A company that manufactures golf balls produces a new type of ball that is supposed to travel significantly farther than the company’s previous golf ball. To determine this, 40 new-style golf balls and 40 original-style golf balls are randomly selected from the company’s production line on a specific day. The balls are then placed in a bag and shaken. A golf pro then selects a ball and hits it using a driver. The distance the ball travels is then measured. The bag is shaken again, and the golf pro selects another ball and hits it with the same driver. He continues this procedure until all 80 of the golf balls are hit.

Which of the following is a benefit of having replication in the experiment?

Repeating the experiment will allow the company to compare the distances traveled by the new type of golf ball from both experiments.
Having a large number of each type of golf ball will allow for a good comparison of the differences in distances traveled for the two types of balls.
Having a large number of each type of golf ball will not allow for a good comparison of the differences in distances traveled for the two types of balls.
Replication is not needed in this experiment. The golf pro hitting one ball of each type is enough to determine if the difference in distance is due to the type of ball.

J would be the correct answer choice.

To put it simply, the number itself will always stay the same. You will always end up with either a negative or positive version of the number.

Answer:

Step-by-step explanation:

To maximize the volume of the box, we need to find the height that will maximize the volume of the box.

Let's start by finding an expression for the volume of the box. The box has dimensions of 14-2x by 10-2x by x, where x is the height of the box. The volume of the box is:

V(x) = (14-2x)(10-2x)(x)

Expanding this expression, we get:

V(x) = 4x^3 - 48x^2 + 140x

To find the value of x that maximizes this expression, we can take the derivative of V(x) with respect to x and set it equal to zero:

V'(x) = 12x^2 - 96x + 140 = 0

We can solve this quadratic equation using the quadratic formula:

x = [96 ± sqrt(96^2 - 4(12)(140))]/(2(12)) = [96 ± 16sqrt(6)]/24

We can simplify this to:

x = 4 ± sqrt(6)/3

Since the dimensions of the box must be positive, we can discard the negative solution:

x = 4 + sqrt(6)/3

So the height of the box that will give a maximum volume is approximately 5.61 inches (rounded to two decimal places).

The area of the composite shape is 155 yd². And the right option is A-155 yd².

A composite shape is any shape that is made up of two or more geometric shapes.

The area of the composite shape can be calculated using the formula below

Note: The composite shape in the question, is made up of a tripezium and a parallelogram.

Formula:

Where:

From the diagram,

Given:

Substitute these values into equation 1

Hence, the area is 155 yd².

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Answer:

x = 73°

Step-by-step explanation:

I added a photo of my notes

An inscribed angle is equal to half the arc on which it rests

So, according to the photo I've added (knowing that a full circle forms 360 degrees) we can write an equation:

2x + 214° = 360°

2x = 360° - 214°

2x = 146° / : 2

x = 73°

I not sure if this is the right answer, though...

According to the given question ∠1 = ∠5 ; Cοrrespοnding angles.

A figure that is created by twο rays οr lines that have the same endpοint is knοwn as an angle in plane geοmetry. Frοm the Latin wοrd "angulus," which means "cοrner," cοmes the English wοrd "angle." The vertex, which is the shared endpοint οf the twο rays, is referred tο as the side οf an angle.

There is nο requirement that an angle in the plane be in Euclidean space. If twο planes intersect in Euclidean οr anοther space tο fοrm an angle, that angle is said tο be a dihedral angle. "" is the symbοl used tο represent angles. Using the Greek letter,,, etc., οne can represent the angle between the twο rays.

Given,

The figure is attached

We have tο prοve that ∠1 = ∠5.

Cοrrespοnding angles;

When twο parallel lines are intersected by anοther line, cοmparable angles are the angles that are created in matching cοrners οr cοrrespοnding cοrners with the transversal (i.e. the transversal).

Here,

∠1 + ∠3 = 180° ; Vertical angles theοrem.

∠5 + ∠6 = 180° ; Linear pair theοrem.

∠1 + ∠3 = ∠5 + ∠6 ; 180° = 180° ; Bοth are supplementary angles.

∠3 = ∠6 ; Cοnsecutive interiοr angles

Nοw,

∠1 = ∠5 ; Cοrrespοnding angles.

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As per the figure provided the exact value of CE will be equivalent to 12.

According to the figure given in the question, Angles ABE and DBC are vertical angles and thus have the same measure. Since the given segment AE is parallel to a segment of CD, angles A and D are of the same distance by the alternate interior angle theorem. As a result, according to the angle-angle theorem, triangles ABE and DBC are equivalent, with vertex A corresponding to vertex B and vertex E to vertex D, respectively.

