Please help and show work pls
noah and emma are standing on opposite sides of a 43.3 ft tree looking up at noah’s cat, which is perched at the very top. they are separated by a horizontal distance of 100 ft. the distance from emma to the cat is 50 ft and her angle of elevation to see the cat is 60°. the distance from noah to the cat is 86.6 ft and his angle of elevation to see the cat is 30°.

(a) identify three different trigonometric ratios that could have been used to find the distance x between emma and the base of the tree. for each trigonometric ratio, determine the distance. round to the nearest whole number.

(b) use the pythagorean theorem to find the distance x between emma and the base of the tree. round to the nearest whole number.

Answer:

9, 10, 11, 12, 13.

Step-by-step explanation:

All possible values for the number of tables that the store must sell in order to meet the requirements are 9, 10, 11, 12, 13

the repeating decimal 0.216 can be written as the geometric series: 0.216 = 216/990.

To write the repeating decimal 0.216 as a geometric series, we first need to express it in the form of a sum of a geometric series.

The decimal 0.216 repeats every three digits, so we can break it down as follows:

0.216 = 0.2 + 0.01 + 0.006 + 0.0002 + 0.00001 + 0.000006 + ...

Now, we can write this as a sum of a geometric series with the first term (a) and the common ratio (r):

a = 0.2
r = 0.01 (because each term is 1/100 of the previous term)

Thus, the geometric series for the repeating decimal 0.216 is:

0.216 = 0.2 + 0.2(0.01) + 0.2(0.01)^2 + 0.2(0.01)^3 + ...

The formula for the sum of an infinite geometric series is S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio.

Using the values for a and r, we can find the sum of the series:

S = 0.2 / (1 - 0.01) = 0.2 / 0.99 = 216/990.

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To calculate the percent error, we need to use the following formula:

percent error = (|measured value - actual value| / actual value) x 100%

1. Determine the difference between the actual value (8 laps) and the estimated value (5 laps).
Actual value = 8 laps
Estimated value = 5 laps
Difference = Actual value - Estimated value = 8 - 5 = 3 laps

2. Divide the difference by the actual value:
Percent error (decimal) = Difference / Actual value = 3 laps / 8 laps = 0.375

3. Convert the decimal to a percentage by multiplying by 100:
Percent error = 0.375 * 100 = 37.5%

4. Round to the nearest percent:
Percent error ≈ 38%

So, Michael's percent error in estimating his laps around the track was approximately 38%.

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The speed of the train is 60 miles per hour.

We calculate in two steps:

To calculate the speed of the train, we need to use the formula:

Speed = Distance / Time

Here, the distance travelled by the train is 480 miles, and the time taken is 8 hours (from 10:30 A.M. to 6:30 P.M.). So, we can calculate the speed of the train as:

Speed = 480 miles / 8 hours

Speed = 60 miles per hour

Therefore, the speed of the train is 60 miles per hour.

Sabine rode on a passenger train for 480 miles between 10:30 A.M. and 6:30 P.M.

To calculate the speed of the train, we used the formula Speed = Distance / Time, where Distance is 480 miles and Time is 8 hours (since the journey was between 10:30 A.M. and 6:30 P.M.).

Substituting the values, we get the speed of the train as 60 miles per hour.

This means that the train travelled at a speed of 60 miles per hour throughout the journey, covering a distance of 480 miles in 8 hours.

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To use logarithmic differentiation to find the derivative of the function y = x²/x, we first take the natural logarithm of both sides:

ln(y) = ln(x²/x)

Using the properties of logarithms, we can simplify this to:

ln(y) = 2 ln(x) - ln(x)

Now we differentiate both sides with respect to x using the chain rule:

1/y * y' = 2/x - 1/x

Simplifying this expression, we get:

y' = y * (2/x - 1/x²)

Substituting back in the original expression for y, we have:

y' = x²/x * (2/x - 1/x²)

Simplifying further, we get: y' = 2x - 1/x

Therefore, the derivative of the function y = x²/x using logarithmic differentiation is y' = 2x - 1/x.

