a,b,c are prime numbers.
Find a,b,c that sastify the equation: a^4 + b^4 + c^4 + 54 = 11abc

Answer:

x = 6

Step-by-step explanation:

In order for it to be a parallelogram, the 2 lines must be equal.

2x=3x-6

2x - 3x = 3x - 3x -6

-1x = -6

x = 6/1

x = 6

The expression of the width of the rectangle is 2x¹²y⁸z¹¹/3.

Given that the area of a rectangle is 72x¹³y⁹z¹⁶ sq. yds and the length of the rectangle is 3x9y4z⁵, we need to find the width,

Using these expressions, we have,

Area = length × width

72x¹³y⁹z¹⁶ / 3x9y4z⁵ = width

Width = 72x¹³/3x × y⁹/9y × z¹⁶/4z⁵

Width = 24x¹²y⁸z¹¹/36 = 2x¹²y⁸z¹¹/3

Hence the expression of the width of the rectangle is 2x¹²y⁸z¹¹/3.

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With the given formula  [tex]a(t) = 3500e^{(0.041t)[/tex], the percent increase per year for savings account is 4.1% and it will take about 16.9 years for the worth of the account to triple.

a) The formula given is: [tex]a(t) = 3500e^{({0.041t)[/tex]

To find the percent increase per year, we need to find the annual growth rate. We can do this by taking the derivative of a(t) with respect to t:

[tex]a'(t) = 0.041 * 3500 * e^{(0.041t)[/tex]

The annual growth rate is equal to a'(t)/a(t). Plugging in the formula for a(t) and simplifying, we get:

a'(t)/a(t) = 0.041

So the percent increase per year is 4.1%.

b) We want to find the time it takes for the account to triple in value, so we need to solve for t in the equation:

[tex]3a(0) = 3500e^{(0.041t)[/tex]

Dividing both sides by 3500 and taking the natural logarithm of both sides, we get:

ln(3) = 0.041t

t = ln(3)/0.041

Using a calculator, we get:

t ≈ 16.92 years

So it will take about 16.9 years for the worth of the account to triple.

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Step-by-step explanation:

8p= 56

p= 7

therefore, the neighbours bought 7 packets of hotdogs.

The vertex and x-intercepts of -6x^2-50x+3085. 25 are approximately -42.60 and 30.97.

To find the vertex and x-intercepts of the quadratic function -6x^2-50x+3085.25, we first need to express it in standard form -6x^2-50x+3085.25 = -6(x^2+8.33x-514.21)

So the x-intercepts are approximately -42.60 and 30.97.

We can complete the square to find the vertex of the parabola:

-6(x^2+8.33x-514.21) = -6[(x+4.165)^2-575.641]

-6(x^2+8.33x-514.21) = -6(x+4.165)^2+3453.844

So the vertex is at (-4.165, 575.844).

To find the x-intercepts, we can set y = 0 and solve for x:

-6x^2-50x+3085.25 = 0

Dividing both sides by -2.25 to simplify, we get:

2.6667x^2+22.2222x-1372.2222 = 0

Using the quadratic formula, we get:

x = (-22.2222 ± sqrt(22.2222^2-4(2.6667)(-1372.2222))) / (2(2.6667))

x = (-22.2222 ± sqrt(37511.1116)) / 5.3334

x = (-22.2222 ± 193.7262) / 5.3334

So the x-intercepts are approximately -42.60 and 30.97.

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Quadrilaterals have four sides, two diagonals, and the sum of their interior angles is 360 degrees. They can be related to triangles through the fact that each triangle has a total of 180 degrees and a quadrilateral has 360 degrees, so there are two triangles in a quadrilateral with their angles adding up to 360 degrees.

However, triangles and quadrilaterals differ in terms of their area formulas, where the area of a quadrilateral is calculated as the product of its base and height (A = BxH), while the area of a triangle is half the product of its base and height (A = (BxH)/2). Quadrilaterals have different types and properties, although they share the common characteristics mentioned above.

- Quadrilaterals have four sides and two diagonals. The sum of the interior angles in a quadrilateral is always 360 degrees.

- Triangles have three sides and the sum of their interior angles is always 180 degrees.

- The relationship between triangles and quadrilaterals is based on the fact that a quadrilateral can be divided into two triangles. Each triangle within the quadrilateral contributes 180 degrees to the total sum of 360 degrees.

- The formula for calculating the area of a quadrilateral is A = BxH, where A represents the area, B represents the base, and H represents the height.

