Place an inequality symbol between each fraction pair. State reasoning or rationale. (8)/(9),(10)/(12) -(5)/(6),-(6)/(8) Circle fractions that are completely simplified. State how this was determined.

The value of x is (12 ± √141) / 3. The solution has been obtained by solving the algebraic equation.

A mathematical expression is said to be "algebraic" if it includes variables, constants, and algebraic operations (addition, subtraction, etc.). The equals sign is a requirement for the expression to satisfy the algebraic equation.

We are given an equation as 3x (x - 6) - 8x = -(2x + 1)

On solving the given equation, we get

⇒3x (x - 6) - 8x = -(2x + 1)

⇒3x² - 18x - 8x = -2x - 1

⇒3x² - 26x = -2x - 1

⇒3x² - 24x = - 1

⇒3x² - 24x + 1 = 0

Using quadratic formula, we get

x = (12 ± √141) / 3

Hence, the value of x is (12 ± √141) / 3.

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One possible value of m is 53.

A possible value of m can be found using the Chinese Remainder Theorem. The Chinese Remainder Theorem states that if two numbers a and b are relatively prime, then there exists an integer x such that:
x ≡ a (mod m)
x ≡ b (mod n)

In this case, we have:
m ≡ 3 (mod 5)
m ≡ 1 (mod 7)

Both 5 and 7 are prime so we can start by finding the smallest positive integer that is congruent to 3 (mod 5) and 1 (mod 7). This integer is 18:

18 ≡ 3 (mod 5)
18 ≡ 1 (mod 7)

Now, we can add multiples of 5*7 = 35 to 18 until we find a value of m that is greater than 40 but less than 80:

18 + 35 = 53

53 is greater than 40 but less than 80, so one possible value of m is 53.

Therefore, one possible value of m is 53.

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(a) Revenue function: R = 13x
Cost function: C = 2x + 1540
(b) Break-even point: 2x + 1540 = 13x
x = 1540/11
x = 140 units

(c) Break-even point as a percent of capacity: 140/2500 x 100% = 5.6%
(d) Break-even point in sales dollars: 13 x 140 = $1820
(a) The revenue function is R(x) = 13x, where x is the number of units sold. The cost function is C(x) = 2x + 1540, where x is the number of units sold.
(b) To find the break-even point, we set R(x) = C(x):
13x = 2x + 1540
11x = 1540
x = 140
So the firm needs to sell 140 units to break even.
(c) The break-even point as a percent of capacity is 140/2500 = 0.056 or 5.6%.
(d) The break-even point in sales dollars is 140 units * $13/unit = $1820.

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According to the information we can infer that the volume of the prism would be 63cm³

To find the volume of the prism we must perform the following mathematical operation:

First of all we must find the cube root of each of the options.
[tex]\sqrt[3]{84} = 4,3795191398878892657\\\sqrt[3]{63} = 3,9790572078963918596\\\sqrt[3]{42} = 3,4760266448864497867\\\sqrt[3]{7} = 1,9129311827723891012[/tex]

Once we find these values we must check them with the formula for the volume of a pyramid. In this case, it would be the area of the base times the height. Then we must take into account that the cube root of these numbers would be the equivalent to the side length of the figures.

In the case of 63, the procedure would be as follows:

Then we can establish that this value corresponds to the length of the sides and the height of the pyramid because as a result it gives us the volume of the pyramid. So to find the volume of the prism we must do the following operation:

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Answer: 372 units²

Step-by-step explanation:

     We can use this formula to solve for the surface area:

Surface Area = 2lw + 2lh + 2hw

     We will substitute the known values and solve:

Surface Area = 2lw + 2lh + 2hw

Surface Area = 2(12)(3) + 2(12)(10) + 2(10)(3)

Surface Area = 372 units²

If 2x-1 is a factor of 24x⁴ - 2x² +2x +f , the value of f is  1 - 2x.

Synthetic division is a method of polynomial division used to divide a polynomial by a linear expression of the form x - a, where a is a constant. It is a shorthand way of performing long division for polynomials and is particularly useful when the divisor is of the form x - a, as it eliminates the need for writing out the full divisor in each step of the division process.

