The volume of this cone is 339. 12 cubic feet. What is the height of the cone?

The value of x that makes line A and B parallel is 13.

Two Angles are Supplementary when they add up to 180 degrees.

From the diagram:

Angle 1 = 9x + 24

Angle 2 = 3x

Angle 1 and angle 2 are supplementary as their sum equals 180 degrees making line A and B parallel.

Hence:

Angle 1 + Angle 2 = 180°

Plug in the values

9x + 24 + 3x = 180

Solve for x

Collect like terms

9x + 3x = 180 - 24

12x = 156

Divide both sides by 12

12x/12 = 156/12

x = 56/12

x = 13

Therefore, the value of x is 13.

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The possible amount of time for Dinah to drive 420 miles on the highway is between 6 & 7 hours.

To get the possible time, we need to consider the range of speeds she can drive at.

Because she must drive at least 60 miles per hour and at most 70 miles per hour, we can calculate the possible time ranges using "Time = Distance / Speed"

At 60 miles per hour:

= 420 miles / 60 miles per hour

= 7 hours

At 70 miles per hour:

= 420 miles / 70 miles per hour

= 6 hours.

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To determine if a table or a set of ordered pairs can be modeled by a quadratic function, you should look for the following characteristics:

1. Consistent differences: Examine the differences between consecutive y-values. If there's a constant second difference (i.e., the differences between consecutive first differences remain the same), it's likely that the data can be modeled by a quadratic function.

2. Parabolic shape: Graph the ordered pairs. If the graph resembles a parabola (a U-shaped or inverted U-shaped curve), it indicates that the data can be modeled by a quadratic function.

By analyzing the ordered pairs and their differences, as well as examining the shape of the graph, you can determine if a quadratic function is the best fit for the data.

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The 95% confidence interval for the population mean 30-year fixed mortgage rate is (6.091%, 6.709%), and the 99% confidence interval is (5.993%, 6.807%).

To estimate the mean 30-year fixed mortgage rate for a home loan in the United States using a random sample of 10 recent loans with an average of 6.40% and a population standard deviation of 0.5%, you can compute the 95% and 99% confidence intervals as follows:

1: Identify the sample mean (x), sample size (n), and population standard deviation (σ).

x = 6.40%

n = 10

σ = 0.5%

2: Calculate the standard error (SE) using the formula SE = σ/√n.

SE = 0.5%/√10 ≈ 0.158%

3: Determine the critical z-values for 95% and 99% confidence intervals.

For a 95% confidence interval, z = 1.96 (from the z-table)

For a 99% confidence interval, z = 2.576 (from the z-table)

4: Calculate the margin of error (ME) using the formula ME = z * SE.

For 95% CI,

ME = 1.96 * 0.158% ≈ 0.309%

For 99% CI,

ME = 2.576 * 0.158% ≈ 0.407%

5: Compute the confidence intervals by adding and subtracting the ME from the sample mean.

95% CI:

(6.40% - 0.309%, 6.40% + 0.309%) = (6.091%, 6.709%)
99% CI:

(6.40% - 0.407%, 6.40% + 0.407%) = (5.993%, 6.807%)

So, the 95% confidence interval and 99% confidence interval is (6.091%, 6.709%) and (5.993%, 6.807%) respectively.

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To synthesize information regarding when books were written, Alex should create a timeline.  So, the correct option is A.

A timeline is a visual representation of chronological events, which can be used to arrange information in order based on their time of occurrence. In this case, Alex can plot the publication dates of the books on a timeline, allowing him to see how the dates relate to each other and to other events. This will help him to identify patterns and trends in the publication history of the books.

A Venn diagram, on the other hand, is a tool used to compare and contrast two or more sets of information. It is not well-suited for presenting chronological information.

A chart may be useful in presenting data in a visual manner, but it may not be as effective as a timeline in showing the order of events over time.
Therefore, the best answer from the choices provided is A. timeline.

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The first derivative of the implicit function given by x^(2/3) + y^(2/3) = 14 can be found using implicit differentiation. We take the derivative of both sides with respect to x and use the chain rule to differentiate the terms involving y:(2/3)x^(-1/3) + (2/3)y^(-1/3) * dy/dx = 0Then, we solve for dy/dx:dy/dx = -(x/y)^(1/3)This is the first derivative of the implicit function. To evaluate it at a specific point, we need to substitute the coordinates of that point into the equation above.

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The continuous compound rate of decay of this isotope is approximately 1.41%.

In order to find the continuous compound rate of decay of the radioactive isotope, we can use the formula:

N(t) = N0 * e^(-λt)

Where N(t) is the amount of the isotope present at time t, N0 is the initial amount of the isotope, λ is the continuous decay constant, and t is the time elapsed (in this case, 6 years).

