Show that equation (5.3) is true by considering an investment in the asset combined with a short position in a futures contract. Assume that all income from the asset is reinvested in the asset. Use an argument similar to that in footnotes 2 and 4 of this chapter and explain in detail what an arbitrageur would do if equation (5.3) did not hold.
5.3 equation:
F0= S0e(r-q)T
Footnote 2:
For another way of seeing that equation (5.1) is correct, consider the following strategy: buy one unit of the asset and enter into a short forward contract to sell it for F0 at time T. This costs S0 and is certain to lead to a cash inflow of F0 at time T. Therefore S0 must equal the present value of F0; that is, S0= F0erT, or equivalently F0= S0erT
Equation 5.1 F0= S0erT.
Footnote 4:
For another way of seeing that equation (5.2) is correct, consider the following strategy: buy one unit of the asset and enter into a short forward contract to sell it for F0 at time T. This costs S0 and is certain to lead to a cash inflow of F0 at time T and an income with a present value of I. The initial outflow is S0. The present value of the inflows is I+ F0e-rT. Hence, S0=I + F0e-rT, or equivalently F0= (S0 – I)erT
Equation 5.2 F0= (S0 – I)erT

Function to calculate amount of coffee in Dave's system after x hours

= y = 135(0.6)ˣ

An exponential function is a Mathematical function in the form f (x) = aˣ, where “x” is a variable and “a” is a constant which is called the base of the function and it should be greater than 0. The most commonly used exponential function base is the transcendental number e, which is approximately equal to.

Given,

Dave drinks 12-ounce cup of coffee each morning

One cup has 136 mg of caffeine

caffeine in his system decreases about 40% every hour
In x hours,

Amount of caffeine in his system

y = a(1 - 40/100)ˣ

or

y = 136(1-0.4)ˣ

y = 135(0.6)ˣ

Hence, y = 135(0.6)ˣ is the function to calculate amount of caffeine in system of Dave after x hours.

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the coach can form 4 teams of 9 boys and 9 girls each, and there will be 9 players left over who cannot form a complete team.

The GCD (Greatest Common Divisor), also known as the HCF (Highest Common Factor), is the largest positive integer that divides two or more integers without leaving a remainder.

For example, the GCD of 12 and 18 is 6, because 6 is the largest positive integer that divides both 12 and 18 without leaving a remainder.

Given by the question.

To form teams with an equal number of boys and girls, we need to find the greatest common divisor (GCD) of 36 and 45.

The prime factorization of 36 is 2^2 x 3^2, and the prime factorization of 45 is 3^2 x 5.

The common factors are 3^2 = 9. Therefore, the GCD of 36 and 45 is 9.

This means that the coach can form teams of 9 boys and 9 girls each.

To find out how many teams he can form, we need to divide the total number of boys and girls by the number of players on each team:

Number of teams = Total amount of players / Number of players per team

Number of teams = (36 + 45) / 18

Number of teams = 81 / 18

Number of teams = 4 with a remainder of 9

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Simplifying the equations will give 3x -1 = 4x, 3x + 18 = 4x + 7 and 7x = 1

Lets simplify each of the equations in order to match with the  equivalent equation from the second set

3x+6=4x+7

Collecting like terms

3x +6 - 7 = 4x

3x -1 = 4x

The second equation

3(x+6)=4x+7

expanding the bracket

3x + 18 = 4x + 7

The third equation

4x+3x=7-6

simplifying the equation

7x = 1

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The composite figure consists of two rectangular pyramids, volume is 45 unit³.

Pyramids are three-dimensional structures having triangle faces and have an encompassing polygon shape at its base. If the base of a pyramid is rectangular, then it is called a rectangular pyramid.

Given,

Total height of two rectangular pyramids = 10 units

height of each pyramid is 5 unit.

Base lengths are 9units and 1.5 units.

Both are connected at their bases.

