Four identical cubes are joined to make a new shape.

What is the new shape’s surface area?

Answer:

26.0 years

Step-by-step explanation:

It will take approximately 16.79 years (to the nearest hundredth) for the initial mass of 20 grams to decay to four grams, given a constant decay rate of 6% per year.

To determine how long it will take for the 20 grams of the element to decay to four grams, we need to calculate the time using the given decay rate.

Given:

Initial mass (M₀) = 20 grams

Decay rate: 6% per year

Let's set up an equation to represent the decay of the element:

M(t) = M₀ * (1 - r)^t

Here, M(t) represents the mass at time t in years, M₀ is the initial mass, r is the decay rate (expressed as a decimal), and t is the time in years.

We want to find the value of t when M(t) = 4 grams:

[tex]4 = 20 * (1 - 0.06)^t[/tex]

Dividing both sides by 20:

[tex]0.2 = (1 - 0.06)^t[/tex]

Taking the natural logarithm of both sides:

[tex]ln(0.2) = ln[(1 - 0.06)^t][/tex]

Using logarithmic properties, we can simplify further:

[tex]ln(0.2) = t * ln(1 - 0.06)[/tex]

Now, we can solve for t by dividing both sides of the equation by ln(1 - 0.06):

t =[tex]ln(0.2) / ln(1 - 0.06)[/tex]

Using a calculator, we find:

t ≈ 16.79 years

Therefore, it will take approximately 16.79 years (to the nearest hundredth) for the initial mass of 20 grams to decay to four grams, given a constant decay rate of 6% per year.

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Therefore, the solution to the system of linear equations is (x, y) = (-1, 4).

In mathematics, an equation is a statement that two expressions are equal. It typically contains one or more variables (unknowns) and specifies a relationship between those variables. Equations are used to model real-world phenomena, solve problems, and make predictions. There are many types of equations in mathematics, including linear equations, quadratic equations, polynomial equations, exponential equations, trigonometric equations, and many more. Each type of equation has its own set of methods and techniques for solving it. Solving an equation means finding the values of the variables that satisfy the equation. This may involve manipulating the equation algebraically, using graphical methods, or employing numerical techniques such as iteration or approximation. Equations play a central role in many areas of mathematics and science, as well as in everyday life. They are used to model physical systems, optimize processes, design products, and analyze data, among many other applications.

Here,

We are given the system of linear equations:

17x + 4y = -1 ...(1)

y - 4x - 8 = 0 ...(2)

We can use the substitution method to solve this system. Solving equation (2) for y, we get:

y = 4x + 8

Substituting this expression for y into equation (1), we get:

17x + 4(4x + 8) = -1

Simplifying this equation, we get:

17x + 16x + 32 = -1

33x = -33

x = -1

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

y - 4(-1) - 8 = 0

y + 4 - 8 = 0

y = 4

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The area of trapezoid ABCD is 39 ft².

A trapezοid is a quadrilateral with twο sides that are parallel. It has fοur sides, fοur angles, and twο pairs οf οppοsite sides that are equal in length. The nοn-parallel sides, οr legs οf the trapezοid, are usually referred tο as the bases οf the trapezοid. The twο parallel sides are referred tο as the bases, the nοn-parallel sides are referred tο as the legs, and the line cοnnecting the twο parallel sides is called the midline. The angles created by the bases and the legs are called the base angles

Area of trapezoid = [tex]\dfrac{a + b}{2} h[/tex]

Here, a = 5ft

b = 21 ft

h = 3ft

Area of trapezoid = [tex]\dfrac{5 + 21}{2} (3)[/tex]

Area of trapezoid = 39 ft²

Thus, The area of trapezoid ABCD is 39 ft².

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The statement "The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9" is false.

In mathematics, an equation is a statement that two expressions are equal. It typically contains one or more variables (unknowns) and specifies a relationship between those variables. Equations are used to model real-world phenomena, solve problems, and make predictions. There are many types of equations in mathematics, including linear equations, quadratic equations, polynomial equations, exponential equations, trigonometric equations, and many more. Each type of equation has its own set of methods and techniques for solving it.

Here,

To determine which statements are true about the circle whose equation is x2 + y2 – 2x – 8 = 0, we can use the standard form of the equation of a circle, which is:

(x - h)² + (y - k)² = r²

where (h, k) is the center of the circle and r is the radius.