We are given that 91.9% of the isotope is still present after 6 years, so:

0.919 = e^(-λ * 6)

To solve for λ, we can take the natural logarithm of both sides:

ln(0.919) = -6λ

Now, we can solve for the decay constant:

λ = -ln(0.919) / 6 ≈ 0.0141

So, the continuous compound rate of decay of this isotope is approximately 1.41%.

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Using metric system, we can find that Miss Marge's fish tank holds 100 litres of water.

Everything around us has a quantitative value, from the amount of sugar you put in a cake to the size of a football pitch. Each thing is measured differently based on its length, weight, volume, or duration. With these measurements, the idea of "Metric System" is introduced. The collection of standard units designed to measure length, weight, area, and capacity is known as the metric system of measurement in mathematics. As it uses numbers in powers of 10, it is based on the decimal system.

A litre (L) and millilitres (mL) are two units for measuring capacity in the metric system. A bottle of water holds 1 Liter of water. A millilitre is about 20 drops of water.

So, 100 litres of water will be the correct answer.

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The given statement "In an equation with two x’s, the solution is the number that makes the two sides equalwhen put in for both x’s is false because  in an equation with two x's, the solution is the number that makes the two sides equal when put in for one or both of the x's.

For example, consider the equation 2x + 3 = 5x - 1. To find the solution, we need to find the value of x that makes both sides of the equation equal. We can do this by simplifying the equation:

2x + 3 = 5x - 1

2x - 5x = -1 - 3

-3x = -4

x = 4/3

So the solution to this equation is x = 4/3. Notice that we only substituted the value of x once in the equation, but we still found the solution.

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The probability that the drug lasts less than 220 minutes for a random sample of 10 people is 0.0571, or about 5.71%.

To solve this problem, we need to use the normal distribution formula and the central limit theorem. The formula for the standard normal distribution is:

z = (x - μ) / σ

where z is the z-score, x is the observed value, μ is the mean, and σ is the standard deviation.

In this case, we want to find the probability that the drug lasts less than 220 minutes (3 hours and 40 minutes) for a random sample of 10 people. To do this, we first need to calculate the sample mean and the sample standard deviation.

The sample mean is the same as the population mean, which is 240 minutes:

μ = 240 minutes

The sample standard deviation is given by the formula:

σ = population standard deviation / sqrt(sample size)

σ = 40 minutes / sqrt(10) = 12.65 minutes (rounded to the nearest hundredth)

Now, we can calculate the z-score for a drug duration of 220 minutes:

z = (220 - 240) / 12.65 = -1.58 (rounded to the nearest hundredth)

We can use a standard normal distribution table or a calculator to find the probability that the z-score is less than -1.58. The probability is approximately 0.0571 (rounded to the nearest ten-thousandth).

Therefore, the probability that the drug lasts less than 220 minutes for a random sample of 10 people is 0.0571, or 5.71%.

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Answer:

f(x) = g(x - 9)

Step-by-step explanation:

The transformation from g(x) to f(x) is a translation of 9 units to the right.

A horizontal translation of h units takes place when x is replaced by x - h.

Here, replace x by x - 9.

f(x) = g(x - 9)

o, the correct sequence of transformations that would result in figure M' is: A) Reflection over the y-axis, then translation down 5 units.

In mathematics, a transformation is a process that changes the position, size, or shape of a geometric figure. Transformations are often used to study the properties of geometric objects and to solve problems in various areas of mathematics, such as geometry, algebra, and calculus.

Here,

Looking at the coordinates of the vertices of M and M', we can see that M' is obtained from M by reflecting it over the y-axis and then translating it down 5 units. So, the correct sequence of transformations that would result in figure M' is:

A) Reflection over the y-axis, then translation down 5 units.

This means that we first reflect M over the y-axis to obtain a new rectangle, which we can call M''. Then, we translate M'' down 5 units to obtain M'. Therefore, the sequence of transformations is:

Reflection over the y-axis

Translation down 5 units

This sequence of transformations will map M onto M'.

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The absolute maximum value is 3 at x=0, and the absolute minimum value is -2 at x=-1.

To find the absolute maximum and minimum values of the function f(x) = 2x³- 2x² - 2x + 3 over the interval (-1, 0], we'll first find the critical points and then evaluate the function at the endpoints of the interval.

1: Find the derivative of f(x) and set it equal to zero. f'(x) = 6x² - 4x - 2

2: Solve the equation f'(x) = 0 for x to find the critical points. 6x² - 4x - 2 = 0

This quadratic equation does not have rational roots, so there are no critical points in the given interval.