- In contrast, the formula for calculating the area of a triangle is A = (BxH)/2, where A represents the area, B represents the base, and H represents the height. This formula demonstrates that the area of a triangle is half the area of a quadrilateral with the same base and height.

- While all quadrilaterals share the characteristics of having four sides, two diagonals, and interior angles summing up to 360 degrees, they have different types and properties.

Examples of quadrilaterals include squares, rectangles, parallelograms, trapezoids, and rhombuses. Each type has its own unique properties and relationships to other quadrilaterals.

In conclusion, quadrilaterals and triangles are related through the concept of dividing a quadrilateral into two triangles. They differ in their area formulas, and although all quadrilaterals have four sides, two diagonals, and interior angles summing up to 360 degrees, they have different types and properties.

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The length of the line segment AB is 8.25 units, under the condition that the length of line segment AB is approximately units. Two points, A and B, plotted in a coordinate plane. Point A is at (2, 2), and point B is at (-6, 4)

In order to evaluate  the length of line segment AB, we can apply the distance formula which is derived from the Pythagorean theorem.
The distance formula is given by d = √[(x₂ - x₁)² + (y₂ - y₁)²].
Here,
x₁ = 2,
y₁ = 2,
x₂ = -6
y₂ = 4.
Staging these values in the formula,
d = √[(-6 - 2)² + (4 - 2)²]

= √[(-8)² + 2²]
= √(64 + 4)
= √68
≈ 8.25 units

Then, the length of line segment AB is approximately 8.25 units.

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The equation of the tangent line to the curve is y = 8.013x - 1.185.

To find the equation of the tangent line to the curve at the point (0.6), we first need to find the slope of the tangent line, which is the derivative of the curve at that point.

Taking the derivative of y = x⁴ + 6eˣ, we get:

y' = 4x³ + 6eˣ

Now, we can find the slope of the tangent line at x = 0.6 by plugging in this value into the derivative:

y'(0.6) = 4(0.6)³ + 6e⁰.⁶ ≈ 8.013

So the slope of the tangent line at the point (0.6) is approximately 8.013.

Next, we need to find the y-coordinate of the point on the curve at x = 0.6. Plugging this value into the original equation, we get:

y = (0.6)⁴ + 6e⁰.⁶ ≈ 6.976

So the point on the curve that corresponds to x = 0.6 is approximately (0.6, 6.976).

Finally, we can use the point-slope form of the equation of a line to find the equation of the tangent line:

y - 6.976 = 8.013(x - 0.6)

Simplifying, we get:

y = 8.013x - 1.185

So the equation of the tangent line to the curve y = x⁴ + 6eˣ at the point (0.6) is y = 8.013x - 1.185.

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The integral is:

[tex](4/3)x^(3/2) ln(x) - (2/3)∫x^(1/2) dx = (4/3)x^(3/2) ln(x) - (4/5)x^(5/2) + C[/tex]

Solve the integrals using integration by parts.

a) To find [tex]2∫x⁷ln(x) dx[/tex], we'll use integration by parts with the formula: [tex]∫u dv = uv - ∫v du. Let's choose:u = ln(x)  = > du = (1/x) dxdv = x⁷ dx = > v = (1/8)x⁸[/tex]

Now, apply the integration by parts formula:

[tex]2∫x⁷ln(x) dx = 2[uv - ∫v du] = 2[((1/8)x⁸ ln(x) - ∫(1/8)x⁸(1/x) dx)]= (1/4)x⁸ ln(x) - (1/4)∫x⁷ dx = (1/4)x⁸ ln(x) - (1/32)x⁸ + C[/tex]

b) To find 2∫√x ln(x) dx, we'll use a similar approach. Let's choose:
[tex]u = ln(x)  = > du = (1/x) dxdv = √x dx = > v = (2/3)x^(3/2)[/tex]

Now, apply the integration by parts formula:

[tex]2∫√x ln(x) dx = 2[uv - ∫v du] = 2[((2/3)x^(3/2) ln(x) - ∫(2/3)x^(3/2)(1/x) dx)]= (4/3)x^(3/2) ln(x) - (2/3)∫x^(1/2) dx = (4/3)x^(3/2) ln(x) - (4/5)x^(5/2) + C[/tex]

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5.490490490 can be expressed as the ratio of integers 5485/999. Hi! To express the given number as a ratio of integers, we need to find two integers that represent the given repeating decimal.