Hence, AB ÷ DB = EB ÷ CB

10 ÷ 5 = 8 ÷ CB

Since, CB=4 and CE= CB+BE

CE = 4 + 8

CE=12.

Therefore CE is equal to 12.

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Answer:

m∠ 1 = 165°

m∠ 2 = 15°

Step-by-step explanation:

We Know

∠1 + ∠2 = 180°

Let's solve

5x + (x - 18) = 180

5 + x - 18 = 180

6x - 18 = 180

6x = 198

x = 33

Now we put 33 in for x and solve for ∠1 and ∠2 !

m∠ 1 = 5x°

m∠ 1 = 5(33)

m∠ 1 = 165°

m∠ 2 = (x - 18)

m∠ 2 = 33 - 18

m∠ 2 = 15°

Answer:

m∠1 = 165°

m∠2 = 15°

Step-by-step explanation:

Note that, when the angle measurements are combined, the total measurement is 180°, based on the definition of a straight line.

It is given that m∠1 = 5x°, and m∠2 = (x - 18)°. Set the two angle measurements equal to the total measurement:

[tex]5x + (x - 18) = 180[/tex]

First, simplify. Combine like terms. Like terms are terms that share the same amount of the same variables:

[tex](5x + x) - 18 = 180\\(6x) - 18 = 180[/tex]

Next, isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS.

PEMDAS is the order of operations, and stands for:

Parenthesis

Exponents

Multiplications

Divisions

Additions

Subtractions

~

First, add 18 to both sides of the equation:

[tex]6x - 18 = 180\\6x - 18 (+18) = 180 (+18)\\6x = 180 + 18\\6x = 198\\[/tex]

Next, divide 6 from both sides of the equation:

[tex]6x = 198\\\frac{6x}{6} = \frac{198}{6}\\ x = \frac{198}{6} = 33[/tex]

x = 33. Next, plug in 33 for x for both measurements:

[tex]m\angle1 = 5x\\m\angle1 = 5 * (33)\\m\angle1 = 165\\\\m\angle2 = x - 18\\m\angle2 = (33) - 18\\m\angle2 = 15[/tex]

Check. Both, when combined, should make 180°

165 + 15 = 180

180 = 180 (True)

~

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the two ordered pairs that can be added to the given set and still have it represent a function are (3,4) and (-3,-2).

In order for a set of ordered pairs to represent a function, each input (x-coordinate) must be paired with exactly one output (y-coordinate).

Looking at the given set of ordered pairs, we can see that this condition is met:

The input -2 is paired with the output -1.

The input -1 is paired with the output 0.

The input 0 is paired with the output 1.

The input 1 is paired with the output 2.

The input 2 is paired with the output 3.

To add two more ordered pairs to this set and still have it represent a function, we need to make sure that the new pairs each have a unique x-coordinate that is not already in the given set.

For example, we could add the pairs (3,4) and (-3,-2):

The input 3 is paired with the output 4, which is not already in the original set.

The input -3 is paired with the output -2, which is not already in the original set.

Therefore, the two ordered pairs that can be added to the given set and still have it represent a function are (3,4) and (-3,-2).

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Answer:

Rounding to the nearest hundred thousand, the volume of the cone is approximately 19,400,000 cubic meters.

Answer:

Step-by-step explanation:

Rahul correctly solved the problem, because it changed the sign in places when there was a "-" in front of the bracket.

( x² + 3x - 5) - (4x - 5) = x² + 3x - 5 - 4x + 5 =  x²-x

-(a+b) = -a-b

-(a-b+c-a+c)= -a+b-c+a-c

Answer:

it is given that diagram shows part of the Wheel of Theodorus. graphics for exercise 20 page 428 geometry part (a). It is required to identify which ...

Step-by-step explanation:

The equation that represents the proportional relationship is [tex]y=\frac{1}{4}x[/tex].

What is proportion?

Proportion is a mathematical concept that describes the equality of two ratios. In other words, it is a statement that two ratios or fractions are equal.

For example, if we have two fractions [tex]\frac{a}{b}[/tex] and [tex]\frac{c}{d}[/tex], we can say that they are in proportion if:

=> [tex]\frac{a}{b}=\frac{c}{d}[/tex]

Now the ratio between x and y = 4:1

Here x = 28 then y = [tex]\frac{1}{4}\times28 = 7[/tex]

Then, x = 12 then [tex]y=\frac{1}{4}\times12 = 3[/tex]

We can see that the ratio between X and Y is always 4:1, which means that Y is one-fourth of X. We can write this as:

=>[tex]y=\frac{1}{4}x[/tex]

Therefore, the equation that represents the proportional relationship is:[tex]y=\frac{1}{4}x[/tex].