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The slope of the line on a coordinate plane that makes an angle of depression of 32 degrees is approximately 0.625.

To find the slope of the line on a coordinate plane that makes an angle of depression of 32 degrees,:

Step 1: Determine the angle of elevation. Since the angle of depression is 32 degrees, the angle of elevation is also 32 degrees, because they are alternate angles.

Step 2: Use the tangent function to find the slope. The tangent of an angle in a right triangle is equal to the ratio of the side opposite the angle (rise) to the side adjacent to the angle (run). In this case, the tangent of the angle of elevation (32 degrees) is equal to the slope of the line.

Step 3: Calculate the tangent of 32 degrees. Using a calculator or a trigonometric table, you can find that tan(32°) ≈ 0.625.

So, the slope of the line on a coordinate plane that makes an angle of depression of 32 degrees is approximately 0.625.

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1. The statement "Each exterior angle of a regular hexagon is acute" is True.

2. The statement "The sum of the interior angles of a polygon is always a multiple of 180" is False.

3. The statement "In any polygon, the larger the number of vertices, the smaller the measure of an exterior angle" is True.

1. TRUE: Each exterior angle of a regular hexagon is acute.
A regular hexagon has six equal sides and six equal interior angles. The sum of the interior angles of a hexagon is (6-2) * 180 = 720 degrees. Since it's a regular hexagon, each interior angle is 720/6 = 120 degrees. The exterior angles are supplementary to the interior angles, so each exterior angle is 180 - 120 = 60 degrees. Since 60 degrees is less than 90 degrees, each exterior angle is acute.

2. FALSE: The sum of the interior angles of a polygon is always a multiple of 180.
The formula for the sum of the interior angles of a polygon is (n-2) * 180, where n is the number of vertices (or sides). As you can see, the result is always a multiple of 180.

3. TRUE: In any polygon, the larger the number of vertices, the smaller the measure of an exterior angle.
For a regular polygon, the measure of an exterior angle can be calculated as 360/n, where n is the number of vertices (or sides). As the number of vertices increases, the measure of an exterior angle decreases, since they are inversely proportional.

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The sum of all edges of the regular quadrilateral prism is 108 cm.

To find the sum of all edges of a regular quadrilateral prism with height h = 11 cm and base edges b = 8 cm, follow these steps:

1. Determine the number of base edges: A quadrilateral has 4 edges, so there are 4 base edges for the top and 4 for the bottom, totaling 8 base edges.
2. Determine the number of height edges: There are 4 vertical edges connecting the top and bottom bases.
3. Add the number of base and height edges: 8 base edges + 4 height edges = 12 edges in total.
4. Calculate the sum of all edge lengths: (8 base edges (8 cm)) + (4 height edges (11 cm)) = 64 cm + 44 cm = 108 cm.

So, the sum of all edges of the regular quadrilateral prism is 108 cm.

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The relative frequency of people who strongly agree with the statement "I feel respected by those I work for" is 0.312, or 31.2%.

This means that out of the 500 employees surveyed, 156 strongly agreed with the statement. To find the relative frequency, you simply divide the number of people who strongly agree by the total number of people surveyed (156/500).

This result suggests that the majority of employees feel respected by their employers, which is a positive sign for the workplace culture.

However, it's important to note that there are still a significant number of employees who either disagree or feel neutral about this statement, indicating that there may be room for improvement in terms of fostering a more respectful and supportive work environment.

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To solve the problem, we first need to graph the equation y = 3 / (√36-9x^2) and find the points where it intersects the x-axis and y-axis.

To find the x-intercept, we set y = 0 and solve for x:
0 = 3 / (√36-9x^2)
0 = 3
This has no solution, which means that the graph does not intersect the x-axis.

To find the y-intercept, we set x = 0 and solve for y:
y = 3 / (√36-9(0)^2)
y = 3 / 6
y = 1/2
So the graph intersects the y-axis at (0, 1/2).

