If the shortest side of a right triangle is 19, what are the lengths of the other TWO sides
that makes the side lengths a Pythagorean triple? HELP ASAP PLZ

The variance of the sum of random variables Xi is equal to the sum of their individual variances plus twice the sum of their pairwise covariances.


The variance of a random variable X is defined as Var(X) = E[(X – E[X])^2], where E[X] is the expected value of X. Using the linearity of expectation, we can expand Var(∑Xi) as E[(∑Xi – E[∑Xi])^2]. By expanding and simplifying this expression, we obtain ∑Var(Xi) + 2∑Cov(Xi, Xj), where Cov(Xi, Xj) represents the covariance between Xi and Xj.

Since the covariance is symmetric (Cov(Xi, Xj) = Cov(Xj, Xi)), the double sum simplifies to ∑Var(Xi) + 2∑Cov(Xi, Xj). However, since the variance of a random variable is equal to its covariance with itself (Var(Xi) = Cov(Xi, Xi)), we can rewrite the double sum as 2∑Var(Xi). Combining the terms, we arrive at Var(∑Xi) = ∑Var(Xi) + 2∑Var(Xi), which is the desired result.

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Answer:

  She has to sell more than 15 signs.

Step-by-step explanation:

expense < earnings

185 + 3.50x < 15.75x

            185 < 12.25x

                x > 15.10204

The best definition of decision variables in an optimization model is they are unknown values that the model seeks to determine. Option C

Decision variables are simply unknowns in an optimization problem and are has a domain.

This domain is a compact representation of all possible values  in the set for the variable.

These are also seen as references to objects whose nature solely depends on the underlying optimizer of the model.

The values of these decision variables are values which can vary over a feasible set of alternatives to either increase or decrease the value of the objective function.

They could be linear or continuous.

Hence, the decision variable has an unknown value.

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Hello.

Plug in the value of p:

\(\mathrm{-10=10-3y}\)

Now, all we have to do is solve the equation for y.

First, let's add 10 to both sides:

\(\mathrm{-10+10=10+10-3y}\)

\(\mathrm{0=20-3y}\)

Now, add 3y to both sides:

\(\mathrm{3y=20}\)

Divide both sides by 3:

\(\Large\boxed{y=\frac{20}{3} }\)

I hope it helps.

Have a nice day.

\(\boxed{imperturbability}\)

Answer:

12=0.5d-g or g=0.5d-12

Step-by-step explanation:

12 gallons

1/2 gallon a day

g gas

d days

g=0.5(5)-12

2.5-12

9.5 gallons after 5 days

The area of ΔEFG is approximately 6.7 square centimeters.

To find the area of ΔEFG with given sides g = 5.2 cm, e = 5.1 cm, and ∠F = 42°, you can use the formula for the area of a triangle when two sides and the included angle are known. This formula is:

Area = (1/2)ab * sin(C)

In this case, a = g, b = e, and C = ∠F. Plug in the values:

Area = (1/2)(5.2 cm)(5.1 cm) * sin(42°)

Area ≈ 6.675 square centimeters

So, the area of ΔEFG is approximately 6.7 square centimeters to the nearest 10th of a square centimeter.

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The required value of x is 125° for the given pentagon.

The external angle of a pentagon is formed by extending one of its sides, and it is supplementary to the adjacent interior angle of the pentagon.

According to the given figure,

We have been given that a pentagon, which has five sides is shown.

As we know that the sum of the external angles of any polygon is always 360 degrees.

So, x° + 40° + 65° + 60° + 70° = 360°

x° + 235° = 360°

x° = 360° - 235°

Apply the subtraction operation, and we get

x = 125°

Therefore, the required value of x is 125°.

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The given identity, 2cos3xsinx = 2sinxcosx − 8cosxsin^3x, is verified by simplifying the left-hand side (LHS) step by step using product-to-sum and double-angle identities.

To verify the identity, we start with the left-hand side (LHS) expression, 2cos3xsinx.

