Suppose that the relation H is defined as follows.
H = {(−3, 8) , ( 0, -3) , (1, 3) , (1, -1)}
Give the domain and range of H.
Write your answers using set notation.
Domain = ?
Range = ?

The gas has a capacity of 22 litres at 0 degrees Celsius and increases by 225 litres for each degree above that.

Volume is the amount of space a thing takes up, whereas capacity is the amount of substance it can hold, such as a solid, liquid, or gas. Volume is typically measured in cubic units, whereas capability can be measured in virtually any other unit, such as litres, gallons, pounds, and so on.

According to the problem statement, the volume of the gas and the temperature T in degrees Celsius have a direct relationship. Furthermore, we know that the gas expands by 225 litres for every degree of temperature rise. With this knowledge, we can create the equation shown below:

V = 22 + 225T

where T is indeed the temperature is degrees Celsius and V is the gas's volume in liters.

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Answer: $75

Step-by-step explanation:

The future population of the rabbit in the first year is 290.

Logistic growth is defined as the population growth where the growth rate gets smaller as the size of the population approaches the maximum. This maximum point is called carrying capacity.

Here given that,

Carrying capacity, K = 2000

Initial population, P₀ = 200

Growth rate, r = 50% = 0.5

Population in the first year, P₁ = P₀ + r [(1 - P₀/K) P₀]

                                                 = 200 + 0.5 [(1 - 200/2000) 200]

                                                 = 200 + 90

                                                 = 290

Hence the population in the first year is 290.

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The value of  (p o g)(-1) is -2 and the value of (q o p)(-1) is 15 which represents the composite of the function.

It is defined as a special type of relationship, and they have a predefined domain and range according to the function every value in the domain is related to exactly one value in the range.

We have a function p and q:

p(x) = 2x - 2

q(x) = x² - 1

= (p o q)(-1)

Find the value of q(-1):

q(-1) = 0

Plug the above value in the function p(x):

(p o q)(-1) = p(q(-1)) = -2

= (q o p)(-1)

Find the value of p(-1):

p(-1) = -4

(q o p)(-1) = q(p(-1)) = 15

Thus, the value of  (p o g)(-1) is -2 and the value of (q o p)(-1) is 15 which represents the composite of the function.

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If point M is the largest integer less than √ 28, the value of point M is found as 5.

A number that contains both positive and negative integers, including zero, is called an integer. There are no fractional or decimal parts in it. Integers include things like -5, 10, 1, 45, 8, 97, and 3,43.

Point M is the largest integer less than √ 28.

√ 28 = 5.29

largest integer which is less than 5.92 is 5.

Thus, if point M is the largest integer less than √ 28, the value of point M is found as 5.

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In linear equation, 3.99 does the amount of money left on the gift card change each month.

What in mathematics is a linear equation?

A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept. The variables in the previous sentence, y and x, are referred to as a "linear equation with two variables" at times.

                          Equations whose variables have a power of one are called linear equations. A single variable example with real values for a and b and x as the variable is ax+b = 0.

new difference = [tex]a_{m + 1} - a_{m}[/tex] = answer

            let m = 1

  that's month 1 & 2

         50 - 3.99( m + 1) - (50 - 3.99(m))

         50 - 3.99 - 50

  hence  0 + 0 - 3.99

            = - 3.99

the amount of money left on the gift card change each month by  - 3.99

{that is it reduce by 3.99 }

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P(sum of 8 | sum > 7) = 2/36 = 1/18

So the probability of rolling a sum of 8 given that the sum is greater than 7 is 1/18 or approximately 0.056.

We must first ascertain the total number of outcomes where the sum is larger than 7 in order to calculate the likelihood that the sum of two fair dice will be 8.

A sum larger than 7 can only result in the following outcomes: (2,6), (3,5), (4,4), (5,3), and (6,2). As there are six alternative outcomes for each die roll, there are 36 permutations in which these five events can occur.

We must now count the number of these 36 outcomes that have a sum of 8. There are just two outcomes that could satisfy this requirement: (2,6) and (3,5). Hence, provided that the sum is more than 7, the following is the likelihood of rolling an 8:

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Using equations, we can find that the total cards sold on Mother's Day = 1100.