Next, we need to find the point where the graph intersects the vertical line x = 3. To do this, we substitute x = 3 into the equation y = 3 / (√36-9x^2):
y = 3 / (√36-9(3)^2)
y = 3 / (√-243)
This is undefined, which means that the graph does not intersect the line x = 3.

Now we can draw a rough sketch of the graph and the region bounded by the x-axis, the line x = 0, and the curve y = 3 / (√36-9x^2):

           |
    _______|
   /       |
  /        |
 /         |
/_________|
|         |

The area we want to find is the shaded region, which is bounded by the x-axis, the line x = 0, and the curve y = 3 / (√36-9x^2). To find the area, we need to integrate the equation y = 3 / (√36-9x^2) with respect to x from x = 0 to x = 3:

A = ∫(0 to 3) 3 / (√36-9x^2) dx

We can simplify this integral by using the substitution u = 3x, du/dx = 3, dx = du/3:

A = ∫(0 to 9) 1 / (u^2 - 36) du/3

Next, we use partial fractions to break up the integrand into simpler terms:

1 / (u^2 - 36) = 1 / (6(u - 3)) - 1 / (6(u + 3))

So we have:

A = ∫(0 to 9) (1 / (6(u - 3))) - (1 / (6(u + 3))) du/3

A = (1/6) [ln|u - 3| - ln|u + 3|] from 0 to 9

A = (1/6) [ln(6) - ln(12) - ln(6) + ln(6)]

A = (1/6) [ln(1/2)]

A = (-1/6) ln(2)

Therefore, the exact area bounded by y = 3 / (√36-9x^2), x = 0, y = 0, and x = 3 is (-1/6) ln(2).
To find the area bounded by y = 3 / (√36-9x^2), x = 0, y = 0, and x = 3, we can set up an integral to compute the definite integral of the function over the given interval [0, 3]. The integral will represent the area under the curve:

Area = ∫[0, 3] (3 / (√(36-9x^2))) dx

To solve the integral, perform a substitution:

Let u = 36 - 9x^2
Then, du = -18x dx

Now, we can rewrite the integral:

Area = ∫[-√36, 0] (-1/6) (3/u) du

Solve the integral:

Area = -1/2 [ln|u|] evaluated from -√36 to 0

Area = -1/2 [ln|0| - ln|-√36|]

Area = -1/2 [ln|-√36|]

Since the natural logarithm of a negative number is undefined, there's an error in the original problem. Check the problem's constraints and the given function to ensure accuracy before proceeding.

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The estimate converges to x ≈ 3.14159265358979, the solution to the equation tan(x) = 0 to that many decimal places as well.

To use Newton's method to solve the equation tan(x) = 0 with an initial estimate of xo = 3, we need to follow these steps:

1. Find the derivative of the function f(x) = tan(x): f'(x) = sec^2(x).
2. Use the formula for Newton's method: xn+1 = xn - f(xn)/f'(xn)
3. Substitute f(x) = tan(x) and f'(x) = sec^2(x) into the formula: xn+1 = xn - tan(xn)/sec^2(xn)
4. Plug in xo = 3 and use your calculator to find xn+1:
x1 = xo - tan(xo)/sec^2(xo) = 3 - tan(3)/sec^2(3) ≈ 3.1425465430743
x2 = x1 - tan(x1)/sec^2(x1) ≈ 3.14159265358979
x3 = x2 - tan(x2)/sec^2(x2) ≈ 3.14159265358979

We can see that the estimate converges to x ≈ 3.14159265358979, which is the value of pi to 14 decimal places. Therefore, we can estimate the solution to the equation tan(x) = 0 to that many decimal places as well.

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The "equivalent-expression" for the given expression "b²c⁵b¹ - 2c¹b²" is b²c(bc⁴ - 2).