Comparing the given equation to the standard form, we can complete the square by adding 1 to both sides:

x² - 2x + y² - 8 = -1

(x - 1)² + y² = 9

From this, we can see that the center of the circle is (1, 0) and the radius is 3 units. Therefore, the statements that are true are:

The center of the circle does not lie on the x-axis, so the statement "The center of the circle lies on the x-axis" is false.

The center of the circle does lie on the y-axis, so the statement "The center of the circle lies on the y-axis" is true.

The standard form of the equation is (x - 1)² + y² = 3, so the statement "The standard form of the equation is (x - 1)² + y² = 3" is true.

The radius of this circle is not the same as the radius of the circle whose equation is x² + y² = 9, so the statement "The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9" is false.

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Thus, we can be 90% confident that the true proportion of men in the college is between 8.8% and 14.0%.

Percent, denoted by the symbol "%", is a way of expressing a number as a fraction of 100. For example, 50% is equivalent to the fraction 50/100, which can be simplified to 1/2. Percentages are commonly used to express ratios, proportions, or rates in various contexts such as finance, statistics, and everyday life. For instance, an interest rate of 3% means that for every $100 borrowed, the borrower will be charged $3 per year. Similarly, a score of 80% on an exam means that the student has answered 80 out of 100 questions correctly.

by the question.

To find the confidence interval of the true proportion of men in the college, we need to use the formula:

[tex]CI = p ± z\sqrt((p(1-p))/n)[/tex]

where p is the sample proportion (80/700 = 0.114), z is the z-score associated with the confidence level (90% in this case), and n is the sample size (700).

To find the z-score, we can use a standard normal distribution table or a calculator. For a 90% confidence level, the z-score is approximately 1.645.

Plugging in the values, we get:

[tex]CI = 0.114 ± 1.645\sqrt((0.114(1-0.114))/700)[/tex]

Simplifying, we get:

[tex]CI = 0.114 ± 0.026[/tex]

Therefore, the 90% confidence interval for the true proportion of men in the college is:

[tex]CI = 0.114 ± 0.026[/tex]

Rounding to the nearest tenth of a percent, we get:

CI = (8.8%, 14.0%)

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Answer:

the answer is b because 16km is resolved by 19.2

The square root of 6,57,55,881 using long division method is 8563.

Group the digits starting from the units place into pairs of two, from right to left. If there are any remaining digits, add a zero to the left of them. So for 65755881:

65 | 75 | 58 | 81

Find the largest number whose square is less than or equal to the leftmost group (in this case, 65). The square of 8 is 64, so the first digit of the square root must be 8. Write 8 as the first digit of the answer and subtract 64 from 65 to get 1.

Bring down the next group (75) to the right of the remainder (1) to get 175.

Double the first digit of the current partial answer (which is 8) to get 16, and place a blank digit to its right. We will be filling in this blank digit with the next digit of the square root.

Find the largest digit d such that the number 16d, when multiplied by d and then by 10, is less than or equal to 175. To do this, we can start with d=9 and work downwards until we find a suitable value. We find that d=5 works, since 165510=8250 is less than 17500. So the next digit of the square root is 5.

Append the digit 5 to the partial answer, so we have 85 as the current partial answer.

Subtract 165 (the product of 16 and 5) from 175 to get the new remainder, which is 10.

Bring down the next group (58) to the right of the remainder (10) to get 1058.

Double the first digit of the current partial answer (which is 85) to get 170, and place a blank digit to its right.

Find the largest digit d such that the number 170d, when multiplied by d and then by 10, is less than or equal to 1058. To do this, we can start with d=9 and work downwards until we find a suitable value. We find that d=6 works, since 1706610=102360 is less than 105800. So the next digit of the square root is 6.

Append the digit 6 to the partial answer, so we have 856 as the current partial answer.

Subtract 1706 (the product of 170 and 6) from 1058 to get the new remainder, which is 402.

Bring down the last group (81) to the right of the remainder (402) to get 40281.

Double the first digit of the current partial answer (which is 856) to get 1712, and place a blank digit to its right.

Find the largest digit d such that the number 1712d, when multiplied by d, is less than or equal to 40281. To do this, we can start with d=9 and work downwards until we find a suitable value. We find that d=3 works, since 17123×3=51369 is less than 40281. So the next digit of the square root is 3.

Append the digit 3 to the partial answer, so we have 8563 as the final answer.

Therefore, the square root of 65755881 is 8563.