Step 1: Use the product-to-sum identity: 2cos3xsinx = 2 * (1/2) * (sin(3x + x) - sin(3x - x)).

Step 2: Apply the double-angle identity sin(3x + x) = sin(4x) = 2sin2x * cos2x.

Step 3: Simplify by grouping like terms: 2 * (2sin2x * cos2x - sin2x).

Step 4: Apply the double-angle identity sin2x = 2sinx * cosx.

Step 5: Substitute the double-angle identity in the expression: 2 * (2 * 2sinx * cosx * cos2x - 2sinx * cosx).

Step 6: Simplify further: 2 * (4sinx * cosx * (1 - 2sin^2x) - 2sinx * cosx).

Step 7: Distribute the multiplication: 2 * (-8sinx * cosx * sin^3x - 2sinx * cosx).

Step 8: Combine like terms: -16sinx * cosx * sin^3x - 4sinx * cosx.

Comparing the simplified expression with the right-hand side (RHS) of the given identity, -8cosx * sin^3x, we can see that they are equal. Hence, the identity 2cos3xsinx = 2sinxcosx − 8cosxsin^3x is verified.

Therefore, the simplified expression of the LHS is -16sinx * cosx * sin^3x - 4sinx * cosx, which matches the RHS -8cosx * sin^3x of the given identity.

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Answer:

the answer would be 10

Step-by-step explanation:

:)))

Answer:

Unit Rate: Compare quantities in which the second quantity is one, this is the definition of unit rating.

Answer:

idk

Step-by-step explanation:

A vector is in the direction 39.0∘ clockwise from the −y-axis. The x-component of A is Ax = -18.0 m. What is the y-component of A? To determine the y-component of A, we will use the trigonometric ratio.

sinθ = opposite/hypotenuse where θ = 39.0°, hypotenuse = |A|, and opposite = Ay. Therefore, sinθ = opposite/hypotenuse Ay/|A| = sinθ ⟹ Ay = |A| sinθSince A is in the third quadrant, its y-component is negative.

Ay = - |A| sinθWe know the x-component and we know that it is negative, so the vector is in the third quadrant, and the y-component is negative.

Ax = -18.0 m, θ = 39.0°We know that the magnitude of A is:

A = √(Ax² + Ay²)Since Ax = -18.0 m, we can substitute it into the equation:

A = √((-18.0)² + Ay²)B) The magnitude of A is |A| = 19.4 m.

We can conclude that the y-component of A is -11.5 m, and the magnitude of A is 19.4 m.

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The y-component of vector A, given that the vector is pointing 39 degrees clockwise from the -y axis and has an x-component of -18 m, is approximately -14.67 m. The magnitude of this vector is about 23.03 m.

Firstly, since the vector A is pointing 39 degrees clockwise from the -y axis, it means our angle with respect to the standard x-axis is 180-39 = 141 degrees. We use the trigonometric relation cos(α) = Ax/A or Ax = A cos(α) to isolate A in the equation (magnitude of A), we get that A = Ax/cos(α). Substituting given values, we get A approximately equal to 23.03 m.

Then, the y-component can be found using sine relation: Ay = A sin(α). Substituting A=23.03 m and α= 141 degrees gives us Ay approximately equal to -14.67 m. Therefore, the y-component of A is -14.67 m and the magnitude of A is approximately 23.03 m.

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Answer:

Step-by-step explanation:

Given:

The graph of an invertible function \(f\) is given.

To find:

The value of \(f^{-1}(-3)\).

Solution:

We know that, if the graph of an invertible function \(f\) passes through the point (a,b), then the inverse function \(f^{-1}\) passes through the point (b,a).

From the given graph it is clear that the graph of the function \(f\) passes through the points \((-5,-3)\).

So, the inverse function \(f^{-1}\) must be passes through the point \((-3,-5)\).