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Erica covered a total distance of 4,500 meters in 60 minutes (1 hour).

Distance traveled is the total length of the path covered by an object in motion over a given period of time. It is a scalar quantity.

Erica ran at a rate of 3.5 meters per second for 1/2 hour, which is 30/2 = 15 minutes.

During this time, she covered a distance of:

distance = rate x time

distance = 3.5 m/s x 900 s

distance = 3,150 meters

Next, Erica walked at a rate of 1.5 meters per second for 1/2 hour, which is also 15 minutes.

During this time, she covered a distance of:

distance = rate x time

distance = 1.5 m/s x 900 s

distance = 1,350 meters

Adding the distances Erica ran and walked, we get:

total distance = 3,150 meters + 1,350 meters

total distance = 4,500 meters

Therefore, Erica covered a total distance of 4,500 meters in 60 minutes (1 hour).

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Answer:12

Step-by-step explanation:

have a good day

A test estimate of 16,900 is required to appraise the extent of the populace having a particular genetic marker.

To decide the test estimate required to assess the extent of the populace with a characterized edge of mistake and certainty level, we will utilize the equation:

[tex]n = (z^2 * p * (1 - p)) / (E^2)[/tex]

Or:

n is the sample size

z is the z-score related to the level of confidence

p is the estimated ratio of the population to the genetic marker

1 - p is the estimated proportion of the population with no genetic markers

E is the desired margin of error (in decimal)

By substituting the values ​​given in the formula, we get:

[tex]n = (z^2 * p * (1 - p)) / (E^2)[/tex]

[tex]n = (2.33^2 * 0.5 * 0.5) / (0,01^2)[/tex]

[tex]n = 1.69 * 10^4[/tex]

therefore, a test estimate of 16,900 is required to appraise the extent of the populace having a particular hereditary marker with an edge of mistake of 1% and a certainty level of 98. %, expecting 50% of the populace has the marker.

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The solution for differential equation is the negative square root, since y(π) = -5. Thus, the final solution is;  y = 3 - √(9 - 6 tan(x))

A differential equation is a mathematical equation that relates a function or a set of functions with their derivatives.

Given differential equation;  dy/dx = sec²(x) (2+y)²

separate the variables and integrate both sides:

∫ 1/(2+y)² dy = ∫ sec²(x) dx

Using the substitution u = 2+y, du/dy = 1, we can rewrite the left-hand side as:

∫ 1/u² du = -1/u + C₁

Similarly, we can integrate the right-hand side using the identity ∫ sec²(x) dx = tan(x) + C₂, Substituting these expressions back into the original equation, we get:

-1/(2+y) + C₁ = tan(x) + C₂

To determine the values of C₁ and C₂, we use the initial condition y(π) = -5, which implies x = π. Substituting these values, we get:

-1/(2-5) + C₁ = tan(π) + C₂

-1/(-3) + C₁ = 0 + C₂

C₁ = C₂ + 1/3

putting the value of C₁ and C₂ into the previous expression, So,

-1/(2+y) + C₁ = tan(x) + C₁ - 1/3

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

Multiplying both sides by (2+y)², we get:

-(2+y) = (2+y)² tan(x) - (2+y)²/3

Simplifying and solving for y, we get:

y² - 6 - 6 tan(x) = 0

Solve it for y by using the quadratic formula,

y = 3 ± √(9 - 6 tan(x))

Therefore, the solution to the differential equation dy/dx = sec²(x) (2+y)² with the initial condition y(π) = -5 is:    y = 3 ± √(9 - 6 tan(x))

We choose the negative square root, since y(π) = -5. Thus, the final solution is:     y = 3 - √(9 - 6 tan(x))

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Answer:

3 more girls than boys.

Step-by-step explanation:

Let's change 4:5 to 4x(boys) and 5x(girls).

Since there are 27 students, the equation would be:

4x+5x=27

Then, you simplify the equation:

9x=27

x=3

We are not finished yet!

Since there is 4x boys and 5x girls,

do 4 times 3 which gets you 12 boys

and 5 times 3 which gets you 15 girls.

15 girls minus 12 boys is 3 more girls than boys!

