Conduct the hypothesis test and provide the test statistic and the critical​ value, and state the conclusion. A person drilled a hole in a die and filled it with a lead​ weight, then proceeded to roll it 200 times. Here are the observed frequencies for the outcomes of​ 1, 2,​ 3, 4,​ 5, and​ 6, respectively: 29​, 31​, 49​, 37​, 26​, 28. Use a 0.025 significance level to test the claim that the outcomes are not equally likely. Does it appear that the loaded die behaves differently than a fair​ die?

Answer:

statmen there for this should me solved and abcd are proof

reason bcz this is not equal square

The indicated operation is the composition of functions. To perform this operation, we substitute the expression for g(x) into f(x). The composition of f(g(x)) is given by f(g(x)) = (4x - 1)^2 + 2.

To compute f(g(x)), we first evaluate g(x) by substituting x into the expression for g(x): g(x) = 4x - 1. Next, we substitute this result into f(x): f(g(x)) = f(4x - 1).

Now, let's expand and simplify f(g(x)):

f(g(x)) = (4x - 1)^2 + 2

        = (4x - 1)(4x - 1) + 2

        = 16x^2 - 8x + 1 + 2

        = 16x^2 - 8x + 3.

The domain of f(g(x)) is the same as the domain of g(x) since the composition involves g(x). In this case, g(x) is defined for all real numbers. Therefore, the domain of f(g(x)) is also all real numbers.

In summary, the composition of f(g(x)) is given by f(g(x)) = 16x^2 - 8x + 3, and the domain of f(g(x)) is all real numbers.

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The final answer after evaluating the expression 3.14([tex]a^{2}[/tex] + ab) (by putting the value a = 3 and b = 4) is 65.94.

When a = 3 and b = 4, we substitute the supplied values into the expression to assess 3.14([tex]a^{2}[/tex] + ab):

3.14([tex]3^{2}[/tex] + 3 * 4)

We begin by solving the exponent:

[tex]3^{2}[/tex] = 3 * 3 = 9

The values are then entered into the expression:

3.14(9 + 3 * 4)

Inside the brackets, multiply the result:

3.14(9 + 12)

The numbers in the brackets are added:

3.14(21)

The decimal number is now multiplied by 21:

3.14 * 21 = 65.94

The evaluated expression is 65.94 as a result.

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.

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The answer is:

Work/explanation:

We're asked to evaluate the expression [tex]\sf{3.14(a^2+ab)}[/tex] for a = 3 and b = 4.

Plug in the data:

[tex]\sf{3.14(3^2+3*4)}[/tex]

[tex]\sf{3.14(9+12)}[/tex]

[tex]\sf{3.14(21)}[/tex]

[tex]\bf{65.94}[/tex]

Answer:

a+2bc/3a....4+2(--5×--7)/3(4)....4+2(35)/12.....4+70/12...74/12..answer =6⅙..option C

Answer:

perimeter ≈ 58.3 units

Step-by-step explanation:

the perimeter of the sector includes 2 radii and the arc

perimeter = 4 + 4 + 16π = 8 + 16π ≈ 58.3 ( to 1 decimal place )

i) the original distance when the lighthouse is due West of the ship is 7 km

ii) The time in minutes when the lighthouse is due West of the ship is 21 minutes

iii) The distance in km of the ship from the lighthouse when the lighthouse is due West of the ship is 29.97 km

To solve this problem, we'll use the concepts of relative velocity and trigonometry. Let's break down the problem into three parts:

i) Finding the original distance when the lighthouse is due West of the ship:

The ship's velocity is given as 21 km/h at a bearing of 052°. Since the ship observed the lighthouse due North, we know that the angle between the ship's initial heading and the lighthouse is 90°.