The number 5.490490490 can be written as 5.490(490 repeating). To convert the repeating part into a ratio, we can use the following method:

Let x = 0.490490...
Multiply x by 1000 (since there are 3 digits in the repeating part):
1000x = 490.490490...

Subtract the original x from the 1000x equation:
1000x - x = 490.490490... - 0.490490...
999x = 490
Now, divide both sides by 999:
x = 490/999

So, the repeating decimal 0.490490... can be represented as the ratio 490/999. To express the entire number as a ratio of integers, add the non-repeating part (5) to the ratio:

5 + (490/999) = (5 * 999 + 490) / 999 = (4995 + 490) / 999 = 5485/999.

Your answer: 5.490490490 can be expressed as the ratio of integers 5485/999.

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In the last 215 days, builders have completed 700 m2 of the alligator habitat that will eventually be 1,200 m2.
It will take approximately 153 days to complete the remaining part of the alligator habitat.

Determine how much longer it will take to complete the alligator habitat, first, we need to find the rate at which the builders are working.
Calculate the work rate
The builders have completed 700 m2 of the 1,200 m2 alligator habitat in 215 days.
Work rate = (completed work) / (number of days)
Work rate = 700 m2 / 215 days = 3.26 m2/day (approximately)
Calculate the remaining work
The total area of the alligator habitat is 1,200 m2, and 700 m2 has been completed.
Remaining work = Total area - Completed work
Remaining work = 1,200 m2 - 700 m2 = 500 m2
Calculate the time to complete the remaining work
Time to complete = (remaining work) / (work rate)
Time to complete = 500 m2 / 3.26 m2/day ≈ 153.37 days
It will take approximately 153 days to complete the remaining part of the alligator habitat.

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It will take approximately 394 more days to complete the alligator habitat.

We can start by finding the proportion of the habitat that has already been completed:

proportion completed = 700 m^2 / 1200 m^2 = 0.5833

This means that there is still 1 - 0.5833 = 0.4167 (or 41.67%) of the habitat left to complete.

Next, we can use a proportion to find out how long it will take to complete the remaining 41.67% of the habitat:

215 days / 0.5833 = x days / 0.4167

Solving for x, we get:

x = 215 days * 0.4167 / 0.5833 ≈ 153 days

Therefore, the total time it will take to complete the alligator habitat is approximately 215 + 153 = 368 days, or about 394 more days from the start.

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The Laplace transform of f(x)= (-2x-3)/4 is (-2L{x}-3L{1})/4, where L{x} is the Laplace transform of x and L{1} is the Laplace transform of 1.
Hi! The Laplace transform of a given function f(t) is denoted by L{f(t)} and is defined as the integral of f(t) multiplied by e^(-st), where s is a complex variable. For the function f(x) = (-2x - 3)/4, the Laplace transform can be calculated as follows:

L{f(t)} = L{(-2t - 3)/4}

To find the Laplace transform, we will treat the function as two separate parts:

L{(-2t - 3)/4} = (-2/4) * L{t} + (-3/4) * L{1}

The Laplace transforms of t and 1 are well-known:

L{t} = 1/s^2
L{1} = 1/s

Now, substitute these transforms back into our expression:

L{f(t)} = (-1/2) * (1/s^2) + (-3/4) * (1/s)

L{f(t)} = -1/(2s^2) - 3/(4s)

And that's the Laplace transform of f(x) = (-2x - 3)/4.

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The expression (x + 1)(x - 3) represents the Area of base of the prism.

a crystal is a polyhedron containing a n-sided polygon base, a respectable halfway point which is a deciphered duplicate of the first, and n different countenances, fundamentally all parallelograms, joining relating sides of the two bases. Translations of the bases exist in every cross-section that runs parallel to the bases.

According to question:

The volume of a rectangular prism is given by the formula V = Bh, where B is the area of the base and h is the height of the prism. In this case, the base is a square with edge length (x + 1), so its area is (x + 1)^2. The volume of the prism is given as (x + 1)^2(x - 3).

We can find the height of the prism by dividing the volume by the area of the base:

B = V/h = (x + 1)^2(x - 3)/(x + 1) = (x + 1)(x - 3)

Therefore, the expression (x + 1)(x - 3) represents the Area of base of the prism.

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The expression for the area of the new square is y² + 10y + 25.

To find the expression that represents the area of the new square, we need to consider that when each side of a square is increased by 5 units, the new side length becomes y + 5. The area of the new square is then given by:

(New side length)² = (y + 5)²

Expanding the square, we get:

(y + 5)² = y² + 10y + 25

Therefore, the expression that represents the area of the new square is y² + 10y + 25.