Hope this helps :)

(a) According to the null hypothesis, there is evidence that the average telephone survey lasts less than 15 minutes and that the premium rate is not appropriate. According to the alternative hypothesis, there is evidence that the premium rate is appropriate and that the average telephone survey lasts longer than 15 minutes.

(b) The value of the test statistic is 2.959.

(a)  Based on the available data, Fowle Marketing Research Inc. is requesting a basic fee from a customer under the presumption that the average telephone survey will last 15 minutes or less. The following are the alternative and null hypotheses:

Write down the null hypothesis.

Null hypothesis:

H₀ : μ ≤ 15

Alternative hypothesis:

H₁ : μ > 15

(b) The test statistic's value is as follows:

Given the information, μ = 11, σ = 4 and n=35.

x' = Σx/n

x' = (17 + 11 + 12 + 23 + 20 + 23 + 15 + 16 + 23 + 22 + 18 + 23 + 25 + 14 + 12 + 12 + 20 + 18 + 12 + 19 + 11 + 11 + 20 + 21 + 11 + 18 + 14 + 13 + 13 + 19 + 16 + 10 + 22 + 18 + 23)/35

x' = 595/35

x' = 17

Therefore,

z = (x' - μ)/(σ/√n)

z = (17 - 15)/(4/√35)

z = 2/(4/5.92)

z = 2/0.676

z = 2.959

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The complete question is:

Fowle Marketing Research Inc., bases charges to a client on the assumption that telephone surveys can be completed in a mean time of 15 minutes or less. If a longer mean survey time is necessary, a premium rate is charged. A sample of 35 surveys provided the following survey times in minutes:

17; 11; 12; 23; 20; 23; 15; 16; 23; 22; 18; 23; 25; 14; 12; 12; 20; 18; 12; 19; 11; 11; 20; 21; 11; 18; 14; 13; 13; 19; 16; 10; 22; 18; 23.

Based upon past studies, the population standard deviation is assumed known with s = 4 minutes. Is the premium rate justified?

a. Formulate the null and alternative hypotheses for this application.

b. Compute the value of the test statistic.

The probability that at most 50 people out of 2400 who have not taken drugs will test positive on a drug test that is accurate 98% of the time is approximately 0.109

To calculate the probability, we need to first determine the parameters of the binomial distribution. Let's define success as testing negative on the drug test, since that is what we want to happen. Therefore, the probability of success is 0.98, and the probability of failure (testing positive) is 0.02.

Next, we need to determine the number of trials (n) and the number of successes (x) we are interested in. In this case, n = 2400 (the number of people taking the test) and x = 0, 1, 2, ..., 50 (the number of people who test positive).

Using the binomial distribution formula, we can calculate the probability of getting at most 50 people testing positive as follows:

P(X ≤ 50) = Σ(i=0 to 50) [(n choose i) * p^i * (1-p)^(n-i)]

where (n choose i) = n! / (i! * (n-i)!) is the binomial coefficient.

Plugging in the values, we get:

P(X ≤ 50) = Σ(i=0 to 50) [(2400 choose i) * 0.98^i * 0.02^(2400-i)]

Using a computer program or calculator, we can evaluate this sum to get P(X ≤ 50) ≈ 0.109.

Therefore, the probability that at most 50 people out of 2400 who have not taken drugs will test positive on a drug test that is accurate 98% of the time is approximately 0.109.

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It will take both machines approximately 1.71 hours or 1 hour and 42 minutes to produce widgets together.

To solve this problem, we can use the formula:

1 / time taken by machine 1 + 1 / time taken by machine 2 = 1 / time taken by both machines

Let's denote the time taken by the old machine as x hours and the time taken by the new machine as y hours.

From the problem statement, we know that the old machine can produce widgets in 3 hours, so we have:

1 / x = 1 / 3

Solving for x, we get:

x = 3

Similarly, we know that the new machine can produce widgets in 4 hours, so we have:

1 / y = 1 / 4

Solving for y, we get:

y = 4

Now, we can plug in the values of x and y into the formula and solve for the time taken by both machines:

1 / 3 + 1 / 4 = 1 / t

Multiplying both sides by 12t, we get:

4t + 3t = 12

7t = 12

t = 12 / 7

Therefore, it will take both machines approximately 1.71 hours or 1 hour and 42 minutes to produce widgets together.

In conclusion, we used the formula for the combined work rate of two machines to calculate the time taken by both machines to produce widgets. We first found the individual work rates of the old and new machines and then substituted those values into the formula to solve for the time taken by both machines working together.