To find the distance, we'll consider the ship's velocity in the North direction only. Using trigonometry, we can determine the distance as follows:

Distance = Velocity * Time = 21 km/h * (20 min / 60 min/h) = 7 km (to three significant figures).

ii) Finding the time in minutes when the lighthouse is due West of the ship:

To find the time, we need to consider the change in angle from 052° to 312°. The difference is 260° (312° - 052°), but we need to convert it to radians for calculations. 260° is equal to 260 * π / 180 radians. The ship's velocity in the West direction can be calculated as:

Velocity in West direction = Velocity * cos(angle) = 21 km/h * cos(260 * π / 180) ≈ -19.98 km/h (negative because it's in the opposite direction).

To find the time, we can use the formula:

Time = Distance / Velocity = 7 km / (19.98 km/h) = 0.35 h = 0.35 * 60 min = 21 minutes (to three significant figures).

iii) Finding the distance in km of the ship from the lighthouse when the lighthouse is due West of the ship:

We can use the formula for relative velocity to find the distance:

Relative Velocity = sqrt((Velocity in North direction)² + (Velocity in West direction)²)

Using the values we calculated earlier, we have:

Relative Velocity = sqrt((21 km/h)² + (-19.98 km/h)²) ≈ 29.97 km/h (to three significant figures).

Therefore, the ship is approximately 29.97 km away from the lighthouse when the lighthouse is due West of the ship.

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Jacob bought 0.7 yards of ribbon.

To find out how many yards of ribbon Jacob bought, we need to determine 4/5 of the length of ribbon that Cindy bought.

Cindy bought 7/8 yard of ribbon. To find 4/5 of this length, we multiply 7/8 by 4/5:

(7/8) * (4/5) = (7 * 4) / (8 * 5) = 28/40

To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor, which is 4:

(28/4) / (40/4) = 7/10

Therefore, Jacob bought 7/10 yard of ribbon.

However, we can convert this fraction to a mixed number or decimal to express it in yards.

To convert 7/10 to a mixed number, we divide the numerator (7) by the denominator (10):

7 ÷ 10 = 0 with a remainder of 7

So, 7/10 is equivalent to 0 7/10 or 0.7 yards.

Therefore, Jacob bought 0.7 yards of ribbon.

In summary, Jacob bought 7/10 yard or 0.7 yards of ribbon.

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Currency exchange rates are primarily based on each country's economy.

The exchange rate of a currency is influenced by various factors, such as inflation rates, interest rates, political stability, economic performance, trade balances, and market supply and demand.

These factors reflect the overall strength or weakness of a country's economy and play a significant role in determining the value of its currency relative to other currencies in the foreign exchange market.

While historical exchange rate systems, such as the gold standard, had an impact on currency values in the past, the modern exchange rate regime is primarily determined by market forces and economic fundamentals.

Bartering and one-for-one exchange are not directly related to currency exchange rates in the context of global currency markets, as exchange rates involve the relative value of one currency against another.

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Answer:

? = 220°

Step-by-step explanation:

the measure of the inscribed angle QRS is half the measure of its intercepted arc QS , then

QS = ? = 2 × QRS = 2 × 110° = 220°

The value of a is 8 ln 9 - 36. Given an arithmetic sequence that has the first term Ina and a common difference In 3. The 13th term in the sequence is 8 ln 9.

We need to find the value of a.

Step 1: Finding the 13th term. Using the formula for the nth term of an arithmetic sequence: an = a1 + (n - 1)d, where an is the nth term, a1 is the first term, n is the number of terms, and d is the common difference.

Substituting the given values, we get:an = a1 + (n - 1)d 13th term, a 13 = a1 + (13 - 1)3a13 = a1 + 36 a1 = a13 - 36 ...(1)Given that a13 = 8 ln 9.

Substituting in equation (1), we get: a1 = 8 ln 9 - 36.

Step 2: Finding the value of a. Using the formula for the nth term again, we can write the 13th term in terms of a as: a13 = a + (13 - 1)3a13 = a + 36a = a13 - 36.

Substituting the value of a13 from above, we get:a = 8 ln 9 - 36. Therefore, the value of a is 8 ln 9 - 36.