Since 2x-1 is a factor of the given polynomial, we know that it will divide exactly into the polynomial, leaving no remainder. Therefore, we can use polynomial long division to find the quotient when 24x⁴ - 2x² + 2x + f is divided by 2x-1:

           12x³ + 5x² - 3x - f/2 + 1/4

      ____________________________________

2x - 1 |   24x⁴ + 0x³ - 2x² + 2x + f

         - (24x⁴- 12x³)

----------------------

12x^3

- (12x³ - 6x^2)

----------------------

5x^2

- (5x² - 2.5x)

----------------------

2.5x

- (2.5x - f/4)

-----------------------

f/4 + 2x/4 - 1/4

Since there is no remainder, the last term in the quotient must be zero. Therefore:

f/4 + 2x/4 - 1/4 = 0

Simplifying and solving for f, we get:

f/4 = 1/4 - 2x/4

f = 1 - 2x

Therefore, f = 1 - 2x.

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The box of largest volume the company can build for $96 is the box that costs exactly $96.

The company can build a box of the largest volume for $96 by calculating the area of the bottom and sides of the box. To calculate the area of the bottom, use the formula A=l x w, where l is the length of one side and w is the width of one side. Since the bottom is square, l=w. Thus, A=l^2. The area of the bottom is then multiplied by the cost of the material for the bottom, which is $2 per cm2, to calculate the total cost of the bottom.

Similarly, to calculate the cost of the sides, use the formula A=2lh + 2wh, where h is the height of the box. The area of the sides is then multiplied by the cost of the material for the sides, which is $1 per cm2, to calculate the total cost of the sides.

The total cost of the box is the cost of the bottom plus the cost of the sides. Since the company has budgeted $96 to build one box, the total cost of the box must not exceed $96. Therefore, the box of largest volume the company can build for $96 is the box that costs exactly $96.

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The points F(8,-1), G(2,1), H(1, -2), and I(7,-4) form a quadrilateral. The desired slopes and lengths are follows:

The type of quadrilateral that BEST identifies the given points is a parallelogram.

To find the slopes and lengths of the quadrilateral, we will use the slope formula and the distance formula.

Slope formula: m = (y2 - y1)/(x2 - x1)
Distance formula: d = √((x2 - x1)² + (y2 - y1)²)

Slopes:
FG: m = (1 - (-1))/(2 - 8) = -1/3
GH: m = (-2 - 1)/(1 - 2) = 3
HI: m = (-4 - (-2))/(7 - 1) = -1/3
IF: m = (-1 - (-4))/(8 - 7) = 3

Lengths:
FG: d = √((2 - 8)² + (1 - (-1))²) = √(40) = 2√(10)
GH: d = √((1 - 2)²+ (-2 - 1)²) = √(10)
HI: d = √((7 - 1)^2 + (-4 - (-2))²) = √(40) = 2√(10)
IF: d = √((8 - 7)² + (-1 - (-4))^2) = √(10)

From the above calculations, we can see that the slopes of FG and HI are equal, and the slopes of GH and IF are equal. This means that the opposite sides of the quadrilateral are parallel. Additionally, the lengths of FG and HI are equal, and the lengths of GH and IF are equal. This means that the opposite sides of the quadrilateral are also congruent.

Therefore, the type of quadrilateral that BEST identifies the given points is a parallelogram.

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Answer:

Area = 15.6

Perimeter = 20.6

Step-by-step explanation:

Both smaller triangles are right triangles

The left triangle is isosceles right triangle since 2 angles are 45o so AB is hypotenuse and the other 2 sides are equal

so c^2 = a^2 + b^2

c^2 = 2a^2

(3√2)^2 = 2a^2

2a^2 = 18

a^2 = 9

a = 3

For the right triangle the angle at B is 60o, hypotenuse is BC

so cos(60) = adjacent/hypotenuse

1/2 = 3/BC

=> BC = 6

sin(60) = opposite/hypotenuse

√3/2 = opp/6

opp = 3√3

so AC = 3 + 3√3 = 6√3

Area of triangle = A = 1/2bh = 1/2(6√3)(3) = 9√3 = 15.6

Perimeter = 3√2 + 6√3 + 6 = 20.6

Answer:

y=3/2x-7

Step-by-step explanation:

The population two years ago was 28,080.

