Find the critical mumbers for g(x) = 2 sin r- r on (0,7). Then find the absolute maximum and minimum values for g(x) on (0,7). Give exact answers, not decimal approximations."

Two sides of a plot measure 32 m and 24 m and the angle between them is a perfect right angle. The other two sides measure 25 m each and the other three angles are not right angles. The area of the plot is 384 sq meters.

The Pythagorean theorem is a fundamental geometric idea that deals with the connections between the sides of right triangles. The square of the length of the hypotenuse (c) of a right triangle is equal to the sum of the squares of the lengths of the other two sides, according to the theorem (a and b). This may be stated mathematically as follows:

c² = a² + b²

Pythagoras, the ancient Greek mathematician who is credited with inventing the theorem, is named for him. It is employed in domains like physics, astronomy, and surveying and has extensive applications in mathematics, science, and engineering.

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Answer:

Step-by-step explanation:

The plot is in the shape of a trapezium with two sides measuring 32 m and 24 m, and two other sides measuring 25 m each.

To find the area of the plot, we need to first find the height of the trapezium. We can use the Pythagorean theorem to do this.

The side opposite to the right angle is the hypotenuse of the right-angled triangle formed by the two sides measuring 25 m each. So,

h² = 25² - 24²

h² = 625 - 576

h² = 49

h = 7

Therefore, the height of the trapezium is 7 m.

The area of a trapezium is given by the formula:

Area = (sum of parallel sides) x (height) / 2

In this case, the sum of the parallel sides is:

32 + 24 = 56

So, the area of the plot is:

Area = 56 x 7 / 2

Area = 196 m²

Therefore, the area of the plot is 196 square meters.

Carmen needs (6/7) cups of strawberry frosting and (1 1/7) cups of lemon frosting to make 2 cups of the frosting mixture.

To determine the amount of strawberry frosting and lemon frosting that Carmen needs to make 2 cups of the frosting mixture, we need to use a proportion.

Let x be the amount of strawberry frosting needed in cups, and y be the amount of lemon frosting needed in cups.

From the given information, we know that Carmen mixed 1/4 cup of strawberry frosting with 1/3 cup of lemon frosting. Thus, the ratio of the amounts of strawberry frosting to lemon frosting is:

x/y = (1/4)/(1/3)

Simplifying this ratio, we get:

x/y = 3/4

We also know that the total amount of frosting needed is 2 cups, so:

x + y = 2

Using substitution, we can solve for x:

x + (4/3)x = 2

(7/3)x = 2

x = (6/7) cups

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Answer:

6, 9, 10, 10, 11, 11, 12, 13, 15, 15, 18

Q1 (the first quartile) represents the data point that separates the lowest 25% of the data from the rest of the data. To find Q1, we can use the formula:

Q1 = (n + 1) / 4

where n is the total number of data points.

In this case, n = 11, so:

Q1 = (11 + 1) / 4 = 3rd data point

So, Q1 is 10.

Q3 (the third quartile) represents the data point that separates the highest 25% of the data from the rest of the data. To find Q3, we can use the formula:

Q3 = 3(n + 1) / 4

In this case:

Q3 = 3(11 + 1) / 4 = 9th data point

So, Q3 is 15.

D4 represents the fourth decile, which is the data point that separates the lowest 40% of the data from the rest of the data. To find D4, we can use the formula:

D4 = (n + 1) / 10 * 4

In this case:

D4 = (11 + 1) / 10 * 4 = 5th data point

So, D4 is 11.

D Assimilation represents the data point that is closest to the mean (average) of the data. To find D Assimilation, we first need to find the mean of the data:

Mean = (6 + 9 + 10 + 10 + 11 + 11 + 12 + 13 + 15 + 15 + 18) / 11 = 12

The data point closest to the mean is 12, so:

D Assimilation = 12

Pss (the range) represents the difference between the largest and smallest data points. In this case:

Pss = 18 - 6 = 12
6  9  10 10 11 11 12 13 15 15 18

                             Dss=12

   Q1=10       Q3=15

       D4=11

Step-by-step explanation:

It should be noted that m⁸m⁻⁶ is equivalent to (m²)¹, which is equal to m².

Using the product of powers rule for exponents, we can simplify m⁸m as follows:

m⁸m⁻⁶ = m⁸⁻⁶) = m²

Therefore, m⁸⁻⁶ is equal to m².

Now, we can further simplify by expressing m² as (m²)¹. Multiplying the exponents, we get:

(m²)¹ = m²

We can say that m⁸m⁻⁶ is equivalent to (m²)¹, which is equal to m².