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Answer:

for what value of y is the rational expression y+6÷2y-3 be yndefined?

Answer:

5) A = 70°

6) X = 55°

7) Y = 8°

8) A = 51°

Step-by-step explanation:

There really no way to explain. All you ahve to do is inverse function to solve for the angle. Use a calculator.

5) sin(A) = 0.9397

[tex]A=sin^-1(0.9397)\\\\A=69.701...\\\\A = 70\\[/tex]

6) cos(X) = 0.5763

[tex]X=cos^-1(0.5763)\\\\X=54.998...\\\\X=55\\[/tex]

7) tan(Y) = 0.1405

[tex]Y=tan^-1(0.1405)\\\\Y=7.997...\\\\Y = 8\\[/tex]

8) sin(A) = 0.7771

[tex]A=sin^-1(0.7771)\\\\A=50.995...\\\\A = 51\\[/tex]

Answer:

the length of IJ is 7.2 units.

Step-by-step explanation:

Since IJ is parallel to LM, we can use the property of similar triangles to solve for IJ.

Triangles KIJ and KLM are similar, so we have:

IJ/KL = KI/LM

Substituting the given values, we get:

IJ/8 = 18/20

IJ/8 = 9/10

Multiplying both sides by 8, we get:

IJ = (8)(9/10)

IJ = 7.2

Therefore, the length of IJ is 7.2 units.

Answer:

Step-by-step explanation:

Yes, that is correct. The measure of the exterior angle of a triangle is equal to the sum of the two remote interior angles. In other words, if you extend one side of a triangle to form an exterior angle, then the measure of that exterior angle is equal to the sum of the measures of the two interior angles that are not adjacent to the exterior angle. This property is known as the Exterior Angle Theorem.

Mathematically, we can express this as:

Exterior angle = Sum of remote interior angles

Or, in symbols:

∠ABC = ∠A + ∠C

where ∠ABC is the exterior angle formed by extending side BC of triangle ABC, and ∠A and ∠C are the two remote interior angles.

Here are two more examples that illustrate the Exterior Angle Theorem:

Example 1:

Consider a triangle ABC with angles A, B, and C as shown below. If we extend side AC to form an exterior angle, then we can label the exterior angle as ∠DCE. The remote interior angles are ∠B and ∠A.

css

Copy code

      C

     / \

    /   \

   /     \

  /       \

 /         \

/______\

A             B

      D

According to the Exterior Angle Theorem, we have:

∠DCE = ∠B + ∠A

In response to the given question, we can state that As a result, Igbo expressions experienced a N50.00 loss.

In mathematics, you can multiply, divide, add, or subtract. An expression is formed as follows: Expression, number, and math operator Numbers, parameters, and functions make up a mathematical expression. It is possible to contrast phrases and expressions. Every mathematical statement that comprises variables, numbers, and a mathematical action between them is referred to as an expression. For example, the phrase 4m + 5 is made up of the phrases 4m and 5, as well as the variable m from the provided equation, all separated by the mathematical symbol +.  

Igbo spent N40.00 on 5 books, for a total of N40.00.

N200.00 = 5 books x N40.00 each

Igbo lost money since he sold the books for N150.00. To find the amount of the loss, subtract the selling price from the cost price:

Cost Price - Selling Price = Loss

N200.00 - N150.00 = Loss

Loss = N50.00

As a result, Igbo experienced a N50.00 loss.

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The relationship between the variables x and y is a direct variation as the constant of variation 1/3 is the same for all values of x and y

If the values of y varies directly as does of x, then there is a constant say "k" that is true for them, else y does not vary directly as x.

For x = 3 when y = 9, we can solve for the constant k as follows:

3 = k9

we divide through by 9

3/9 = k9/9

k = 1/3

The constant 1/3 is same for x = 9 when y = 27, x = 13 when y = 39, and x = 20 when y = 60.

Therefore, the relationship between the variables x and y is a direct variation as the constant of variation 1/3 is the same for all values of x and y.

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Option C. Al spends approximately 21 euros on his lunch in France.

Conversion rate is the percentage of website visitors or potential customers who take a desired action, such as making a purchase, signing up for a newsletter, or filling out a form. In other words, it is the rate at which website visitors are converted into customers or subscribers.

