7.8 feet and the area enclosed is 5 feet is given as 40/7 feet.

Area refers to the field on the ground which is enclosed by the closed polygon. It is defined within the boundary of the closed polygon. Open polygons cannot have a deterministic area. In the given problem, the length of one side is 7/8 feet and the area of the rectangle is 5 feet.

It is known that area of rectangle is equal to the product of its length and its width.

∴Area of rectangle = Length × width

⇒ 5 = 7/8 × width

⇒ width = (5 × 8)÷7 = 40/7 feet

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well, we know the discount is just 30 - 21 = 9, so hmm if we take 30(origin amount) to be the 100%, what's 9 off of it in percentage?

[tex]\begin{array}{ccll} Amount&\%\\ \cline{1-2} 30 & 100\\ 9& x \end{array} \implies \cfrac{30}{9}~~=~~\cfrac{100}{x} \\\\\\ 30x=900\implies x=\cfrac{900}{30}\implies x=30[/tex]

The star Proxima Centauri is 4.2 light-years away from Earth, making it the sun's nearest rival. The word "nearest" means "nearest" in Spanish.

"A method to find a single unit value from a multiple unit value and to find a multiple unit value from a single unit value."

We always count the unit or amount value first and then calculate the more or less amount value.

For this reason, this procedure is called a unified procedure.

Many set values ​​are found by multiplying the set value by the number of sets.

A set value is obtained by dividing many set values ​​by the number of sets.

Hence, The star Proxima Centauri is 4.2 light-years away from Earth, making it the sun's nearest rival. The word "nearest" means "nearest" in Spanish.

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the answer to the problem that you need to is 1024

Therefore represented by the point of intersection, which is (10, 7). All products are sold at this price and quantity, clearing the market.

A function is a mathematical concept that links an input (usually denoted by the variable x) to an output (usually denoted by the variable y or f(x)) in a well-defined way. A function can be compared to a computer that processes inputs and outputs in accordance with rules or guidelines. Each input value (also known as the domain) in a function is connected to precisely one output value (also known as the range). For instance, the linear function f(x) = 2x multiplies the input value x by two to create the output value.

given

f(x) = -0.5x + 12

The supply role is also provided by:

g(x) = 0.5x + 2

We must put these two functions equal to one another and then solve for x to determine where these two functions intersect:

-0.5x + 12 = 0.5x + 2

By multiplying both parts by 0.5, we get:

12 = x + 2

x = 10

The position at which the supply and demand functions intersect is therefore (10, f(10)) or (10, g(10)). We can enter the value x = 10 into either of the following functions to obtain the appropriate y-coordinate:

f(10) = -0.5(10) + 12 = 7

g(10) = 0.5(10) + 2 = 7

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Answer:

Total number bags is 36

1:2:3 the answer

You can make all four sides of each trapezoid different. Since a trapezoid has four sides, we can change the length of each of the remaining two sides to create different trapezoids. option B is correct.

A trapezoid is a four-sided polygon with two parallel sides and two non-parallel sides. It is sometimes called a trapezium outside of North America. The parallel sides of a trapezoid are called bases, while the non-parallel sides are called legs. The height of a trapezoid is the perpendicular distance between the bases. The area of a trapezoid is calculated by taking the average of the bases and multiplying it by the height. Trapezoids are commonly encountered in geometry and are used in many real-world applications, such as in construction and engineering.

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The point-slope form of the line equation that passes through the points (- 9, - 9) and (- 8, 1) is equals to the y= 10x + 81.

The standard equation of a line is, y = mx + b where m is the slope and b is the y-intercept. This is a very useful form of the linear equation when comparing lines. We have to determine the equation of the line that passes through the points (- 9, - 9) and (- 8, 1). First we determine the value of slope of line. The formula for slope of line that passing through the points (x₁,y₁),(x₂,y₂) is [tex]m = \frac{y_2 - y_1}{x_2- x_1} [/tex]

here, x₁ = -9, y₁ = -9 , x₂ = -8, y₂ = 1, so

[tex]m = \frac{1-(-9)}{-8-(-9)} = \frac{10}{1} = 10[/tex]

The point-slope form for line passing through points (x₁,y₁),(x₂,y₂) is y − y₁ = m(x − x₁)

=> y - (-9) = 10( x -(-9))

=> y + 9 = 10( x + 9)

=> y + 9 = 10x + 90

=> y = 10x + 81

Hence required equation is y = 10x + 81.

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Side BC is the hypotenuse of the right triangle ABC. Against A: The side that is opposed to angle A is side BC. Adjacent A: Side AB is the side that faces angle A. The side that is in opposition to angle C is side AB.

