Franco has enough paint to cover an area of approximately 226 square feet.
He needs to
paint a cylinder that is 5 feet long. What is the maximum
possible radius of the cylinder that can be covered by the paint?
A. 14.4 feet
B. 4 feet
C. 9 feet
D. 3.8 feet

The number of terms in the expansion of {(2x - 3y)² (2x + 3y)²}² are 5

Consider given mathematical expression {(2x - 3y)² (2x + 3y)²}²

We need to find the number of terms in the expansion of {(2x - 3y)² (2x + 3y)²}²

consider, {(2x - 3y)² (2x + 3y)²}²

= {[(2x - 3y) × (2x - 3y)] × [(2x + 3y) × (2x - 3y)]}²

= {[(2x - 3y) × (2x + 3y)] × [(2x - 3y) × (2x - 3y)]}²   ......(commutative property of multiplication)

We know that the expansion of a² - b² = (a + b)(a - b)

Using this identity (2x - 3y) × (2x + 3y) = (2x)² - (3y)²

So given expression becomes,

{(2x - 3y)² (2x + 3y)²}²

=  {[(2x)² - (3y)² ] × [(2x)² - (3y)² ]}²

= {[(2x)² - (3y)² ]²}²

= [(2x)² - (3y)² ]⁴            .........(using properties of exponents)

We know that the number of terms in the binomial expansion of (a + b)^n are (n + 1)

Here, a = (2x)², b = - (3y)² and n = 4

Thus, the number of terms in the expansion of  [(2x)² - (3y)² ]⁴   are 4 + 1 = 5

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The complete question is:

What is the number of terms in the expansion of  {(2x - 3y)² (2x + 3y)²}² ?

We can calculate limits considering the following rules:

Constant Rule The constant rule of limit calculus is utilized when a function is given in which there is no identical variable available. This rule states that the limit of the constant function remains as it is.

Limu→a k = k

Constant time function rule This rule is utilized when there are stagnant coefficients along with the identical variables of the function. This rule states that the constant coefficient will be written outside the limit notation.

Limu→a kf(u) = k Limu→a f(u)

Sum Rule The sum rule is utilized when there are two or more terms or functions given along with the additional sign among them. This rule states that the denotation of the limit will apply to every function separately.

Limu→a [f(u) + g(u)] = Limu→a [f(u)] + Limu→a [g(u)]

Difference Rule The difference rule is utilized when two or more terms or functions are mentioned along with the minus sign among them. This rule expresses that the notation of the limit will be implied to each function segregation.

Limu→a [f(u) – g(u)] = Limu→a [f(u)] – Limu→a [g(u)]

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Answer:

Step-by-step explanation:

We can calculate limits considering the following rules:

Constant Rule The constant rule of limit calculus is utilized when a function is given in which there is no identical variable available. This rule states that the limit of the constant function remains as it is.

Limu→a k = k

Constant time function rule This rule is utilized when there are stagnant coefficients along with the identical variables of the function. This rule states that the constant coefficient will be written outside the limit notation.

Limu→a kf(u) = k Limu→a f(u)

Sum RuleThe sum rule is utilized when there are two or more terms or functions given along with the additional sign among them. This rule states that the denotation of the limit will apply to every function separately.

Limu→a [f(u) + g(u)] = Limu→a [f(u)] + Limu→a [g(u)]

Difference RuleThe difference rule is utilized when two or more terms or functions are mentioned along with the minus sign among them. This rule expresses that the notation of the limit will be implied to each function segregation.

Limu→a [f(u) – g(u)] = Limu→a [f(u)] – Limu→a [g(u)]

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The two data sets for which a linear regression model is reasonable include the following: a) Aaliyah’s and Julio’s data sets.

In Mathematics, a positive correlation is used to described a scenario in which two variables move in the same direction and are in tandem.

This ultimately implies that, a positive correlation exist when two variables have a linear relationship or are in direct proportion. This ultimately implies that, when one variable increases, the other increases, as well.