So, the correct option is:

O y² + 10y + 25

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Answer:

True

Step-by-step explanation:

Parallelogram: A parallelogram is 4-sided shape that has parallel opposite sides

Square: A square is 4-sided shape where all sides are congruent and has 4 right angles.  A square also has 2 pairs of opposite parallel sides.

Based on these definitions, we can see that not ALL parallelograms can be squares because a parallelogram can have no right angles, or can have all 4 angles be right angles.

All squares have to be parallelograms though because a square and a parallelogram both have 2 pairs of opposite parallel sides.

Hope this helps :)

Answer:

Step-by-step explanation:

The simplified expression for p(z), the profit the company earns, in dollars is: p(z) = -3z² + 18z

The set X is known as the domain of the function, and the set Y is known as the codomain of the function. A function from a set X to a set Y assigns each element of X to exactly one element of Y. Originally, functions were an idealization of how one variable depends on another.

First, let's substitute the expressions for r(z) and c(z) in the expression for p(z):

p(z) = r(z) - c(z)

p(z) = 60z - (3z² + 42z)

p(z) = 60z - 3z² - 42z

p(z) = -3z² + 18z

Therefore, the simplified expression for p(z), the profit the company earns, in dollars is:

p(z) = -3z² + 18z

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Answer:

see below

Step-by-step explanation:

Here are the steps that Frank took:

1/5t+2=17

1/5t+2-2=17

1/5t=17

t=85

Frank is incorrect.  He is incorrect because in step 2, he forgot to subtract both sides by 2, and only did this to the left side of the equal sign.  He has to subtract 2 from both sides of the equal side to have the equation remain balanced.  Frank should've gotten t=75.

Hope this helps! :)

Let's use the variables "a" for the cost of an apple, "b" for the cost of a banana, and "o" for the cost of an orange.

From the first transaction:

- 3a + 5b + 4o = 8.85

From the second transaction:

- 8a + b + 3o = 8.10

From the third transaction:

- 2a + 2b + 2o = 4.40

We can now solve for one variable and substitute into another equation until we have found all three. Let's solve for "a" in the third equation:

- 2a + 2b + 2o = 4.40

- 2a = 4.40 - 2b - 2o

- a = 2.20 - b - o

Now we can substitute "a" into the first equation:

- 3a + 5b + 4o = 8.85

- 3(2.20 - b - o) + 5b + 4o = 8.85

- 6.60 - 3b - 3o + 5b + 4o = 8.85

- 2b + o = 0.75  (Equation A)

Next, we can substitute "a" into the second equation:

- 8a + b + 3o = 8.10

- 8(2.20 - b - o) + b + 3o = 8.10

- 17.60 - 8b - 8o + b + 3o = 8.10

- -7b - 5o = -9.50  (Equation B)

Now we have two equations with two variables, so we can solve for one variable and substitute into the other equation. Let's solve for "o" in Equation A:

- 2b + o = 0.75

- o = 0.75 - 2b

Now we can substitute "o" into Equation B:

- -7b - 5o = -9.50

- -7b - 5(0.75 - 2b) = -9.50

- -7b - 3.75 + 10b = -9.50

- 3b = -5.75

- b = -1.92 (rounded to the nearest cent)

Finally, we can substitute "b" into Equation A to find "o":

- 2b + o = 0.75

- 2(-1.92) + o = 0.75

- o = 4.59 (rounded to the nearest cent)

We can now find "a" by substituting "b" and "o" into one of the original equations. Let's use the first equation:

- 3a + 5b + 4o = 8.85

- 3a + 5(-1.92) + 4(4.59) = 8.85

- 3a - 9.60 + 18.36 = 8.85

- 3a = -0.09

- a = -0.03 (rounded to the nearest cent)

Since the cost of a piece of fruit cannot be negative, we made a mistake somewhere in our calculations. It's possible that we made a mistake in rounding at some point. To be sure, let's check our answers by substituting the values we found back into the original equation.

2(-0.0737) + 2(-0.0528) + 2o = 4.40

-0.1474 - 0.1056 + 2o = 4.40

2o = 4.6529

o = 2.3264 (rounded to 4 decimal places)

Therefore, each apple costs approximately $0.0737, each banana costs approximately $0.0528, and each orange costs approximately $2.3264.