Equations are logical assertions in mathematics that have two algebraic expressions on either side of an equals (=) sign. It is demonstrated that the expressions on the left and right are equivalent to one another.

In all mathematical equations, LHS = RHS (left hand side = right hand side) is a constant. The value of an unknown variable, or unknown quantity, can be determined through solving equations.

Here,

Let the total no. of cards sold be = x

Now 2% of cards sold = 22

We can say, 2% of x = 22

⇒ 2/100 × x = 22

⇒ x = (22 × 100)/2

⇒ x = 1100

Therefore, the total number of cards sold on Mother's Day = 1100.

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Answer:

[tex]310 ^\circ F[/tex]

Step-by-step explanation:

The formula relating surface temperature in °F and time in t minutes after removal from the oven is given as

[tex]S(t)= 72 + 238 (0.97^t)\\\\\text{The initial temperature is when t = 0}\\\\\text{At t = 0}\\\\S(0) = 72 + 238(0.97^0)\\\\0,97^0 = 1 \text{ since any number raised to 0 = 1}\\\\\text{Therefore}\\S(0) = 72 + 238 \cdot 1\\= 72 + 238\\\\= 310 ^\circ F[/tex]

She earned a total of $44 from doing chores for her neighbors.

Amount is a numerical quantity that is used to measure, count, or indicate the size, magnitude, or extent of something. It can be used to refer to a variety of things, including money, people, time, temperature, and even abstract concepts. Amounts are often expressed as a numerical value, but may also be expressed as a word or phrase.

Alice earned $44 doing chores for her neighbors. She spent $15 on a book, leaving her with $29. She then spent half of that, or $14.50, on the necklace, leaving her with $14.50. Therefore, she earned a total of $44 from doing chores for her neighbors.

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The shape οf the graph is similar tο the letter W; hοwever, the regiοn οf interest will lie οn the right οf the οrigin, which means οnly the right half οf the W(a negative number οf cans cannοt be prοduced).

The prοfit prοduced will be maximized if the number οf cans prοduced is greater than 3. The cοmpany will be at a lοss if the number οf cans prοduced is near 2.5. The break even situatiοns are present if the cοmpany prοduces 2 οr 3 cans.

A pοlynοmial functiοn is a functiοn that can be expressed as a sum οf pοwer functiοns in οne variable, where each term has a cοnstant cοefficient and a nοn-negative integer expοnent.Tο sketch the graph οf the functiοn [tex]P(x) = x^4-x^3-11x^2+9x+18[/tex], we can start by analyzing the end behaviοrs οf the functiοn and identifying where the functiοn crοsses the x-axis (i.e., where P(x) = 0).

Tο find the x-intercepts, we need tο sοlve fοr the rοοts οf the equatiοn P(x) = 0. This can be dοne by factοring the expressiοn οr by using numerical methοds. After sοme algebraic manipulatiοn, we can factοr the expressiοn as fοllοws:

[tex]P(x) = (x-3)(x+1)(x^2-8x-6)[/tex]

Therefοre, the rοοts οf the equatiοn P(x) = 0 are x = 3, x = -1, and x = 4 + √(7) οr x = 4 - √(7). The first twο rοοts indicate that the cοmpany breaks even when x = 3 οr x = -1, which means that the cοmpany needs tο prοduce and sell at least 3 οr -1 cans οf sοda tο make a prοfit οf zerο.

With this infοrmatiοn, we can sketch the graph οf the functiοn by plοtting the x-intercepts and the end behaviοrs. The graph will be symmetric abοut the vertical line x = 1.5 (the average οf 3 and -1) and will have a glοbal minimum at the pοint (1.5, -19.5). The graph will alsο have twο additiοnal rοοts at x = 4 + 2√(7) and x = 4 - 2²(7).

In summary, the graph οf the functiοn [tex]P(x) = x^4-x^3-11x^2+9x+18[/tex] has a "U" shape with a glοbal minimum at (1.5, -19.5). The cοmpany will break even when it prοduces and sells at least 3 οr -1 cans οf sοda. By analyzing the end behaviοrs and the x-intercepts, we can sketch the graph withοut using technοlοgy.