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To find the values of a and b, we need to ensure that the function is differentiable at x = 1. Thus, the function defined by f(x) = x^2 + 3x + a, x <= 1 and bx + 2, x >= 1 differentiable at x = 1 are a = 3 and b = 5.

First, we need to check that the function is continuous at x = 1. Since the function has different definitions for x <= 1 and x >= 1, we need to check that the limit of the function as x approaches 1 from both sides is the same.

Limit as x approaches 1 from the left (x <= 1):

f(x) = x^2 + 3x + a

lim x->1- f(x) = lim x->1- (x^2 + 3x + a) = 1^2 + 3(1) + a = 4 + a

Limit as x approaches 1 from the right (x >= 1):

f(x) = bx + 2

lim x->1+ f(x) = lim x->1+ (bx + 2) = b + 2

For the function to be continuous at x = 1, these two limits must be equal.

4 + a = b + 2

a = b - 2

Now we need to check that the derivative of the function at x = 1 exists and is equal from both sides.

Derivative of the function for x <= 1:

f(x) = x^2 + 3x + a

f'(x) = 2x + 3

f'(1) = 2(1) + 3 = 5

Derivative of the function for x >= 1:

f(x) = bx + 2

f'(x) = b

f'(1) = b

For the function to be differentiable at x = 1, these two derivatives must be equal.

5 = b

Substituting b = 5 into the equation we found earlier for a, we get:

a = 5 - 2 = 3

Therefore, the values of a and b that make the function defined by f(x) = x^2 + 3x + a, x <= 1 and bx + 2, x >= 1 differentiable at x = 1 are a = 3 and b = 5.
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The projected stock price of Ghost Rider Co. in 4 years is $45.24.

First, we need to calculate the future EPS of Ghost Rider Co. in 4 years. We can do this using the formula for the future value of an annuity:

[tex]FV = PV x (1 + r)^n[/tex]

where FV is the future value, PV is the present value, r is the growth rate, and n is the number of years.

Using this formula, we get:

[tex]FV = $1.65 x (1 + 0.085)^4 = $2.36[/tex]

Next, we can use the following formula to determine the anticipated stock price:

Estimated stock price = EPS x PE ratio

When we enter the values we have, we obtain:

Projected stock price = $2.36 x 19.15 = $45.24

Therefore, the projected stock price of Ghost Rider Co. in 4 years is $45.24.

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Trapezoids can be split into two triangles and a rectangle.

Trapezoid is also known as a trapezium which is a closed shape having 4  sides with one pair of parallel sides. Trapezium is quadrilateral with 4 sides The parallel sides of a trapezium are known as the bases, and its non-parallel sides are called legs. A trapezium can also have parallel legs. The parallel sides can be horizontal, vertical, or slanting. Few real-life objects example of trapezium is a lamp, popcorn box etc.

A trapezoid consists of two triangles and one rectangle figure shows how can we cut the trapezium to split the trapezium.

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Answer:

To convert the equation 3x - 15y = 11 into slope-intercept form, we need to solve for y.

First, we'll subtract 3x from both sides:

-15y = -3x + 11

Next, we'll divide both sides by -15:

y = (3/15)x - (11/15)

Simplifying the fraction:

y = (1/5)x - (11/15)

This is the slope-intercept form, where the slope is 1/5 and the y-intercept is -11/15.

To find the general solution of the linear system y' = [2 2 -4; 2 -1 -2; 4 2 -6] y, we need to first find the eigenvectors and eigenvalues of the coefficient matrix A = [2 2 -4; 2 -1 -2; 4 2 -6].

Using the characteristic equation, we can find the eigenvalues:

det(A - λI) = 0

=> det([2-λ 2 -4; 2 -1-λ -2; 4 2 -6-λ]) = 0

=> (2-λ)[(-1-λ)(-6-λ) - 4] - 2[(-2)(-6-λ) - 8] + 4[2(-1-λ) - 4] = 0

=> λ^3 -  + 8λ - 4 = 0

=> (λ-1)(λ-2[tex])^2[/tex] = 0

Thus, λ = 1, 2 (with multiplicity 2). For each eigenvalue, we need to find a corresponding eigenvector.