To calculate this, we need to use the following formula: Population two years ago = Population now - (Population increase rate x Number of years). Therefore, the population two years ago was 31,200 - (10% x 2) = 28,080.

We can also look at this another way. To calculate the 10% increase per year, we can use the following formula: Population increase rate = (Population now - Population two years ago) / Number of years. Therefore, 10% = (31,200 - 28,080) / 2 = 3,120.

We can use this same formula to calculate the population three years ago. Using the same formula, the population three years ago was 25,960.

To summarize, the population two years ago was 28,080 and the population three years ago was 25,960. The population has been increasing by 10% each year.

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The European company will earn 128,750 Euros in profit for selling 1,500 headphones.

To find the profit for selling 1,500 headphones, we need to plug in the value of h into the profit function E(h) and simplify.

Since h is measured in thousands of headphones, we need to divide 1,500 by 1,000 to get h = 1.5.

Now we can plug in h = 1.5 into the profit function and simplify:

E(1.5) = -4(1.5)^(4) + 7(1.5)^(3) - 7(1.5) + 20

E(1.5) = -4(5.0625) + 7(3.375) - 10.5 + 20

E(1.5) = -20.25 + 23.625 - 10.5 + 20

E(1.5) = 12.875

So the profit for selling 1,500 headphones is 12.875 ten thousands of Euros, or 128,750 Euros.

Therefore, the European company will earn 128,750 Euros in profit for selling 1,500 headphones.

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The volume of the cone is  1004.8 cm³.

Let's calculate the volume of the cone with radius 8 cm and the slant height is 17 cm. Therefore, let's find the volume of the cone as follows:

volume of a cone = 1 / 3 πr²h

where

Therefore,

h² = 17² - 8²

h = √289 - 64

h = √225

h = 15 cm

Therefore,

volume of a cone = 1 / 3 × 3.14 × 8² × 15

volume of a cone = 1 / 3 × 3.14 × 64 × 15

volume of a cone = 3014.4 / 3

volume of a cone = 1004.8 cm³

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Answer:

Degree: The degree of the function is 3.

X-Intercepts with multiplicity greater than 1: x = -2 and x = 0

How many distinct x-intercepts?: There are two distinct x-intercepts.

How many zeros are there?: There are three zeros, one at x = -2 and two at x = 0 (with a multiplicity of 2).

The demand for nurses in the country can be expressed as the function D(x) = 2.8 + 0.044x. The year in which the demand for nurses in the country will reach 3.5 million is in 2027.

a. The demand for nurses in the country can be expressed as a function of the number of years after 2011, x, by using the equation D(x) = 2.8 + 0.044x. This equation represents the initial demand in 2011 (2.8 million) plus the expected increase in demand each year (0.044 million) multiplied by the number of years after 2011 (x).

b. To find the year in which the demand for nurses will reach 3.5 million, we can set D(x) equal to 3.5 and solve for x:

3.5 = 2.8 + 0.044x

0.7 = 0.044x

x = 0.7 / 0.044

x = 15.9

Since x represents the number of years after 2011, we can add 15.9 to 2011 to find the year in which the demand will reach 3.5 million:

2011 + 15.9 = 2026.9

Therefore, the demand for nurses in the country will reach 3.5 million during the year 2027.

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The number of fruits that Kim bought in all, given the apples and oranges was ;

Let's call the number of oranges that Kim bought "O" and the number of apples that Kim bought "A".

From the problem, we know that Kim bought fewer than 20 pieces of fruit, so we can write:

A + O < 20

A = O + 3

Simplifying:

2O + 3 < 20

2O < 17

O < 8.5

Since O represents the number of oranges, we know that Kim bought fewer than 8.5 oranges. Since she can't buy a fraction of an orange, this means that she must have bought either 8 oranges or fewer.

Now we can use the equation A = O + 3 to find the number of apples she bought:

A = 8 + 3 = 11

So Kim bought 8 oranges and 11 apples.

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The interval giving the middle 68% of the lifetime of bulbs in hours is given as follows:

(1350, 1450).