Hence, the graph of the function would be given below:

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Answer: Let x be the side length of each square cut out from the corners.

After cutting out the squares, the dimensions of the resulting rectangle will be:

length = 360 - 2x

width = 240 - 2x

The height of the box will be equal to the side length of the squares cut out, which is x.

The volume of the box is given by the formula:

V = length × width × height

Substituting the expressions for length, width, and height, we get:

V = (360 - 2x)(240 - 2x)x

Expanding this expression, we get:

V = x(86400 - 120x + 4x^2)

To find the maximum volume, we need to find the value of x that maximizes the expression for V.

We can do this by finding the critical points of the function V(x) and then determining whether these points correspond to a maximum or a minimum.

Taking the derivative of V(x) with respect to x, we get:

V'(x) = 86400 - 240x + 8x^2

Setting V'(x) = 0 to find the critical points, we get:

8x^2 - 240x + 86400 = 0

Dividing both sides by 8, we get:

x^2 - 30x + 10800 = 0

Using the quadratic formula to solve for x, we get:

x = (30 ± √(30^2 - 4(1)(10800))) / 2

x = (30 ± 210) / 2

We can discard the negative solution, since x represents a length and therefore must be positive. Thus, we get:

x = (30 + 210) / 2

x = 120

Therefore, the side length of the squares cut out from the corners should be 120 mm in order to maximize the volume of the box.

To find the maximum volume, we can substitute x = 120 into the expression for V:

V = x(86400 - 120x + 4x^2)

V = 120(86400 - 120(120) + 4(120)^2)

V ≈ 27648000 mm^3

Therefore, the maximum volume of the box is approximately 27,648,000 mm^3, and this maximum is achieved when the side length of each square cut out from the corners is 120 mm.

Step-by-step explanation:

The drawings, class of the polygons, perimeters and areas are presented as follows;

1. The polygon is a kite

Please find attached the drawing of the kite created with MS Excel

The perimeter is about 15.2 units

The area of the kite is 12 square units

3. The polygon is a scalene triangle

Please find attached the drawing of the scalene triangle created with MS Excel

The perimeter of the scalene triangle is about 15.2 units

The area of the triangle is 7 square units

4. The polygon is a rectangle

Please find attached the drawing of the rectangle created with MS Excel

The perimeter of the rectangle is about 14.1 units

Area of the rectangle is 12 square units

A polygon is a geometric figure with three or more straight sides forming a loop

1. Please find attached the drawing of the polygon created with MS Excel

The points on the quadrilateral are; (-2, 3), (3, 1), (-2, -1),  (-3, 1)

The slope of the segment between (-2, 3), and (-3, 1) is (1 - 3)/(-3 - (-2)) = 2

Slope of the segment between (3, 1), (-2, -1) is (-1 - 1)/(-2 - 3) = 0.4

Slope of segment between (-3, 1) and (-2, -1), is (-1 - 1)/(-2 - (-3)) = -2

Slope of the segment between  (-2, 3) and (3, 1) is (1 - 3)/(3 - (-2)) = -2/5 =  -0.4

The length of the sides are;

Distance between (-2, 3) and (-3, 1) = √5

Distance between (-2, -1) and (-3, 1) = √5

Distance between (-2, -1) and (3, 1) = √(29)

Distance between (-2, -3 and (3, 1) = √(29)

3. Please find attached the drawing of the polygon created with MS Excel

The  points on the figure are;

E(-4, 1), F(-2, 3), G(-2, -4)

The length of the segments are;

Distance between (-4, 1), and (-2, 3) = 2·√2

Distance between (-2, -4), and (-2, 3) = 7

Distance between (-2, -4), and (-4, 1)= √(29)

4. Please find attached the drawing of the polygon created with MS Excel

The points on the figure are;

T(1, -2), U(4, 1), V(2, 3), W(-1, 0)

Distance between T(1, -2), and U(4, 1) = 3·√2

Distance between U(4, 1), and V(2, 3) = 2·√2

Distance between V(2, 3) and W(-1, 0) = 3·√2

Distance between T(1, -2) and W(-1, 0) = 2·√2

The slope of the sides are;

Slope between T(1, -2), and U(4, 1) = 1

Slope between U(4, 1), and V(2, 3) = -1

Slope between V(2, 3) and W(-1, 0) = 1

Slope between T(1, -2) and W(-1, 0) = -1

Therefore;

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Based on proportional multiplication, the quantity of medication in grams that the patient, who weighs 63 kg needs in 12 hours is 1,224.3 mg.