Dimensions are same of both pyramids

Composite figure consists of two rectangular pyramids.

Volume of the composite figure

= 2(length × width × height)/3

= 2(9×1.5×5)/3

= 2(13.5×5)/3

= 2(67.5)/3

= 135/3

= 45 unit³

Hence, 45 unit³ is volume of composite figure, that consists of two rectangular pyramids.

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Answer:A D

Step-by-step explanation:

The correct answers are: a reflection across the x-axis and a translation 1, unit right,  a rotation of 180° about the origin and a translation of 1 unit left.

To check for congruence, we need to verify that the two triangles have the same shape and size, which means that one triangle can be mapped onto the other using a sequence of transformations.

For the first sequence of transformations (a reflection across the x-axis and a translation 1 unit right), we can reflect triangle ABC across the x-axis, which will map (-6, 2) to (-6, -2), (-4, 6) to (-4, -6), and (-2, 2) to (-2, -2). We can then translate the reflected triangle 1 unit to the right, which will map (-6, -2) to (-5, -2), (-4, -6) to (-3, -6), and (-2, -2) to (-1, -2). This mapped triangle is triangle ADEF, and since it was obtained by a sequence of transformations, we can conclude that ABC is congruent to ADEF.

For the second sequence of transformations (a rotation of 180° about the origin and a translation of 1 unit left), we can rotate triangle ABC 180° about the origin, which will map (-6, 2) to (6, -2), (-4, 6) to (4, -6), and (-2, 2) to (2, -2). We can then translate the rotated triangle 1 unit to the left, which will map (6, -2) to (5, -2), (4, -6) to (3, -6), and (2, -2) to (1, -2). This mapped triangle is also ADEF, and since it was obtained by a sequence of transformations, we can conclude that ABC is congruent to ADEF.

Therefore, the correct answers are:

a reflection across the x-axis and a translation 1 unit right

a rotation of 180° about the origin and a translation of 1 unit left.

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When one and seventy one hundredth is written in decimal form, it would be 1. 71

A single part of something that has been evenly divided into one hundred parts is referred to as a hundredth in mathematics. The tenths place is represented by the first digit following the decimal. The hundredths place is represented by the digit that follows the decimal.

This means that when given the number one and seventy One Hundredths in words, then the number, " one " would be written before the decimal and the 71 would be 0. 71 as a hundredth figure.

When put together, this becomes:

= 1 + 0. 71

= 1. 71

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The diameter of the pie is approximately 9 inches

What is a diameter ?

Diameter can be defined as the double of the radius

To determine the diameter of the pie, we need to use the formula for the circumference of a circle:

C = 2πr

where C is the circumference, π (pi) is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.

In this case, we are given that the circumference (circumstances) of the pie is 28.26 inches, so we can set up the equation:

28.26 = 2πr

To solve for the radius, we can divide both sides of the equation by 2π:

r = 28.26 / (2π)

r ≈ 4.5

So the radius of the pie is approximately 4.5 inches. To find the diameter, we just need to double the radius:

d = 2r

d ≈ 9

Therefore, the diameter of the pie is approximately 9 inches.

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Answer:

k = 23/7

Step-by-step explanation:

To solve the problem, we can use the fact that the product of the slopes of two perpendicular lines is -1. We can start by finding the slope of the line passing through the points (3, 5) and (k, 12).

slope = (12 - 5) / (k - 3)

We can simplify this expression by multiplying both numerator and denominator by -1:

slope = (-7) / (3 - k)

Now, let's find the slope of the line passing through the points (0, 7) and (2, 10):

slope = (10 - 7) / (2 - 0) = 3 / 2

Since the two lines are perpendicular, we can set the product of their slopes equal to -1:

(-7) / (3 - k) * (3 / 2) = -1

Simplifying this equation, we get:

(7/2) * (3 - k) = 1

Multiplying both sides by 2/7, we get:

3 - k = 2/7

Subtracting 3 from both sides, we get:

-k = 2/7 - 3 = -21/7 - 2/7 = -23/7

Finally, dividing by -1, we get:

k = 23/7

Therefore, the value of k that makes the statement true is k = 23/7.