An "Equivalent-Expression" is an expression which has the same-value as the original expression, but may look different. The two expressions are equivalent if they simplify to the same result.

We have to solve the expression : "b²c⁵b¹ - 2c¹b²",

To simplify this expression, we first combine the "like-terms" by adding the exponents of b and c;

= b²c⁵b¹ - 2c¹b²,

Now we add the exponents having the same-base;

= b²⁺¹c⁵ - 2b²c¹;

= b³c⁵ - 2b²c

= b²c(bc⁴ - 2).

Therefore, the required "equivalent-expression" is b²c(bc⁴ - 2).

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The given question is incomplete, the complete question is

Write an equivalent expression for the given expression "b²c⁵b¹ - 2c¹b²".

Using basic mathematical procedures, we can determine that the total cost of the 8 bags of popcorn and 5 drinks is $53.

An operation, in mathematics, is a mathematical function that transforms zero or more input values into a precisely defined output value.

The quantity of operands affects the operation's arity.

The rules that specify the order in which we should carry out the operations required to solve an equation are referred to as the order of operations.

PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. (from left to right).

The whole price is thus:

Popcorn costs $3.50 for one bag.

One beverage costs $5.

Total cost for 5 beverages and 8 bags of popcorn:

(8 × 3.50) + (5 × 5) 28 + 25 $53

Therefore ,Using basic mathematical procedures, we can determine that the total cost of the 8 bags of popcorn and 5 drinks is $53.

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The appropriate response is provided below:

Zahra is going to the cinema with a few of her friends. A bag of popcorn costs $3.50 and a drink costs $5 at the theatre. How much would it cost if they purchased 5 beverages and 8 bags of popcorn?

By subtracting this from the dividend, the next step would be:

[tex](2x^4 - 3x^3 - 3x^2 + 7x - 3) - (-5x^3 + 10x^2 - 5x) = 2x^4 + 2x^3 - 13x^2 + 12x - 3[/tex]

This error occurs because he forgets to distribute the -2 in [tex]-2(x^2 - 2x + 1)[/tex]when subtracting from [tex]2x^4[/tex]. This leads to a mistake in the next step when he subtracts [tex]x^3 - 2x^2[/tex] from [tex]x^3 - 5x^2[/tex] to get [tex]-3x^2[/tex]instead of [tex]-3x^2 + 6x[/tex]. This error then leads to the incorrect constant term in the quotient.

Therefore, the error Randy makes is a subtraction error in the first step of the long division. It is important to pay attention to signs and distribute coefficients correctly when performing long division with polynomials.

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Answer: A. x + 2

Step-by-step explanation:

Edge 2023

Answer:

h ≈ 3.39

Step-by-step explanation:

V = πr^2h

h = V/πr^2

h = 154/ π · 3.8^2

h ≈ 3.39472

The system has no solution because the equations are parallel

Janelle is correct in saying that the system of equations has no solution. She can tell by looking at the coefficients of the variables in the two equations.

Both equations have the same coefficients for x and y, which means that they are parallel lines in the xy-plane.

Since parallel lines never intersect, there are no values of x and y that would satisfy both equations simultaneously, meaning that the system has no solution.

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Step-by-step explanation:

you can solve cos(u) by

cos(u) = adjecent / hypotenes...general formula of cos

cos(u) = √44 / 12

cos(u) = 2√11 / 12 ..... √44 = √4×11 = 2√11

cos(u) = √11 / 6

u = cos^-1 ( √11 / 6 ) ..... divided both aide by cos ( multiple by cos invers )

u = 56.442 .... so we get it's angle

Answer:

[tex]cos(U)=\frac{\sqrt{11} }{6}[/tex]

Step-by-step explanation:

In a right triangle, the cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse. Therefore, we have:

cos(U) = adjacent/hypotenuse = TU/SU

We are given that TU = sqrt(44) and SU = 12, so:

cos(U) = sqrt(44)/12

To simplify this expression, we can first factor 44 into 4 * 11, since 4 is a perfect square and a factor of 44:

cos(U) = sqrt(4 * 11) / 12

cos(U) = (sqrt (4) * sqrt (11)) / 12

cos (U) = (2 * sqrt (11)) / 12

Simplifying the fraction by dividing both the numerator and denominator by 2, we get:

cos(U) = sqrt(11)/6

Therefore, the exact value of cos(U) in simplest radical form is sqrt(11)/6

Furthermore, if you want another way to write the answer, dividing by 6 is the same as multiplying by 1/6 so you can do cos (U) = 1/6 * sqrt (11)

Although the other individual was correct that you use inverse trig (cos ^ -1) to find the measure of U, getting an exact answer requires us to leave it in simplest radical form since the number is so large and at best will yield an approximation if you don't keep it in simplest radical form.

The value of the function at that point is equal to the z-coordinate of the point on the plane.

A horizontal tangent plane is a plane that is parallel to the x-y plane and tangent to a surface at a point where the slope in the horizontal direction is zero.

To use the tangent plane equation to find a horizontal tangent plane, we need to find the partial derivatives of the function with respect to x and y, evaluate them at the point of interest, and check if they are both zero.

If they are both zero, then the tangent plane is horizontal and the equation simplifies to f(a,b) = z.

The tangent plane equation is given by:

f(a,b) + f(1)(x-a) + f(2)(y-b) = z

where (a,b) is the point where the tangent plane intersects the surface, and f(1) and f(2) are the partial derivatives of the function with respect to x and y, evaluated at (a,b).

To use this equation to find the horizontal tangent plane, we first find the partial derivatives f(1) and f(2), and evaluate them at the point where we want to find the tangent plane. If f(1) and f(2) are both zero at that point, then the tangent plane is horizontal and the equation simplifies to:

f(a,b) = z

This means that the value of the function at that point is equal to the z-coordinate of the point on the plane.

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An equation that could be used to determine M, the unknown side length of the logo is X = (36/5) x M

Let's assume that the unknown side length of the logo is 'M'. The logo is a rectangle, and the area of a rectangle is given by multiplying its length and width. Since we know the length of the logo is 7(1)/(5) feet, we can write the equation:

A = L x W

where A is the area of the logo, L is the length of the logo, and W is the width of the logo.

Substituting the given values, we get:

A = (7(1)/(5)) x M

or

A = (36/5) x M

Now, we know that the area of the logo is exactly the maximum area allowed by the building owner. Let's assume this maximum area is 'X'. So, we can write another equation:

A = X

Combining both equations, we get:

X = (36/5) x M

This is the required equation that could be used to determine the unknown side length 'M' of the logo if we know the maximum area allowed by the building owner 'X'.

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Let x be the number of pinon nuts each animal has collected. To make the chipmunk have ten more pinon nuts than the squirrel, the squirrel would have to give the chipmunk 10 pinon nuts.

So, after the exchange, the squirrel would have x - 10 pinon nuts, and the chipmunk would have x + 10 pinon nuts.

Since they are each giving an equal amount, the total number of pinon nuts remains the same. Therefore, we can set up the equation:

x + (x - 10) = 2x - 10

Simplifying and solving for x, we get:

2x - 10 = 2x

-10 = 0

This is a contradiction, so there is no solution that satisfies the conditions of the problem.

Therefore, the problem is not well-defined and there is no answer.

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For the piecewise-defined function, f(-4) = 42.

The given function is a piecewise-defined function, which means that it is defined differently depending on the value of x. In this case, we have two different formulas for the function depending on whether x is less than or greater than 3. For values of x less than 3, the function is given by f(x) = 22 - 5x, while for values of x greater than 3, the function is given by f(x) = 2x + 5.

To find f(-4), we need to determine which part of the function applies to the value of x = -4. Since -4 is less than 3, we use the first part of the function, which gives us f(-4) = 22 - 5(-4) = 22 + 20 = 42. This means that if x is equal to -4, the function f(x) evaluates to 42.