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The proper way to move on coordinate surface from point A's location to point b is to move 6 units to the right and 1 unit to the up. The correct option is a.

What does the word "coordinate" mean in mathematics?

Coordinates are a pair of numbers (also known as Cartesian coordinates), or occasionally a letter and an integer, that identify a particular point on a grid,also known as a coordinate system. The [tex]x[/tex] -axis (horizontal) and [tex]y[/tex] -axis are the two vectors on a coordinate plane, which has four quadrants. (vertical).

On the coordinate surface, we can use the following methods to move from Point A to Point B:

Point A, which is two units to the right of the Centre and nine units up, is a good place to start.

Point C, which is situated at Move five floors to the toward the right and five units up to get there. (5, 5).

Move three units to your right and three units up from Point C to Point B, which is situate

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Therefore, the width of the rectangle is 10.

So, the answer is (O) 10.

To find the width of a rectangle, you will need to have at least one of the following pieces of information:

Length and area: If you know the length and area of the rectangle, you can use the formula for area (A = length x width) and solve for the width (W = A/length).

Perimeter and length: If you know the perimeter and length of the rectangle, you can use the formula for perimeter (P = 2 x length + 2 x width) and solve for the width (W = (P - 2 x length)/2).

Diagonal and length: If you know the diagonal and length of the rectangle, you can use the Pythagorean theorem ([tex]a^2 + b^2 = c^2[/tex]) to solve for the width. [tex](W = \sqrt(c^2 - length^2)),[/tex] where c is the diagonal and a and b are the length and width (in some order).

Ratio of length to width: If you know the ratio of length to width, you can use this ratio to solve for the width. For example, if the ratio of length to width is 3:2, and the length is 15, then the width would be 10 (because 15/3 = 5 and 5 x 2 = 10).

Measurements: If you have physical access to the rectangle, you can measure the width directly using a ruler or tape measure.

Let's assume the length of the rectangle as "L" and the width as "W".

From the given information, we know that:

The area of the rectangle is 40, so we have the equation:

[tex]L x W = 40[/tex]

The width is 2 less than 3 times the length, so we have the equation:

[tex]W = 3L - 2[/tex]

Now we can substitute the second equation into the first equation and get:

[tex]L x (3L - 2) = 40[/tex]

Expanding the left-hand side, we have:

3L^2 - 2L - 40 = 0

Dividing both sides by 1, we get:

[tex]3L^2 - 2L - 40 = 0[/tex]

Using the quadratic formula, we get:

[tex]L = 4 or L = -10/3[/tex]

Since the length of a rectangle cannot be negative, we take L = 4.

Then, we can use the equation W = 3L - 2 to find the width:

[tex]W = 3 x 4 - 2 = 10[/tex]

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For part A the length of the perpendicular is approximately 156.18. Length of GH, the area of the rectangle is approximately 12336.22 square units.

FOR Part B the length of the perpendicular is approximately 182.48.

the area of the triangle is approximately 8954.96 square units.

How to solve the question?

To find the perpendicular of a right triangle, we need to use the Pythagorean theorem, which states that the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

Let the length of the perpendicular be h, and the length of the other leg of the triangle be b. Then, we can set up the equation:

175²= h²+ 79²

Simplifying, we get:

30,625 = h²+ 6,241

Subtracting 6,241 from both sides, we get:

24,384 = h²

Taking the square root of both sides, we get:

h ≈ 156.18

Therefore, the length of the perpendicular is approximately 156.18.

The area of a rectangle is given by the formula:

Area = length × width

Substituting the given values, we get:

Area = 156.18 × 79

Area = 12336.22 square units

Therefore, the area of the rectangle is approximately 12336.22 square units.

To find the perpendicular, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In this case, we know the length of the hypotenuse and one of the other sides, so we can solve for the length of the perpendicular.

Using the Pythagorean theorem:

h²= b²+ p²

where h is the length of the hypotenuse, b is the length of the base, and p is the length of the perpendicular.

Substituting the given values:

207²= 98²+ p²

Simplifying:

42849 = 9604 + p²

Subtracting 9604 from both sides:

33245 = p²

Taking the square root of both sides (since p is a length, we only take the positive square root):

p = sqrt(33245)

p ≈ 182.48

Therefore, the length of the perpendicular is approximately 182.48.