The opposite side of any right-angled triangle, ABC, is referred to as the hypotenuse. Here, we adhere to the tradition that the side opposite angle A is designated with the letter a. Both sides opposing B and C are given the letters b and c, respectively.

When a and b if the lengths of the legs of a right triangle are and the hypotenuse's length is c, then the sum of the squares of the legs' lengths equals the square of the hypotenuse's length.

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Answer:

58.24

Step-by-step explanation:

1456 ÷ 25 = 58.24

Answer:

we bought 12 $3 clearance mystery books and 5 $8 regular-priced science fiction books.

Step-by-step explanation:

Each pair of children gets to bat for 7.5 minutes.

To find out how much time each pair gets to bat, we need to divide the total session time by the number of pairs of children who bat.

Number of pairs of children who bat = 6 groups x 1 pair/group = 6 pairs

Total time for the session = 45 minutes

Time per pair of children who bat = Total time / Number of pairs of children who bat

= 45 minutes / 6 pairs

= 7.5 minutes per pair

Therefore, each pair of children gets to bat for 7.5 minutes.

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If the set of expressions represents measures of the sides of a triangle x, 4, 6 , the range of possible measures of x is 2 < x < 10.

To determine the range of possible measures of X if the set of expressions represents measures of the sides of a triangle x, 4, 6, we need to use the triangle inequality theorem. According to this theorem, in a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Mathematically, this can be expressed as:

x + 4 > 6

x + 6 > 4

4 + 6 > x

Simplifying these inequalities, we get:

x > 2

x > -2

x < 10

The first two inequalities indicate that x must be greater than 2, since the sum of any two sides of a triangle must be greater than the third side. The third inequality indicates that x must be less than 10, since the longest side of a triangle cannot be greater than the sum of the other two sides.

This means that x can take any value between 2 and 10, but not including 2 or 10, in order for the set of expressions to represent the measures of the sides of a triangle.

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The volume of the rectangular prism is approximately 0.2996 cubic meters.

Six rectangular faces that are congruent and parallel to one another make up the three-dimensional solid known as a rectangular prism. A rectangular cuboid is another name for it.

By multiplying the prism's length, width, and height, one may get the volume of a rectangular prism, which is the volume of the interior space of the prism.

Given that, area of the square base is 7 1/2 sq. m.

The side of the square base is thus,

√(7 1/2) ≈ 2.7386 m

The volume of the rectangular prism is given as:

Volume = length x width x height

Volume ≈ 2.7386 x 2.7386 x 1/5 m

Volume ≈ 0.2996 cubic meters.

Hence, the volume of the rectangular prism is approximately 0.2996 cubic meters.

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Answer:

0.4

Step-by-step explanation:

Subtract 2.4m on each side

-1.2 = -3.0m

divide by -3.0

you get 0.4

Answer:

To solve the integral, we can use partial fractions and then integrate each term separately. The integrand can be written as:

[tex]\dfrac{x^4(1-x)^4}{1+x^2} = \dfrac{x^4(1-x)^4}{(x+i)(x-i)}[/tex]

Using partial fractions, we can write:

[tex]\dfrac{x^4(1-x)^4}{(x+i)(x-i)} = \dfrac{Ax+B}{x+i} + \dfrac{Cx+D}{x-i}[/tex]

Multiplying both sides by (x+i)(x-i), we get:

[tex]x^4(1-x)^4 = (Ax+B)(x-i) + (Cx+D)(x+i)[/tex]

Substituting x=i, we get:

[tex]i^4(1-i)^4 = (Ai+B)(i-i) + (Ci+D)(i+i)[/tex]

Simplifying, we get:

[tex]16 = 2Ci + 2B[/tex]

Substituting x=-i, we get:

tex^4(1+i)^4 = (Ci+D)(-i-i) + (Ai+B)(-i+i)[/tex]

Simplifying, we get:

[tex]16 = 2Ai + 2D[/tex]

Substituting x=0, we get:

[tex]0 = Bi + Di[/tex]

Substituting x=1, we get:

[tex]0 = A+B+C+D[/tex]

Solving these equations simultaneously, we get:

A = -22/7 + π

B = 0

C = 22/7 - π

D = -2/5

Therefore, the integral can be written as:

[tex]\int_0^1 \dfrac{x^4(1-x)^4}{1+x^2}dx = \int_0^1 \left[\dfrac{-22/7+\pi}{x+i} + \dfrac{22/7-\pi}{x-i} - \dfrac{2/5}{1+x^2}\right]dx[/tex]

Integrating each term separately, we get:

[tex]\int_0^1 \dfrac{-22/7+\pi}{x+i}dx = [-22/7+\pi]\ln(x+i) \bigg|_0^1 = [\pi-22/7]\ln\left(\dfrac{1+i}{i}\right)[/tex]