3: Evaluate the function at the endpoints of the interval.

f(-1) = 2(-1)³ - 2(-1)² - 2(-1) + 3 = -2 f(0) = 2(0)³ - 2(0)² - 2(0) + 3 = 3

Since there are no critical points in the interval, the absolute maximum and minimum values occur at the endpoints.

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Based on these calculations, option 1 gives you the cheaper price per square inch of pizza.

To determine which option will give you a cheaper price per square inch of pizza, you will need to compare the prices of the two options and calculate their respective prices per square inch.

Let's say option 1 is a large pizza with a diameter of 18 inches and costs $20, while option 2 is a medium pizza with a diameter of 14 inches and costs $15.

To calculate the area of each pizza, you need to use the formula for the area of a circle:

A = πr^2

For option 1, the radius is 9 inches (half of the diameter), so the area is:

A = π(9)^2 = 81π square inches

For option 2, the radius is 7 inches (half of the diameter), so the area is:

A = π(7)^2 = 49π square inches

Now you can calculate the price per square inch for each option by dividing the cost by the area:

Option 1:

Price per square inch = $20 / (81π) = $0.078/square inch

Option 2:

Price per square inch = $15 / (49π) = $0.098/square inch

Based on these calculations, option 1 gives you the cheaper price per square inch of pizza.

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The domain of (g o f o g)(x),  where "o" denotes composition of functions is the set of all x in F such that f(x) is in the domain of g, and g(f(x)) is in the domain of g.

To determine the domain of the function (g o f o g)(x), we need to consider the domains of the functions g(x) and f(x), as well as the composition of these functions.

Let's assume that the domain of g(x) is G and the domain of f(x) is F.

Since (g o f o g)(x) means that we apply the function g to f(x), and then apply g to the result again, we need to ensure that the output of f(x) is a valid input for g(x).

Therefore, the domain of (g o f o g)(x) is the set of all values of x in F such that g(f(x)) is in G, and g(y) is in G for all y in the range of g(f(x)).

In other words, the domain of (g o f o g)(x) is the set of all x in F such that f(x) is in the domain of g, and g(f(x)) is in the domain of g.

Thus, the domain of (g o f o g)(x) is the set of all x in F such that f(x) is in the domain of g, and g(f(x)) is in the domain of g.

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The vector describing the cat's final position is <7,0.5>.

The cat first travels along the vector <-1,2> and then chases the ball along the vector <2,-6>. So the cat's position after these two movements is:

<-1,2> + <2,-6> = <1,-4>

The cat's position after this movement is:

1.5 * <4,3> = <6,4.5>

Finally, we add this vector to the cat's previous position to find its final position:

<1,-4> + <6,4.5> = <7,0.5>

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The parabola has an axis of symmetry x=22, vertex at (22, 2), and opens downward.

Given the focus (22, 3) and directrix y=1, we can determine the following:

a. Axis of symmetry: Since the parabola is vertical (directrix is horizontal), the axis of symmetry will be a vertical line passing through the focus. So, x=22 is the axis of symmetry.

b. Vertex: The vertex is the midpoint between the focus and the directrix. To find the vertex, average the y-coordinates of the focus and the directrix. Vertex = (22, (3+1)/2) = (22, 2).

c. Direction: If the focus is above the directrix, the parabola opens upward. If the focus is below the directrix, the parabola opens downward. In this case, the focus (22, 3) is above the directrix y=1, so the parabola opens downward.

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Answer:

Part 1: 73 periods

Part 2: 73 periods in 1 year - 365/5=73

Part 3: 0.075 periodic interest rate - 180/2400=0.075

The ordered pairs for the data in the table include the following:

In Mathematics and Geometry, an ordered pair is sometimes referred to as a coordinate and it can be defined as a pair of two (2) elements or data points that are commonly written in a fixed order within parentheses as (x, y), which represents the x-coordinate (abscissa) and the y-coordinate (ordinate) on the coordinate plane of any graph.

Based on the table shown in the image attached below, we can reasonably infer and logically deduce that all of the coordinate points or ordered pairs would be located in quadrant 1.

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

To get the center of mass of the system with masses m1 = 4, m2 = 8, and m3 = 9 located at points P1 = (-6, -8), P2 = (3, 1), and P3 = (6, 2), the center of mass of the system is approximately (2.57, -0.29).