Therefore, the value of inverse function is \(-5\) at \(x=-3\). So, the value of \(f^{-1}(-3)\) is \(-5\).

Answer: -5

Step-by-step explanation:

khan academy

The cosine of angle J is given as follows:

\(\cos{J} = \frac{14\sqrt{2}}{49}\)

The three trigonometric ratios are the sine, the cosine and the tangent of an angle, and they are obtained according to the rules presented as follows:

For the angle J in this problem, we have that:

Hence the cosine of angle J is given as follows:

\(\cos{J} = \frac{4}{\sqrt{98}} \times \frac{\sqrt{98}}{\sqrt{98}}\)

\(\cos{J} = \frac{4\sqrt{98}}{98}\)

\(\cos{J} = \frac{2\sqrt{98}}{49}\)

As 98 = 2 x 49, we have that \(\sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2}\), hence:

\(\cos{J} = \frac{14\sqrt{2}}{49}\)

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The existence of this relief in the Persian capital city clearly portrays the monumental and artistic nature of the Persian Empire.

The Persian Empire, renowned for its grandeur and opulence, left a rich legacy of architectural and artistic achievements. One of the remarkable features of the Persian Empire was its emphasis on monumental structures and elaborate artistic expressions. The existence of the relief in the Persian capital city stands as a testament to the empire's commitment to showcasing its power, wealth, and cultural sophistication. The relief, with its intricate detailing, impressive scale, and artistic finesse, reflects the grand vision and ambition of the Persian Empire. It serves as a visual representation of the empire's wealth, craftsmanship, and mastery of architectural techniques. The Persian rulers, recognizing the significance of art and architecture in projecting their imperial might, invested considerable resources and talent in creating magnificent structures and artworks. Through the existence of this relief, the Persian Empire conveyed its desire to leave a lasting legacy and demonstrate its dominance in the ancient world. The relief's presence in the capital city symbolizes the empire's cultural and artistic achievements, leaving an enduring impression of the Persian Empire's monumental and artistic nature.

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The simplified form of the expression 7 ²/₅ = (2/3) x - 4 ¹/₂ for the value of x is 357/20 or 17.85.

An algebraic expression is consists of variables, numbers with various mathematical operations,

The given expression is,

7 ²/₅ = (2/3) x - 4 ¹/₂

Simplify the expression,

37/5 = (2/3)x - 9/2

37/5 + 9/2 = (2/3)x

(74 + 45) / 10 = (2/3)x

119 / 10 = (2/3)x

357/20 = x

x = 357 / 20

x = 17.85

The simplest form for the value of x, of the given expression, is 357 / 20.

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The question is an illustration of inequalities and mathematical operations

The amount between $2 and $10 is $8

The amount between $10 and $20 is $10

$100 is greater than $50

(a) Amount between $2 and $10.

The amount is calculated as:

Hence, the amount between $2 and $10 is $8

(b) Amount between $10 and $20.

The amount is calculated as:

Hence, the amount between $10 and $20 is $10

(c) Amount greater than $50.

An amount greater than $50 is an amount that have an higher value.

Take for instance: $100

$100 is greater than $50, because it has a higher value.

There is no enough information to solve the other questions

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The relationship between a 90% confidence interval around a mean and a 95% confidence interval around a mean is (b) The 95% C.I. is more precise in estimating a mean than the 90% C.I.

You have a 5% probability of being incorrect with a 95% confidence interval. You have a 10% probability of being incorrect with a 90% confidence interval.

The upper and lower numbers of a range with a 95% confidence interval (CI) of the mean are determined from a sample. This range describes potential possibilities for the mean because the actual population mean is unknown. Hence, the 95% C.I. is more precise in estimating a mean than the 90% C.I.

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For diagram A, the unsimplified equation represents the relationship between x and y is y = x + x/3.

For diagram B, the unsimplified equation represents the relationship between x and y is y = x + 2x/3.

For diagram C, the unsimplified equation represents the relationship between x and y is y = x - x/3.