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The total surface area of the pyramid is option c [tex]72 cm^2[/tex].

The total surface area of a pyramid is given by the formula;S= ½Pl + BWhere B is the area of the base and P is the perimeter of the base.

To find the perimeter, add the length of all the sides of the base. Here, the base of the pyramid is a square with sides measuring 6 cm each.Therefore, its perimeter = 6 + 6 + 6 + 6 = 24 cm.

Now, to find the total surface area, we need to find the area of all four triangular faces. To find the area of one of the triangular faces, we can use the formula:

A = 1/2bhWhere b is the base of the triangle and h is the height.

To find the height, we can use the Pythagorean theorem:

[tex]h = \sqrt(6^2 - 3^2) = \sqrt(27) = 3 \sqrt(3)[/tex]

Therefore, the area of one of the triangular faces is:

A = 1/2bh = [tex]1/2(6)(3\sqrt(3)) = 9\sqrt(3)[/tex]

We have four triangular faces, so the total area of the triangular faces is:

[tex]4(9\sqrt(3)) = 36\sqrt(3)[/tex]

Finally, we can find the total surface area by adding the area of the base and the area of the triangular faces:

S = ½Pl + B = [tex]1/2(24)(3\sqrt(3)) + 6^2 = 36\sqrt(3) + 36 = 36(\sqrt(3) + 1).[/tex]

Therefore, the total surface area of the pyramid is 36(sqrt(3) + 1) cm², which is approximately 72 cm². Hence, the correct option is C. [tex]72 cm^2[/tex].

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Answer:

3x - x²

Step-by-step explanation:

x(3 - x)

each term in the parenthesis is multiplied by the x outside. This is called the Distributive law

= (x × 3) + (x× - x)

= 3x + (- x²)

= 3x - x²

The answer is:

Work/explanation:

To simplify this expression, we will use the distributive property:

[tex]\sf{x(3-x)}[/tex]

Distribute the x:

[tex]\sf{x\cdot3-x\cdot x}[/tex]

[tex]\sf{3x-x^2}[/tex]

Step-by-step explanation:

To solve this problem, we need to use the product rule of differentiation and some trigonometric identities. Let's start by finding the derivative of y with respect to x:

y = (sin 2x) √(3+2x)

Using the product rule, we get:

dy/dx = (sin 2x) d/dx(√(3+2x)) + (√(3+2x)) d/dx(sin 2x)

To find these derivatives, we need to use the chain rule and the derivative of sin 2x:

d/dx(√(3+2x)) = (1/2√(3+2x)) d/dx(3+2x) = (1/√(3+2x))

d/dx(sin 2x) = 2cos 2x

Substituting these values, we get:

dy/dx = (sin 2x) / √(3+2x) + 2cos 2x (√(3+2x))

Now, we need to simplify this expression to the desired form. To do that, we can use the trigonometric identity:

sin 2x = 2sin x cos x

Substituting this value, we get:

dy/dx = 2sin x cos x / √(3+2x) + 2cos 2x (√(3+2x))

Now, we can use the trigonometric identity:

cos 2x = 1 - 2sin^2 x

Substituting this value, we get:

dy/dx = 2sin x cos x / √(3+2x) + 2(1 - 2sin^2 x)(√(3+2x))

Simplifying further, we get:

dy/dx = (2cos x - 4cos x sin^2 x) / √(3+2x) + 2√(3+2x) - 4sin^2 x√(3+2x)

Now, we can see that this expression matches the desired form:

dy/dx = sin 2x + (4 + Bx)cos 2x / √(3+2x)

where A = -4 and B = -2. Therefore, we have shown that:

dy/dr = sin 2x + (4 - 2x)cos 2x / √(3+2x)

where A = -4 and B = -2.

"1/49," accurately represents the result of Urvi's work.The correct answer is option D.

Urvi's work is accurate and follows the correct rule of "multiply by the reciprocal" to solve the fraction division problem.

