The slope of the curve ƒ(x) = x2 + 2 at x = 3 is _______.

6
9
3
0

The polar angle test is the equivalent of the vertical line test in polar coordinates.

Pοlar cοοrdinates are a system οf cοοrdinates used in mathematics tο describe the pοsitiοn οf a pοint in a plane using a distance and an angle.

In mathematics, the vertical line test is a tοοl used tο determine whether a relatiοn is a functiοn, based οn whether a vertical line intersects the graph οf the relatiοn at mοre than οne pοint.

When working with polar coordinates, a similar tool can be used to determine whether a curve is a function of angle, called the "polar angle test" or the "polar line test." The polar angle test states that if every ray emanating from the pole intersects the curve at most once, then the curve represents a function of angle.

In other words, if we consider the polar coordinates (r, θ) of points on the curve, and we draw a ray from the origin at angle θ for each possible value of θ, then the polar angle test says that if each ray intersects the curve at most once, then the curve represents a function of angle.

For example, the curve r = 1 + sin(θ) passes the polar angle test, since every ray from the origin intersects the curve at most once. On the other hand, the curve [tex]r^2[/tex] = cos(2θ) fails the polar angle test, since some rays intersect the curve at two points, such as the rays at θ = π/4 and θ = 5π/4. Therefore, the curve [tex]r^2[/tex] = cos(2θ) does not represent a function of angle in polar coordinates.

Therefore, The polar angle test is the equivalent of the vertical line test in polar coordinates.

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Answer:

y = (5e + 5sqrt(3) + 5) / (1 + (e + sqrt(3)) * e^(-t))

Step-by-step explanation:

To find the formula for a function in the form y = a/(1 + be^(-t)) with a y-intercept of 5 and an inflection point at t = 1, we can use the following steps:

Step 1: Find the value of a

Since the y-intercept of the function is 5, we know that the point (0, 5) lies on the graph of the function. So, we substitute t = 0 and y = 5 into the equation and solve for a:

5 = a / (1 + be^(0))

5 = a / (1 + b1)

5 = a / (1 + b)

a = 5*(1 + b)

Step 2: Find the value of b

To find the value of b, we use the fact that the function has an inflection point at t = 1. The inflection point is where the concavity of the function changes from upward to downward or vice versa. It is also the point where the second derivative of the function is zero or undefined.

The first derivative of the function is:

y' = -abe^(-t) / (1 + b*e^(-t))^2

The second derivative of the function is:

y'' = abe^(-t)(be^(-t) - 2) / (1 + b*e^(-t))^3

Setting t = 1 and y'' = 0, we get:

0 = abe^(-1)(be^(-1) - 2) / (1 + b*e^(-1))^3

Simplifying and using the value of a from Step 1, we get:

0 = 5be^(-1)(be^(-1) - 2) / (1 + b*e^(-1))^3

Multiplying both sides by (1 + be^(-1))^3 and simplifying, we get:

0 = 5b^2e^(-2) - 10b*e^(-1) + 1

Solving for b using the quadratic formula, we get:

b = (10e^(-1) ± sqrt(100e^(-2) - 451e^(-2))) / (25*e^(-2))

b = (2e + sqrt(4e^2 - e^2)) / (2e^(-1))

b = e + sqrt(3)

Step 3: Write the function in the form y = a/(1 + be^(-t))

Using the values of a and b from Steps 1 and 2, we get:

a = 5*(1 + b) = 5*(1 + e + sqrt(3)) = 5e + 5sqrt(3) + 5

b = e + sqrt(3)

So, the function in the form y = a/(1 + be^(-t)) with a y-intercept of 5 and an inflection point at t = 1 is:

y = (5e + 5sqrt(3) + 5) / (1 + (e + sqrt(3)) * e^(-t))

Answer:

[tex]12x + 19 + 22x - 9 = 180[/tex]

[tex]34x + 10 = 180[/tex]

[tex]34x = 170[/tex]

[tex]x = 5[/tex]

Answer:

x = 5

Step-by-step explanation:

The angles shown on the image are supplementary angles because they make a straight line.

