What is the distance between point T (-5,1) and point I (-1,1)

Answer:

Step-by-step explanation:

Year 1: $850 * 0.91 = $773.50

Year 2: $773.50 * 0.91 = $704.69

Year 3: $704.69 * 0.91 = $641.95

...

Year 34: (continue the pattern)

We can continue this calculation for each year, but to save time, we can use an exponential decay formula:

Remaining Amount = Initial Amount * (1 - rate)^years

Substituting the values:

Remaining Amount = $850 * (1 - 0.09)^34

Calculating this expression:

Remaining Amount ≈ $850 * (0.91)^34 ≈ $255.88

After 34 years, approximately $255.88 will be left with Sally.

Julia Roberts would be 5ft 8.5 inches tall, or 68.1 inches tall.

Using the parameters:

The average height of a woman in the United States= 63 inches

The standard deviation of height = 2.7 inches.

Z score of 2.16 means that Julia Roberts is 2.16 standard deviations above the average height. This means that she is taller than 98.4% of women in the United States.

Height = Average Height + z Score * Standard Deviation

Height = 63 + 2.16 * 2.7 = 68.1

Therefore, Julia Roberts height is 68.1 inches

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In the first two sets of statements, the conclusion is not necessarily true. For example, a lion has fur and is an animal, but it does not have claws.

In the third set of statements, the conclusion is true. A rhombus is a type of square, so it has four congruent angles. In the fourth set of statements, the conclusion is not necessarily true. For example, the number 12 is divisible by 12 and 6, but it is not divisible by 3.

In the fifth set of statements, the conclusion is true. A number that is divisible by 12 is also divisible by 6 and 3. In the sixth set of statements, the conclusion is not necessarily true. For example, the number 9 is divisible by 9 and 3, but it is not divisible by 6.

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Answer:

(7, 3 )

Step-by-step explanation:

3y - 2 = x → (1)

- 2x - y = - 17 → (2)

substitute x = 3y - 2 into (2)

- 2(3y - 2) - y = - 17

- 6y + 4 - y = - 17

- 7y + 4 = - 17 ( subtract 4 from both sides )

- 7y = - 21 ( divide both sides by - 7 )

y = 3

substitute y = 3 into (1)

3(3) - 2 = x

9 - 2 = x , that is

x = 7

solution is (7, 3 )

A relation is a general term that defines the relationship between inputs and outputs, regardless of whether each input has a unique output is: True.

A relation is a general term that defines the relationship between inputs and outputs, regardless of whether each input has a unique output.

On the other hand, a function is a specific type of relation in which each input value is associated with exactly one output value. Therefore, all functions can be considered as relations, but not all relations qualify as functions.

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Answer:

a. 396cm².

b. 414.69 km²

Step-by-step explanation:

a.

The surface area of a cuboid is the total area of all the faces of the cuboid. A cuboid is a six-sided object with each face being a rectangle. The formula for the surface area of a cuboid is:

Surface area = 2lw + 2lh + 2wh=2(lw+lh+wh)

where:

Here, l=7cm, w=6cm and h=12 cm

substituting value:

Surface area : 2(7*6+7*12+6*12)=396 cm²

Therefore, Surface area is 396cm².

b.

The surface area of a cylinder is the total area of all the faces of the cylinder. It can be calculated using the following formula:

Surface area = 2πr² + 2πrh=2πr(r+h)

Where:

In this case, we have:

r = 6 km

h = 5 km

Therefore, the surface area of the cylinder is:

Surface area =2π*6(6+5)=132π or 414.69 km²

Therefore, the surface area of the cylinder is approximately 414.69square kilometers.