The Empirical Rule states that, for a normally distributed random variable, the symmetric distribution of scores is presented as follows:

The middle 68% of scores is within one standard deviation of the mean, hence the bounds of the interval are given as follows:

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The comparison between the distance he biked and swam is 3 : 1.

A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.

From the given information the comparison of the distance he swam to the distance he biked can be represented in the form of ratios.

Given, He rode his bike 3/4 miles and swam for 1/4 mile.

Therefore, The ratio of biking : swimming = 3/4 : 1/4.

Now, Multiplying by 4 to get rid of the denominator we have,

Biking : Swimming = 3 : 1.

As the days in both comparisons increase in the same manner the ratio will remain the same.

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Debra's two area baking plates are 17 square inches different in size.

The area surrounding a shape forms its perimeter. Area is a unit of measurement for interior space. Surface area is a measurement of a solid shape's exposed surface, whereas area is a two-dimensional measurement of the size of a flat surface (three-dimensional).

The square baking dish has the following surface area with an 8-inch side length:

Area of first square = (side length)² = 8² = 64 square inches

Similarly, the surface area of a square baking dish with 9-inch sides is:

Area of second square = (side length)² = 9² = 81 square inches

Difference in area = |Area of second square - Area of first square|

Difference in area = |81 - 64|

Difference in area = 17 square inches

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Option D i.e. number line with closed circle on point negative 7 over 22 and arrow shaded to the left represents the solution to the inequality.

A comparison between two or more mathematical expressions is known as an inequality.

We are given an inequality as one seventh m is less than or equal to negative one twenty second.

So, from the above information, we get the inequality as

m/7 ≤ −1/22

Now, on solving the above inequality, we get the following

⇒m/7 ≤ −1/22

⇒m ≤ −7/22

Hence, Option D i.e. number line with closed circle on point negative 7 over 22 and arrow shaded to the left represents the solution to the inequality.

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Answer:

97271.1577

Step-by-step explanation:

multiply 3374.60 by 2882.45%

The process to find the scale factor of a shape and the concept of sequence of transformations is explained below.

In order to find the scale factor of a shape, we need to compare the corresponding side lengths of the original shape to the corresponding side lengths of the new shape after a dilation.

The scale factor is the ratio of any two corresponding side lengths.

The sequence of transformations is defined as the order in which different transformations are applied to a shape.

For example, if we have a triangle and we first translate it, then rotate it, and then dilate it, the sequence of transformations would be : translate → rotate → dilate.

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The given question is incomplete, the complete question is

How do you find the scale factor of a shape? And What is the sequence of transformations?

The total number of popsicle sticks needed to make one snowman using the given conditions for system of equations is equal to 9.

Let us consider 'x' is the number of popsicle sticks required to make one pencil cup.

And 'y' represents the number of popsicle sticks required to make one snowman.

The required set up two equations are,

8x + 15y = 271  ___(1)

14x + 15y = 373 ___(2)

Multiplying both sides of equation (1) by 14 and both sides of equation (2) by -8 we get,

112x + 210y = 3794

-112x - 120y = -2984

Add both the above equations to eliminate the x variable we get,

⇒ 112x + (-112x) + 210y + (-120y) = 3794 + (-2984)

Simplifying the equation we get,

⇒90y = 810

Dividing both sides by 90 we get,

⇒ y = 9

Therefore, number of popsicle sticks required to make one snowman is equal to 9.

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(a) The permutations can be written as a product of disjoint cycles as follows:
α = (1 5 4 3 2)(6 7 8)
β = (1)(2 3 8 4 7)(5 6)
γ = (1 2 3 5)(4 5 6 8)

(b) The order of each permutation is the least common multiple of the lengths of the disjoint cycles. Therefore:
Order of α = lcm(5, 3) = 15
Order of β = lcm(1, 5, 2) = 10
Order of γ = lcm(4, 4) = 4

(c) The permutations can be written as a product of 2-cycles as follows:
α = (1 2)(2 3)(3 4)(4 5)(5 1)(6 7)(7 8)(8 6)
β = (1 1)(2 3)(3 8)(8 4)(4 7)(7 1)(5 6)(6 5)
γ = (1 2)(2 3)(3 5)(5 1)(4 5)(5 6)(6 8)(8 4)
Since α and β have an even number of 2-cycles, they are even permutations. γ has an odd number of 2-cycles, so it is an odd permutation.