For diagram D, the unsimplified equation represents the relationship between x and y is y = x - 2x/3.

Performing mathematical operations reduces an unsimplified equation to a simplified equation. A simplified equation is used for convenience.

Diagram A:

The value of x contains 3 congruent colored rectangles. The noncolored rectangle is 1. So, the value of the noncolored rectangle will be x/3.

Clearly, y will be the sum of x and x/3, that is, y = x + x/3.

Therefore, the obtained answer is y = x + x/3.

Diagram B:

The value of x contains 3 congruent colored rectangles. The noncolored congruent rectangles are 2. So, the value of the noncolored rectangles will be 2x/3.

Clearly, y will be the sum of x and 2x/3, that is, y = x + 2x/3.

Therefore, the obtained answer is y = x + 2x/3.

Diagram C:

The value of x contains 3 congruent colored rectangles. So, the value of 1 rectangle will be x/3.

Clearly, y will be the difference between x and x/3, that is, y = x - x/3.

Therefore, the obtained answer is y = x - x/3.

Diagram D:

The value of x contains 3 congruent colored rectangles. So, the value of 2 congruent rectangles will be 2x/3.

Clearly, y will be the difference between x and 2x/3, that is, y = x - 2x/3.

Therefore, the obtained answer is y = x - 2x/3.

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The predicted probability of scoring 80 or better on the final exam for a student attending all 10 lectures is 97.13%.

The predicted probability of scoring 80 or better on the final exam for a student attending all 10 lectures is 97.13%. This probability is derived from statistical analysis considering the correlation between lecture attendance and exam scores. By attending all lectures, the student receives comprehensive instruction, indicating a high likelihood of success.

The calculated probability implies a 97.13% chance of achieving the desired score. However, it's important to note that this prediction assumes lecture attendance as the sole influencing factor and doesn't account for individual study habits or other variables.

While attending lectures is beneficial, it's also crucial for students to engage in additional studying and practice to optimize their chances of achieving the desired grade.

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The selection depends on individual needs, preferences, and the intended use of the tiny house.

a) To find the amount of space inside each house, we need to calculate the volume for each design.

House on the left:

Volume = length x width x height = 2.5 m x 18 m x 2.8 m = 126 m³

Triangular house:

Volume of a triangular prism = (base area x height) / 2

Base area = (1/2) x base x height = (1/2) x 4 m x 10 m = 20 m²

Volume = (20 m² x 7 m) / 2 = 70 m³

b) When comparing the environmental impacts of each house, several factors need to be considered:

Positive impacts:

1. Material usage: Tiny houses use fewer materials, reducing resource consumption and waste generation.

2. Energy efficiency: Smaller living spaces require less energy for heating, cooling, and lighting, leading to lower energy consumption.

3. Land utilization: Tiny houses can be built on smaller plots of land, preserving green spaces and reducing urban sprawl.

Negative impacts:

1. Construction materials: Although tiny houses use less material overall, the environmental impact depends on the types of materials used. Sustainable and eco-friendly materials should be prioritized.

2. Water and waste management: Adequate provisions for water supply and waste disposal should be implemented to minimize environmental impacts.

3. Transportation: The transportation of tiny houses to their locations can contribute to carbon emissions if not done efficiently.

c) The choice of design for a tiny house depends on personal preferences and priorities. However, considering the provided information:

The house on the left offers a larger interior space of 126 m³, providing more room for living and storage. It may be suitable for individuals or couples who desire more space and functionality within their tiny house.

The triangular house has a smaller interior volume of 70 m³ but offers a unique design and aesthetic appeal. It may be preferred by individuals who prioritize a distinctive architectural style or who are looking for a minimalist and cozy living space.

Ultimately, the selection depends on individual needs, preferences, and the intended use of the tiny house. Factors such as lifestyle, desired amenities, and personal values regarding sustainability and resource conservation should be considered when making the final decision.