In order to determine if a data set has a good linear relationship, we would have to interpret the correlation coefficient (r), which typically ranges between -1 to 1 as depicted by both Aaliyah’s and Julio’s data sets.

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Complete Question:

Aaliyah, Julio, Dan, and Miriam are each studying sets of data. Aaliyah is working with 14 data pairs with a correlation coefficient of 0.9. Julio is working with 13 data pairs with a correlation coefficient of –0.8. Dan is working with 10 data pairs with a correlation coefficient of 0.1. Miriam is working with 15 data pairs with a correlation coefficient of –0.2. For which two data sets is a linear regression model reasonable?

a)Aaliyah’s and Julio’s data sets

b)Julio’s and Miriam’s data sets

c)Dan’s and Miriam’s data sets

d)Miriam’s and Aaliyah’s data sets

Answer:

I'm unable to plot graph, but I can give you the steps to graph the equation y = -²- 10x - 16 on the given set of axes.

Plot the vertex of the parabola: The vertex of the parabola is the point where the parabola changes direction. To find the vertex, you can use the formula (-b/2a, f(-b/2a)), where a, b, and c are the coefficients of the equation y = ax² + bx + c. In this case, a = -1, b = -10, and c = -16, so the vertex is (-10/2(-1), -(-10/2(-1))² - 10(-10/2(-1)) - 16) = (5,16).

Plot the roots: The roots of the equation are the values of x that make y = 0. To find the roots, you can set y = 0 and solve for x. In this case, y = 0 = -x² - 10x - 16. This equation can be solved by factoring or by using the quadratic formula. The roots are x = 4 and x = -2.

Plot five points on the parabola, including the vertex and the roots. To do this, you can substitute different values of x into the equation y = -x² - 10x - 16 and find the corresponding y-values. For example, you could plot the point (4,0), which is one of the roots, and (5,16), which is the vertex.

Connect the points to form a parabola.

Keep

Step-by-step explanation:

An inequality in one variable that models this scenario is 2(3.00) + 1.90x + 2.50(x + 2) ≤ 27.

The maximum number of whole pounds of grapes he can buy is equal to 4.

In order to write an inequality that represents or models this scenario, we would assign a variable to the number of pounds of grapes, and then translate the word problem into algebraic equation as follows:

From the information provided above, we have the following parameters:

The rate of grapes per pound = $1.90.

The rate of pineapples per pound = $3.00.

The rate of strawberries per pound =  $2.50.

Pounds of strawberries = g + 2.

Therefore, an inequality that represents or models this scenario is given by;

p + x + s ≤ 27

2(3.00) + 1.90x + 2.50(x + 2) ≤ 27

Next, we would solve the above inequality algebraically as follows;

6 + 1.90x + 2.50x + 5 ≤ 27

4.4x ≤ 27 - 11

4.4x ≤ 16

x ≤ 16/4.4

x ≤ 3.64 ≈ 4

x ≤ 4.

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The graph of a linear equation is always a straight line.

Linear equations in two variables, for example, are known when they have two variables in them. A, B, and C are constants in the generic linear equation in two variables, which is written as axe + by + c = 0.

Simply assume a set of values for x and compute the corresponding set of values for y to discover the graph of axe + by + c = 0. We now possess a set of coordinates (x, y). Now connect the spots on the graph. A straight line will be generated on the graph.

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A parabola is a u shaped curve. It is a plane curve whose all points are at a fixed distance form a point called focus.

A parabola is formed by cutting a right circular cone in half along a plane perpendicular to the cone's generator. It is a locus of a moving point whose separation from the focus is equal to its separation from a stationary line (directrix)

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Answer:

5/8 × 4 = 5/8 × 4/1 = 20/8 = 5/2

Step-by-step explanation:

hope this helps

Answer:

The solution is 5/2

Step-by-step explanation:

[tex] \sf \frac{5}{8} \times 4 \\ = \sf \frac{5}{ \cancel8} \times \cancel4 \\ = \sf \frac{5}{2} [/tex]

Answer:16

Step-by-step explanation:

15+22+13+14=64

64:4=16

the median is 64

please give me the best answer i really need it

Answer:

123°

Step-by-step explanation:

x + 32 + 25 = 180

x = 123

We can solve for log bases without a calculator by using  logarithmic formulas and identities.