The sum of supplementary angles would be equal to 180 so we can write the following equation to find the value of x:

12x + 19 + 22x - 9 = 180

34x + 10 = 180

34x = 170

x = 5

Answer:

[tex] \frac{5x {}^{2} }{2} - 8x + 6[/tex]

Step-by-step explanation:

That's your answer

To get the number of tiles needed, we must first determine the surface area of the box that must be covered. Surface area of the square box = top area + front area + rear area + left side area + right side area.

Top area = length x width = 6 cm x 4 cm = 24 cm2 Front and back area = height x breadth = 8 cm x 4 cm = 32 cm2 (for each) Left and right side area = height x length = 8 cm x 6 cm = 48 cm2 (for each) Surface area total = 24 + 2(32) + 2(48) = 232 cm2. To get the number of tiles needed, divide the total surface area by the area of each tile. each's area 5 mm × 5 mm = 0.25 cm2 tile Total surface area / Area of each tile = 232 / 0.25 = 928 tiles required As a result, Dmitri need 928 glass tiles to cover the top and sides of the box.

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The required inequality is  5x + 2 ≤ 12 and the solution is 2

Inequalities are mathematical expressions that compare two values or expressions, indicating that one is less than, greater than, or equal to the other.

In this problem, we used a linear inequality to represent the relationship between the amount of water used and the total cost of the water bill.

We then used algebra to rearrange the inequality and solve for x, which gave us the maximum amount of water that the family can use without going over budget.

Here we have

The cost of the water bill is made up of two components: a fixed monthly connection fee of $2, and a variable cost based on the amount of water used.

The variable cost is $5 per thousand gallons, and the total cost, C, in dollars can be expressed as:

C = 5x + 2

where x is the number of thousands of gallons used.

The problem states that the family has budgeted $12 for their water bill this month. Hence the cost must not exceed $12  

=> 5x + 2 ≤ 12

The above expression can be rewritten as follows

=>  5x + 2 ≤ 12  

Subtract 2 from both sides

=> 5x ≤ 10

Divide by 2 into both sides

=> x ≤ 2

Therefore,

The required inequality is  5x + 2 ≤ 12 and the solution is 2

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Answer:

x< 4 or x>4

Interval Notation  [tex]\left(-\infty \:,\:4\right)\cup \left(4,\:\infty \:\right)\\[/tex]

Step-by-step explanation:

its D

Answer: Can you take a closer picture?-

Step-by-step explanation:

Thanks!!

Answer:

Lines intersecting at a single point, One Solution

Step-by-step explanation:

1.) The first step to problems like these is to rewrite both of the equations in slope-intercept form. Slope-intercept form is a form in which y is isolated (By itself).

2.) Since we now know what to do, we can start with the first equation. You have to first get 7y by itself. To do this, you add 4x on both sides to move it to the other side. You would get 7y = 4x + 7. Now, divide by 7 on both sides to get y by itself. The final equation would be y = 4/7x + 1.

3.) Now, let's move on to the second equation. You first move x to the other side by subtracting by x on both sides. This would get you 8y = -x + 7. Finally, divide by 8 on both sides, which gets you y = -1/8x + 7/8.

4.) We got our two equations in slope-intercept form, so now we just have to do a comparison. If the slopes (The number next to the x) are the same, they are parallel lines. Parallel lines have no solutions. If the entire equation is the same, you have identical lines, which have infinite solutions. If the slopes are not the same, you have an intersection at a single point and one solution.

5.) By this criteria, since the slopes are not the same, you have an intersection at a single point. Since you have an intersection at a single point, that means you have a single solution.

The polynomial 6x² - 8x is factored to (3x - 4)(2x 1). The polynomial 6x² - 3x 8x - 4 is factored to (2x - 1)(3x 4).

When an algebraic expression is factored, it is written as the product of its factors. Any algebraic statement can be factored to remove a rational number, a negative number, or the greatest common factor (GCF).

The polynomial 6x² - 8x is factored to (3x - 4)(2x 1).