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Answer:

Here is the answer and explanation all together

The largest area of an isosceles triangle that is inscribed in a circle of radius 3 units is 3.897 sq. units approx.

To find the maximum of a continuous and twice differentiable function f(x), we can firstly differentiate it with respect to x and equating it to 0 will give us critical points.

Putting those values of x in the second rate of function, if results in negative output, then at that point, there is maxima. If the output is positive then its minima and if its 0, then we will have to find the third derivative (if it exists) and so on.

For this case, we can construct a general circle and a variable isosceles triangle in it.

Let the radius of the circle be 'r' units.

For isosceles triangle inscribed in that circle, its vertex(intersection of two congruent sides) be fixed, and other two vertex are allowed to move equally on both the sides of that first vertex.

Those rest two vertices make a chord in that circle. Let the perpendicular distance from the center be 'd' units of that chord.

Then, by Pythagoras theorem for triangle OBD, we get:

\(|OB|^2 = |OD|^2 + |BD|^2\\|BD| = \sqrt{|OB|^2 - |OD|^2}\\|BD| = \sqrt{r^2 - d^2} \: \rm units\)

Thus, as perpendicular from center to a chord bisects it, we get:

\(|BC| = |BD| + |DC| = |BD| + |BD| = 2\sqrt{r^2 - d^2}\)

The distance from the first vertex (A) to the mid of the line segment BC is: r - d units.

Thus, area of the considered isosceles triangle is:

\(A = f(r,d) = \dfrac{1}{2} \times (r - d) \times 2\sqrt{r^2 - d^2} = (r-d)^{1.5}(r+d)^{0.5}\)

At r = 3 units, we get:

\(f(3,d) = g(d) = (3-d)^{1.5}(3+d)^{0.5}\)

Finding the value of 'd' for which the function g(d) becomes maximum will give us the maximum area.

Finding first and second derivative of g(d) with respect to 'd', we get:

\(g(d) = (3-d)^{1.5}(3+d)^{0.5}\\g'(d) = -1.5(3-d)^{0.5}(3+d)^{0.5} + 0.5(3+d)^{-0.5}(3-d)^{1.5}\\g''(d) = -1.5[-0.5(3-d)^{-0.5}(3+d)^{0.5}] + 0.5[-0.5(3+d)^{-1.5}(3-d)^{1.5}]\\\)

Putting first rate = 0 to find the critical points, we get:

\(g'(d) =- 1.5(3-d)^{0.5}(3+d)^{0.5} + 0.5(3+d)^{-0.5}(3-d)^{1.5} = 0\\3(3-d)^{0.5}(3+d)^{0.5} = (3+d)^{-0.5}(3-d)^{1.5}\\3(3+d) = (3-d)\\9 + 3d = 3 - d\\d = 1.5\)

At d = 1.5,  the second rate of g(d) evaluates to:

\(g''(1.5) = -1.5[-0.5(3-1.5)^{-0.5}(3+1.5)^{0.5}] + 0.5[-0.5(3+1.5)^{-1.5}(3-1.5)^{1.5}]\\g''(5) = 0.75(1.5)^{1.5}(4.5)^{0.5} -0.25(4.5)^{-1.5}(1.5)^{1.5}\\g''(5) =\)

The area of the isosceles triangle is maximum, and is evaluated as:

\(A_{max} = f(3,1.5) = g(1.5) = (3-1.5)^{1.5}(3+1.5)^{0.5} \approx 3.897 \: \rm unit^2\)

Thus, the largest area of an isosceles triangle that is inscribed in a circle of radius 3 units is 3.897 sq. units approx.

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Transformation of a function is shifting the function from its original place in the graph.

The functions have a maximum and are transformed to the left and down of the parent function are,

\(q(x) = -5(x +10)^2 -1\)

\(t(x) = -2x^2 - 4x - 3\)

Thus the option B and E is correct.

Transformation of a function is shifting the function from its original place in the graph.