Answer:

a)  396 cm²

c)  132π km²

Step-by-step explanation:

The formula for the surface area of a rectangular prism is:

[tex]\boxed{\begin{minipage}{5.5cm}\underline{Surface area of a rectangular prism}\\\\$SA=2(wl+hl+hw)$\\\\where:\\ \phantom{ww}$\bullet$ $w$ is the width.\\\phantom{ww}$\bullet$ $l$ is the length.\\\phantom{ww}$\bullet$ $h$ is the height.\\\end{minipage}}[/tex]

From inspection of the given diagram:

Substitute the values of w, l and h into the formula:

[tex]\begin{aligned}\textsf{Surface area}&=2(6 \cdot 7+12 \cdot 7+12 \cdot 6)\\&=2(42+84+72)\\&=2(198)\\&=396\; \sf cm^2\end{aligned}[/tex]

Therefore, the surface area of the given rectangular prism is 396 cm².

[tex]\hrulefill[/tex]

The formula for the surface area of a cylinder is:

[tex]\boxed{\begin{minipage}{6.5cm}\underline{Surface area of a cylinder}\\\\$SA=2\pi rh+2\pi r^2$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius of the circular base.\\\phantom{ww}$\bullet$ $h$ is the height.\\\end{minipage}}[/tex]

From inspection of the given diagram:

Substitute the values of r and h into the formula:

[tex]\begin{aligned}\textsf{Surface area}&=2 \pi (6)(5)+2 \pi (6)^2\\&=2 \pi (30)+2 \pi (36)\\&=60 \pi+72 \pi\\&=132 \pi \; \sf km^2\end{aligned}[/tex]

Therefore, the surface area of the given rectangular prism is 132π km².

EG = (19/5)FG = (19/5)(550/43) = 418.Hence, EG is Equal to 418.

Let's assume the length of EF is x.

According to the given information, the ratio of EF / FG / GH is 4/5 / 2.

This can be written as EF / FG = 4/5 and FG / GH = 2.

We know that the sum of the ratios in a proportion is equal to 1.

So, (EF / FG) + (FG / GH) = 1.

Substituting the given values, we have (4/5) + 2 = 1.

Simplifying, we get (4 + 10) / 5 = 1.

This implies that EF is equal to 14/5 times FG.

Now, let's consider the length of EG.

EG is equal to the sum of EF and FG.

So, EG = EF + FG = (14/5)FG + FG = (19/5)FG.

We are also given that EH = 55.

EH is equal to the sum of EG and GH.

So, EH = EG + GH.

Substituting the values, we get 55 = (19/5)FG + GH.

Since FG / GH = 2, we can write GH = (1/2)FG.

Substituting this into the equation, we have 55 = (19/5)FG + (1/2)FG.

To solve for FG, we need to find a common denominator.

Multiplying 19/5 by 2/2, we get 38/10.

So, 55 = (38/10)FG + (1/2)FG.

Combining like terms, we have 55 = (38/10 + 5/10)FG = (43/10)FG.

To isolate FG, we divide both sides by (43/10):

55 / (43/10) = FG.

Simplifying, we get FG = 550 / 43

Therefore, EG = (19/5)FG = (19/5)(550/43) = 418.

Hence, EG is equal to 418.

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Answer: 72 cm³

Step-by-step explanation:

Volume = Bh    

B = Area of base, triangle = 1/2 bh

B = 1/2 (6)(8)

B = 3(8)

B= 24

h= height of figure = 3

V = Bh

V = 24(3)

V = 72 cm³

Answer:

72 cm³

Step-by-step explanation:

The formula for the volume of a triangular prism is:

[tex]\boxed{\sf Volume\;of\;a\;triangular\;prism=base\;area \times height}[/tex]

The bases of a triangular prism are the triangles.

The area of a triangle is half the product of its base and height.

[tex]\boxed{\sf Area\;of\;a\;triangle=\dfrac{1}{2}bh}[/tex]

From inspection of the given diagram:

Therefore, the volume of the given triangular prism is:

[tex]\begin{aligned}\sf Volume&=\sf area\;of\;base\;triangle \times height\\\\&=\dfrac{1}{2}(6)(8) \times 3\\\\&=(3)(8) \times 3\\\\&=24 \times 3\\\\&=72\; \sf cm^3\end{aligned}[/tex]

A repeating decimal is acceptable to use during calculations when you require a certain level of precision

A repeating decimal is acceptable to use during calculations when you require a certain level of precision, but it's important to keep in mind that the result will be an approximation rather than an exact value.

It is commonly used in situations where a rounded value is sufficient and the level of precision doesn't significantly impact the final result.

On the other hand, you should convert a repeating decimal to a fraction for calculations when you need an exact value or when further mathematical operations or comparisons are required.

Converting a repeating decimal to a fraction allows for precise calculations and eliminates any potential rounding errors associated with working with decimal approximations.

Converting a repeating decimal to a fraction involves identifying the repeating pattern and expressing it as a fraction. This allows for exact calculations and maintains the mathematical properties of fractions.

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Answer:

y = 55 , y = 100

Step-by-step explanation:

9

the sum of the 3 angles in a triangle is 180°

sum the angles and equate to 180

y + 75 + 50 = 180

y + 125 = 180 ( subtract 125 from both sides )

y = 55

10

the sum of the exterior angles of a triangle is 360°

sum the exterior angles and equate to 360

y + 120 + 140 = 360

y + 260 = 360 ( subtract 260 from both sides )

y = 100

Answer:

1) ∠Q  = 110

2) ∠a  + ∠b  + ∠c  + ∠d = 360

Step-by-step explanation:

1) By kite property, the sum of angles is 360 and ∠Q = ∠O

⇒ ∠N + ∠P + ∠Q + ∠O = 360

⇒ 85 + 55 + 2∠Q  = 360

⇒ 140 + 2∠Q  = 360

⇒ 2∠Q  = 360-140

⇒ 2∠Q  = 220

⇒ ∠Q  = 110

2)By property, the sum of exterior angles is 360

Proof:

∠a + 120 = 180 (straight line)

⇒ ∠a  = 60

∠b + 70 = 180

⇒ ∠b  = 110

∠c + 80 = 180

⇒ ∠c  = 100

∠d + 90 = 180

⇒ ∠d  = 90

⇒ ∠a  + ∠b  + ∠c  + ∠d = 60 + 110 + 100 + 90 = 360

The ship started at the Position 0.14 units.

To determine the starting position of the ship, we can consider the distance sailed to the left and the distance to the treasure.

Given that you sailed 0.055 units to the left and found the treasure at 0.085 units, we can represent this situation mathematically as follows:

Starting position + Distance sailed to the left = Distance to the treasure

Let's assign a variable, "x," to represent the starting position of the ship.

The equation becomes:

x - 0.055 = 0.085

To find the value of x, we can solve this equation by isolating x on one side:

x = 0.085 + 0.055

x = 0.14

Therefore, the ship started at the position 0.14 units.

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a. Find the sum of the interior angles of a decagon.

[tex]\sf {\text{To find the sum of the interior angles of a polygon, we can use the formula:}} \\[/tex]

[tex]\sf S = (n - 2) \times 180^\circ \\[/tex]

[tex]\sf \text{where S represents the sum of the interior angles and n represents the number of sides of the polygon.} \\[/tex]

[tex]\sf \text{For a decagon (10 sides), the sum of the interior angles can be calculated as follows:} \\[/tex]

[tex]\sf S = (10 - 2) \times 180^\circ \\[/tex]

[tex]\sf S = 8 \times 180^\circ \\[/tex]

[tex]\sf S = 1440^\circ \\[/tex]

c. If the sum of the interior angles of a regular polygon is 10,800°, how many sides does the polygon have?

[tex]\sf \text{Using the same formula as before,} \\[/tex]

[tex]\sf S = (n - 2) \times 180^\circ \\[/tex]

[tex]\sf \text{we can solve for n, the number of sides.} \\[/tex]

[tex]\sf 10,800^\circ = (n - 2) \times 180^\circ \\[/tex]

[tex]\sf \frac{10,800^\circ}{180^\circ} = n - 2 \\[/tex]

[tex]\sf 60 = n - 2 \\[/tex]

[tex]\sf n = 60 + 2 \\[/tex]

[tex]\sf n = 62 \\[/tex]

[tex]\sf \text{Therefore, the polygon has 62 sides.} \\[/tex]

e. What is the exterior angle of a regular 90-gon?

[tex]\sf \text{To find the measure of each exterior angle of a regular polygon, we can use the formula:} \\[/tex]

[tex]\sf \text{Each exterior angle } = \frac{360^\circ}{n} \\[/tex]

[tex]\sf \text{where n represents the number of sides of the polygon.} \\[/tex]

[tex]\sf \text{For a regular 90-gon (90 sides), we can calculate the measure of each exterior angle as follows:} [/tex]

[tex]\sf \text{Each exterior angle } = \frac{360^\circ}{90} \\[/tex]

[tex]\sf \text{Each exterior angle } = 4^\circ \\[/tex]

[tex]\sf \text{Therefore, the exterior angle of a regular 90-gon measures } 4^\circ. \\[/tex]

The relationship between the area rate of coverage and the speed is approximately four times as large. This means that doubling the speed leads to approximately four times the rate of coverage, making option B

To determine the relationship between the area rate of coverage (acre/hour) and the speed (mph) based on the given chart, we can compare the values at different speeds.

According to the chart, as the speed increases, the area rate of coverage also increases. For example, at 3 mph, the rate is 5.96 acres per hour, while at 3.5 mph, it is 6.95 acres per hour. This pattern continues as the speed increases: 4 mph yields 7.95 acres per hour, 4.5 mph gives 8.94 acres per hour, and so on.

By comparing the values, we can observe that as the speed doubles, the area rate of coverage also approximately doubles. For instance, at 3 mph, the rate is 5.96 acres per hour, and at 6 mph, it is 11.92 acres per hour. This indicates that doubling the speed results in approximately twice the rate of coverage.

Similarly, as the speed quadruples, the area rate of coverage also approximately quadruples. For instance, at 3 mph, the rate is 5.96 acres per hour, and at 6.5 mph, it is 12.92 acres per hour. This suggests that quadrupling the speed results in approximately four times the rate of coverage.

Option B

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1. Yes, f(x) = x² + 3x + 1 is continuous on the closed interval [0, 7].

2. The value of f at f(0) is 1 and f(7) is 71.

3. We can say A. Yes, the Intermediate Value Theorem can be applied to f on the closed interval [0, 7].

4. The value of C is {-6, 3}.

1. A function that is continuous on a closed interval does not have any sudden jumps or breaks within that interval. It flows smoothly without any interruptions or gaps between its values.

The function f(x) = x²+ 3x + 1 is continuous on the closed interval [0, 7] because as a polynomial, it satisfies the conditions for continuity, like the existence of limits, finiteness of limits, and equality of left-hand and right-hand limits.

Therefore, f(x) is continuous on [0, 7].

2. To find the values of f at x = 0 and x = 7, we have:

f(0) = (0)² + 3(0) + 1 = 0 + 0 + 1 = 1

f(7) = (7)² + 3(7) + 1 = 49 + 21 + 1 = 71

So, f(0) = 1 and f(7) = 71.

3. To determine whether the Intermediate Value Theorem can be applied to f on the closed interval [0, 7], we shall check if f(0) < f(c) < f(7) for some value c between 0 and 7.

f(0) = 1 and f(7) = 71. Since 19 is between 1 and 71, f(0) < f(c) < f(7) is estimate.

So, the Intermediate Value Theorem can be applied.

4. To find the value of c, we know that f(c) = 19.

So, we shall solve the equation f(x) = 19 for x.

x² + 3x + 1 = 19

Rearranging the equation:

x² + 3x + 1 - 19 = 0

x² + 3x - 18 = 0

Solve the quadratic equation.

(x + 6)(x - 3) = 0

Setting each factor to 0:

x + 6 = 0 or x - 3 = 0

x = - 6 or x = 3

Therefore, the values of c are -6 and 3.

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Let's call the weight of the boxer in question as Boxer A and the weight of his opponent as Boxer B.

To convert deciagrams to kilograms, we divide the weight in deciagrams by 1000 (since there are 1000 grams in a kilogram).

So, the weight difference between the two boxers is 100 dag, which is equal to 100/1000 = 0.1 kg.

Let's denote the weight of the opponent as 'x' kg.

According to the information given

The weight of the first boxer (91 kg) is equal to the weight of the opponent (x kg) plus the weight difference (0.1 kg):

We are given that Boxer A weighs 91 kg and is 100 dag (or 10 kg) heavier than his opponent.

To find the weight of Boxer B, we can subtract the weight difference from Boxer A's weight. The weight of Boxer B can be calculated as follows:

Weight of Boxer B = Weight of Boxer A - Weight Difference

Weight of Boxer B = 91 kg - 10 kg

Weight of Boxer B = 81 kg

Therefore, Boxer B weighs 81 kg.

In summary, Boxer A weighs 91 kg and is 100 dag (10 kg) heavier than his opponent, Boxer B, who weighs 81 kg.

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The inverse of f(x) = 5x/3 + 5 is f-1(x) = 3(x - 5)/5.

To find the inverse of f(x), we need to interchange the roles of x and f(x) and solve for x. Let's go step by step:

1: Start with the given function f(x) = 5x/3 + 5.

2: Replace f(x) with y. The equation becomes y = 5x/3 + 5.

3: Swap x and y. The equation becomes x = 5y/3 + 5.

4: Solve for y. Subtract 5 from both sides to isolate the term with y. We get x - 5 = 5y/3.

5: Multiply both sides by 3/5 to solve for y. We obtain (x - 5)(3/5) = y.

6: Simplify the right side. Distribute the 3/5 to the terms inside the parentheses: (3/5)(x - 5) = y.

7: Simplify further. (3/5)(x - 5) = 3(x - 5)/5 = (3x - 15)/5 = f-1(x).

8: Therefore, the inverse of f(x) = 5x/3 + 5 is f-1(x) = 3(x - 5)/5.

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When the unit of labor is 6, the marginal product is 796.

The marginal production function (MP) can be obtained by taking the derivative of the production function with respect to labor (L). In this case, the production function is Q = 4L + 3L² + 7L³. Taking the derivative of this equation with respect to L will give us the marginal production function.

The derivative of the production function with respect to L is MP = 4 + 6L + 21L².

To find the marginal product when the unit of labor is 6, we substitute L = 6 into the marginal production function:

MP = 4 + 6(6) + 21(6)²

  = 4 + 36 + 21(36)

  = 4 + 36 + 756

  = 796.

In summary, the marginal production function is MP = 4 + 6L + 21L², and the marginal product when the unit of labor is 6 is 796. The marginal production function represents the additional output produced when one additional unit of labor is employed.

In this case, the marginal production function is a quadratic function with positive coefficients, indicating increasing returns to scale. The marginal product of labor represents the change in output resulting from a one-unit increase in labor input, given the level of other inputs.

When the unit of labor is 6, the marginal product is found by substituting L = 6 into the marginal production function, resulting in a value of 796. This means that when the firm employs an additional unit of labor (6th unit), the output increases by 796 units.

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Answer:

Carole will need to save for about 4 weeks to afford the coat.

Step-by-step explanation:

$80 - $25 = $55

$55 / $15 = 3.67 (approximately)

Rounded up: 4

Answer:

Aircraft A: 403 seats

Aircraft B: 298 seats

Step-by-step explanation:

Let's x represent the seats in Aircraft B

We have the equations:

Aircraft A: (x + 105)

Aircraft B: x

We Know

Their total number of seats is 701; find the number of seats for each aircraft.

We Take

x + (x + 105) = 701

2x + 105 = 701

2x = 596

x = 298 seats

So, there are 298 seats in Aircraft B

Aircraft A has how many seats?

We Take

701 - 298 = 403 seats

So, there are 403 seats in Aircraft A

Final Answers:

Aircraft A: 403 seats

Aircraft B: 298 seats

Rounded to the nearest tenth, Bip can ride his moped at approximately 12.3 miles per hour.

To convert yards per second to miles per hour, we need to apply the appropriate conversion factors:

1 yard = 0.000568182 miles (approximately)

1 hour = 3600 seconds

Given that Bip rides his moped at a rate of 6 yards per second, we can calculate his speed in miles per hour:

6 yards per second * 0.000568182 miles per yard * 3600 seconds per hour ≈ 12.2727 miles per hour

Rounded to the nearest tenth, Bip can ride his moped at approximately 12.3 miles per hour.

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Answer:

see below

Step-by-step explanation:

ASA congruence theorem: If two angles and the side between two angles of one triangle are congruent to the corresponding angles and side of another triangle, then the two triangles are congruent.

△ABG and △IBG has same side BG, ∠ABG congruent with ∠IBG, ∠AGB congruent with ∠IGB

Therefore △ABG is congruent with △IBG

then IG is congruent with AG

Answer:

In ΔABG and ΔIBG

∡AGB=∡IGB given

BG=BG reflexive property

∡ABG=∡IBG given

AB=BC given

ΔABG ≅ ΔIBG by ASA (Angle-Side-Angle) axiom.

Since,

The corresponding side of the congruent triangle theorem states that if two triangles are congruent, then all of their corresponding sides are also congruent.

IG=AG being corresponding side of congruent triangle.

Hence Proved.

The inequality value that would be able to show/x/ ≤ 5 is option A

In mathematics, inequality refers to a mathematical statement that expresses a relationship between two quantities, indicating that one quantity is greater than, less than, or not equal to the other.

The symbol that we have  is telling us that the value of the x would be less than  or it would be seen to be equal to five.

We can see that /x/ ≤ 5 would exclude the negative values and this can be seen in the option that is marked option A

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If Gavin contributes towards a Pension fund (7,5% of his Basic Salary), then his net salary will be reduced by the amount he contributed towards pension fund i.e. R652.5.

1.1 Name of employee is Gavin Pillay. 1.2 UIF stands for Unemployment Insurance Fund. UIF is a benefit to provide short-term relief to workers who lose their jobs, or can't work due to illness, maternity or adoption leave. 1.3 The UIF paid by Gavin is calculated correctly.

Firstly, we need to calculate Gross Salary of Gavin, which is given as: Basic Salary + Overtime + M = R8700 + R1200 + R3201 = R13101Then, calculate total deductions from gross salary.

Total Deductions = PAYE + UIF = R1200 + 87 = R1287Hence, Gavin's net salary = Gross salary - Total Deductions = R13101 - R1287 = R11814Therefore, UIF paid by Gavin is calculated correctly as R87.1.4 This payslip is for the month of May i.e.

31/05/2023.1.5 Basic salary of Gavin = R8700. Deduct 7.5% of his basic salary to calculate his pension fund contribution. Pension Fund Contribution = 7.5/100 x R8700 = R652.5.1.6

Total deductions of Gavin includes PAYE, UIF and pension fund contribution.

Therefore, Total Deductions = PAYE + UIF + Pension Fund Contribution = R1200 + R87 + R652.5 = R1939.5.1.7

Gavin's age can not be determined as his date of birth is not provided.1.8 Address of Gavin's workplace is 21 Halsted Road, Pietermaritzburg?

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The cotangent of the angle θ  is 2√3/3

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

sec θ = √7/2

To calculate the cotangent of the angle, we use the following identity

1 + tan² θ = sec² θ

So, we have

1 + tan² θ = (√7/2)²

Evaluate the exponent

1 + tan² θ = 7/4

Subtract 1 from both sides

tan² θ = 3/4

So, we have

tan θ = √3/2

Take the inverse of both sides

cot θ = 2/√3

Rationalize

cot θ = 2√3/3

Hence, the cotangent of the angle is 2√3/3

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Answer:

11x +12y = 13

22x + 24y = 26

Step-by-step explanation:

Infinite many solutions means that if you did something to the equations, they would end up being the same line.  A line on top of a line intersect infinitely many times that's why infinite solutions.

11x +12y = 13

22x + 24y = 26

Look at the last pair of equations.  If i divided the second equation by 2(all of the terms):

[tex]\frac{22x + 24y = 26}{2}[/tex]

=11x +12y = 13

You get the same equation as the first.  So this is the system that has infinite solutions.