A logarithm is basically a mathematical operation which is able to determine how many times a particular number, which is known as the base, will be multiplied by itself to reach the other number. We can solve for log bases even without a calculator.

For example, we need to find the value of log 128 to the base 8, which we can write as log₈28.

We know that,

[tex]log_{a} b^{x} = 1/b(log_{a}x)[/tex]

[tex]log_{a} x^{b} = b(log_{a}x)[/tex]

Using this property of logs,

log₈128 = log₂³2⁷

7/3 (log₂2)

Another property is,

[tex]log_{a}a = 1[/tex]

Using this property of logs,

⇒ 7/3(log₂2) = 7/3×1

=7/3

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A limit in mathematics is a point at which a function approaches the output for the specified input values.

The idea of the limit of a geometric net expands the definition of the limit of a sequence and relates it to the limit and direct limit in theory category. In general, there are two sorts of integrals: definite integrals and indefinite integrals.

Limit is said to be exits when f(x) = [tex]\lim_{h \to 0\0} f(h-x) = \lim_{h \to0 \0} f(h+x)[/tex]

It means that when f(x) = L.H.L = R.H.L

LHL means left hand limit and RHL means right hand limit.

Some Property of limits are as follows

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Answer:

In this example the maximum is the one on the top while the minimum is the one on the curve.

Step-by-step explanation:

Answer:

d1=50 and c2 21/25

Step-by-step explanation:

The vertex of the graph of  y = 3x² + 2x + 1 is (-0.33, 1.33)

We know that the general form of a parabola is given by an equation y = ax² + bx + c.

the x coordinate of the vertex of the quadratic equation is given by the formula -b/(2a)

To find y-coordinate we substitute value of x in given quadratic equation and solve it for x.

Consider given quadratic equation y = 3x² + 2x + 1

Comparing given quadratic equation y = 3x² + 2x + 1 with y = ax² + bx + c we get, a = 3, b = 2 and c = 1

The x-coordinate of vertex would be,

-b/(2a)

= -2/(2 × 3)

= -2/6

= -1/3

= - 0.33

substitute x = -1/3 in given quadrtic equation in order to find the y-coordinate.

 y = 3(-1/3)² + 2(-1/3) + 1

y = (-1)² - 2/3 + 1

y = 2 - 2/3

y = 4/3

y = 1.33

So, (-1/3, 4/3) is the vertex.

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To find log7 to the base 2 on a calculator, you will need to use the change of base formula that I mentioned earlier: log_b(x) = log_c(x) / log_c(b).

If you want to find log7 to the base 2 on a calculator, you can use the following steps:

Press the log button on your calculator to change to the logarithm mode (usually represented by "log" or "ln").

Enter the number 7

Press the "log" or "ln" button again to switch to the base 2 logarithm mode.

Enter the number you want to find the logarithm of.

Press the division button (/)

Press the log or ln button again and enter the number 7

Press the equals button (=)

The result will be the logarithm of the number with respect to base 2.

Alternatively, you can find the logarithm to base 2 of 7 by dividing the logarithm to base 10 of 7 by logarithm to base 10 of 2 (or natural logarithm of 7 by natural logarithm of 2)

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The intensity of earthquakes is measured on the Richter scale. It is a device which compares earthquakes.

Earthquake strength is measured using instruments such as the Richter scale. The Richter scale was invented in the 1930s to help us understand the magnitude and intensity of earthquakes measured by seismographs. Ground surveillance equipment. Standard scale. This is a logarithmic scale measuring a factor of 10. It is measured using information collected by seismographs. It has a base 10 logarithmic scale.