For this polynomial, first identify the greatest common factor (GCF).

This is GCF  2x.

Divide the polynomial by the GCF to get 3x - 4 and 2x 1.

These two terms can be multiplied by the factor (3x - 4) (2x 1).

The polynomial 6x² - 3x 8x - 4  (2x - 1)(3x 4) is factored. For this polynomial, we must first identify the greatest common factor (GCF).

In this case, the GCF is 2x - 1. To do this, the given polynomial is divided by the GCF, resulting in 3x 4 and 2x - 1.

These two terms can be multiplied to give the quotient (2x - 1)(3x4)

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There are 26.52 cubic feet of sand in the container and the container can hold an additional 14.56 cubic feet of sand.

What is the volume of a rectangular figure?

The volume of the figure is given by:

Volume = length × width × height

Given, a rectangular container 6.5 ft long, 3.2 ft wide, and 2 ft high is filled with sand to a depth of 1.3 ft.

The volume of the container is given by:

Volume = length × width × height = 6.5 ft × 3.2 ft × 2 ft = 41.6 cubic feet

Volume of sand = length × width × depth of sand = 6.5 ft × 3.2 ft × 1.3 ft = 26.52 cubic feet

Therefore, there are 26.52 cubic feet of sand in the container.

To calculate how much more sand the container can hold, you need to find the remaining volume of the container. The remaining depth of the container is:

Remaining depth = height - depth of sand = 2 ft - 1.3 ft = 0.7 ft

The remaining volume of the container is:

Remaining volume = length × width × remaining depth = 6.5 ft × 3.2 ft × 0.7 ft = 14.56 cubic feet

Therefore, the container can hold an additional 14.56 cubic feet of sand.

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If line m parallel to line n, the value of y in terms of x is y = 3x + 10.

Let's consider the given information: a diagram with line m parallel to line n, and an angle (3x + 10)°.
Since line m is parallel to line n, we can use the properties of parallel lines and their transversals to find the relationship between x and y.

In this case, we'll use the property called alternate interior angles, which states that if two parallel lines are cut by a transversal, their alternate interior angles are congruent.
Identify the alternate interior angles.
Let's say the angle (3x + 10)° is an alternate interior angle to angle y°.
Use the property of alternate interior angles.
Since alternate interior angles are congruent, we can set up the equation:
(3x + 10)° = y°
Express y in terms of x.
Now that we have the equation, we can express y in terms of x:
y = 3x + 10
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Question -

A diagram with line m parallel to line n is shown express the value of y in terms of x

Answer: Im trying to find out too

Step-by-step explantion i dont know

Answer:

Step-by-step explanation:

You want to know if the vertex is a minimum or maximum for functions ...

The sign of the leading coefficient is the sign of the function values when x has a large magnitude.

If the leading coefficient is positive, the parabola opens upward. Hence the vertex is a minimum. If the leading coefficient is negative, the parabola opens downward, so the vertex is a maximum.

The coefficient of the (x+1)² term is +1, so the vertex is a minimum.

The coefficient of the (x-2)² term is -1, so the vertex is a maximum.

The vertex οf f(x) is (-1, -2) is a minimum pοint, and the vertex οf g(x) is (2, 1) is a maximum pοint, based οn the cοefficients οf the squared terms in their respective vertex fοrms.

In mathematics, a vertex is a pοint where twο οr mοre lines, curves, οr edges meet. It can alsο refer tο the highest οr lοwest pοint οn a curve οr surface.

The vertex fοrm οf a quadratic functiοn is given by [tex]f(x) = a(x - h)^2 + k[/tex]where (h,k) is the vertex οf the parabοla. In this fοrm, the value οf "a" determines the shape and οrientatiοn οf the parabοla, and whether the vertex is a minimum οr a maximum.

Let's first rewrite the given functiοns in vertex fοrm:

[tex]f(x) = (x + 1)^2 - 2= 1(x - (-1))^2 - 2= 1(x - (-1))^2 + (-2)[/tex]

Therefore, the vertex of f(x) is (-1, -2), and since the coefficient of the squared term is positive, the parabola opens upward. This means that the vertex (-1, -2) is a minimum point.