Proportional multiplication involves the cross-multiplication of two fractional ratios, which are equated to each other.

Proportions involve the equation of two ratios.

The dosage required by a patient in 4 hours with a weight of 9 kg = 58.3 mg

Proportionately, the dosage required by a patient in 12 hours with a weight of 63 kg = 1,224.3 mg (58.3 x 63/9 x 12/4).

Thus, if a patient weighing 9 kg receives 58.3 mg of medicine in 4 hours, proportionately, in 12 hours, a patient weighing 63 kg will receive 1,224.3 mg.

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Answer:

Step-by-step explanation:

7 + 28 + 7 = 42 units^2     see image below

Therefore, we can estimate that having a 14 ft by 24 ft brick patio installed would cost around $3,033.33.

Cost is a key concept in economics. The price that was paid in return for any products or services is referred to. Cost is a monetary evaluation of the materials, labour, risks, expenses, and time required to purchase goods and services.

We can use proportions to estimate the cost of having a 14 ft by 24 ft brick patio installed based on the given information.

Let's use x to represent the estimated cost of having a 14 ft by 24 ft brick patio installed.

We can set up a proportion:

15/20 = 2275/x

To solve for x, we can cross-multiply and simplify:

15x = 20 x 2275

15x = 45500

x = 3033.33

So the estimated cost of having a 14 ft by 24 ft brick patio installed is approximately $3,033.33.

Therefore, we can estimate that having a 14 ft by 24 ft brick patio installed would cost around $3,033.33.

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a) The total cost of purchasing the car that is worth $32,000 with the harmonized sales tax (HST) of 13% is $36,160.

b) By leasing the car instead of purchasing it outright, the customer will save $14,760.

c) The lease options for the lessee after returning the car include:

Lease buyout

Extending the lease

Signing a new lease agreement

Buying out the car and then reselling it.

A lease is a financing option for the use of capital assets.

With a lease, the lessee uses the asset for a specific period and the lessor receives periodic payments.

here, we have,

There are operating and finance (capital) leases.  Though, under new accounting standards, all leases are to be classified as finance leases.

Value of the car = $32,000

Harmonized Sales Tax (HST) = 13%

Cost of the car with HST = $36,160 ($32,000 x 1.13)

Lease Arrangement:

Down payment = $1,000

Financing part = $35,160 ($36,160 - $1,000)

Installment payments = 48

Periodic payments = $425

Total lease cost = $21,400 ($1,000 + 48 x $425)

Savings (advantages) from leasing = $14,760 ($36,160 - $21,400)

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The closest answer is C. between $40.00 and $45.00.

What is the arithmetic operations?

The four basic mathematical operations are Addition, subtraction, multiplication, and division.

Jennifer bought 14.75 gallons of gasoline at a cost of $2.95 a gallon.

The amount she paid for the gasoline is:

14.75 gallons * $2.95/gallon = $43.56

Rounded to the nearest whole dollar, the amount she paid is $44.

Hence, the closest answer is C. between $40.00 and $45.00.

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Answer: To calculate the tip amount, we need to multiply the bill amount by the percentage tip as a decimal:

Tip amount = 0.20 x $76 = $15.20

To find the total amount to pay, we add the bill amount and the tip amount:

Total amount = $76 + $15.20 = $91.20

Rounding this to the nearest dollar gives:

Total amount = $91

Therefore, you need to pay $91 in total if you leave a 20% tip on a $76 bill.

Step-by-step explanation:

Daffynition Decoder is a fun and challenging word puzzle that requires both creativity and logical thinking to solve. It is often found in puzzle books, newspapers, and online puzzle websites.

Daffynition Decoder is a word puzzle game that involves decoding phrases or sentences by using a letter-substitution method. The game provides a list of numbered words, where each word is a cryptic definition of a common word or phrase. The solver's goal is to figure out what each word represents and then use the letters from those words to decode a secret message. The game is typically played by filling in a chart or grid, where each row corresponds to a numbered word in the list, and each column represents a letter in the decoded message. The solver uses the numbered words to fill in the corresponding letters in the grid, eventually revealing the hidden message.