To find the area of the triangle, we can use the formula:

Area = (1/2) x base x height

Substituting the given values:

Area = (1/2) x 98 x 182.48

Simplifying:

Area = 0.5 x 98 x 182.48

Area ≈ 8954.96

Therefore, the area of the triangle is approximately 8954.96 square units.

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Answer:

V = 4,423.4

SA = 1,508.0

Step-by-step explanation:

r = d/2
V = πr²h

(3.14)(8)²(22) = 4,423.362456

SA = 2πr(h + r)

2(3.14)(8)(22 + 8) = 1,507.964474

The rate of change in the area of the circle is 22 cm²/s when the radius of the circle is increasing at a rate of 0.5 cm/s.

It is half of the diameter of the circle or sphere. The radius is commonly denoted by the letter "r".

The area of a circle is given by the formula A = πr², where A is the area and r is the radius.

If the radius is increasing at a rate of 0.5 cm, then the rate of change of the radius with respect to time is dr/dt = 0.5 cm/s.

Using the chain rule of differentiation, we can find the rate of change of the area with respect to time:

dA/dt = dA/dr * dr/dt

By varying the area formula, we can determine dA/dr:

dA/dr = 2πr

Plugging in the values of r and dr/dt, we get:

dA/dt = (2πr)(0.5) = πr

At the initial radius of 7cm, the rate of change in the area is:

dA/dt = π(7) = 22/7 * 7 = 22 cm²/s

Therefore, the rate of change in the area of the circle is 22 cm²/s when the radius of the circle is increasing at a rate of 0.5 cm/s.

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The value of x for the given polynomial that are similar in nature through which the relation is satisfied is x = 21.

What about similar character?

In mathematics, similarity refers to the property of having the same shape but not necessarily the same size. Two geometric figures are said to be similar if they have the same shape and their corresponding angles are congruent, and the ratio of their corresponding side lengths is constant. This constant ratio is called the scale factor of the similarity.

For example, two triangles are similar if their corresponding angles are congruent, and their corresponding sides are in proportion. That is, if we take one side of the first triangle and divide it by the corresponding side of the second triangle, we get the same ratio as if we took another pair of corresponding sides and divided them. This ratio is the scale factor of the similarity.

According to the given information:

Similar polygons have congruent corresponding angles and proportionate corresponding sides.

To find the lengths of another polygon that is comparable, multiply or divide a polygon's side lengths by a scale factor.

Here, we use the similarity operation  in which the ratio of side are equal in nature.

⇒[tex]\frac{x-5}{12} = \frac{18}{13.5} = \frac{20}{15}[/tex]

⇒ [tex]\frac{x-5}{12} = \frac{4}{3} = \frac{4}{3}[/tex]

⇒ [tex]\frac{x-5}{12} = \frac{4}{3}[/tex]

⇒ [tex]x = 21[/tex]

So, the value of x for which the given relation is satisfied is x  = 21.

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Answer:

Step-by-step explanation:

A

The roots of the given quadratic equation are x = -3± √27

Quadratics can be defined as a polynomial equation of a second degree, which implies that it comprises a minimum of one term that is squared. It is also called quadratic equations.

Given is a quadratic equation x² + 6x = 18,

x² + 6x = 18

Comparing the equation with the standard form,

b = 6, c = -18

(x + b/2)² = -(c - b²/4)

So,

(x+6/2)² = -(-18-6²/4)

(x+3)² = -(-18-9)

(x+3)² = 27

x+3 = ±√27

x = ±√27-3

x = -3± √27

Hence, the roots of the given quadratic equation are x = -3± √27

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Hi! I can help you with that.

Each package weighs 30 oz. So, if Paul buys 4 packages, he will be buying:

30 oz x 4 = 120 oz

To convert this into pounds, we need to divide the number of ounces by 16 (as there are 16 ounces in 1 pound). So,

120 oz ÷ 16 = 7.5 pounds

Therefore, Paul will be buying 7.5 pounds of hot dogs for his camping trip.

No, the square piece of plywood cannot enter through the door because its diagonal is longer than the height of the door.

To see if the plywood would enter through the door, we can use the Pythagorean theorem to find the length of the diagonal of the plywood:

diagonal^2 = side1^2 + side2^2

So, we have

diagonal^2 = 2.2^2 + 2.2^2

diagonal^2 = 4.84 + 4.84

diagonal^2 = 9.68

Take the square root

diagonal ≈ 3.11

Therefore, the diagonal of the plywood is approximately 3.11 meters.

Since the height of the door is only 3 meters, the plywood is too big to fit through the door. So we can conclude that the plywood cannot enter through the door.

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