Mathematically, If M is magnitude and I is intensity of earthquake then formula,

M = log(I/Iₛ)

where, Iₛ --> standard intensity

The strength of an earthquake is measured or estimated by two things: amplitude and energy..

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Complete question:

How is the intensity of an earthquake measured ?

Answer:

to the left of the zero mark   at the -63.2 mark

Step-by-step explanation:

The correct pairs pf points and axis of symmetry  are  (-1,-8) & y=-8 respectively.

The vertex of a parabola is the point at the intersection of the parabola and its line of symmetry. For a parabola whose equation is given in standard form , the vertex will be the minimum (lowest point) of the graph if and the maximum (highest point) of the graph if .

Given here; A parabola with upwards curve.

Now we know, The vertex is the highest point if the parabola opens downward and the lowest point if the parabola opens upward. The axis of symmetry is the line that cuts the parabola into 2 matching halves and the vertex lies on the axis of symmetry.

Thus we can clearly observe from the graph that (-1,-8) is the vertex of the parabola and we know the parabola is symmetric around its vertex.

Therefore the axis of symmetry of the parabola is y=-8

Hence, The correct pairs pf points and axis of symmetry  are  (-1,-8) & y=-8 respectively.

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The process of solving systems by elimination with multiplication for the equation is explained below.

In math the term equation is known as a mathematical statement that is made up of two expressions connected by an equal sign.

Here we need to write down the steps to solve systems by elimination with multiplication.

Here the first steps id to out the equations in Standard Form.

And then here we need to determine which variable to eliminate.

Finally we have to Multiply the equations and solve.

And this process that allows us to multiply both sides of an equation by the same constant while still having an equivalent expression.

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Answer:

y = 5.2

Step-by-step explanation:

We can use the laws of sine to solve for this, the laws of sine state that:

[tex]\displaystyle{\dfrac{\sin A}{a}=\dfrac{\sin B}{b} = \dfrac{\sin C}{c} = 2R}[/tex]

In this diagram, we can rewrite the general form of sinA, sinB, sinC as:

[tex]\displaystyle{\dfrac{\sin F}{f}=\dfrac{\sin E}{e} = \dfrac{\sin D}{d} = 2R}[/tex]

We can use the equation of [tex]\diplaystyle{\dfrac{\sin F}{f} = \dfrac{\sin E}{e}}[/tex] to solve for the value of e. In this case, f is 7 since sides are opposite to measurements, and the opposite of measurement F is side length 7 units. The opposite side of measurement E is side length y units. Hence, f = 7 and e = y.

[tex]\diplaystyle{\dfrac{\sin F}{7} = \dfrac{\sin E}{y}}[/tex]

We know that the measurement E is 48°.

[tex]\diplaystyle{\dfrac{\sin F}{7} = \dfrac{\sin 48 \textdegree}{y}}[/tex]

We can find the measurement F by using the euclidean triangle theory that all sides will add up to 180°. This will result in 42° + 48° + F = 180. Solve the equation:

[tex]\displaystyle{90 \textdegree + F = 180 \textdegree}\\\\\displaystyle{F = 180 \textdegree - 90 \textdegree}\\\\\displaystyle{F = 90 \textdegree}[/tex]

Hence, the measurement of F will equal to 90°.  

[tex]\diplaystyle{\dfrac{\sin 90 \textdegree}{7} = \dfrac{\sin 48 \textdegree}{y}}[/tex]

Solve the equation for y by firstly multiplying both sides by 7y to clear off denominators.

[tex]\diplaystyle{\dfrac{\sin F}{7} \cdot 7y = \dfrac{\sin 48 \textdegree}{y} \cdot 7y}\\\\\displaystyle{y\sin 90 \textdegree = 7\sin 48 \textdegree}[/tex]

We know that sin90° = 1 so we will have [tex]\displaystyle{y=7\sin 48 \textdegree}[/tex]. Now input the value in a calculator since we cannot evaluate sin48° with an ordinary method.