(d) The inverse of each permutation can be found by reversing the order of the elements in each cycle. The inverses can be written as a product of disjoint cycles as follows:
[tex]α^-1 = (1 2 3 4 5)(6 8 7)β^-1 = (1)(2 7 4 8 3)(5 6)γ^-1 = (1 5 3 2)(4 8 6 5)[/tex]

(e) The products αβ and βα can be found by performing the permutations in the given order. The products can be written as a product of disjoint cycles as follows:
αβ = (1 6 8 4 7 2 3 5)(5 1)
βα = (1 7 4 8 6 5 3 2)(2 1)
The order of αβ is lcm(8, 2) = 8, and the order of βα is lcm(8, 2) = 8.

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Pythagorean theorem: Show that ⟨x, y⟩ = 0

Let V be an Euclidean space and let x, y ∈ V.(a) If ∥x + y∥ = ∥x − y∥ then determine ⟨x, y⟩.We know that ∥x + y∥² = ⟨x + y, x + y⟩ = ⟨x, x⟩ + ⟨x, y⟩ + ⟨y, x⟩ + ⟨y, y⟩ = ∥x∥² + 2⟨x, y⟩ + ∥y∥²And ∥x − y∥² = ⟨x − y, x − y⟩ = ⟨x, x⟩ − ⟨x, y⟩ − ⟨y, x⟩ + ⟨y, y⟩ = ∥x∥² − 2⟨x, y⟩ + ∥y∥²Since ∥x + y∥ = ∥x − y∥, we have ∥x∥² + 2⟨x, y⟩ + ∥y∥² = ∥x∥² − 2⟨x, y⟩ + ∥y∥²⟨x, y⟩ = 0(b) Show that 2⟨x, y⟩ ≤ ∥x∥² + ∥y∥².We know that ∥x + y∥² = ∥x∥² + 2⟨x, y⟩ + ∥y∥²And ∥x + y∥² ≥ 0So ∥x∥² + 2⟨x, y⟩ + ∥y∥² ≥ 0⟨x, y⟩ ≤ (∥x∥² + ∥y∥²)/2(c) If (z, y) = 0 for all z ∈ V then show that y = 0.Let z = y, then (y, y) = 0⟨y, y⟩ = ∥y∥² = 0∥y∥ = 0So y = 0(d) If ∥x∥ = ∥y∥ for some x, y ∈ V then ⟨x + y, x − y⟩ = 0.We know that ∥x + y∥² = ∥x∥² + 2⟨x, y⟩ + ∥y∥²And ∥x − y∥² = ∥x∥² − 2⟨x, y⟩ + ∥y∥²Since ∥x∥ = ∥y∥, we have ∥x∥² = ∥y∥²So ∥x + y∥² − ∥x − y∥² = 4⟨x, y⟩ = 0⟨x, y⟩ = 0(e) Pythagorean theorem: Show that ⟨x, y⟩ = 0 if and only if ∥x + y∥² = ∥x∥² + ∥y∥².We know that ∥x + y∥² = ∥x∥² + 2⟨x, y⟩ + ∥y∥²If ⟨x, y⟩ = 0, then ∥x + y∥² = ∥x∥² + ∥y∥²If ∥x + y∥² = ∥x∥² + ∥y∥², then 2⟨x, y⟩ = 0⟨x, y⟩ = 0

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Therefore , the solution of the given problem of equation comes out to be julian must  sell at least 28 wallets in order to make even.

Mathematical formulas frequently use the same variable letter to try to impose unity between two assertions. Many academic numbers are shown to be equal using mathematical fraction equations, also known as assertions. Using y + 6 as an illustration, the normalise does not divide 12 into two parts, but instead b + 6. It is possible to determine the connection between each sign part and the number of lines. The significance of a symbol usually contradicts itself.