To convert US dollars to euros, we multiply the US dollar amount by the exchange rate of 0.69 euro per US dollar:

30.00 US dollars × 0.69 euro/US dollar = 20.70 euros ≅ 21 euros

Therefore, Al can spend approximately 21 euros on his lunch in France.

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If the company sells more than 3,000 pairs of socks per month, it will make a profit.

What is Algebraic expression ?

Algebraic expression can be defined as combination of variables and constants.

To analyze the profit of the Sock Company, we need to consider the total cost and total revenue.

Let's start with the total cost:

The cost per pair of socks is $6.

If x is the number of pairs of socks sold, then the total cost is 6x.

There is also a fixed cost of $12,000 per month.

So, the total cost can be expressed as:

C(x) = 6x + 12,000

Now, let's move to the total revenue:

The selling price per pair of socks is $10.

If x is the number of pairs of socks sold, then the total revenue is 10x.

So, the total revenue can be expressed as:

R(x) = 10x

The profit is the difference between the total revenue and total cost:

P(x) = R(x) - C(x) = 10x - (6x + 12,000) = 4x - 12,000

To find the range of x where the company makes a profit, we need to set P(x) greater than 0:

4x - 12,000 > 0

4x > 12,000

x > 3,000

Therefore, if the company sells more than 3,000 pairs of socks per month, it will make a profit.

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This means that option A, "4 < d < 26," is the correct answer.

The triangle inequality can be used to calculate various distances between Sherry and Amy. The lengths of any two sides of a triangle must add up to more than the length of the third side in order for the triangle inequality to hold.

If we assume that d is the separation between Amy and Sherry in this instance, we can then create a triangle with sides that are 11, 15, and d in length. The inequality of the triangle indicates that:

11 + 15 > d

or

d < 26

and

15 + d > 11

or

d > 4

So, d might have any of the following values:

4 < d < 26

This indicates that "4 d 26" in option A is the right response.

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Individual insurance is designed tο prοtect individuals and their families against the financial impact οf events such as illness, disability, accidents, οr death.

Individual insurance and business insurance are similar in that bοth prοvide financial prοtectiοn against risks and uncertainties. Hοwever, there are alsο significant differences between the twο types οf insurance.

Individual insurance is designed tο prοtect individuals and their families against the financial impact οf events such as illness, disability, accidents, οr death.

Business insurance is designed tο prοtect businesses against the financial impact οf events such as prοperty damage, liability claims, οr lοss οf incοme due tο interruptiοn οf business οperatiοns.

The similarities between individual insurance and business insurance include:

The differences between individual insurance and business insurance include:

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Answer:Answer:

median=10, 6 numbers, so median is between 3rd and 4th

.. 9 11 . .

mode=11

.. 9 11 11 .

range=4

7 8 9 11 11 11

Step-by-step explanation:

Two ancient mathematics concepts are given as follows:

The Egyptians used fractions extensively in their calculations and were the first civilization to develop a notation for fractions. Their notation used a symbol for the numerator placed above a symbol for the denominator, similar to our modern fraction notation.

Euclid's Elements is one of the most influential works in the history of mathematics. It provided a systematic and rigorous approach to geometry, which has been the basis of mathematical education for over two thousand years.

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On solving the provided question we cans ay that As a result, x + 4 function denotes the number of years the principal is older than the instructor, where x is the student's age in years.

Mathematicians examine numbers with their modifications, equations and associated structures, forms and their locations, and feasible positions for these things. The term "function" indicates the connection between a collection of inputs, each of which has a corresponding output. A function is a connection of inputs and outputs where each supply leads to a specific, identifiable outcome. Each function is assigned a domain, a codomain, or a scope. The letter f is widely used to denote functions (x). The symbol for entry is an x. The four primary types of accessible capabilities are on functions, yet another skills, so multiple capabilities, in capabilities, and on operations.

Let's start by writing down the phrases from the problem:

• The teacher's age is six years older than the student's age: T = 2x + 6, where T is the age of the instructor.

• The principal's age is ten years older than the student's age: P = 3x + 10, where P is the age of the principal.

To determine how many years the principal is older than the instructor, subtract the teacher's age from the principal's age:

P - T = (3x + 10) - (2x + 6)

When we simplify the equation, we get:

P - T = x + 4

As a result, x + 4 denotes the number of years the principal is older than the instructor, where x is the student's age in years.

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The cοrrelatiοn cοefficient οf data is 1.

A statistical measure knοwn as cοrrelatiοn expresses hοw clοsely twο variables are related linearly (meaning they change tοgether at a cοnstant rate).