For λ = 1, we need to find the null space of the matrix (A - λI):

A - λI = [1 2 -4; 2 -2 -2; 4 2 -7]

=> R2 <- R2 - 2R1, R3 <- R3 - 4R1

[1 2 -4; 0 -6 6; 0 -6 9]

=> R3 <- R3 - R2

[1 2 -4; 0 -6 6; 0 0 3]

So, we have a basic eigenvector of the form [4,-2,1]^T. To obtain a linearly independent eigenvector, we use the method of generalized eigenvectors. We need to find a vector v such that (A - λI) v = u, where u is the basic eigenvector.

(A - λI) v = u

=> [1 2 -4; 2 -2 -2; 4 2 -7] v = [4; -2; 1]

=> R2 <- R2 - 2R1, R3 <- R3 - 4R1

[1 2 -4; 0 -6 6; 0 -6 9] v = [4; -2; 1]

=> R3 <- R3 - R2

[1 2 -4; 0 -6 6; 0 0 3] v = [4; -2; 1]

=> -6v2 + 6v3 = -2

=> 3v3 = 1

=> 2v2 - 4v3 = -2

=> v2 = 0

So, we have v = [0; 1/3; 2/3[tex]]^T[/tex] as the second eigenvector corresponding to λ = 1.

For λ = 2, we need to find the null space of the matrix (A - λI):

A - λI = [0 2 -4; 2 -3 -2; 4 2 -8]

=> R1 <-> R2

[2 -3 -2; 0 2 -4; 4 2 -8]

=> R3 <- R3 - 2R1

[2 -3 -2; 0 2 -4; 0 8 -12]

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The measurement closest to the length of the piece of washi tape needed is approximately 21.36 feet.

The circumference of the circular mirror can be calculated using the formula C = 2πr, where r is the radius. Plugging in the given radius of 3.4 feet, we get C = 2π(3.4) = 21.36 feet (rounded to two decimal places). Since Rosalinda is decorating the edge of the mirror with washi tape, she needs a piece of tape that is equal in length to the circumference of the mirror. Therefore, the length of the piece of washi tape needed is closest to 21.36 feet.

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The total area of the soccer field is approximately 8150 square yards.

We'll find the total area of the soccer field in square meters first, and then convert it to square yards using the conversion factor provided.

Find the area in square meters (m²):
Area = Length × Width
Area = 102.9 m × 66.3 m
Area ≈ 6816.47 m²

Convert the area to square yards (yd²):
Use the conversion factor: 1 yard = 0.9144 meters
1 m² = (1/0.9144)² yd²
1 m² ≈ 1.19599 yd²
Now, multiply the area in m² by the conversion factor to get the area in yd²:
Area ≈ 6816.47 m² × 1.19599 yd²/m²
Area ≈ 8150 yd² (rounded to the nearest whole number).

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The answer is option B: 1.116%.

To find the APR(Annual Percentage Rate) for Jason's loan, we first need to calculate the total amount of interest he paid.

The finance charge of $55 is the interest paid for the 360-day term.

To find the total interest, we can use the formula:

Total interest = (finance charge / loan amount) x (days in a year / loan term in days)

Plugging in the values, we get:

Total interest = (55 / 5000) x (365 / 360)

Total interest = 0.011 x 1.01389

Total interest = 0.01116 or 1.116%

Therefore, the APR for Jason's loan is 1.116%.

The answer is option B: 1.116%.

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Answer:

Rate: 36:3

Unit Rate: 12:1

Step-by-step explanation:

The correct answer is e. 2/7.

To evaluate this line integral, we need to parameterize the curve given by the parabola y = x from (1, 1) to (-1, 1).

Let's let x = t and y = t, where t goes from 1 to -1. Then we can rewrite the integral as follows:

[tex]\int\ C (x - y)\dx + (y \sin y)\dy[/tex]

[tex]= \int\limits^1_{-1} {[(t - t)dt + (t sin t)}\,dt}[/tex]

[tex]= \int\limits^1_{-1} { (t \sin t)} \, dt[/tex]

We can evaluate this integral using integration by parts:

Let u = t and [tex]dv = sin t\ dt[/tex]. Then [tex]du/dt = 1[/tex] and v = -cos t.

Using the formula for integration by parts, we have:

[tex]\int\limits^1_{-1} { (t \sin t)}\, dt = -t \cos t |_{-1}^{1} + \int\limits^1_{-1} { cos t}\, dt[/tex]

= -cos(-1) + cos(1) + sin(-1) - sin(1)

= 2sin(1) - 2cos(1)

Therefore, the value of the line integral is:

[tex]S_c(x - y)dx + (y \sin y)dy = 2\sin(1) - 2\cos(1)[/tex]

Hence, the correct answer is e. 2/7.