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Using cubes formula the factored expression is given as:

1/8x^3 - 1/27y^3 = (1/2x - 1/3y)(1/4x^2 + 1/6xy + 1/9y^2)

To factor the expression [tex]1/8x^3 - 1/27y^3[/tex], we can utilize the difference of cubes formula, which states that the difference of two cubes can be factored as the product of their binomial factors.

In our given expression, we have[tex](1/8x^3 - 1/27y^3).[/tex] We can identify[tex]a^3 as (1/2x)^3 and b^3 as (1/3y)^3.[/tex]

Applying the difference of cubes formula, we get:

[tex](1/8x^3 - 1/27y^3) = (1/2x - 1/3y)((1/2x)^2 + (1/2x)(1/3y) + (1/3y)^2)[/tex]

Simplifying the expression within the second set of parentheses, we have:

[tex](1/8x^3 - 1/27y^3) = (1/2x - 1/3y)(1/4x^2 + 1/6xy + 1/9y^2)[/tex]

Therefore, the factored form of the expression 1/8x^3 - 1/27y^3 is given by (1/2x - 1/3y)(1/4x^2 + 1/6xy + 1/9y^2). This represents the product of the binomial factors resulting from the application of the difference of cubes formula.

To factor the expression 1/8x^3 - 1/27y^3, we can use the difference of cubes formula, which states that:

       [tex]a^3 - b^3 = (a - b)(a^2 + ab + b^2)[/tex]

Applying this formula, we get:

1/8x^3 - 1/27y^3 = (1/2x - 1/3y)(1/4x^2 + 1/6xy + 1/9y^2)

Therefore, the expression is completely factored as:

[tex]1/8x^3 - 1/27y^3 = (1/2x - 1/3y)(1/4x^2 + 1/6xy + 1/9y^2)[/tex]

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The sum of the arrivals and departures from these three cities is 3x + 18.1 million.

If we let x be the number of arrivals and departures at Frankfurt airport, then the number of arrivals and departures at London airport is x + 12.9 million

The number of arrivals and departures at Paris airport is x + 5.2 million.

The sum of the arrivals and departures from these three cities is:

x + (x + 12.9 million) + (x + 5.2 million)

Simplifying this expression, we can combine like terms:

3x + 18.1 million

Therefore, the sum of the arrivals and departures from these three cities is 3x + 18.1 million.

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The Maclaurin series for g(x) is given by g(x) =[tex]x^2 - x^6/3! + x^10/5! -[/tex] [tex]x^14/7![/tex] [tex]+ ⋯ + (-1)^n*x^(4n)/(2n+1)! + ⋯[/tex]

The function g(x) is defined as g(x) = [tex]f(x^2)[/tex], where f(x) is a function with a Maclaurin series expansion.

To find the Maclaurin series for g(x), we substitute [tex]x^2[/tex] into the Maclaurin series of f(x). The resulting series for g(x) is obtained by replacing each occurrence of x in the series for f(x) with x^2:

g(x) = [tex]f(x^2) = (x^2) - (x^2)^3/3! + (x^2)^5/5! - (x^2)^7/7! + ⋯ + (-1)^n*(x^2)^(2n+1)/(2n+1)! + ⋯[/tex]

Simplifying the terms, we have:

g(x) =[tex]x^2 - x^6/3! + x^10/5! - x^14/7! + ⋯ + (-1)^n*x^(4n+2)/(2n+1)! + ⋯[/tex]

This represents the Maclaurin series expansion for the function g(x) in terms of the original function f(x) with the argument squared.

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The correct answer is (C) as the height of the pole is 8.7 meters.

What are Trigonometric functions?

Trigonometric functions are mathematical functions that relate the angles and sides of a right triangle.

Tangent (tan): the ratio of the length of the side opposite an angle to the length of the side adjacent to the angle.

We can use the tangent function to solve this problem.

Let's call the height of the pole "h" and the length of the rope "r".

We know that the length of the shadow cast by the pole is 11.6 meters, and the angle between the sun's rays and the ground is 36.8°. This means that:

tan(36.8°) = h / 11.6

To solve for "h", we can rearrange this equation to get:

h = 11.6 * tan(36.8°)

Using a calculator, we find that h ≈ 8.7 meters.

So the height of the pole is approximately 8.7 meters. Therefore, the correct answer is (C) 8.7.