[tex]g(x) = -(x-2)^2+1[/tex]

[tex]= -1(x - 2)^2 + 1[/tex]

Therefore, the vertex of g(x) is (2, 1), and since the coefficient of the squared term is negative, the parabola opens downward. This means that the vertex (2, 1) is a maximum point.

In summary, the vertex of f(x) is a minimum point, and the vertex of g(x) is a maximum point, based on the coefficients of the squared terms in their respective vertex forms.

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The lateral area οf a cylinder is calculated as L = 2rh, where r is the radius οf the base and h is the height οf the cylinder.

In several current branches οf geοmetry and tοpοlοgy, a cylinder can alsο be defined as an infinitely curvilinear surface. There is sοme ambiguity in terminοlοgy due tο the change in the basic meaning οf the wοrds—sοlid versus surface (as in ball and sphere).

By cοmparing sοlid cylinders and cylindrical surfaces, the twο ideas can be separated. Bοth οf these οr a mοre specialised οbject, the right circular cylinder, may be referred tο simply as "cylinder" in literary wοrks.

First, we must calculate the radius οf the cylinder, which is half its diameter.

Sο the radius is 14/2 = 7 meters.

Then, we can plug in the values we have intο the fοrmula: L = 2 × 3.14 × 7 × 13 ≈ 572 square meters.

Therefοre, the answer is A) 572 square meters, rοunded tο the nearest square meter.

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The expression (6x10⁻³)(1.4x10¹) have the solution in scientific notation as 8.4 x 10⁻². Very large or small numbers can be more easily understood when written in scientific notation.

A number can be expressed using scientific notation if it cannot be conveniently expressed in decimal form due to its size or shape, or if doing so would require writing out an abnormally long string of digits. In the UK, it is also referred to as standard form, standard index form, and standard form.

Although we are aware that complete numbers can never end, yet we are unable to put such massive sums of data on paper. The numbers that appear at the millions place after the decimal also have to be represented using a more straightforward approach. A small number of integers might be difficult to portray in their larger form because of this. We use scientific notation as a result.

Given:

= (6x1.4)(10⁻³x10¹) = (8.4x10¹)(10⁻²)

= 8.4x (10¹ x 10⁻²)

= 8.4x (10¹ x 10⁻²)

= 8.4 x 10⁻²

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Answer:

w=2

Step-by-step explanation:

1. You need to distribute your -3 and 6

2. After you do your distributing the numbers, the equation would be -15w+12=12w-42

3. You need to make x to the right or left and it look like 27w=54

4. Solve for x

5. The answer is w=2

Answer:

The two figures are not equal because 3/5 doesn't does not equal 6/9

Step-by-step explanation:

The numbers just aren't equal.

it would be 8000 because of the interception

Answer:

(a) Missing x:8 Missing y:23

Step-by-step explanation:

The length of XZ in adjoining figure  is 80/3 cm

Rectangular surfaces refer to flat two-dimensional shapes that have four straight sides and four right angles, where opposite sides are parallel and equal in length. They are called rectangular because they can be formed by taking a rectangle and flattening it out into a plane. Examples of rectangular surfaces include sheets of paper, computer screens, tabletops, and walls of rectangular rooms. The area of a rectangular surface is calculated by multiplying the length and width of the rectangle.

Let the length, width, and height of the rectangular prism be x, y, and z, respectively. Then, we have:

xy = 720 (since the area of the rectangular surfaces of the prism is 720 sq. cm)

XX' = 20

Let XY = 5k, XZ = 4k, and YZ = 3k (since XY : XZ: YZ = 5:3:4).

Using the Pythagorean theorem, we can find the value of k as follows:

XX²+ XZ²= ZZ²

20²+ (4k)²= (5k)²

400 + 16k² = 25k²

9k² = 400

k² = 400/9

k = 20/3

Therefore, XZ = 4k = 80/3 cm.

Hence, the length of XZ is 80/3 cm.