When you put 7sin48° in a calculator, you will get 5.20201377... then you round the value to nearest tenth. Hence, y is 5.2 units.

solved, for  x^2+2x=35,   x=5 or x= -7

x^2+2x=35

Subtract 35 from both sides, and we get

x^2 +2x−35=0

To solve the equation, factor

x^2 +2x−35 using the formula  

x^2 +(a+b)x+ab=(x+a)(x+b).

To find a and b, set up a system to be solved.

a+b= 2

ab=−35

Since ab is negative, a and b have opposite signs. Since a+b is positive, the positive number has a greater absolute value than the negative. List all such integer pairs that give product −35.

−1, 35

−5, 7

Calculate the sum for each pair.

−1+35=34

−5+7=2

The solution is the pair that gives sum 2, which are

a=−5

b=7

Rewrite factored expression (x+a)(x+b) using the obtained values.

(x−5)(x+7)

To find equation solutions, solve x−5=0 and x+7=0.

x=5

x= −7

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The expression to represent each ATM transaction for the month is

10 + 2x.

Where x is the number of ATM transactions.

The total cost for 5 ATM transactions is $20.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

Monthly account fee = $10

Fee for ATM transaction = $2

The cost of x ATM transaction for a month.

= 10 + 2x

The total cost of 5 ATM transactions for a month.

= 10 + 2 x 5

= 10 + 10

= $20

Thus,

The expression is 10 + 2x.

The total cost is $20.

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Answer:The American colonies declared independence from British rule in 1776 because they were seeking to establish themselves as a sovereign nation, separate from British control and governance.

The colonies had grown increasingly discontent with British rule and policies, particularly those related to taxation and trade. The colonists felt that they were not being fairly represented in British government and that the laws and taxes imposed on them by Britain were unjust and burdensome.

In 1765, the British Parliament passed the Stamp Act, which placed a tax on printed materials such as newspapers, legal documents, and books. The colonists saw this as a violation of their rights as British citizens, as they had no representation in Parliament and felt that they were being taxed without their consent. This led to widespread protests and resistance to the tax, known as the Stamp Act riots.

In 1773, the British government passed the Tea Act, which granted a monopoly on the sale of tea to the British East India Company and imposed taxes on tea imported to the colonies. This led to the famous Boston Tea Party, in which colonists dumped 342 chests of tea into the Boston Harbor as a protest against the tax.

These and other events, such as the Quartering Acts and the Intolerable Acts, which were designed to punish the colonies for their resistance, further fueled the colonists' anger and resentment towards British rule.

In 1774, the First Continental Congress was held, in which representatives from the colonies met to discuss a unified response to British policies. The Congress adopted a Declaration of Rights and Grievances, in which they demanded that their rights as British citizens be respected and that they be given fair representation in Parliament.

In 1775, the American Revolutionary War broke out between the colonies and Great Britain. On July 4, 1776, the Second Continental Congress adopted the Declaration of Independence, which stated that the colonies were no longer subject to British rule

Step-by-step explanation:

Answer: The colonists joined together to fight Britain and gain independence.

Step-by-step explanation: Many colonists were angry because no one represented their needs in the British government. Colonists believed they did not have self-government. The British forced colonists to allow British soldiers to sleep and eat in their homes.

The distance between two points in a two-dimensional coordinate system represents the length of the shortest path between those two points. This length is also called the Euclidean distance, it is a measure of the straight-line distance between two points.

In other words, it is the length of the line segment connecting the two points. It is calculated using the distance formula which is derived from the Pythagorean theorem, it is calculated as the square root of the sum of the squares of the differences of the coordinates of the two points.

The distance between two points is always a positive value and it is a scalar quantity, it does not have any direction associated with it. It is a measure of the separation between two points, regardless of the direction of the line connecting them.