In the given problem, she is dividing 14 by the fraction 2/7. According to the rule, to divide a fraction by another fraction, we multiply the first fraction by the reciprocal of the second fraction.

In Urvi's work, she first takes the reciprocal of 2/7, which is 7/2. Then, she multiplies 14 by the reciprocal, which gives us (14 * 7/2).

Simplifying the multiplication, we get 98/2, which simplifies further to 49. Therefore, the correct answer to the fraction division problem is 1/49.

Option D, which states "1/49," accurately represents the result of Urvi's work. This option is the most accurate description of her work because it correctly shows the final simplified fraction after applying the "multiply by the reciprocal" rule.

Overall, Urvi's work demonstrates a correct understanding and application of the rule for solving fraction division problems.

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The Probable question may be:
Urvi solved a fraction division problem using the rule “multiply by the reciprocal.” Her work is shown below.

A StartFraction 14 divided by StartFraction 2 Over 7 EndFraction.

B. StartFraction 1 Over 14 EndFraction times

C. StartFraction 2 Over 7 EndFraction = StartFraction 2 Over 98

D. EndFraction or StartFraction 1 Over 49 EndFraction

Which is the most accurate description of Urvi’s work?

The volume of the cube with a diagonal of 4√2 is 64 cubic units.

To find the volume of a cube, we need to know the length of its side. In this case, we are given the length of the diagonal, which we can use to find the side length.

Let's assume the side length of the cube is represented by "s". We know that the diagonal of a cube forms a right triangle with two sides of equal length, which are the sides of the cube.

Using the Pythagorean theorem, we can set up the equation:

s² + s² = (4√2)²

Simplifying the equation:

[tex]2s² = 32[/tex]

Dividing both sides by 2:

[tex]s² = 16[/tex]

Taking the square root of both sides:

[tex]s = 4[/tex]

Now that we have the side length of the cube, we can find the volume by cubing the side length:

Volume = s³ = 4³ = 64 cubic units.

Therefore, the volume of the cube with a diagonal  of 4√2 is 64 cubic units.

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The volume of the cube whose diagonal is 4√2 is 128√2 cubic units.

To find the volume of a cube, we need to know the length of its side. Given that the diagonal of the cube is 4√2, we can use this information to determine the side length.

In a cube, the diagonal is related to the side length (s) by the equation:

diagonal = s√3

The given diagonal is 4√2. So we can set up the equation:

4√2 = s√3

To find s, we can divide both sides of the equation by √3:

s = (4√2) / √3

To simplify this expression, we can rationalize the denominator by multiplying both the numerator and denominator by √3:

s = (4√2 ׳ √3) / (√3 × √3)

s = (4√6) / √3

Now, let's calculate the value of s:

s = (4√6) / √3

s = (4/√3) × √6

s = (4/√3) × (√3 × √2)

s = 4√2

So the side length of the cube is 4√2.

Now, to calculate the volume of the cube, we use the formula:

Volume = side^3

Volume = (4√2)³

Volume = 4³ × (√2)³

Volume = 64 × 2√2

Volume = 128√2

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Answer:

127

Step-by-step explanation:

Angle C+Angle D=Angle ABC

Since C+D+CBD=180 and ABC+CBD=180

subtract getting C+D-CBD=0 and C+D=CBD

so 67+60=127 which is your answer

*Just to clarify, when i said C and D, i meant angle C and angle D

Answer:

(a)  $556 billion

(b)  $581 billion

(c)  $693 billion

Step-by-step explanation:

The given function is:

[tex]\boxed{A(x)=314e^{0.044x}}[/tex]

where A(x) is the assets (in billions of dollars) for a financial firm .

If x = 7 corresponds to the year 2007 then:

Therefore, to find the assets for each of the given years, substitute the corresponding value of x into the function.