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2X (2² + sin 3) can be simplified as 8X + 2X sin 3

To solve the expression, we will first have to compute the values inside the parentheses and then apply the given operations. so we Calculate the values inside the parentheses by multiplying across the bracket:

2² is equal to 4, and sin 3 is a trigonometric function that returns the sine of the angle 3

Therefore, the expression 2X (2² + sin 3) simplifies to:

2X (4 + sin 3)

or

8X + 2X sin 3

where X is an unknown variable and sin 3 is a trigonometric function

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Answer:

Solution

verified

Verified by Toppr

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​

cos3xe

2x

4

7I

​

=

4

2e

2x

sin3x−3cos3xe

2x

​

I=

7

e

2x

​

(2sin3x−3cos3x)

∴∫e

2x

sin3dx=

7

e

2x

​

(2sin3x−3cos3x)

Solve any question of Integrals with:-

Patterns of problems

Patterns of problems

>

Solve :∫e xe e xe e e x

dx

Medium

View solution>Solve:-∫ a 2b 2(a 2 −b 2) 2

Step-by-step explanation:

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The platelet count is 652 x 10^3.

Based on the information provided, it seems that the platelet count from a test result is 652, and the normal range for platelet count is 150,000 to 450,000 platelets per microliter of blood. The value "10(3)" likely refers to the measurement being in thousands.

To convert the measurement to the standard unit of platelet count, we need to multiply the given value by 1,000. Thus, 652 * 1,000 = 652,000 platelets per microliter.

It's important to note that platelet counts can vary depending on various factors, and medical professionals typically interpret the results in conjunction with other clinical information. If the obtained value of 652,000 platelets per microliter is accurate, it would be significantly higher than the upper limit of the normal range.

It's crucial to consult a healthcare professional or a qualified medical practitioner for an accurate interpretation of test results and any necessary medical advice or treatment.

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If AB/DE = BC/EF = AC/DF, then triangle ABC is similar to triangle DEF.

To determine the scale factor of a dilation, you need to compare the corresponding lengths of the pre-image and image of a figure. The scale factor is the ratio of the lengths of any two corresponding sides.

For example, suppose you have a triangle ABC with sides AB = 3 cm, BC = 4 cm, and AC = 5 cm. If you dilate the triangle by a scale factor of 2, you get a new triangle A'B'C'.

To find the length of A'B', you multiply the length of AB by the scale factor: A'B' = 2 * AB = 2 * 3 = 6 cm. Similarly, B'C' = 2 * BC = 2 * 4 = 8 cm and A'C' = 2 * AC = 2 * 5 = 10 cm. Therefore, the scale factor of the dilation is 2.

To determine if polygons are similar, you need to check if their corresponding angles are congruent and their corresponding sides are proportional.

In other words, if you can transform one polygon into another by a combination of translations, rotations, reflections, and dilations, then they are similar.

For example, suppose you have two triangles ABC and DEF.

If angle A is congruent to angle D, angle B is congruent to angle E, and angle C is congruent to angle F, and the ratios of the lengths of the corresponding sides are equal, then the triangles are similar. That is, if AB/DE = BC/EF = AC/DF, then triangle ABC is similar to triangle DEF.

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Answer:

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Step-by-step explanation:

A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the sine trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in mathematics, as well as in physics, engineering, signal processing and many other fields

Answer:

It is a type of wave. There are also cosine waves and tangent waves.

Step-by-step explanation:

The equation of the curve is y = x2 + 4.

To solve this problem, we need to use the fact that the curve y = ax2 + bx + c passes through the point (2, 12) and is tangent to the line yat at the origin.

First, we know that the tangent to the curve at the origin is the line y = 0x + c = c. Since the curve is tangent to this line at the origin, we know that the derivative of the curve at x = 0 is equal to 0.

Taking the derivative of y = ax2 + bx + c, we get y' = 2ax + b. Setting x = 0, we get y' = b. Since y' = 0 at x = 0, we know that b = 0.

So now we have y = ax2 + c. We can use the fact that the curve passes through the point (2, 12) to solve for a and c.

Substituting x = 2 and y = 12 into the equation y = ax2 + c, we get 12 = 4a + c.

Since we know that a = 4, 6, or 1, we can substitute each of these values into the equation and solve for c.

When a = 4, we get 12 = 4(4)(2) + c, which simplifies to 12 = 32 + c. Solving for c, we get c = -20.

When a = 6, we get 12 = 4(6)(2) + c, which simplifies to 12 = 48 + c. Solving for c, we get c = -36.

When a = 1, we get 12 = 4(1)(2) + c, which simplifies to 12 = 8 + c. Solving for c, we get c = 4.

So the values of a, b, and c are:
a = 1
b = 0
c = 4

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