PLEASE HELP ME
Use the quadratic formula to find both solutions to the quadratic equation
given below.

The correct conclusion based on the given statements is:

If the time is between 8:00 a.m. and 6:00 p.m., then people may swim or dive into the pool.

A conditional statement, also known as an "if-then" statement, is a type of logical statement that consists of two parts: a condition (the "if" part) and a result (the "then" part).

It expresses a cause-and-effect relationship, stating that if a certain condition is true, then a specific result or action will follow.

"If the time is between 8:00 a.m. and 6:00 p.m., then people may swim or dive into the pool." This conclusion follows from the given information that if the time is within the specified range, the public swimming pool is open, and if the pool is open, people are allowed to swim or dive into it.

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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

Answer:

see explanation

Step-by-step explanation:

the sum of the interior angles of a polygon is

sum = 180° (n - 2) ← n is the number of sides

a hexagon has 6 sides , then

sum = 180° × (6 - 2) = 180° × 4 = 720°

12

the polygon has 5 sides , so

sum = 180° × (5 - 2) = 180° × 3 = 540°

sum the interior angles and equate to 540

y + 90 + 120 + 90 + 110 = 540

y + 410 = 540 ( subtract 410 from both sides )

y = 130

13

the polygon has 7 sides , so

sum = 180° × (7 - 2) = 180° × 5 = 900°

sum the interior angles and equate to 900

p + 90 + 141 + 130 + 136 + 123 + 140 = 900

p + 760 = 900 ( subtract 760 from both sides )

p = 140

The ratio 2 to 7 written as a fraction in its lowest term is 2/7

To write the ratio 2 to 7 as a fraction in lowest terms, we can simply express it as a fraction thus :

Required Fraction = Numerator/ denominator = 2/7

Since there are no common factors other than 1 between 2 and 7, the fraction 2/7 is already in its lowest terms.

Hence, the fraction is 2/7

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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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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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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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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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The Wronskian of the set {x, sin(2x), cos(2x)} is det(W) = 8x + 16sin^2(2x) + 4cos^2(2x). The Wronskian of a set of functions is a determinant that can determine linear independence or dependence of the functions.

To compute the Wronskian of the set {x, sin(2x), cos(2x)}, we arrange the functions in a matrix as follows:

W = | x sin(2x) cos(2x) |

| x' sin(2x)' cos(2x)' |

| x'' sin(2x)'' cos(2x)'' |

Here, the derivatives of the functions with respect to x are taken in the second and third rows. Taking the derivatives, we get:

W = | x sin(2x) cos(2x) |

| 1 2cos(2x) -2sin(2x) |

| 0 -4sin(2x) -4cos(2x) |

To compute the determinant of W, we expand it along the first row:

det(W) = x(2cos(2x)(-4cos(2x)) - (-2sin(2x))(-4sin(2x)))

- sin(2x)(1(-4cos(2x)) - 0(-4sin(2x)))

+ cos(2x)(1(-4sin(2x)) - 0(-4cos(2x)))

Simplifying the determinant, we get:

det(W) = 8x + 16sin^2(2x) + 4cos^2(2x).

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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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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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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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Cubic measurements are used to measure the volume of a cubic box, while volume measurements are used to measure the capacity of a three-dimensional object, whether solid or liquid.

Cubic measures and their relationship with volume measures. The terms cubic measures and volume measures are used in mathematics to represent three-dimensional magnitudes. Cubic measures are the units of measurement of a three-dimensional volume in a cubic box.

On the other hand, volume measurements are the units of measurement of the capacity of an object, whether it is a solid or a liquid.

The relationship between cubic measurements and volume measurements is explained by the fact that both are used to measure the amount of space an object occupies. In most cases, volume measurements are expressed in units such as cubic centimeters, cubic meters, milliliters, liters, gallons, and ounces.

In solids, volume measurements can be expressed in units such as cubic feet, cubic inches, and cubic yards. In summary, the relationship between cubic measurements and volume measurements lies in the fact that both are used to measure occupied space. for an object.

Cubic measurements are used to measure the volume of a cubic box, while volume measurements are used to measure the capacity of a three-dimensional object, whether solid or liquid.

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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.

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:

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

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 = 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:

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

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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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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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.

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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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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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:

(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 )

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]

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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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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