The cοnstant cοefficient (οr cοnstant term) is the cοefficient nοt attached tο variables in an expressiοn. Fοr example, the cοnstant cοefficients οf the expressiοns abοve are the number 3 and the parameter c, respectively.

The cοefficient attached tο the highest degree οf the variable in a pοlynοmial is referred tο as the leading cοefficient. Fοr example, in the expressiοns abοve, the leading cοefficients are 2 and a, respectively.

In excel, use fοllοwing cοmmand tο find r

=>cοrrel(select data οf grade, select data οf Reading level)  

press enter

=> 1  

i.e. 1.00

The required cοrrelatiοn cοefficient r is 1.00

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If I were to run this company, I would be concerned about the discrepancy between the allocated budget and the actual spending in the graph.

Graph is a mathematical structure used to represent relationships between objects. It consists of vertices (also known as nodes) which represent the objects and edges (also known as arcs) which represent the relationships between the objects. Graphs are often used to represent networks, such as social networks, transportation networks, and communication networks. Graphs can also be used to represent data structures, such as trees and linked lists.

In particular, the finance, admin, marketing, HR, and developer departments all have allocated budgets that are greater than their actual spending. This could be a sign that the company is not utilizing its resources efficiently, since it is not spending up to its allocated budget. The discrepancy between allocated budget and actual spending could also indicate that the company is wasting money on unnecessary projects, or that its budget is not allocated correctly.

Furthermore, the customer support department has allocated budget of $100k, yet its actual spending is only $50k. This could indicate that the company is not providing adequate customer support, which could lead to customer dissatisfaction and a decrease in customer loyalty. It could also mean that the company is not taking advantage of its resources and not utilizing them to their fullest potential.

Overall, the discrepancy between the allocated budget and the actual spending could be a sign of inefficiencies and mismanagement within the company. As the company’s leader, I would be concerned about the implications of this discrepancy and would take steps to ensure that the company is utilizing its resources correctly and efficiently.

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Answer:

for any regular polygon, the sum of the exterior angles is always 360 degrees. In a regular polygon, all exterior angles have the same measure, which is equal to 360 degrees divided by the number of sides.

Let n be the number of sides of the polygon. Then we have:

360/n = 21.2

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

360 = 21.2n

n = 360/21.2

n ≈ 16.98

Since n must be a whole number (the number of sides in a polygon must be a whole number), the closest option is D) 17. Therefore, the polygon has 17 sides.

The area of Gabriel's father's garden is 62/3 square units, which is approximately 20.67 square units.

The question's purpose is obscure. But, based on its measurements of 7 3/4 by 2 2/3, I'm assuming you're interested in learning how to locate the location of Gabriel's father's garden.

We must multiply the length by the breadth in order to determine the garden's area. To make the calculation simpler, we can convert the mixed values to improper fractions as follows:

7 3/4 = (7 × 4 + 3)/4 = 31/4

2 2/3 = (2 × 3 + 2)/3 = 8/3

The garden is thus 8/3 in width and 31/4 in length. We multiply these values to determine the area:

Area is equal to 31/4 x (8/3), or 62/3 square units.

The size of Gabriel's father's garden is therefore 62/3 square units, or roughly 20.67 square units.

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The translation that maps K onto K' is a horizontal translation of 2.8 ft to the right, followed by a vertical translation of 5.6 ft downwards.

We first need to know the distance and direction of the two reflections in order to find the single translation that maps K onto K'.

When figure K is mirrored across line p, figure K" is the resultant image:

K'  K''

      |  |

      |  |

p ------   ------ d

      |  |

      |  |

     K   K'

We can see that the distances between K and p and K" and p, as well as K and d and K" and d, are the same. Because of this, K and K" are separated by a distance that is 2 2.8 feet longer than K and d.

When positioned above line D, figure K"s reflected image is K". This reflection, being perpendicular to p, has no impact on the horizontal placement of K" in regard to K'. K" is now beneath K', although the vertical axis has changed.

K'  K''

      |  |

      |  |

p ------   ------ d

      |  |

      |  |

     K'' K'

Consequently, a horizontal translation of 2.8 feet to the right, then a vertical translation of 5.6 feet downward, is the translation that maps K onto K'. The straightforward translation is:

(x, y) → (x + 2.8, y - 5.6).