Answer:

Step-by-step explanation:

m1=y2-y1/x2-x1=12-5/k-3=7/k-3

m2=10-7/2-0=3/2

two lines with slopes m1 and m2 are perpendicular if m1×m2 = -1.

(7/k-3) *3/2=-1

3(7)=-2(k-3)

21=-2k+6

-2k=15

k=-15/2

The kinetic energy (in joule) of the asteroid is, 33 million

The kinetic energy is the energy associated with the particle having mass m and moving with the velocity v.

To find the kinetic energy of the asteroid, we need to use the formula:

KE = (1/2)mv²

where KE is the kinetic energy, m is the mass of the asteroid, and v is its velocity.

The mass of the asteroid can be calculated using its density and volume:

V = (4/3)πr³

where V is the volume of the asteroid, r is its radius, and π is the mathematical constant pi.

Since the asteroid is spherical, its radius is half its diameter, which is 0.54 km.

So,

V = (4/3)π(0.54 km)³

  = 0.65 km³

The mass of the asteroid is then:

m = density × volume

   = 3000 kg/m³ × 10⁹ m³/km³ × 0.65 km³

   = 1.95 × 10¹² kg

Now we can calculate the kinetic energy:

KE = (1/2)mv²  

     = (1/2) × 1.95 × 10¹² kg × (23.0 km/sec)²

     = 1.32 × 10²³ joule

To convert this to megatons of TNT, we divide by the energy of one megaton:

= 1.32 × 10³ joule / (4 × 10¹⁵ joule/megaton)

= 3.3 × 10⁷ megatons

Hence, the KE of asteroid is 33 million

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3x+3n+20 is the answer

Answer:

Step-by-step explanation:

1. measurement of length

2. measurement of area

3. measurement of volume

4. measurement of weight

5. measurement of length

6. measurement of weight

7.measurement of volume

8. measurement of area

To find out how much interest will be paid on a $155,000 house if a monthly payment of $850 is made for 30 years, we can use the formula for the total interest paid on a fixed-rate mortgage loan:

Total Interest = (Monthly Payment x Number of Payments) - Loan Amount

where Monthly Payment is $850, Number of Payments is 30 years x 12 months/year = 360 months, and Loan Amount is $155,000.

Plugging in these values, we get:

Total Interest = ($850 x 360) - $155,000

Total Interest = $306,000 - $155,000

Total Interest = $151,000

Therefore, the total interest paid on the $155,000 house over 30 years with a monthly payment of $850 is $151,000.

According to the figure of the right triangle

When a right triangle has one of the angles as 45 degrees the angle must be 45 degrees two. Also, the sides are equal, hence we have that

a = b

Using trigonometry we solve for b as follows

cos 45 = b / 10√2

multiplying both sides by 10√2

10√2 * cos 45 = 10√2 * b / 10√2

10√2 * cos 45 = b

rearranging

b = 10√2 * cos 45

where cos 45 = 1/√2 = √2/2

b = 10√2 * 1/√2

b = 10

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The function is decreasing on the interval (-∞, -2), has a local minimum at x = -2, is increasing on the interval (-2, 2), and is increasing on the interval (2, ∞). The function has a relative minimum at x = 3 (f(3) = -27) and no relative maximum.

Begin by finding the derivative of y.

y' = 3x^2 - 12

Find the critical numbers. (Enter your answers as a comma-separated list.)

To find the critical numbers, we need to solve y' = 0 for x:

3x^2 - 12 = 0

x^2 - 4 = 0

(x - 2)(x + 2) = 0

So the critical numbers are x = -2 and x = 2.

Test the intervals to determine the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.)

To test the intervals, we can use a sign chart:

Interval (-∞, -2) (-2, 2) (2, ∞)

y' (-)(-) (-)(+) (+)(+)

y Decreasing Local Min Increasing

So the function is decreasing on the interval (-∞, -2), has a local minimum at x = -2, is increasing on the interval (-2, 2), and is increasing on the interval (2, ∞).

f(x) = x^4 - 4x^3

To find the relative extrema, we need to find the critical numbers and then evaluate the function at those points.

Find the derivative of f(x).

f'(x) = 4x^3 - 12x^2

Find the critical numbers by solving f'(x) = 0 for x.

4x^3 - 12x^2 = 0

4x^2(x - 3) = 0

x = 0 or x = 3

So the critical numbers are x = 0 and x = 3.

Evaluate the function at the critical numbers and the endpoints of the interval to find the relative extrema.

f(-∞) = ∞

f(0) = 0

f(3) = -27

f(∞) = ∞

So the function has a relative minimum at x = 3 (f(3) = -27) and no relative maximum.

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

Find the critical numbers and the open intervals on which the function is increasing or decreasing. Use a graphing utility to verity your results.

y = x^3 − 12x + 2

Step 1: Begin by finding the derivative of y.

y'=

Step 2: Find the critical numbers. (Enter your answers as a comma-separated list.)

x=

Step 3: Test the intervals to determine the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.)

Increasing (___)

Decreasing (____)

2) Find all relative extrema of the function. (If an answer does not exist, enter DNE.)

f(x) = x4 − 4x3

relative max:(____)

relative min:(_____)

A Reasonable prediction for the number of times a white or red marble will be drawn is 3/10 x 180 = 54. Thus, option b is correct

If a marble is drawn and not replaced, the probability of drawing an orange marble on the second draw depends on the color of the marble drawn on the first draw. If an orange marble was drawn on the first draw, then there will be one less orange marble in the bag, and the probability of drawing another orange marble on the second draw will be 4 out of the remaining 29 marbles. If a non-orange marble was drawn on the first draw, then there will be 5 orange marbles in the bag, and the probability of drawing an orange marble on the second draw will be 5 out of 29.

There are a total of 30 marbles in the bag.

The probability of drawing a white or red marble is

2/30 + 7/30

= 9/30

= 3/10.

Therefore, a reasonable prediction for the number of times a white or red marble will be drawn is 3/10 x 180 = 54. Answer B.

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The required solutions of the given system of equation are  x = 1 and y = 1.

The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.

To find the values of x and y that satisfy the system, we need to solve the equations simultaneously:

y = -2x + 3 ... (1)

y = 5x - 4 ... (2)

Setting the right-hand sides of (1) and (2) equal to each other, we get:

-2x + 3 = 5x - 4

Simplifying this equation, we get:

7x = 7

Dividing both sides by 7, we get:

x = 1

Substituting x = 1 into either equation (1) or (2), we get:

y = -2(1) + 3 = 1

Therefore, the only solution that satisfies both equations is x = 1 and y = 1.

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The discrete probability distribution is given as follows:

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes.

There are 153 nests, hence the distribution in this problem is constructed dividing each of the amounts by 153, as follows:

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The worst month for the team in terms of results was October, with a total of three losses in five matches.

Number is a mathematical entity used to represent a computer magnitude it can be symbol or a combination of simple you should in order to take quantity such as the two, five, seven, three or eight number are used to verify the context including counting measuring and computing.


The team won only one match and drew one, resulting in a total of five points. This was the lowest points total of any month during the season, a stark contrast to the other months where the team achieved much higher totals. The team's form during October was a far cry from their form during the rest of the season, with only one win in five matches. This poor form contributed to the team's overall lack of success during the season and led to their eventual relegation from the top division.

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Work Shown:

5r + p

5*4 + 3

20 + 3

23

The possible set of inequalities are;

x = -3,

x = -1,

x = 1,

x = 3,

x = 5.