[tex]\int_0^1 \dfrac{22/7-\pi}{x-i}dx = [22/7-\pi]\ln(x-i) \bigg|_0^1 = [22/7-\pi]\ln\left(\dfrac{1-i}{-i}\right)[/tex]

[tex]\int_0^1 \dfrac{-2/5}{1+x^2}dx = -\frac{2}{5}\tan^{-1}(x)\bigg|_0^1 = -\frac{2}{5}\tan^{-1}(1) + \frac{2}{5}\tan^{-1}(0) = -\frac{2}{5}\tan^{-1}(1)[/tex]

Therefore, the correct options are:

A) [tex]\pi-\frac{22}{7}[/tex]

B) [tex]\frac{2}{105}[/tex]

C) 0

D) [tex]\frac{71}{15}-\frac{3\pi}{2}[/tex]

The Answer: The first step is Rearranging the Equation

v=9

v=-5

Step-by-step explanation:

STEP

1

:

Rearrange this Absolute Value Equation

Absolute value equalitiy entered

     4|-v+2|-3 = 25

Another term is moved / added to the right hand side.

     4|-v+2| = 28

STEP

2

:

Clear the Absolute Value Bars

Clear the absolute-value bars by splitting the equation into its two cases, one for the Positive case and the other for the Negative case.

The Absolute Value term is 4|-v+2|

For the Negative case we'll use -4(-v+2)

For the Positive case we'll use 4(-v+2)

STEP

3

:

Solve the Negative Case

     -4(-v+2) = 28

    Multiply

     4v-8 = 28

    Rearrange and Add up

     4v = 36

    Divide both sides by 4

     v = 9

STEP

4

:

Solve the Positive Case

     4(-v+2) = 28

    Multiply

     -4v+8 = 28

    Rearrange and Add up

     -4v = 20

    Divide both sides by 4

     -v = 5

    Multiply both sides by (-1)

     v = -5

    Which is the solution for the Positive Case

STEP

5

:

Wrap up the solution

v=9

v=-5

Step-by-step explanation:

remember, the sum of all angles in a triangle is always 180°.

the law of sine :

a/sin(A) = b/sin(B) = c/sin(C)

with a, b, c being the sides, and A, B, C being the three corresponding opposite angles.

so, the angle at Q is

180 = 48 + 48 + angle Q = 96 + angle Q

84° = angle Q

5mm/sin(48) = PR/sin(84)

PR = 5×sin(84)/sin(48) = 6.691306064... mm

The length of PR is approximately 10.33 mm.

what is isosceles triangle ?

An isosceles triangle is a triangle with at least two sides that have equal length, and thus two corresponding angles that are also equal in measure. The third side and angle of an isosceles triangle may or may not be of different length or measure. The two sides that are equal in length are called the legs, and the third side is called the base. The angle opposite the base is called the vertex angle, while the angles adjacent to the legs are called the base angles. In an isosceles triangle, the two base angles are equal in measure.

Since the sum of the angles in a triangle is 180 degrees, we can find the measure of angle PQR as follows:

PQR = 180 - QPR - QRP

PQR = 180 - 48 - 48

PQR = 84 degrees

Since angles QRP and QPR have the same measure, we know that sides OP and OR have equal length (they are opposite those angles). Therefore, triangle POR is an isosceles triangle.

To find the length of PR, we can use the Law of Cosines:

PR^2 = OP^2 + OR^2 - 2(OP)(OR)cos(POR)

Since OP and OR are equal in length, we can simplify this equation to:

PR^2 = 2(OP^2) - 2(OP^2)cos(POR)

We know that POR is 180 - PQR = 96 degrees. We also know that OP = OR, and that QP = 5 mm. Using the Law of Cosines, we can find the length of OP:

OP^2 = QP^2 + OR^2 - 2(QP)(OR)cos(QPR)

OP^2 = 5^2 + OR^2 - 2(5)(OR)cos(48)

OP^2 = OR^2 - 5ORcos(48) + 25

Since OP = OR, we can substitute OP for OR in the above equation:

OP^2 = OP^2 - 5OPcos(48) + 25

5OPcos(48) = 25

OP = 25/(5cos(48))

OP ≈ 6.25 mm

Now we can substitute this value into the equation we derived earlier to find PR:

PR^2 = 2(OP^2) - 2(OP^2)cos(POR)

PR^2 = 2(6.25^2) - 2(6.25^2)cos(96)

PR ≈ 10.33 mm

Therefore, the length of PR is approximately 10.33 mm.  

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Answer:

(3480 - 600)/2 = 1440cm

Step-by-step explanation: