A spring gun at ground level fires a golf ball at an angle of 45 degrees. The ball lands 10 m away.
a) What was the ball's initial speed?
b) For the same initial speed, find the two firing angles that make the range 6 m.
Recall that the Ideal Projectile Motion Equation is
r=(vo*cos(theta))ti+((vo*sin(theta)t-1/2*g*t^2)j.
Answer: a) vo=sqrt(10g)
b) theta=1/2*arcsin(3/5),
theta=pi-1/2*arcsin(3/5).
And is arcsin the same thing as sin^-1?

The critical points are (t, y) = (t, 0) and (t, -4c/5).

To find the partial derivatives, we need to differentiate f(t, y) with respect to each variable separately.

f1(t, y) = ∂f/∂t = 0 (since there is no t term in the function)

fy(t, y) = ∂f/∂y = -12cy^3 + 15y^4

fz(t, y) = ∂^2f/∂t∂z = 0 (since there is no z term in the function)

fy(t, y) = ∂^2f/∂y∂z = 0 (since there is no z term in the function)

So, 2. - f1(,y) fy(x, y) = = frz(, y) = fry(x, y) = 0 - 12cy^3 + 15y^4 = 0  (since f1(,y) and frz(, y) and fry(x, y) are all 0)

Therefore, -12cy^3 + 15y^4 = 0

Factor out y^3:

y^3(-12c + 15y) = 0

This gives us two solutions: y = 0 or -4c/5.

So, the critical points are (t, y) = (t, 0) and (t, -4c/5).

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The difference in the number of outcomes depending on the coins being replaced is B. 10 outcomes.

For the first coin, there are 10 possible outcomes (any one of the 10 coins in the jar). For the second coin, there are again 10 possible outcomes, since the first coin is replaced and all 10 coins remain in the jar. Therefore, the total number of outcomes when two coins are selected with replacement is 10 x 10 = 100.

The number of outcomes when two coins are selected without replacement can be calculated as follows:

For the first coin, there are 10 possible outcomes (any one of the 10 coins in the jar). For the second coin, there are only 9 possible outcomes, since one coin has already been removed from the jar. Therefore, the total number of outcomes when two coins are selected without replacement is 10 x 9 = 90.

Difference is:

= 100 - 90

= 10 outcomes

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Step-by-step explanation:

To solve for inverse, utilize the following steps.

Step 1: let f(x)=y so we get

[tex]y = \sqrt{x - 6} + 5[/tex]

Step 2: Swap y and x

[tex]x = \sqrt{y - 6} + 5[/tex]

Solve for y.

[tex]x - 5 = \sqrt{y - 6} [/tex]

[tex](x - 5) { }^{2} + 6 = y[/tex]

Step 4: Let y =f^-1(x)

[tex](x - 5) {}^{2} + 6 = f {}^{ - 1} (x)[/tex]

Answer: [tex]f^{-1}(x) =[/tex] x²-10x+19

Step-by-step explanation:

Let's replace f(x) for y for now.

[tex]y=\sqrt{x-6}+5[/tex]

To find inverse.  make your y into x, and your x into y

[tex]x=\sqrt{y-6}+5[/tex]        >Now you solve for y.  subtract 5 from both sides

[tex]x-5=\sqrt{y-6}[/tex]        >Square both sides to get rid of root

[tex](x-5)^{2} =(\sqrt{y-6})^{2}[/tex]     >drop root and square (x-5)

(x-5)(x-5) = y-6              >FOIL

x²-5x-5x+25 = y-6        > combine like terms

x²-10x+25 = y-6            >add 6 to both sides

x²-10x+19=y            > this is your inverse now put the y into inverse form

[tex]f^{-1}(x) =[/tex] x²-10x+19

The surface area of the hexagonal pyramid is 75240 square millimeters to the nearest square millimeter.

The surface area of the hexagonal pyramid can be calculated by finding the sum of the areas of its faces. The hexagonal pyramid has a base with six equal sides of length 44 millimeters, and six identical triangular faces with a height of 95 millimeters.

Each triangular face is an isosceles triangle, with two sides of length 44 millimeters and a base of length equal to the perimeter of the hexagonal base, which is 6 times 44 millimeters, or 264 millimeters.

To calculate the area of each triangular face, we can use the formula for the area of an isosceles triangle, which is (base x height) / 2. Substituting the values we have, we get: Area of each triangular face = (264 x 95) / 2 = 12540 square millimeters

Since the hexagonal pyramid has six identical triangular faces, we can multiply the area of one triangular face by 6 to get the total surface area of the pyramid: Total surface area = 6 x 12540 = 75240 square millimeters.

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Answer:

The answer to your problem is, 201.06 or 201.1

Step-by-step explanation:

To find the area you use the formula:

A = π [tex]r^2[/tex]

R = Radius

A = Area

We know the radius of the circle is 8

So replace A = π [tex]r^2[/tex]

= π × 8 ≈ 201.06193

Or 201.06 or 201.1

Thus the answer to your problem is, 201.06 or 201.1

Therefore, Sara has 3.5 grams of white beads in her jar.

Based on the information provided, Sara has 2.5 grams of blue beads, 4.9 grams of pink beads, and 7.1 grams of yellow beads. If she weighs her jar this week and finds out she has a total of 18 grams of beads, we can determine the number of grams of white beads she has by following these steps:

Step 1: Add the weights of the blue, pink, and yellow beads together.
2.5 grams (blue) + 4.9 grams (pink) + 7.1 grams (yellow) = 14.5 grams

Step 2: Subtract the total weight of the blue, pink, and yellow beads from the total weight of the jar (18 grams).
18 grams (total weight) - 14.5 grams (blue, pink, and yellow beads) = 3.5 grams
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After 5 years with 8% depreciation, the Audi R8's value will be around $81,249.36.

To determine how much money you must have in your account now, you can use the formula A = Pe^(rt), where A is the final amount, P is the principal (the initial amount invested), e is the constant 2.71828, r is the annual interest rate expressed as a decimal, and t is the time in years. We will calculate using this formula.Plugging in the given values, we get:
A = $23,000
r = 0.068 (6.8% expressed as a decimal)
t = 5 years
So, $23,000 = P*e^(0.068*5)
Solving for P, we get:
P = $16,376.59
Therefore, you must have $16,376.59 in your account now to reach your goal of $23,000 in 5 years with 6.8% continuous compounding interest. To determine how much the Audi R8 will be worth in 5 years, you can use the formula A = P(1 - r)^t, where A is the final amount, P is the initial amount, r is the annual depreciation rate expressed as a decimal, and t is the time in years. Plugging in the given values, we get:
P = $148,700
r = 0.08 (8% expressed as a decimal)
t = 5 years
So, A = $148,700*(1 - 0.08)^5
Simplifying, we get:
A = $81,249.36
Therefore, the Audi R8 will be worth approximately $81,249.36 in 5 years with 8% depreciation.

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Length of the platform at the front of the room whose area is 3x² + 10x - 8 and width is (x+4) m is (3x - 2) m

Area of the rectangular platform = 3x² + 10x - 8

Width of the rectangular platform = x+4

Area = length × width

Length = area/width

Length = [tex]\frac{3x^{2} + 10x - 8}{x+4}[/tex]

By splitting the middle term we get

Length = [tex]\frac{3x^{2} + 12x -2x -8 }{x+4}[/tex]

By taking common we get

Length = [tex]\frac{3x(x+4) - 2(x+4)}{x+4}[/tex]

By taking x+4 common we get

Length = [tex]\frac{(3x-2)(x+4)}{x+4}[/tex]

Cutting the x+4 from denominator and numerator we get

Length = 3x-2

Length of the platform at the front of room is 3x-2

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The  volume of the region U is 16π cubic units.

To find the volume of the region U, we can use cylindrical coordinates. In cylindrical coordinates, a point in space is represented by the coordinates (r, θ, z), where r is the distance from the z-axis, θ is the angle between the x-axis and the projection of the point onto the xy-plane, and z is the height above the xy-plane.

In this case, the region U is defined by 0 ≤ r ≤ 2, 0 ≤ θ ≤ 2π, and 0 ≤ z ≤ 12 - r sin(θ).

To find the volume of U, we can integrate over the cylindrical coordinates. The volume of U is given by the integral:

V = ∫∫∫_U dV

where dV = r dz dr dθ is the volume element in cylindrical coordinates.

Substituting in the limits of integration, we have:

V = ∫₀²π ∫₀² ∫₀^(12-rsinθ) r dz dr dθ

Integrating with respect to z, we get:

V = ∫₀²π ∫₀² r(12-rsinθ) dr dθ

Integrating with respect to r, we get:

V = ∫₀²π [(6r² - (1/3)r³sinθ)] from r=0 to r=2 dθ

Simplifying, we get:

V = ∫₀²π [(24 - 16/3 sinθ)] dθ

Integrating, we get:

V = [24θ + 16/3 cosθ] from θ=0 to θ=2π

Simplifying, we get:

V = 48π/3 = 16π

Therefore, the volume of the region U is 16π cubic units.
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The correct statement regarding the solution to the system of equations is given as follows:

b. Infinite solutions.

The system of equations in the context of this problem is defined as follows:

Replacing the second equation into the first, the value of x is obtained as follows:

3(8 + 2x) - 6x = 24

24 + 6x - 6x = 24

24 = 24.

24 = 24 is a statement that is always true, hence the system has an infinite number of solutions, and thus option B is the correct option for this problem.

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The size of each section in a graph tells in relation to the portion of the data :

The magnitude of a graphic's segment indicates a specific attribute of the data being presented. In charts like pie charts or stacked bar graphs, each component's size denotes the relative frequency or proportion of data points in a particular category.

This implies that larger fragments reflect an increased number of data points for its associated categories whereas smaller ones represent categories with lesser data points.

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Answer: The answer is (A

Step-by-step explanation:

The answer isB because A is constant, C is irrelevant, and D is dependent.

To calculate d²y/dx², we first need to find the first derivative of y, which is dy/dx. For y = 0.5x^-0.2, we can use the power rule of differentiation, which states that the derivative of x^n is n*x^(n-1). Therefore,

dy/dx = -0.1x^-1.2

To find the second derivative, d²y/dx², we need to differentiate dy/dx again. Using the power rule again, we get:

d²y/dx² = 0.12x^-2.2

This is the second derivative of y with respect to x.

In calculus, a derivative is a measure of how a function changes as its input changes. The second derivative is a measure of how the rate of change of the function itself changes as its input changes. It tells us about the curvature of the function at any given point.

In this case, we have calculated the second derivative of y, which gives us information about the rate of change of the slope of the function. If the second derivative is positive, the function is concave up (curving upward), and if it is negative, the function is concave down (curving downward). If the second derivative is zero, the function has an inflection point (a point where the curvature changes direction).

Overall, the second derivative is a powerful tool in calculus that helps us understand the behavior of functions in more detail.

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The mass of the Antarctic iceberg is approximately 2.56 × 10¹more kilograms than the mass of the Rock of Gibraltar.

To find out, we can subtract the mass of the Rock of Gibraltar from the mass of the Antarctic iceberg:

4.55 × 10¹³ kg - 1.78 × 10¹² kg = 4.37 × 10¹³ kg

Therefore, the mass of the Antarctic iceberg is about 2.56 × 10¹ (or 25.6) times greater than the mass of the Rock of Gibraltar.

This is because the mass of the Antarctic iceberg is much larger than the mass of the Rock of Gibraltar, as it is a massive block of ice floating in the ocean while the Rock of Gibraltar is a solid rock formation on land.

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The total area of the figure is 215π square feet

From the question, we have the following parameters that can be used in our computation:

Radius, r = 5 cm

Height, h = 14 cm

So, we have

A1 = πr²

A1 = π * 5² = 25π

A2 = 2πrh

A2 = 2 * π * 5 * 14 = 140π

A3 = 1/2(4πr²)

A3 = 1/2(4π * 5²) = 50π

So, we have

Total area = 25π + 140π + 50π

Evaluate

Total area = 215π

Hence, the total area is 215π

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If he visited all the cities in order A(O,4), B(4,1). C(3. 1) and D(-1. 2). then he traveled 12.3 units distance ( nearest tenth).

John visited four cities that form a rectangle on a coordinate grid at A(0, 4), B(4, 1), C(3, 1), and D(-1, 2). If he visited all the cities in order and ended up where he started, the distance he traveled can be found by calculating the perimeter of the rectangle.

Calculate the distance between consecutive points.
AB = √[(4-0)^2 + (1-4)^2] = √[16 + 9] = √25 = 5
BC = √[(3-4)^2 + (1-1)^2] = √[1 + 0] = √1 = 1
CD = √[(-1-3)^2 + (2-1)^2] = √[16 + 1] = √17 ≈ 4.1 (rounded to nearest tenth)
DA = √[(0-(-1))^2 + (4-2)^2] = √[1 + 4] = √5 ≈ 2.2 (rounded to nearest tenth)

Calculate the total distance traveled (perimeter of the rectangle).
Total Distance = AB + BC + CD + DA = 5 + 1 + 4.1 + 2.2 = 12.3

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A. angle CXD = 140 degrees

Angle XCD and Angle XDC are congruent since the triangle is isosceles. Remember that the sum of the interior angles of a triangle is 180 degrees.

20 + 20 + x = 180

x = 140

B. 18 sides

To find the number of sides of the polygon, we need to know the measure of one interior angle. One interior angle is angle BCD. We can easily find the measure of this angle because it is on a straight angle of which we are given part of (angle XCD).

Angle XCD + Angle BCD = 180

20 + BCD = 180

BCD = 160

Now that we know the measure of an interior angle, we can use the formula to find the measure of an interior angle and algebraically solve for the number of sides.

[ (n - 2) x 180 ] / n = 160

(n - 2) x 180 = 160n

180n - 360 = 160n

-360 = -20n

n = 18 sides

C. 2880 degrees

The formula for the sum of the interior angles of a regular polygon is (n - 2) x 180, where n is the number of sides.

(18 - 2) x 180

16 x 180

2880

D. 140 degrees

If angle XCD is 20 degrees, then angle BED is also 20 degrees. Angle BED and Angle BEF make up one of the interior angles of the regular polygon. We know that one interior angle is equal to 160 degrees.

Angle BED + Angle BEF = 160

20 + BEF = 160

BEF = 140

Hope this helps!

Answer:

The answer is C. x=z

Step-by-step explanation:

The correct answer is C. x=z.

Since x = y and y = z, then x = z. This is the transitive property of equality.

Here is a more detailed explanation:

The transitive property of equality states that if a = b and b = c, then a = c.

In this case, x = y and y = z. Therefore, x = z.

He must sell the car at R80,625 to make an overall profit of 25%

To find the selling price of the car that gives a 25% profit, we need to use the following steps:

Calculate the total cost of buying and transporting the car to Durban:

Total cost = Cost of car + Cost of transportation

Total cost = R60,000 + R4,500

Total cost = R64,500

Calculate the desired profit:

Profit = 25% of total cost

Profit = 0.25 x R64,500

Profit = R16,125

Calculate the total amount that the car needs to be sold for:

Total amount = Total cost + Profit

Total amount = R64,500 + R16,125

Total amount = R80,625

Therefore, the man needs to sell the car for R80,625 to make an overall profit of 25%.

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The length of the main span of the George Washington Bridge is 3500 feet.

Let x be the length of the main span of the George Washington Bridge.

We know that the main span of the Walt Whitman Bridge is 600 feet longer than two-fifths of the length of the main span of the George Washington Bridge, so we can write the equation:

2000 = (2/5)x + 600

To solve for x, we can start by isolating the term with x on one side of the equation:

(2/5)x = 2000 - 600

(2/5)x = 1400

Then, we can solve for x by multiplying both sides by the reciprocal of (2/5):

x = 1400 / (2/5)

x = 3500

Therefore, the length of the main span of the George Washington Bridge is 3500 feet.

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Thus, the coordinates of ΔF'I'G' after a rotation of 90° clockwise about the origin. are - F'(4,-2), I'(4,-5) and G'(2,-3).

A rotation is a turn made about a specific axis. Both clockwise and anticlockwise rotations are possible. Whereas the image is really the rotating image, the pre-image is the original item.

From the pre-image point, calculate the image. The listed pre-image point is (x , y). Change the x and y coordinates, then multiply this same previous y coordinate by -1 to get a 90 degree anticlockwise rotation. Use the guidelines mentioned below to calculate each rotation.

Clockwise :

Given :

F(2, 4), I(5, 4) and G(3, 2)

After 90 degree rotation: (x , y) ----> (y , -x)

F'(4,-2), I'(4,-5) and G'(2,-3).

Thus, the coordinates of ΔF'I'G' after a rotation of 90° clockwise about the origin. are - F'(4,-2), I'(4,-5) and G'(2,-3).

Graphs for the both triangles are obtained.

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Correct question:

ΔFIG has vertices at F(2, 4), I(5, 4) and G(3, 2). Graph ΔFIG and ΔF'I'G' after a rotation of 90° clockwise about the origin.​

The basal area of the tree after 10 years of growth would be approximately 279.7 square inches.

Given the initial basal area of a tree, which is 154 square inches, and the annual growth rate, which is 6%. To find out what the basal area of the tree will be after 10 years of growth.

By using the formula for compound interest, which can be applied to the growth of the basal area over time. The formula is:

A = P(1 + r)ⁿ

where:

A is the final amount

P is the initial amount

r is the annual growth rate

n is the number of years

To find A, the final basal area of the tree after 10 years of growth. We know that P is 154 square inches, r is 6% or 0.06 and n is 10.

By applying these values in the formula, we get:

A = 154(1 + 0.06)¹⁰

A = 154(1.06)¹⁰

A = ≈ 279.7

Therefore, the basal area of the tree after 10 years of growth would be approximately 279.7 square inches.

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Each of the three sisters owes $75 to cover the cost of hiring the karaoke deejay for Taylor's graduation party.

To determine how much each sister owes, we need to first calculate the total cost of the party and then divide that cost by three, since there are three sisters splitting the cost.

1. Tent rental: $45
2. Microphone system: $60
3. Deejay cost: $30/hour × 4 hours = $120

Now, we'll add these costs together to find the total cost:
Total cost = $45 (tent) + $60 (microphone) + $120 (deejay) = $225

Finally, we'll divide the total cost by the number of sisters (3) to find out how much each sister owes:
Amount owed per sister = $225 (total cost) ÷ 3 (sisters) = $75

So, each sister owes $75 for the karaoke deejay at Taylor's graduation party.

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The true statements are Numbers with same absolute value are opposites because they are same distance from each other and from 0 on the number line. The |7.1| = 7.1. So, correct options are B, C and E.

b) Numbers with the same absolute value are opposites because they are the same distance from each other. This is true because absolute value is the distance from a number to zero on the number line, and if two numbers have the same distance from zero, then they must be equidistant from zero and therefore, they are opposite in sign.

c) The |7.1| = 7.1 because the distance from 7.1 to 0 on the number line is 7.1 units. This is true because the absolute value of a number is always positive, and it represents the distance of that number from zero on the number line.

d) The |-8.4| = 8.4 because the distance from -8.4 to 8.4 on the number line is 0 units. This is false, as the distance between -8.4 and 8.4 on the number line is 16.8 units. The correct value of the absolute value of -8.4 is 8.4.

e) Numbers with the same absolute value are opposites because they are the same distance from 0 on the number line. This is true because 0 is the midpoint of the number line, and if two numbers have the same distance from 0, then they must be equidistant from zero and therefore, they are opposite in sign.

Therefore, the correct statements are b, c, and e.

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The minimum force required to move a block of 35 N resting on a steel table with a coefficient of static friction of 0.40 is 14 N.

Friction refers to the force that resists the motion and thus the force acts in the opposite direction of the force applied.

There are the following types of friction:

1. Static Friction

2. Limiting Friction

3. Kinetic Friction

F = μN

where μ is the coefficient of friction

N is the Normal Force

When the object is resting on a table, Normal force is the weight.

N = 35 N

μ = 0.40

F = 0.4 * 35

= 14 N

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The amount of accumulated interest the investor can expect in 8 years is $2,880.

To calculate the accumulated interest on the life insurance policy, we can use the simple interest formula:

I = P x r x t

In this case, the principal amount invested is $4,500, the annual interest rate is 8.0% (or 0.08 as a decimal), and the time period is 8 years. Therefore:

I = 4,500 x 0.08 x 8

I = $2,880

Therefore, the investor can expect to earn $2,880 in accumulated interest at the end of 8 years.

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The value of sides are KL=5.34, JK=16.434, JL=17.29 and ML=22.25.

∵ ΔJLM is a right triangle, as ∠MJL=90°

∴ tan(∠JML)= JL/JM            [∵ tan∅=perpendicular/hypotenuse]

⇒ tan(51°)=JL/14

⇒ JL=14×tan(51°)

       = 14×1.23

       = 17.29

∴ JL=17.29

Again, ΔJKL is a right triangle, with ∠JKL=90°

∴ cos(∠JLK)=KL/JL              [∵ cos∅=base/hypotenuse]

⇒cos(72°)= KL/17.29

⇒KL=17.29×cos(72°)

       = 17.29×0.309

        = 5.34

∴ KL=5.34

Hence, the value of KL is 5.34.

Also, tan(∠JLK)=KJ/KL

⇒tan(72°)=JK/5.34

⇒JK=5.34×tan(72°)

       = 5.34×3.077

       = 16.434

∴ JK=16.434

And, cos(∠JML)=JM/ML

⇒cos(51°)=14/ML

⇒ML=14/cos(51°)

        =14/.629

        =22.25

∴ ML=22.25

Hence, the value of sides are KL=5.34, JK=16.434, JL=17.29 and ML=22.25.

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The solution of the system of equations is given by the ordered pair (-4, 5).

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

x - y = -9    ......equation 1.

3x + 4y = 8  ......equation 2.

Based on the graph shown in the image attached above, we can logically deduce that the solution to this system of equations is the point of intersection of the lines on the graph representing each of them, which lies in Quadrant II, and it is given by the ordered pairs (-4, 5).

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The solutions to the triangle PQR are:

Angle Q ≈ 112°

Side QR ≈ 8.98

Side PR ≈ 13.71

To solve the triangle PQR, we can use the fact that the sum of the angles in a triangle is always 180°. So we can find angle Q by subtracting the measures of angles P and R from 180°:

angle Q = 180° - angle P - angle R

angle Q = 180° - 42° - 26°

angle Q = 112°

Now, we can use the law of sines to find the lengths of the sides QR and PR.

The law of sines states that in any triangle ABC, the following equation holds:

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

where a, b, and c are the side lengths of the triangle, and A, B, and C are the opposite angles, respectively.

Applying this formula to triangle PQR, we can write:

QR/sin(R) = PQ/sin(Q)

QR/sin(26°) = 19/sin(112°)

Solving for QR, we get:

QR = (19 × sin(26°))/sin(112°)

QR ≈ 8.98

Similarly, we can find PR by applying the law of sines to triangle PQR as follows:

PR/sin(P) = PQ/sin(Q)

PR/sin(42°) = 19/sin(112°)

Solving for PR, we get:

PR = (19 × sin(42°))/sin(112°)

PR ≈ 13.71

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