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Complete question:

What is project Daffynition Decoder?

Deck A the safe walk zone must be painted at the points 13.74 units left and right of the edge of the circle, whereas for Deck B it must be painted at 12.68 units.

Maximum Height refers to the vertical distance between the Reference Level and the highest point of a building's roof surface, including parapet walls, mechanical equipment installed on the roof, authorised Signage, and other permissible projections of any type.

The equation of the given circular space station is:

[tex]x^2 + y^2 = 15^2\\\\x^2 + y^2 = 225[/tex]

Given that, a 6-foot person can walk through the circular station.

Substitute the value of y = 6.

[tex]x^2 + 6^2 = 225\\\\x^2 = 225 - 36\\\\x = \sqrt{189}\\ \\x = \pm 13.74[/tex]

Also, Deck B is 2 feet below deck A thus,

[tex]x^2 + 8^2 = 225\\\\x^2 = 225 - 64\\\\x = \sqrt{161}\\ \\x = \pm 12.68[/tex]

Hence, for Deck A the safe walk zone must be painted at the points 13.74 units left and right of the edge of the circle, whereas for Deck B it must be painted at 12.68 units.

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Answer:

-1/5(x-6)^2+3

Step-by-step explanation:

+3 makes the graph go up by 3

-6 makes the graph go right 6

we put a negative outside of (x-6) since this goes down and is a -x^2 parabola

no clue about 1/5 just kept putting numbers outside of (x-6) to eventually get the width.

Step-by-step explanation:

closest to all points on the graph

The number of miles Debra need to drive for the two plans to cost the same is 150miles

The unitary method is a method for solving a problem by the first value of a single unit and then finding the value by multiplying the single value. Unitary method is a technique by which we find the value of a single unit from the value of multiple devices and the value of more than one unit from the value of a single unit. It is a method that we use for most of the calculations in math.

We are given that;

Initial fee of first plan=$47.96

Additional fee of first plan=$0.19 per mile

Initial fee of second plan=$53.96

Additional fee of second plan=$0.15 per mile

Now,

Let's call the number of miles Debra drives "m". Then the cost of the first plan would be:

47.96 + 0.19m

And the cost of the second plan would be:

53.96 + 0.15m

To find the number of miles at which the two plans cost the same, we can set these expressions equal to each other and solve for "m":

47.96 + 0.19m = 53.96 + 0.15m

0.04m = 6

m = 6 / 0.04

m = 150

Therefore, by  the unitary method answer will be 150 miles.

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The graph of both the exponential functions y = 3(11)ˣ  and y = (1/12)ˣ  can be plotted by assuming some arbitrary values of 'x' that correspond to

different values of 'y'.

A function is said to be n exponential function when the independent variable is the exponent.

For example -: aˣ.

The two exponential functions can be graphed by assuming some arbitrary values of 'x'.

In y = 3(11)ˣ.

At x = 1, y = 33.

At x = - 1, y = 3/11.

At x = 2, y = 363.

So, The points are (1, 33), (- 1, 3/11) and (2, 363).

Now we can plot these points and join them by free hand.

Similarly, y = (1/12)ˣ can also be graphed in a similar manner.

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Answer:

6 units to the right

Step-by-step explanation:

Z(3,4) → Z'(9,4)

d = √(9-3)² + (4-4)² = √6²+0² = 6   to the right

The translation of point Z of triangle XYZ from (3, 4) to Z′(9, 4) is a move of 6 units to the right.

In Mathematics, particularly geometry, the translation of a point is the process of moving the point a specific number of units either up, down, left or right. In the case of your triangle XYZ, the original position of point Z is at (3, 4) and Z′ is positioned at (9, 4). The y-coordinate hasn't changed; it remains at 4. This tells us that the translation is not up or down. However, if we observe the x-coordinate, you’ll see that it went from 3 to 9. Changing the x-coordinate moves the point horizontally, so we know our translation is left or right. Lastly, the difference between 3 and 9 is 6. We can therefore determine that the point moved 6 units to the right.