[tex]\begin{aligned}A(13)&=314e^{0.044 \cdot 13}\\&=314e^{0.572}\\&=314(1.77180712...)\\&=556.34743707...\\&=556\; \sf (nearest\;billion)\end{aligned}[/tex]

[tex]\begin{aligned}A(14)&=314e^{0.044 \cdot 14}\\&=314e^{0.616}\\&=314(1.851507181...)\\&=581.3732549...\\&=581\; \sf (nearest\;billion)\end{aligned}[/tex]

[tex]\begin{aligned}A(18)&=314e^{0.044 \cdot 18}\\&=314e^{0.792}\\&=314(2.20780762...\\&=693.2515954...\\&=693\; \sf (nearest\;billion)\end{aligned}[/tex]

Answer:

a. 16

c. 120°

e. 60°

Step-by-step explanation:

Properties of Parallelogram:

For Question:

a.

DC= AB=16 Opposite side is congruent.

c.

m ∡DCB = m ∡DAB=120° Opposite angles are congruent.

e.

m ∡ABC= ​?

m ∡ABC+ m∡DAB =180° Consecutive angles are supplementary.

Substituting value

m ∡ABC + 120°=180°

m ∡ABC =180°-120°=60°

m ∡ABC=60°

Answer:

a)  DC = 16

c)  m∠DCB = 120°

e)  m∠ABC = 60°

Step-by-step explanation:

The opposite sides of a parallelogram are equal in length. Therefore, DC is the same length as AB.

As AB = 16, then DC = 16.

[tex]\hrulefill[/tex]

The opposite angles of a parallelogram are equal in measure. Therefore, m∠DCB is equal to m∠DAB.

As m∠DAB = 120°, then m∠DCB = 120°.

[tex]\hrulefill[/tex]

Adjacent angles of a parallelogram sum to 180°. Therefore:

⇒ m∠ABC + m∠DAB = 180°

⇒ m∠ABC + 120° = 180°

⇒ m∠ABC + 120° - 120° = 180° - 120°

⇒ m∠ABC = 60°

The probability of event A occurring given that event B has not occurred (P(A\B)) is 0.

To find P(A\B), we need to calculate the probability of event A occurring given that event B has not occurred. In other words, we want to find the probability of A happening when B does not happen.

The formula to calculate P(A\B) is:

P(A\B) = P(A∩B') / P(B')

Where B' represents the complement of event B, which is the event of B not occurring.

Given that P(A∩B) = 3/7 and P(B) = 7/8, we can find P(A∩B') and P(B') to calculate P(A\B).

To find P(B'), we subtract P(B) from 1, since the sum of the probabilities of an event and its complement is always equal to 1.

P(B') = 1 - P(B)

      = 1 - 7/8

      = 1/8

Now, to find P(A∩B'), we need to subtract P(A∩B) from P(B'):

P(A∩B') = P(B') - P(A∩B)

        = 1/8 - 3/7

        = 7/56 - 24/56

        = -17/56

Since the probability cannot be negative, we can conclude that P(A∩B') is 0.

Finally, we can calculate P(A\B) using the formula:

P(A\B) = P(A∩B') / P(B')

      = 0 / (1/8)

      = 0

Therefore, the probability of event A occurring given that event B has not occurred (P(A\B)) is 0.

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Answer:

Step-by-step explanation:

x + 2-3

|x ≥ 1

|x + 2 ≤ -3

x + 22 -6

Answer:

Step-by-step explanation:

(a) To find the linear cost function C(x), we need to consider the fixed cost and the marginal cost. The fixed cost is $100, and the marginal cost is $8 per pair of earrings.

The linear cost function can be represented as C(x) = mx + b, where m is the slope (marginal cost) and b is the y-intercept (fixed cost).

In this case, the slope (m) is $8, and the y-intercept (b) is $100. Therefore, the linear cost function is:

C(x) = 8x + 100.

(b) The average cost function (AC) can be found by dividing the total cost (C(x)) by the number of units produced (x):

AC(x) = C(x) / x.

Substituting the linear cost function C(x) = 8x + 100, we have:

AC(x) = (8x + 100) / x.