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The cost of producing 250 candidate posters is K400, and that 660 potential posters can be provided for K2500. This means that K400 + (x - 250) * K5 + K50 = K2500, and that K400 + (x - 250) * K5 + K50 = K2500. Finally, K400 + (x - 250) * K5 + K50 = K2500, and that K400 = 250.

The unitary method is a mathematical approach used to handle proportional and rate issues. We obtain the value of one unit of a quantity and then use it to get the value of any other quantity that is proportional to it using this procedure.

This generally recognized ease, preexisting variables, and any significant elements from the initial Diocesan customizable query may all be used to complete the task. If so, your may have another chance to interact alongside the item. Otherwise, all important influences on how algorithmic proof acts will be eliminated.  

Here,

(a)

=>  C(x) = K400 for 0  x  250.

=>  C(x) = K400 + (x - 250) * K5 + K50 for x > 250

(b)

(i)

=> C(1250) = K400 + (1250 - 250) * K5 + K50

=> K400 + 1000*K5 + K50

=>  K5400

In light of this, the cost to produce 1250 candidate posters is K5400.

We use the first portion of the charging function since 250  250 to determine the cost for printing 250 candidate posters. So:

=> C(250) = K400

Therefore, K400 is required to produce 250 candidate posters.

(ii)

=> K400 + (x - 250) * K5 + K50 = K2500

=>  (x - 250) * K5 = K2050

=>  x - 250 = 410

Therefore, 660 potential posters can be provided for K2500.

=>  C(x) = K400

=>  x = 250

250 applicant posters can be provided for K400 as a result.

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Answer:

B) There is one unique solution, (2, 1).

Attached is a graph of the system of equations.

Step-by-step explanation:

Given system of equations:

[tex]\begin{cases} 3x+6y=12 \\ 3x+2y=8\end{cases}[/tex]

Types of Solutions to a System of Equations

Using a graphing calculator, we can find the solution(s) to the given system of equations. To find the solution(s), we need to look at the point where the lines intersect; this is called the point of intersection.

Answer:

B: There is one unique solution, (2, 1)

Step-by-step explanation:

Took the test.

Answer: To add the polynomials, we simply add the coefficients of the same degree terms. This gives:

(3x^2 - x - 8) + (7x^2 - 9x - 1) = 10x^2 - 10x - 9

Therefore, the sum of the polynomials is 10x^2 - 10x - 9.

Step-by-step explanation:

55.96 more euros Sandy have if she made her trade on the day with the most favorable exchange rate than if she made her trade on the day with the least favorable exchange rate. Thus, option B is correct.

The number of units of a foreign currency that can be purchased with one unit of the domestic currency, or the opposite, is referred to as the exchange rate between two currencies.

Here, we have

Given: The exchange rate between non-fixed currencies continuously fluctuates. Sandy has $829.04 to convert into euros.

We have to find how many more euros would Sandy have if she made her trade on the day with the most favorable exchange rate than if she made her trade on the day with the least favorable exchange rate.

To get a favorable exchange rate we should do:

                                     USD × Exchange Rate

a) 829.04 × 0.7544  ⇒ Most favorable exchange rate (Tuesday)

= 625.42

b)  829.04 × 0.6869 × 0.94  ⇒ Least favorable exchange rate (Friday)

= 569.46

Difference of most favourable day and Least favorable day

=  625.42 - 569.46

= 55.96

Hence, 55.96 more euros Sandy have if she made her trade on the day with the most favorable exchange rate than if she made her trade on the day with the least favorable exchange rate. Thus, option B is correct.

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The vertices of the region are (7.2, -1.2) and (-2, 2).

In mathematics, an inequality is a statement that one quantity is less than or greater than another quantity. Inequalities are expressed using symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).

or example, "x > 5" means that x is greater than 5, while "y ≤ 10" means that y is less than or equal to 10.

Inequalities can be solved by isolating the variable on one side of the inequality symbol, just as with equations. However, there are some important differences between solving equations and solving inequalities. When solving an inequality, the direction of the inequality symbol may need to be reversed if both sides of the inequality are multiplied or divided by a negative number. In addition, the solution to an inequality is often expressed using interval notation or a number line.