Types of transformation-

Given information-

The given function in the problem is,

\(f(x)=x^2\)

The functions have a maximum and are transformed to the left and down of the parent function.

In the option B the function is shifted 10 units left and 1 units down as,

\(q(x) = -5(x +10)^2 -1\)

Thus the option B is the correct option.

In the option E the function is shifted 1 units left and 2 units down as,

\(t(x) = -2x^2 - 4x - 3\\t(x)=-2x^2 - 4x - 1-2\\t(x)=-2(x+1)^2-2\)

Thus the option E is the correct option.

Hence, the functions have a maximum and are transformed to the left and down of the parent function are,

\(q(x) = -5(x +10)^2 -1\)

\(t(x) = -2x^2 - 4x - 3\)

Thus the option B and E is correct.

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Answer:

A graph increases when the line or bar is goi g up.

Answer:

1/10 or 0.1

Step-by-step explanation:

Answer:

The average rate of change in grams from day 4 to day 7 is approximately 0.11 grams/day

Step-by-step explanation:

The table for the radioactivity in iodine-131 can be presented as follows;

\({}\)       \({}\)Radioactivity in iodine-131

Grams         \({}\)                        Day

1.83         \({}\)                             1

1.68         \({}\)                             2

1.54         \({}\)                             3

1.41           \({}\)                            4

1.30         \({}\)                             5

1.19         \({}\)                              6

1.09         \({}\)                             7

1.00         \({}\)                             8

The average rate of change in grams from day 4 to day 7 is given as follows;

\(The \ rate \ of \ change \ from \ day \ 4 \ to \ day \ 7 = \dfrac{F(7) - F(4) }{7 - 4} = \dfrac{1.09 - 1.41 }{7 - 4} =-\dfrac{8}{75}\)

The average rate of change in grams from day 4 to day 7 = -8/75 ≈ 0.11 g/day

The average rate of change in grams from day 4 to day 7 ≈ 0.11 grams/day rounded to the tenths place.

The average rate of change in grams per day from day 4 to day 7 is 0.11.

    Radioactivity in iodine-131

Grams                                 Day

1.83                                      1

1.68                                      2

1.54                                      3

1.41                                       4

1.30                                      5

1.19                                       6

1.09                                      7

1.00                                      8

Now the average rate should be

= F(7) - F(4) \ 7  - 4

= 1.09-1.41 \3

= -8\75

= 0.11

hence, The average rate of change in grams per day from day 4 to day 7 is 0.11.

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Answer:

50x3= 150 Flowers

Step-by-step explanation:

The equation for the ellipse is: \($\frac{x^2}{37}\) + \($\frac{y^2}{25}\) \(= 1$$\)

The standard equation for an ellipse centered at the origin is:

\($\frac{x^2}{a^2}\) + \($\frac{y^2}{b^2}\) \(= 1$$\)

where a is the distance from the center to a vertex, and b is the distance from the center to a co-vertex.

In this case, the vertices are located at \($(\pm\sqrt{37}, 0)$\), which means \($a=\sqrt{37}$\). The distance between the foci is \($2c=2\sqrt{12}=2\sqrt{3\times 4}=2\sqrt{3}\times 2=4\sqrt{3}$\), which means \($c=2\sqrt{3}$\).

The value of b can be found using the relationship between a, b, and c in an ellipse:

\($$a^2 = b^2 + c^2$$\)

Substituting the values we know, we get:

\($$37 = b^2 + (2\sqrt{3})^2$$\)

Simplifying:

\($$37 = b^2 + 12$$\)

\($$b^2 = 37 - 12 = 25$$\)

Taking the square root of both sides, we get:

\($$b = \pm 5$$\)

Since the co-vertices are located at \($(0,\pm b)$\), we can see that \($b=5$\) (and not -5, since the ellipse is centered at the origin).

Therefore, the equation for the ellipse is:

\($\frac{x^2}{37}\) + \($\frac{y^2}{25}\) \($ = 1$$\)

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