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The difference in measures of center as a multiple of the measure of variation can be expressed using the coefficient of variation.

To express the difference in measures of center as a multiple of the measure of variation, you can use the coefficient of variation (CV).

The CV is calculated by dividing the standard deviation (measure of variation) by the mean (measure of center), and then multiplying by 100 to express the result as a percentage.

For example, if the standard deviation of one dataset is 5 and the mean is 10, the CV would be 50%. If the standard deviation of another dataset is 2 and the mean is 8, the CV would be 25%.

To express the difference in measures of center as a multiple of the measure of variation between these two datasets, you would calculate the difference in their means (10-8=2) and divide it by the CV of the combined dataset ((5/10 + 2/8)/2 = 47.5%).

Therefore, the difference in measures of center is approximately 0.042 times the measure of variation (2/47.5%).

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2/9 as a percentage is approximately 22.2%.

To convert the fraction 2/9 to a percentage, you simply need to divide the numerator (2) by the denominator (9) and then multiply the result by 100.

1. Divide the numerator by the denominator: 2 ÷ 9 ≈ 0.2222
2. Multiply the result by 100: 0.2222 × 100 = 22.22%

Now, to round the answer to one decimal place, we consider the second digit after the decimal point. In this case, it's 2. Since it's less than 5, we can round down.

So, 2/9 as a percentage rounded to one decimal place is approximately 22.2%.

In summary, converting a fraction to a percentage involves dividing the numerator by the denominator and then multiplying the result by 100. Rounding to a specific decimal place helps in presenting the result in a more easily understandable form, especially when dealing with non-integer values.

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To minimize the cost of the fence, the builder should use the expensive fence along the shorter side of the rectangular shape, as this will require less length of the expensive fence. Let's say the length of the rectangle is x meters and the width is y meters. Then the area of the rectangle is given by:

A = xy = 80000

And the perimeter of the rectangle is:

P = 2x + 2y

We are given that the cost of the fence along the front part of the land will cost $60 per meter, while the fence for the other three sides will cost $20 per meter. So the total cost of the fence is:

C = 60x + 20(2x + 2y)

Simplifying this expression, we get:

C = 100x + 40y

We can now use the area equation to eliminate one of the variables. Solving for y, we get:

y = 80000/x

Substituting this expression for y into the cost equation, we get:

C = 100x + 40(80000/x)

Simplifying this expression, we get:

C = 100x + 3200000/x

To minimize this function, we need to take its derivative and set it equal to zero:

dC/dx = 100 - 3200000/x^2 = 0

Solving for x, we get:

x = sqrt(32000) = 178.89

So the length of the rectangle should be approximately 178.89 meters, and the width should be:

y = 80000/178.89 = 446.68

Therefore, the amount of expensive fence needed is 178.89 meters, and the amount of cheap fence needed is:

2(178.89) + 2(446.68) - 178.89 = 893.36 meters

Finally, the length of the fence along the front part of the land that will be cost $60 per meter is simply the width of the rectangle, which is:

y = 446.68 meters (rounded to two decimal places)

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Answer:

The answer to the question provided is choice A, 4.

The value of m which makes the inequality true is, 4

A relation by which we can compare two or more mathematical expression is called an inequality.

Given that;

The inequality is,

⇒ 3m - 4 < 11

Now,. We can simplify as;

⇒ 3m - 4 < 11

⇒ 3m < 11 + 4

⇒ 3m < 15

⇒ m < 5

Thus, The value of m which makes the inequality true is, 4

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When rt is a diameter of circle p, it divides the circle into two equal halves. Since the sum of angles in a circle is 360 degrees, each half of circle p measures 180 degrees.

Thus, each arc of circle p that is intersected by diameter rt measures half of the circle or 90 degrees.

Therefore, each arc of circle p measures 90 degrees when rt is a diameter.

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Note that the point ( 1 , 92 ) (1,92) represents the estimated number of people in the hall at 7pm.

This is based on the given graph.

Note tha the horizontal axis  = x = 0

which corresponds to 6pm

the vertical is the number of scanned tickets.

Also, the pont (1, 92) is the line of best fit so that means that at 7pm which is a n hour after 6pm, there were about 92 persons still in the hall.

Hence we are correct to state that  the point ( 1 , 92 ) (1,92) represents the estimated number of people in the hall at 7pm.

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Full Question:
See attached image/graph

The current number of classrooms is 24, and the number of students per class is 70 if there were a total of 1200 students.