To graph the system of inequalities, we first need to graph the boundary lines:

6y - x = 6 is equivalent to y = (1/6)x + 1, which has a y-intercept of 1 and a slope of 1/6.

y + 3x = -4 is equivalent to y = -3x - 4, which has a y-intercept of -4 and a slope of -3.

The region that satisfies both inequalities is the shaded region below the blue line and to the left of the red line.

To find the vertices of the region, we need to find the points where the two boundary lines intersect. Solving the system of equations:

y = (1/6)x + 1

y = -3x - 4

We get:

(7.2, -1.2) and (-2, 2)

Therefore, the vertices of the region are (7.2, -1.2) and (-2, 2).

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The vertices of the region are (7.2, -1.2) and (-2, 2).

An inequality in mathematics is a claim that one quantity is either less than or bigger than another. Inequalities are represented using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).

For instance, "y ≤10" denotes that y is less than or equal to 10, whereas "x > 5" indicates that x is larger than 5.

Just like with equations, an inequality can be resolved by isolating the variable on one side of the inequality symbol. But there are some significant distinctions between resolving inequalities and resolving equations. If both sides of an inequality are multiplied or divided by a negative number, the inequality symbol's direction may need to be reversed in order to solve the inequality. In addition, interval notation or a number line are frequently used to indicate the answer to an inequality.

We must first graph the boundary lines before we can graph the system of inequality:

6y - x = 6 is equivalent to y = (1/6)x + 1, which has a y-intercept of 1 and a slope of 1/6.

y + 3x = -4 is equivalent to y = -3x - 4, which has a y-intercept of -4 and a slope of -3.

The shaded area to the left of the red line, below the blue line, meets both inequalities.

We must identify the locations where the two boundary lines connect in order to determine the region's vertices. figuring out the equations in

a system:

y = (1/6)x + 1

y = -3x - 4

We get:

(7.2, -1.2) and (-2, 2)

The vertices of the region are (7.2, -1.2) and (-2, 2).

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The equations that do not have a solution are B, D, E, and G.

We can solve each equation to check if it has a solution or not:

A. 3x+1=2x+5

Subtracting 2x from both sides, we get x = 4. This equation has a solution.

B. 4(x-1)+2x=6x+5

Expanding the left side, we get 4x - 4 + 2x = 6x + 5. Simplifying, we get 6x - 4 = 6x + 5, which is not possible. This equation does not have a solution.

C. x+1=2x

Subtracting x from both sides, we get x = 1. This equation has a solution.

D. 10x+1=5x-6

Subtracting 5x from both sides, we get 5x + 1 = -6. This equation does not have a solution.

E. 3x+7=3x-6

Subtracting 3x from both sides, we get 7 = -6. This equation does not have a solution.

F. 4x=5x

Subtracting 4x from both sides, we get x = 0. This equation has a solution.

G. x=x+7

Subtracting x from both sides, we get 0 = 7. This equation does not have a solution.

H. x+4=2(x+4)

Expanding the right side, we get x + 4 = 2x + 8. Subtracting x from both sides, we get 4 = x + 8. Subtracting 8 from both sides, we get -4 = x. This equation has a solution.

Therefore, the equations that do not have a solution are B, D, E, and G.

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A grouping of at least three distinct rectangles, each with a 28-unit perimeter:2 and 12; 5 and 9; and 8 and 6.

To create the table, we start with the original rectangle with dimensions 10 and 4. To find other rectangles with the same perimeter, we can change the length and width while still ensuring that their sum is 14 (half of the original perimeter).

Since the perimeter of the rectangle is 28 units, we can use the formula:

Perimeter = 2(length + width)

where length and width are the dimensions of the rectangle.

If we solve for length, we get:

length = (Perimeter - 2*width) / 2

Using this formula, we can create a table of at least 3 different rectangles that have a perimeter of 28 units:

Width (units) Length (units)

2 12

5 9

8 6

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