(c) To find C(5), we substitute x = 5 into the linear cost function:

C(5) = 8(5) + 100

= 40 + 100

= 140.

Interpretation: C(5) = 140 means that when the artist produces 5 pairs of earrings, the total cost (including fixed and variable costs) is $140.

(d) To find C(50), we substitute x = 50 into the linear cost function:

C(50) = 8(50) + 100

= 400 + 100

= 500.

Interpretation: C(50) = 500 means that when the artist produces 50 pairs of earrings, the total cost (including fixed and variable costs) is $500.

(e) The horizontal asymptote of C(x) represents the cost as the number of units produced becomes very large. In this case, the marginal cost is constant at $8 per pair of earrings, indicating that as the number of units produced increases, the cost per unit remains the same.

Therefore, the horizontal asymptote of C(x) is $8, indicating that the average cost per pair of earrings approaches $8 as the number of units produced increases indefinitely.

In practical terms, this means that for every additional pair of earrings produced beyond a certain point, the average cost will stabilize and remain around $8, regardless of the total number of earrings produced.

Answer:  the correct answer is 5/2

Step-by-step explanation:

To find the slope of a line perpendicular to a given line, we can use the property that the product of the slopes of two perpendicular lines is equal to -1.

The given line has an equation of y = -2/5x + 4/5.

The slope of this line can be determined by comparing it to the slope-intercept form (y = mx + b), where "m" represents the slope. In this case, the slope of the given line is -2/5.

To find the slope of the line perpendicular to this line, we take the negative reciprocal of the given slope. The negative reciprocal of -2/5 is 5/2.

The solution to the inequality f(x) > -5 is x < -5 or x > -1.

To solve the inequality f(x) > -5 for the quadratic function f(x) = x^2 + 6x, we need to find the values of x that satisfy the inequality.

First, set up the inequality:

x^2 + 6x > -5

Next, move all terms to one side of the inequality to get a quadratic expression:

x^2 + 6x + 5 > 0

To solve this quadratic inequality, we can factor it:

(x + 5)(x + 1) > 0

Now, we need to determine the sign of the expression for different intervals on the x-axis.

a) When x < -5:

If x is less than -5, both (x + 5) and (x + 1) are negative, so their product is positive.

Thus, the inequality is satisfied for x < -5.

b) When -5 < x < -1:

If x is between -5 and -1, (x + 5) is positive, but (x + 1) is negative. The product of a positive and a negative number is negative.

Thus, the inequality is not satisfied for -5 < x < -1.

c) When x > -1:

If x is greater than -1, both (x + 5) and (x + 1) are positive, so their product is positive.

Thus, the inequality is satisfied for x > -1.

Therefore, x -5 or x > -1 is the answer to the inequality f(x) > -5.

In summary:

x < -5 or x > -1.

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Answer:

This equation is always false

Step-by-step explanation:

3(x - 4) + 2x = 5x  - 9

3x - 12 + 2x = 5x -9

5x - 12 = 5x - 9

5x - 5x = -9 + 12

0 = 3

This equation is always false

Answer:

Sure. First, we need to find the inverse function of f(x). We can do this by using the following steps:

1. Let y = f(x).

2. Solve the equation y = 4x3 + 6x - 10 for x.

3. Replace x with y in the resulting equation.

This gives us the following inverse function:

```

f^-1(y) = (-1 + sqrt(1 + 12y)) / 2

```

Now, we need to find f^-1′ (0). This is the derivative of the inverse function evaluated at y = 0. We can find this derivative using the following steps:

1. Use the chain rule to differentiate f^-1(y).

2. Evaluate the resulting expression at y = 0.

This gives us the following:

```

f^-1′ (0) = (3 * (1 + 12 * 0) ^ (-2/3)) / 2 = 1.5

```

Therefore, f^-1′ (0) = 1.5.

Step-by-step explanation:

The equation of the line in standard form is 3x + y - 4 = 0.