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The equation of a sphere if one of its diameters has endpoint (2, 1, 4) and (4, 3, 10) is (x+1)²+(y-1)²+(z+3)²= 11

Using the distance equation to find the distance of the diameter:

d² = (xf-xi)² + (yf-yi)² +(zf-zi)²

d² = (2-4)²+(1-3)²+(4-10)²

d²=4+4+36=44

d=sqrt(44)

The radius is half the diameter,  which is: r=d/2

r=sqrt(44)/2

To find the middle of the diameter we are using the center equation:

(xf-xi)/2, (yf-yi)/2, (zf-zi)/2

(2-4)/2, (1-3)/2, (4-10)/2

(-1,1,-3) Center

Now, using the equation for a sphere:

(x-xc)²+(y-yc)²+(z-zc)²=r²

(x+1)²+(y-1)²+(z+3)²=(sqrt(44)/2)²

(x+1)²+(y-1)²+(z+3)²=44/4

(x+1)²+(y-1)²+(z+3)²= 11

Thus, the equation of a sphere if one of its diameters has endpoint (2, 1, 4) and (4, 3, 10) is (x+1)²+(y-1)²+(z+3)²= 11

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d² = (xf-xi)² + (yf-yi)² +(zf-zi)²

d² = (2-4)²+(1-3)²+(4-10)²

d²=4+4+36=44

d=sqrt(44)

The radius is half the diameter,  which is: r=d/2

r=sqrt(44)/2

(xf-xi)/2, (yf-yi)/2, (zf-zi)/2

(2-4)/2, (1-3)/2, (4-10)/2

(-1,1,-3) Center

Based on the constructed difference table, the next term of this sequence 6, −2, −3, 7, 32, 76, . . . is 143.

In Mathematics, a sequence can be defined as a series of real and natural numbers in which each term or list of elements are ordered in a particular order and repetitions are allowed.

In Mathematics, a difference table can be defined as a type of table that is constructed by subtracting adjacent elements (entries) in a sequence, and then repeating the process with next those elements (entries).

Number            Difference from previous             Difference of differences

6

-2                               -8                                          

-3                               -1                                                7            

7                                 10                                              11            4

32                               25                                             15            4

76                              44                                               19            4

143                            67                                                23           4

First column: -2 - 6 = -8; -3 - (-2) = 1; ....... 143 - 76 = 67.

Second column: -1 - (-8) = 7; 10 - (-1) = 11; .... 67 - 44 = 23.

Third column: 11 - 7 = 4; 15 - 11 = 4 .... 23 - 19 = 4.

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Answer: C. 145 degrees

Step-by-step explanation:

Using the information given above, we are now supposed to figure out the measure of angle C. Considering the two angles are supplementary, or two angles with the sum of 180°, we can finally assume that the answer is C.

Angle B + Angle C = 180°

Angle B = 35°

180° - 35° = 145°

Answer: C. 145 degrees

When the preceding equation can be factored as (a+b)(ab) (a + b) (a b), it can be written as a^2 - b^2.

Example: Think about (y^2-100) (y^2+100). Each term in this sentence can be rendered as a square. Here, (y+10) and (y10) serve as the factors. Let's try to answer this now. A) Since any integer cannot be divided by 0 without the result being an undefined number, 0 cannot be a factor of any number other than 0.

B) One is the factor of every number because when one divides a number exactly, leaving no residue, the quotient is equal to the original number.

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This equation can be solved using the addition and subtraction property of equality, x + 7 = 13.

An equation is an expression containing an equal sign that states the equality of two mathematical expressions. It is a statement that two expressions have the same value. Equations often involve variables, which are placeholders for unknown numbers. An equation contains at least one operation, such as addition, subtraction, multiplication, or division. Equations are used in a variety of mathematical applications, from basic algebra to more advanced topics such as calculus.

Subtract 7 from both sides to isolate x.

x + 7 - 7 = 13 - 7

x = 6

Therefore, the two step equation that equals 13 is x + 7 = 13.

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