Let us assume that the number of classes = x

Number of students per class = 1200/x

Number of classrooms planned = x - 4

Number of students planned per class = 1200/ x+10

Total number of students = 1200

By using the above data, the equations will be written as:

(1200 / x-4) = (1200/x) +10

By multiplying the equation 2 we get:

1200x = 1200x + [tex]x^{2}[/tex] - 4800 - 40x

[tex]x^{2}[/tex] - 480- 4x = 0

(x-24) (x+20) = 0

x = 24

Number of rooms built = x =24

Number of students per class = (1200/24-10) = 60 students

Therefore, we can conclude that the current number of classrooms is 24, and the number of students per class is 60 + 10 =70.

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A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

The dice has eight sides, hence the theoretical probability of rolling a six is given as follows:

1/8 = 0.125 = 12.5%.

The experimental probabilities are obtained considering the trials, hence:

The more trials, the closer the experimental probability should be to the theoretical probability.

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Thus, the area of the shaded part is found to be 50 sq. ft.

The shaded region's area is most frequently found in common geometry problems. Such problems always have a minimum of two forms, and you must determine the area for each shape as well as the darkened zone by deducting the smaller shape's area from the larger.

Rectangle's area :

Area has two dimensions: length and width. Square units like square inches, square feet, or square metres are used to measure area.

Length of shaded part = 5 ft

width of shaded part = 10 ft

Area = 5*10

Area = 50 sq. ft

Thus, the area of the shaded part is found to be 50 sq. ft.

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Correct question:

For the given figure find the area of the shaded region.

Length BC = 18 ft

Length CD = 10 ft

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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No, the relation r is not a partial order relation.

To prove this, we need to show that r is not reflexive, not antisymmetric, or not transitive.

Since r fails to satisfy the antisymmetric property, it is not a partial order relation.

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The two inequalities that best describe the plant heights in millimeters are: h > 32 and h < 52. This is because all the recorded heights fall within this range. The other options do not include all the recorded heights or include heights that are not recorded.

To find the best inequalities that describe the plant heights (h) in millimeters, we need to determine the minimum and maximum heights from the given list.

List of plant heights: 52, 46, 51, 32, 50

Minimum height: 32 mm
Maximum height: 52 mm

Now we can write the inequalities that best describe the plant heights:

h > 32 (heights are greater than 32 mm)
h < 52 (heights are less than 52 mm)

Your answer: h > 32, h < 52

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The measure of the angle x in the circle is 65 degrees

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

The circle

On the circle, we have the angle at the vertex of the triangle to be

Angle = 100/2

Angle = 50

The sum of angles in a triangle is 180

So, we have

x + x + 50 = 180

Evaluate the like terms,

2x = 130

So, we have

x = 65

Hence, the angle is 65 degrees

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The mass can be any value greater than zero.

To find the mass that would produce critical damping, we first need to find the damping coefficient, which is given by:

c = damping constant * 2 * √m

where m is the mass in kg.

In this case, c = 5 * 2 * √m = 10√m.

Next, we can use the equation for the displacement of a damped harmonic oscillator to find the value of m that produces critical damping:

x = e^(-ct/2m) * (A + Bt)

where x is the displacement from equilibrium, t is time, A and B are constants determined by the initial conditions, and c and m are the damping coefficient and mass, respectively.

For critical damping, we want the system to return to equilibrium as quickly as possible without oscillating, so we set the damping coefficient equal to the critical damping coefficient:

c = 2 * √km

where k is the spring constant.

Since the spring can be held stretched 0.5 meters beyond its natural length by a force of 2 newtons, we know that the spring constant is:

k = F/x = 2/0.5 = 4 N/m

Substituting this value into the equation for critical damping, we get:

10√m = 2 * √(4m)

Squaring both sides and simplifying, we get:

100m = 16m

84m = 0

Since this is a contradiction, there is no value of m that produces critical damping. Therefore, the mass can be any value greater than zero.

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Measures of center are statistical tools used to determine the central tendency of a dataset, including mean, median, and mode.

The difference between these tools is how they capture the central tendency.

Variability of grades refers to how much grades fluctuate from week to week, which can be measured using statistical tools such as range, variance, and standard deviation. Without specific information about the test, it is not possible to draw conclusions.

However, analyzing the measures of center and variability can provide insights into student performance and grading consistency.

Further analysis, such as comparing grades to class averages or identifying patterns over time, may reveal additional information.

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