To find the equation of a line passing through two points, we can use the slope-intercept form of a linear equation, which is given by y = mx + b, where m is the slope and b is the y-intercept.

Given the points S(, 1) and T(, 4), we need to determine the slope (m) and the y-intercept (b).

The slope (m) can be found using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points.

Substituting the values, we get:

m = (4 - 1) / ( - ) = 3 / ( - ) = -3

Now that we have the slope, we can substitute it into the equation y = mx + b and use one of the given points to solve for the y-intercept (b).

Using the point S(, 1):

1 = (-3)(1) + b

1 = -3 + b

b = 4

Therefore, the equation of the line passing through the points S and T is:

y = -3x + 4

Converting it to standard form Ax + By + C = 0, we rearrange the equation:

3x + y - 4 = 0

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By applying the algebraic identity, the expression (24)³ - (13)³ - (11)³ can be simplified as (11)(1057) - (11)³ = 10396, without actually calculating the cubes.

The expression (24)³ - (13)³ - (11)³ can be evaluated without calculating the cubes by applying the concept of algebraic identities.

By using the identity (a³ - b³) = (a - b)(a² + ab + b²), we can rewrite the expression as (24 - 13)(24² + 24*13 + 13²) - (11)³.

Simplifying further, we have (11)(576 + 312 + 169) - (11)³.

Combining like terms, the expression evaluates to (11)(1057) - (11)³.

Finally, we can simplify to obtain the result of 11 multiplied by 1057 minus 11 cubed, without actually calculating the cubes. The simplified expression (24)³ - (13)³ - (11)³ evaluates to 10396.

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The derivative dy/dx of the implicit equation[tex]x - 2xy + x^2y + y = 10[/tex] is given by[tex]\frac{(2y - 2 + 2xy)}{(-2x + x^2 + 1)}[/tex]

To find the derivative dy/dx of the implicit equation [tex]x - 2xy + x^2y + y =[/tex]10, we will use the implicit differentiation technique.

Step 1: Differentiate both sides of the equation with respect to x.

For the left-hand side:

[tex]d/dx (x - 2xy + x^2y + y) = d/dx (10)[/tex]

Taking the derivative of each term separately:

[tex]d/dx (x) - d/dx (2xy) + d/dx (x^2y) + d/dx (y) = 0[/tex]

Step 2: Apply the chain rule to the terms involving y.

The chain rule states that if we have y = f(x), then dy/dx = dy/du * du/dx, where u = f(x).

For the term 2xy, we have y = f(x) = xy. Applying the chain rule, we get:

[tex]d/dx (2xy) = d/dx (2xy) * dy/dx[/tex]

= 2y + 2x * dy/dx

Similarly, for the term x^2y, we have [tex]y = f(x) = x^2y.[/tex]Applying the chain rule:

[tex]d/dx (x^2y) = d/dx (x^2y) * \frac{dx}{dy} \\= 2xy + x^2 * \frac{dx}{dy}[/tex]

Step 3: Substitute the derivatives back into the equation.

[tex]d/dx (x) - (2y + 2x * dy/dx) + (2xy + x^2 * dy/dx) + d/dx (y) = 0[/tex]

Simplifying the equation:

[tex]1 - 2y - 2x * \frac{dx}{dy} + 2xy + x^2 * \frac{dx}{dy} + \frac{dx}{dy} = 0[/tex]

Step 4: Group the terms involving dy/dx together and solve for dy/dx.

Combining the terms involving dy/dx:

[tex]-2x * \frac{dx}{dy} + x^2 * \frac{dx}{dy} + dy/dx = 2y - 1 + 2xy - 1[/tex]

Factoring out dy/dx:

[tex](-2x + x^2 + 1) * \frac{dx}{dy} = 2y - 1 + 2xy - 1[/tex]

[tex]dy/dx = \frac{(2y - 2 + 2xy)}{(-2x + x^2 + 1)}[/tex]

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