Answer:

y = 1 , x = - 1

Step-by-step explanation:

2x - 3y = - 5 → (1)

y = 2 + x ( subtract 2 from both sides )

y - 2 = x or x = y - 2

substitute x = y - 2 into (1)

2(y - 2) - 3y = - 5

2y - 4 - 3y = - 5

- y - 4 = - 5 ( add 4 to both sides )

- y = - 1 ( multiply both sides by - 1 )

y = 1

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

substitute y = 1 into y = 2 + x

1 = 2 + x ( subtract 2 from both sides )

- 1 = x

A manufacturer must test that his bolts are 4.00 cm long when they come off the assembly line. He must recalibrate his machines if the bolts are too long or too short. After sampling 196 randomly selected bolts off the assembly line, he calculates the sample mean to be 4.04 cm. He knows that the population standard deviation is 0.26 cm.

Assuming a level of significance of 0.02, is there sufficient evidence to show that the manufacturer needs to recalibrate the machines?

Step 1: Null and Alternative HypothesisThe null hypothesis (H0): The machine does not need recalibration after testing 196 bolts.The alternative hypothesis (H1): The machine needs recalibration after testing 196 bolts.

Step 2: Determine the level of significance level of significance (α) = 0.02

Step 3: Find the critical valueThe level of significance is 0.02, which indicates that the critical value is zα/2 = ±2.33 using the z-table.

Step 4: Calculate the z-test statistics = (x - μ) / (σ/√n) = (4.04 - 4.00) / (0.26 / √196)z = 4.0

Step 5: Determine the p-values-value = P (Z > 4) ≈ 0Using the p-value approach, if the p-value is less than the level of significance, reject the null hypothesis.

In this case, the p-value is almost 0, indicating that there is a statistically significant difference between the sample mean and the hypothesized population mean of 4.0 cm.

Step 6: Conclusion Since the p-value is less than the level of significance, we can reject the null hypothesis and accept the alternative hypothesis. It implies that the machines need recalibration as there is sufficient evidence to suggest that the bolts' mean length is more significant than the required specification of 4.00 cm.

Therefore, the manufacturer should recalibrate the machine to ensure that the bolts' length is within the required specification.

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Using tangent half-angle identity, tan(x/2) = ± √((1 + √(7/16)) / (1 - √(7/16)))

To find the value of tan(x/2) using the given information, we can use the half-angle identity for tangent:

tan(x/2) = ± √((1 - cos(x)) / (1 + cos(x)))

Given that sin(x) = -3/4 and x is in the third quadrant (3π/4), we can determine the value of cos(x) using the Pythagorean identity:

sin(x)² + cos(x)² = 1

(-3/4)² + cos(x)² = 1

9/16 + cos(x)² = 1

cos(x)² = 1 - 9/16

cos(x)² = 16/16 - 9/16

cos(x)² = 7/16

cos(x) = ± √(7/16)

Since x is in the third quadrant (3π/4), the cosine is negative, so cos(x) = -√(7/16).

Now, we can substitute the value of cos(x) into the half-angle identity for tangent:

tan(x/2) = ± √((1 - cos(x)) / (1 + cos(x)))

tan(x/2) = ± √((1 - (-√(7/16))) / (1 + (-√(7/16))))

tan(x/2) = ± √((1 + √(7/16)) / (1 - √(7/16)))

This is the simplified expression for tan(x/2) using the given information.

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Based on the information provided, the inference that a significant number of students in the homeroom class plan to become soldiers is valid.

The teacher surveyed his homeroom class about their post-graduation plans. He found out that most of his students plan to become soldiers.

An inference is a conclusion that can be drawn from a given set of data. In this case, the inference that can be made is that a significant number of students in the homeroom class plan to become soldiers.Therefore, the inference is valid based on the data provided.

However, it is essential to keep in mind that the sample size in the homeroom class is relatively small and may not be representative of the entire population. If the teacher had surveyed a more extensive and diverse population, the results might have been different.

In conclusion, based on the information provided, the inference that a significant number of students in the homeroom class plan to become soldiers is valid. However, this may not apply to the entire population, and it is crucial to consider the sample size when drawing inferences.

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