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The rate of speed of the jet is given by S = 98 miles per hour

Speed is defined as the rate of change of position of an object in any direction. Speed is measured as the ratio of distance to the time in which the distance was covered. Speed is a scalar quantity as it has only direction and no magnitude

Speed = Distance / Time

Given data ,

Let the rate of speed of the jet be represented as S

Now , the value of S is

The distance traveled by the jet D = 490 miles

The time taken by the jet to travel 490 miles T = 5 hours

So , the speed of the jet is given by

Speed = Distance / Time

On simplifying , we get

The speed of the jet S = 490 / 5

The speed of the jet S = 98 miles per hour

Now , the distance traveled by the jet in 11 hours be D₁

The time taken to travel D₁ distance is 11 hours

where D₁ = Speed x Time

On simplifying , we get

D₁ = 98 x 11

D₁ = 1,078 miles

Hence , the distance traveled at a speed of 98 mph for 11 hours is 1,078 miles

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The value of sin (x/2) is -√(26/27).

The value of cos (x/2) is -√(2/27)

The value of tan (x/2) is 1/12.

We know that sin(x) = -2/27 in Quadrant 3, which means that x lies between 180 degrees and 270 degrees.

a. To find sin(x/2), we can use the half-angle formula:

sin(x/2) = ±√[(1-cos(x))/2]

Since x lies in Quadrant 3, we know that cos(x) is negative. To find cos(x), we can use the Pythagorean identity:

sin²(x) + cos²(x) = 1

(-2/27)² + cos²(x) = 1

cos²(x) = 1 - (-2/27)²

cos(x) = -√(1 - (-2/27)²)

cos(x) = -25/27

Now we can substitute cos(x) into the half-angle formula:

sin(x/2) = ±√[(1-(-25/27))/2]

sin(x/2) = ±√[(27/27 + 25/27)/2]

sin(x/2) = ±√[52/54]

sin(x/2) = ±√(26/27)

Since x is in Quadrant 3, sin(x/2) is negative.

Therefore;

sin(x/2) = -√(26/27)

b. To find cos(x/2), we can use the half-angle formula:

cos(x/2) = ±√[(1+cos(x))/2]

Again, we know that cos(x) is negative, so we can use the value we found earlier:

cos(x/2) = ±√[(1-25/27)/2]

cos(x/2) = ±√[2/27]

Since x is in Quadrant 3, cos(x/2) is also negative.

Therefore;

cos(x/2) = -√(2/27)

c. Finally, to find tan(x/2), we can use the identity:

tan(x/2) = sin(x)/(1+cos(x))

We already know sin(x) and cos(x), so we can substitute them in:

tan(x/2) = (-2/27)/(1-25/27)

tan(x/2) = (-2/27)/(-24/27)

tan(x/2) = 1/12

Therefore,

tan(x/2) = 1/12

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In order to resolve the ambiguity and find the correct solution, we need to use additional information such as the Law of Sines or the Law of Cosines.

what is a Triangle ?

Triangle can be defined in which it consists three sides, three angles and sum of three angles is always 180 degrees.

An ambiguous case in geometry refers to a situation where, given certain information about a triangle, there are two possible solutions for one of the unknown angles or sides. This situation arises in trigonometry when we have to solve a triangle using the Law of Sines or the Law of Cosines.

The conditions that create an ambiguous case with two possible solutions are as follows:

Given two sides and an angle opposite one of the sides:

If we are given two sides and an angle opposite one of the sides, there can be two possible solutions for the triangle. For example, if we are given side lengths of 4 and 5, and an angle of 40 degrees opposite the side of length 4, there are two possible triangles that can be formed.

Given two angles and a side opposite one of the angles:

If we are given two angles and a side opposite one of the angles, there can be two possible solutions for the triangle. For example, if we are given angles of 30 degrees and 60 degrees, and a side length of 4 opposite the 30-degree angle, there are two possible triangles that can be formed.

In both of these cases, the ambiguous case arises when the given information is not sufficient to uniquely determine the triangle.

Therefore,  In order to resolve the ambiguity and find the correct solution, we need to use additional information such as the Law of Sines or the Law of Cosines.

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