Piecewise-defined functions can be useful in modeling real-world problems where the relationship between variables changes depending on certain conditions or constraints. By defining the function differently depending on the value of x, we can more accurately capture the behavior of the system being modeled.

In this case, the function could be used to model a situation where the value of a variable has different relationships to other variables depending on whether it is less than or greater than a certain threshold value.

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To satisfy the conditions that the mode is larger than the median, and the median is larger than the mean, one possible set of frequencies is 1 person chooses 1, 3 people choose 2, 4 people choose 3, 1 person chooses 4 and 11 people choose 5 This results in a mode of 5, a median of 4, and a mean of approximately 3.75.

Since we are given that the mode is larger than the median, that means that at least 11 people must choose the same number. Let's assume that 11 people choose the number 5.

Now, since the median is larger than the mean, we want to make sure that the remaining 9 people choose numbers that are smaller than 5. If they all choose 1, 2, or 3, then the median will be 3, which is larger than the mean. Therefore, we need to make sure that at least one person chooses 4.

So one possible set of frequencies could be

1 person chooses 1

3 people choose 2

4 people choose 3

1 person chooses 4

11 people choose 5

This set of frequencies gives us a mode of 5 (since 11 people choose 5), a median of 4 (since the middle value is 4), and a mean of

(11 + 32 + 43 + 14 + 11*5) / 20 = 3.7

Since the median is larger than the mean, this set of frequencies satisfies all the given conditions.

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The value of x in the solution to this system of equations 5x - 4y = 27 and y = 2x + 3 is -13.

To find the value of x in this system of equations, we can use substitution method to find the its solution. Start by isolating x in one of the equations and then substituting that value into the other equation.

Let's start by isolating x in the second equation:

y = 2x + 3

Subtracting 3 from both sides:

y - 3 = 2x

Dividing both sides by 2:

(1/2)y - (3/2) = x

Now we can substitute this expression for x into the first equation:

5x - 4y = 27

5((1/2)y - (3/2)) - 4y = 27

Simplifying:

(5/2)y - 15/2 - 4y = 27

Combining like terms:

-(3/2)y = 69/2

Dividing by -(3/2):

y = -23

Now we can substitute this value of y back into the expression we found for x:

x = (1/2)y - (3/2)

x = (1/2)(-23) - (3/2)

x = -13

Therefore, the solution to this system of equations is x = -13, y = -23.

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Answer:

x = 2

Step-by-step explanation:

A secant is a straight line that intersects a circle at two points.

A segment is part of a line that connects two points.

According to the Intersecting Secants Theorem, the product of the measures of one secant segment and its external part is equal to the product of the measures of the other secant segment and its external part.

The given diagram shows two secant segments that intersect at an exterior point.  

Therefore, according to the Intersecting Secants Theorem:

[tex](6x-1+7) \cdot 7=(x+3+9) \cdot 9[/tex]

Solve for x:

[tex]\begin{aligned}(6x+6) \cdot 7&=(x+12) \cdot 9 \\42x+42&=9x+108\\42x+42-9x&=9x+108-9x\\33x+42&=108\\33x+42-42&=108-42\\33x&=66\\33x\div33&=66\div33\\x&=2 \end{aligned}[/tex]

Therefore, the value of x is x = 2.

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Answer: $2,013.43

Step-by-step explanation:

$135,950 x 1.15% = 1,563.425

Round to $1,563.43

Add in $450

$1,563.43 + $450 = $2,013.43

La proportion de boules vertes dans cette boîte est de 2/3.

Pour déterminer la proportion de boules vertes dans cette boîte, nous devons comparer le nombre de boules vertes au nombre total de boules dans la boîte.

Le nombre total de boules dans la boîte est la somme des boules vertes et des boules bleues, soit 12 + 6 = 18 boules.

Maintenant, pour calculer la proportion de boules vertes, nous divisons le nombre de boules vertes par le nombre total de boules.