Inequalities are the comparison of mathematical expressions, whether one quantity is greater or smaller in comparison to another quantity.

We use these symbols to represent inequalities, '>' , '<', '≥', '≤'

The critical points of a rational inequality are the x-values at which the inequality changes direction (from "<" to ">" or vice versa). In this case, the critical points are -1, 3, and 4.

To find the complete solution to the inequality, we need to test points on either side of the critical points. For example, if we test a value less than -1, such as -2, and find that it satisfies the inequality, then all values less than -1 also satisfy the inequality. On the other hand, if a value less than -1 does not satisfy the inequality, then no values less than -1 satisfy the inequality.

Therefore, a complete solution to the inequality can be found by testing values around each critical point. A common set of points to test for this purpose is the critical points themselves, and values just less than and just greater than each critical point.

So, a possible set of points to test to find the complete solution to the inequality is:

x = -3,

x = -1,

x = 1,

x = 3,

x = 5.

These points bracket the critical values and allow us to determine which intervals of values satisfy the inequality.

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Answer:

The average of all the data is calculated in statistics. 

However, statistics is a field that goes beyond average and is not a complete science of it. 

There are other additional statistical tools. 

which explains how statistics is an extension of mathematics.

The accumulated amount will be ​$133.

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

A student invests ​$70 each month in an ordinary annuity with 3% interest compounded monthly.

We know that;

⇒ Simple interest = P × r × t

Where, P is principle amount , 'r' is rate and t is time.

Here, P = $70

r = 3%

n = 30 years

Substitute all the values, we get;

Hence, We get;

⇒ Interest = P × r × t

⇒ Interest = 70 × 0.03 × 30

⇒ Interest = $63

Hence, The accumulated amount will be ;

⇒ P + Interest

⇒ $70 + $63

⇒ $133

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A system of equations where n is the number of nickels and q is the number of quarters he has is represented by (C) 25q + 5n = 7, n = 2q.

The equals sign is a symbol used in mathematical formulas to denote the equality of two expressions.

An equation is a mathematical statement that contains the symbol "equal to" between two expressions with identical values. as in 3x + 5 = 15, for example.

There are many different types of equations, including linear, quadratic, cubic, and others.

Let's learn more about math equations in this tutorial.

So, we know that nickels are double quarters.

And nickels are represented by n and quarters are represented by q.

Then, we can write as:
n = 2q

Then, the equation that represents the situation will be:

25q + 5n = 7

Therefore, a system of equations where n is the number of nickels and q is the number of quarters he has is represented by (C) 25q + 5n = 7, n = 2q.

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Answer:

100

Step-by-step explanation:

1.) Plug in the values of x and y into the equation

10(20/2)

2.) Divide 20 by 2

10(10)

3.) Multiply 10 by 10

100

a. Original statement: If n is any prime number that is greater than 2, then n+1 is even.

Converse: If n+1 is even, then n is prime and greater than 2.

Counterexample: n = 4, which is not a prime number but 4+1=5 is a prime number greater than 2.

b. Original statement: If m is any odd integer, then 2m is even.

Converse: If 2m is even, then m is odd.

Counterexample: m = 2, which is not an odd integer but 2×2=4 is an even integer.

c. Original statement: If two circles intersect in exactly two points, then they do not have a common center.

Converse: If two circles do not have a common center, then they intersect in exactly two points.

Counterexample: Two identical circles with centers coinciding have a common center and intersect at all points on the circle, which is more than two points.

In mathematics, the converse of a conditional statement is formed by interchanging the hypothesis and conclusion.

For example, if the original statement is "If p then q",

the converse statement is "If q then p".

The converse statement may or may not be true, and it needs to be proven separately.

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The areas of the squares are calculated from the Pythagoras relation of right angled triangle and it is 40 ft², 55 ft², 95 ft²

A triangle is a plane figure or polygon with three sides and three angles.