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The sine of 5π/4 is [tex]\frac{-1}{\sqrt{2} }[/tex] , the cosine of 5π/4 is [tex]\frac{-1}{\sqrt{2} }[/tex] and the angle lies in the third quadrant.

What is the reference angle?

A reference angle is an acute angle formed between the terminal side of an angle in standard position and the x-axis. In other words, it is the smallest angle that can be formed between the terminal side of the angle and the x-axis.

To find the sine and cosine of 5π/4 using the reference angle, we first need to identify the reference angle.

In this case, 5π/4 is in the third quadrant, which means that the terminal side of the angle passes through the point (-1, -1) on the unit circle.

To find the reference angle, we draw a line from (-1, -1) to the x-axis, which forms a right triangle with sides of length 1, 1, and √2

The reference angle θ is the angle formed between the x-axis and the adjacent side of the triangle. By applying the trigonometric functions to this reference angle, we can find the sine and cosine of 5π/4.

sin(θ) = opposite/hypotenuse = [tex]\frac{1}{\sqrt{2} }[/tex]

cos(θ) = adjacent/hypotenuse = [tex]\frac{-1}{\sqrt{2} }[/tex]

Since sine is positive in the third quadrant and cosine is negative, we need to take the negative of the sine value and keep the negative sign for the cosine value:

sin(5π/4) = -sin(θ) = [tex]\frac{-1}{\sqrt{2} }[/tex]

cos(5π/4) = cos(θ) = [tex]\frac{-1}{\sqrt{2} }[/tex]

Therefore, the sine of 5π/4 is [tex]\frac{-1}{\sqrt{2} }[/tex], and the cosine of 5π/4 is [tex]\frac{-1}{\sqrt{2} }[/tex] .

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Answer:

PS=21.3

Step-by-step explanation:

please mark as brainliest

Answer:

(x - 5)^2 + (y - 5)^2 = 25

Step-by-step explanation:

If a circle has no x-intercepts and no y-intercepts, it means that the circle does not intersect either the x-axis or the y-axis. In other words, the center of the circle must lie at some point (h, k) where h and k are not equal to zero.

The general equation of a circle with center (h, k) and radius r is:

(x - h)^2 + (y - k)^2 = r^2

Since we know that the circle has a radius of 5, we can substitute r = 5 into the equation:

(x - h)^2 + (y - k)^2 = 25

Now, we need to find values for h and k such that the circle has no x-intercepts and no y-intercepts. Since the x-intercepts occur when y = 0 and the y-intercepts occur when x = 0, we need to make sure that neither of these values satisfies the equation.

Let's first consider the x-intercepts. We want to make sure that the circle does not intersect the x-axis, which means that the y-coordinate of the center must be equal to the radius. In other words, k = 5. Now, the equation becomes:

(x - h)^2 + (y - 5)^2 = 25

Next, we need to make sure that the circle does not intersect the y-axis, which means that the x-coordinate of the center must be equal to the radius. In other words, h = 5. Now, the equation becomes:

(x - 5)^2 + (y - 5)^2 = 25

Therefore, the equation of the circle that has no x-intercepts, no y-intercepts, and a radius of 5 is:

(x - 5)^2 + (y - 5)^2 = 25

The side x = 16.72 approximately to two decimal places using the cosine rule.

In trigonometry, the cosines rule relates the lengths of the sides of a triangle to the cosine of one of its angles.

Considering the given triangle, we shall calculate for the side x is using the cosine rule as follows;

a² = b² + c² - 2(b)(c)cosA

x² = 7² + 12² - 2(7)(12)cos121°

x² = 49 + 144 - 168(-0.5150)

x² = 193 + 86.52

x² = 279.52

x = 16.7189 {take square root of both sides}

Therefore, using the cosine rule, we can conclude the value of the side x of the triangle is 16.7189.

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The prove to show that ΔABC is similar to ΔMNC is explained below.

Similar triangles are triangles that have the same shape but are not necessarily the same size. More precisely, two triangles are similar if their corresponding angles are congruent and their corresponding sides are proportional.

Considering the given diagram, we can say:

AM = MC    (Given)

BN = NC     (Given)

BC = BN + NC (addition property of a line segment)

AC = AM + MC (addition property of a line segment)

NC/ BC = MC/ AC = constant

∠ACB = ∠MCN  (Reflexive property)

Thus, ΔABC ~ ΔMNC (Side-Angle-Side theorem)

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