Proportion de boules vertes = Nombre de boules vertes / Nombre total de boules

Proportion de boules vertes = 12 / 18

Simplifiant cette fraction, nous obtenons :

Proportion de boules vertes = 2/3

La proportion de boules vertes dans cette boîte est donc de 2/3 ou environ 66.67%.

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The given problem involves evaluating the surface integral of the vector field F(X, y, 2) over the top half of a sphere x^2 + y^2 + z^2 = 1, oriented upwards, using the Divergence Theorem.

The Divergence Theorem states that the flux of a vector field F through a closed surface S is equal to the triple integral of the divergence of F over the region enclosed by S.

In this problem, the given vector field F(X, y, z) is F(X, y, 2) = 3z^2xi + (y^3 + tan(2)J + (3x^2z + 1y^2)k.

The surface S is the top half of the sphere x^2 + y^2 + z^2 = 1, oriented upwards. This means that z is positive on S, and the normal vector points in the positive z-direction.

To use the Divergence Theorem, we need to find the divergence of F. The divergence of F is given by div(F) = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z, where ∂Fx/∂x, ∂Fy/∂y, and ∂Fz/∂z are the partial derivatives of F with respect to x, y, and z, respectively.

Taking the partial derivatives of F with respect to x, y, and z, we get:

∂Fx/∂x = 6xz

∂Fy/∂y = 3y^2 + 2y

∂Fz/∂z = 0

So, the divergence of F is: div(F) = 6xz + 3y^2 + 2y

Now, we can apply the Divergence Theorem, which states that the surface integral of F over S is equal to the triple integral of the divergence of F over the region enclosed by S.

The triple integral of the divergence of F over the region enclosed by S can be written as: ∫∫∫ div(F) dV, where dV is the volume element.

Since the given problem asks for the surface integral of F over S, we only need to consider the part of the triple integral that involves the surface S.

The surface integral of F over S can be written as: ∫∫ F · dS, where dS is the outward-pointing normal vector on S and · represents the dot product.

The dot product F · dS can be expressed as: Fx * dSx + Fy * dSy + Fz * dSz, where Fx, Fy, and Fz are the components of F, and dSx, dSy, and dSz are the components of the outward-pointing normal vector on S.

Since the normal vector on S points in the positive z-direction, we have dSx = 0, dSy = 0, and dSz = 1.

Substituting the components of F and the components of dS into the expression for the dot product, we get: Fx * dSx + Fy * dSy + Fz * dSz = (3z^2x)(0) + (y^3 + tan(2)J + (3x^2z +

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The box plot that would represent the data recorded by Mr. Larson would be B. Second box plot.

To find the correct box plot of the data recorded by Mr. Larson, the math teacher, first order the grades from lowest to highest :

70, 71, 73, 75, 77, 80, 84, 86, 87, 92, 95, 96, 97, 100

There are 14 grades which means that the median position would be the 7th and 8th grades average :

= ( 84 + 86 ) / 2

= 170 / 2

= 85

The position of Q3 would be:

= ( n + 1 ) x 75 %

= ( 14 + 1 ) x 75 %

= 11 th position which is 95

The correct box plot is therefore the second box plot which shows the Q3 as 95.

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The simplified form of the given expression is (x-7)/3x.

The given expression is (2x²-8x-42)/6x² ÷ (x²-9)/(x²-3x)

Here, (x²-4x-21)/3x² ÷ (x-3)(x+3)/x(x-3)

= (x²-4x-21)/3x² ÷ (x+3)/x

= (x²-4x-21)/3x² × x/(x+3)

= (x²-4x-21)/3x × 1/(x+3)

= (x²-4x-21)/3x(x+3)

= (x²-7x+3x-21)/3x(x+3)

= [x(x-7)+3(x-7)]/3x(x+3)

= (x-7)(x+3)/3x(x+3)

= (x-7)/3x

Therefore, the simplified form of the given expression is (x-7)/3x.

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