Center of Mass (x, y) = (Σ (mi * xi) / Σ mi, Σ (mi * yi) / Σ mi)
First, find the sum of the masses: Σ mi = m1 + m2 + m3 = 4 + 8 + 9 = 21
Next, calculate the x and y coordinates of the center of mass: Σ (mi * xi) = (4 * -6) + (8 * 3) + (9 * 6) = -24 + 24 + 54 = 54,
Σ (mi * yi) = (4 * -8) + (8 * 1) + (9 * 2) = -32 + 8 + 18 = -6.
Now divide these sums by the total mass: x-coordinate = 54 / 21 ≈ 2.57, y-coordinate = -6 / 21 ≈ -0.29.
So, the center of mass of the system is approximately (2.57, -0.29).

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Answer:

Step-by-step explanation:

The correct option is (a)  270 degree clockwise rotation because they both result in the same final orientation, just in opposite directions.

A 90-degree counter-clockwise rotational  symmetry  is the same as a 270-degree clockwise rotation. This is because rotations are measured in degrees, and a full circle is 360 degrees. Therefore, rotating something 90 degrees counter-clockwise means turning it a quarter of a full circle, while a 270-degree clockwise rotation means turning it three-quarters of a full circle.

To visualize this, imagine a square with its sides parallel to the x and y axes. If we rotate this square 90 degrees counter-clockwise, its sides will now be parallel to the y and negative x axes.

However, if we rotate the same square 270 degrees clockwise, its sides will also be parallel to the y and negative x axes, as it has been turned three-quarters of the full circle in the opposite direction.

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Answer:

B) Unique

Step-by-step explanation:

J coordinates = (0,6)

K coordinates = ( 9, -9)

L coordinates = ( -9, -9)

Unique rule: 1/3x , 1/3y

Which means 1\3 times the x coordinate and 1/3 times the y coordinate

J,K,L after applying the unique rule equals

J= (0,2)

K= (3,-3)

L= (-3,-3)

Which lines up with J’ and K’ and L’.

Making the “unique” statement true, dilation= (1/3x, 1/3y)

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By the rule of indices the  given expression is equivalent to x⁻² y⁻⁵.(option B)

What is index?

In mathematics, Index in plural indices  is the power or exponent which is raised to a number or a variable. For example, for the number 2⁵, 5 is the index of 2. Which gives the value 32.

Given expression is y⁻⁸ y³ x⁰ x⁻²

Here we use the law of indices.

By the law of indices it can be stated that the indices at the time of multiplication is being added up. That is zᵃ × zᵇ = zᵃ⁺ᵇ.

Using this concept of indices the given expression can be rewritten as

 y⁻⁸⁺³ x⁰⁻²

= y ⁻⁵ x⁻²

= x⁻² y⁻⁵

Hence, the given expression is equivalent to x⁻² y⁻⁵.

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Answer:

one

Step-by-step explanation:

The equation 4p + 7 = 3(4p) can be simplified by distributing the 3 on the right-hand side of the equation:

4p + 7 = 12p

Subtracting 4p from both sides of the equation, we get:

7 = 8p

Dividing both sides of the equation by 8, we get:

p = 7/8

Therefore, the equation has only one solution, which is p = 7/8. Answer: one.

A graph of the image of ABC under a dilation whose center is P and scale factor is 3 is shown in the image below.

In Geometry, dilation is a type of transformation which typically changes the size of a geometric shape, but not its shape. This ultimately implies that, the size of the geometric shape would be increased (enlarged) or decreased (reduced) based on the scale factor applied.

In this exercise, we would use an online graphing calculator to plot the image of ABC after a dilation by a scale factor of 3 centered at P.

Based on the image (see attachment), we can logically deduce that each vertex is 3 times as far from center P as the original vertex and each segment is 3 times as long as the original

Side length AC = (3.0)3 = 9.0

Side length AB = (5.0)3 = 15.0

Side length CB = (4.0)3 = 12.0

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To find sin(∠BAC + 30°), we will use the sine addition formula. The sine addition formula is: sin(A + B) = sin(A)cos(B) + cos(A)sin(B).

Step 1: Identify the angles A and B in our expression. In this case, A = ∠BAC and B = 30°.

Step 2: Apply the sine addition formula:

sin(∠BAC + 30°) = sin(∠BAC)cos(30°) + cos(∠BAC)sin(30°).

Now you have the expression for sin(∠BAC + 30°). To calculate its value, you will need the values of sin(∠BAC) and cos(∠BAC), which require more information about the triangle.

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The analogy that best represents the bold letters for each question is related to relationships and cross-sections.

Some analogies are given below:

Explanation: The relationship between happy and sad is that they are opposite emotions.

Similarly, up and down are opposite directions, so the analogy holds.

Explanation: The relationship between a cat and a kitten is that a kitten is a young cat.