Here,

For Julian to break even, his revenue must match his expenses. Therefore, we can equalise the two equations and find x:

=> 18x - 140 = 7x + 160

7x is subtracted from both lines to yield:

=> 11x - 140 = 160

140 added to both ends results in:

=> 11x = 300

When we multiply both parts by 11, we get:

=> x = 27.27 (rounded to two integer places) (rounded to two decimal places)

We can round up to the nearest whole amount because Julian cannot sell a fraction of a wallet. Julian must therefore sell at least 28 wallets in order to make even.

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The surface area of the cone is 578.1 square feet.

We know that for a cone of radius R and slant height H the surface area is given by the formula:

S = pi*R^2 + pi*R*(H)

with pi = 3.14

On the diagram we can see that H = 19.3ft, and the diameter of the base is 14 ft, then the radius is:

R = 14ft/2 = 7ft

REplacing these values in the formula above we will get:

S = 3.14*(7ft)^2 + 3.14*7ft*(19.3ft) = 578.1 ft²

So the correct option is the second one.

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Note that in the above sketch, ΔABC ≇ ΔDCA. That is they are NOT congruent. The proof is that ∠ADC which is supposed to be = ∠BAC are not equal. Thus, line AC which is supposed to be equal to Line BC are not equal.

When two triangles are congruent, it means they are exactly the same in terms of their shape and size, so all corresponding sides and angles are equal.

In ΔABC,

Line AB = 8ft
Line BC = 11ft

Line AC = 13.6ft

∠ABC = 90°

∠BAC = 53.97°
∠ACB = 36.03°


As you can see,

Where as, ∠ADC = 62° - given

∠BAC = 53.97°

Also

Where as:

AC (The Hypotenuse ΔABC which is also the Adjacent Side of ΔACD) = 13.6ft

BC (The adjacent side of ΔABC) = 11ft.

Note that in order to prove congruence, at least two angles and one side from both Triangles must be equal (Angle Angle Side Theorem). Or

Two sides and one angle from both Triangles must be equal (Side - Angle - Side).

Or All three angles (Angle - Angle - Angle);
Or All three Sides (Side - Side Side).

In this case, only one side from both Triangles AC is common to both Triangles.

On the basis of the above, therefore, ΔABC ≇ ΔDCA.


The calculations showing how we arrived at the missing sides and angles are given below:


ΔABC:

AC = √(AB² + BC²)
AC = √(8² + 11²)
AC = √(64 + 121)
AC = √(185)
AC = 13.6014705087

AC [tex]\approx[/tex] 13.60

∠ ACB = arcsin (AB/AC)
∠ ACB = arcsin (8/13.6014705087)
= arcsin(0.58817169767505)
∠ ACB = 0.6288 rad converted to degrees

∠ ACB  = 36.03°

Thus, since the sum of Angles in a Triangle is 180°
∠BAC = 180-36.03° -90°
∠BAC = 53.97°


ΔACD


y (AD) = AC / sin(β)
y (AD) =  13.6/ sin(62°)
y (AD)  = 13.6/0.88294759285

AD = 15.40°

CD = √(AD² - AC²)

CD = √(15.4029526893712 - 13.62)

CD = √52.290951550999

CD = 7.23

The outstanding angle in ΔACD is ∠CAD

Since the sum of Angles in a Triangle is 180°

∠CAD = 180 - 90 - 62
∠CAD = 28°

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Full Question:

Although part of your question is missing, you might be referring to this full question: See attached Image.

The error made by the student is instead of taking +4 as common he has taken - 4 as common.

To factor an expression by grouping, we need to look for groups of terms that have common factors. Then, we factor out those common factors from each group and see if there is a common factor that can be further factored out.

Here we have

6x² ​ – x​ – 12  

The above expression can be factorized as follows

=> 6x² – x – 12

= 6x² – 9x + 8x – 12        [ Splitting the middle term ]

= 3x(2x – 3) + 4(2x – 3)     [ Grouping the terms ]  

= (3x + 4)(2x – 3)  

Hence, the factors of given expression are (3x + 4) and (2x – 3)  

The error made by the student is in step 3 instead of taking +4 as common he has taken - 4 as common which doesn't give the common factors in the next step taken.    

Therefore,

The error made by the student is instead of taking +4 as common he has taken - 4 as common.

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[tex]5 < - 8[/tex]

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