A Triangle has three vertices and the sum of the interior angles add up to 180°

Let the Triangle be ΔABC , such that

∠A + ∠B + ∠C = 180°

The area of the triangle = ( 1/2 ) x Length x Base

For a right angle triangle

From the Pythagoras Theorem , The hypotenuse² = base² + height²

if a² + b² = c² , it is a right triangle

if a² + b² < c² , it is an obtuse triangle

if a² + b² > c² , it is an acute triangle

Given data ,

Let the right angle triangle be represented as ΔABC

Now , the measure of side AB = √40 feet

The measure of side BC = √55 feet

The measure of side AC = √95 feet

For a right angle triangle

From the Pythagoras Theorem , The hypotenuse² = base² + height²

Substituting the values in the equation , we get

AC² = AB² + BC²

( √95 )² = ( √40 )² + ( √55 )²

On simplifying the equation , we get

95 = 40 + 55

95 = 95

Therefore , the area of squares formed is 40 ft², 55 ft², 95 ft²

Hence , the triangle is having sides √40 feet , √55 feet and √95 feet

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Both measurements are below the minimum area of 169 square feet required.

Area is a measurement of size of two dimensional surface such as Square rectangle or circle. It is calculated by multiplying the length of the surface by width area also can measured in terms of square units such as square feet or square meters area is an important concept in mathematics and is used to measure the size of safe and object it is also used to find the total area of a group of shape.

The measurements of the parking spaces shown do not meet the requirements of the town. The first parking space has an area of 144 square feet, while the second parking space has an area of 128 square feet. Both measurements are below the minimum area of 169 square feet required.

The area of the figure can be found by using the formula for the area of a regular polygon. Since the figure is composed of four 60-degree angles and four sides, the formula can be expressed as:

A = (1/2) * (a * s) * n

Where a is the length of one side of the polygon, s is the length of the apothem (a line perpendicular to the center of the polygon to a corner), and n is the number of sides of the polygon.

Since the figure has four sides of equal length (5 meters each), the length of the apothem can be calculated by using the Pythagorean theorem. The equation for the apothem is:

s = [tex](4^{2} - 5^{2} ) ^{0.5}[/tex]

Plugging these values into the area formula yields:

A = (1/2) * (5 * 4.472) * 4

A = (1/2) * (22.36) * 4

A = 44.72 square meters

Rounding to the nearest tenth yields an area of 44.7 square meters.

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The average rate of change in the number of Americans over the age of 65 from 2000 to 2015 is approximately -2.09 million people per year.

To find the average rate of change in the number of Americans over the age of 65 from 2000 to 2015, we need to calculate the change in the number of people over 65 during that time period and divide it by the number of years:

Number of people over 65 in 2015:

n(15) = 0.00246(15)^2 + 0.118(15) + 0.183

n(15) = 2.781 million people

Change in the number of people over 65 from 2000 to 2015:

2.781 million - 34.42 million = -31.639 million people

Average rate of change:

-31.639 million / 15 years = -2.09 million people per year

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The value of x and y are 13 and 3 respectively and the measure of angle mqr  is 51 degree.

It is an measure of rotation between one extended side(we extend it virtually) of the polygon with its adjacent side which is not extended.

WE are given that in the diagram, Ks, LT, MU are parallel to each other.

Angle knj = (2x-5)°, Angle lpn = (2y + 15), Angle pqu = (3x+4y)°.

Sine, the corresponding angles are equal

(2x-5) = (2y+15)

2x - 2y = 15 + 5

2x - 2y = 20

x - y = 10

x = 10 + y

Now,

(3x+4y) = (2y+15)

3x + 4y - 2y = 15

3x + 2y = 15

3(10 + y) + 2y = 15

30 + 5y = 15

y = 3

x = 13

Angle mqr = (3x+4y) = 51 degree.

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