Similarly, a puppy is a young dog, so the analogy holds.

Explanation: The relationship between circle and sphere is that a circle is a two-dimensional shape, while a sphere is a three-dimensional shape that has a circular cross-section.

Similarly, a square is a two-dimensional shape, while a cube is a three-dimensional shape that has a square cross-section, so the analogy holds.

Explanation: The relationship between the oven and baking is that an oven is a tool used to bake food.

Similarly, a grill is a tool used to grill food, so the analogy holds.

Explanation: The relationship between mother and father is that they are the parents of a child.

Similarly, a daughter and a son are the children of their parents, so the analogy holds.

Explanation: The relationship between a book and a chapter is that a book is divided into chapters.

Similarly, a play is divided into acts, so the analogy holds.

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A region is bounded by the curves  y = sinπ x ,  y = 4 x − 1 , and the x-axis. determine the area of the region. use the area formula for a triangle to expedite the calculation and show all your work.

To find the area of the region bounded by the curves y = sin(πx), y = 4x - 1, and the x-axis, we need to first find the points of intersection of the curves.

Setting y = sin(πx) and y = 4x - 1 equal to each other, we get:

sin(πx) = 4x - 1

Solving for x is difficult algebraically, so we can use numerical methods or graphing to estimate the solutions. A graph of the two curves shows that they intersect at approximately x = 0.161 and x = 1.239.

Next, we can find the area of the region by breaking it up into two parts: a triangle and a region bounded by the curve y = sin(πx), the x-axis, and the vertical lines x = 0.161 and x = 1.239.

The triangle has base 1.239 - 0.161 = 1.078 and height 4(1.239) - 1 = 3.956. Using the formula for the area of a triangle, we get:

Area of triangle = (1/2) * base * height

= (1/2) * 1.078 * 3.956

= 2.148

To find the area of the region bounded by y = sin(πx), the x-axis, and the vertical lines x = 0.161 and x = 1.239, we can use integration:

∫ from 0.161 to 1.239 of sin(πx) dx = [-cos(πx)/π] from 0.161 to 1.239 = [-cos(π(1.239))/π] - [-cos(π(0.161))/π] = (1/π) * (cos(0.161π) - cos(1.239π))

Using a calculator, we get:

(1/π) * (cos(0.161π) - cos(1.239π)) ≈ 0.696

Therefore, the total area of the region is:

Area = 2.148 + 0.696

= 2.844 (rounded to three decimal places)

So the area of the region bounded by the curves y = sin(πx), y = 4x - 1, and the x-axis is approximately 2.844 square units.

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The length of the variable x, obtained from the formula for the surface area of a solid about 10.63 inches

What is the surface area of a solid object?

The surface area of a solid object is the area of the outside surface of the  object.

The dimensions of of the open top box with the square corners 2 in by 2 in cut from the corners are;

Length of the box, L = x - 4 inches

Width of the box, W = x - 4 inches

Height of the box, H = 2 inches

The surface area of the box is therefore;

Volume, A = L × W + 2 × L × H + 2 × W × H

Plugging in the expressions for the length, width and height of the box, the surface area = (x - 4) × (x - 4) + 2 × 2 × (x - 4) + 2 × 2 × (x - 4) = 9·x² - 40·x + 16

The surface area of the box, A = 9·x² - 40·x + 16

Second part;

When the surface area = 609 in², we get;

A = 609 = 9·x² - 40·x + 16

9·x² - 40·x + 16 - 609 = 9·x² - 40·x - 593 = 0

x = (20 + √(5737))/9 ≈ 10.63, and x = (20 - √(5737))/9 ≈ -6.19

The length x ≈ 10.63 inches

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Looking at the plot, we can see that the stems range from 60 to 89, with each stem representing a group of ten pounds. The leaves represent the remaining single digits, indicating the exact weight of each tuna. There are 4 tuna that weigh less than 90 pounds

Based on the stem-and-leaf plot of the weights of yellowfin tuna caught during a fishing contest, we can count the number of tuna that weigh less than 90 pounds.

To determine the number of tuna that weigh less than 90 pounds, we need to look at the stems that are less than 9. This includes stems 6, 7, and 8. The leaves associated with these stems show the weights of the tuna that are less than 90 pounds. We can count the number of leaves associated with these stems to determine the number of tuna that weigh less than 90 pounds.

In this case, there are 4 tuna that weigh less than 90 pounds. Two of them weigh 88 pounds and the other two weigh 87 pounds. Therefore, we can conclude that there are 4 tuna that weigh less than 90 pounds in the fishing contest based on the stem-and-leaf plot.

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