Estimate the instantaneous rate of change at x=1

According to simple interest, Patricio will have $537.50 in his savings account after 5 years.

To calculate the amount of money Patricio will have in his savings account after 5 years, we can use the formula for simple interest, which is I = prt. "I" stands for the amount of interest earned, "p" stands for the principal amount deposited, "r" stands for the interest rate per year (as a decimal), and "t" stands for the time period in years.

In this case, the principal amount (p) is $500, the interest rate (r) is 1.5% or 0.015 as a decimal, and the time period (t) is 5 years. Using the formula I = prt, we can calculate the amount of interest earned over 5 years:

I = prt

I = $500 x 0.015 x 5

I = $37.50

So, Patricio will earn $37.50 in simple interest over 5 years. To find out the total amount of money he will have in his savings account after 5 years, we simply add the interest earned to the principal amount:

Total amount = Principal amount + Interest earned

Total amount = $500 + $37.50

Total amount = $537.50

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The solution yn+1 is always less than or equal to yn, which means that the solution is stable for a < 0. Therefore the solution is of trapezoidal method.

The model problem (4.3) with a real constant a < 0 is given by:
[tex]y' = ay, y(0) = 1[/tex]
The trapezoidal method is given by:
[tex]yn+1 = yn + (h/2)(f(tn, yn) + f(tn+1, yn+1))[/tex]
where h is the step size, tn and yn are the current time and solution values, and  [tex]f(t, y) = ay[/tex]  is the right-hand side of the model problem.

To show that the solution of the trapezoidal method is stable for a < 0, we can substitute the right-hand side into the trapezoidal method and solve for yn+1:
[tex]yn+1 = yn + (h/2)(ayn + ayn+1)yn+1 - (h/2)ayn+1 = yn + (h/2)aynyn+1(1 - (h/2)a) = yn(1 + (h/2)a)yn+1 = yn(1 + (h/2)a)/(1 - (h/2)a)[/tex]

Since a < 0, the denominator [tex](1 - (h/2)a)[/tex] is always positive, and the numerator [tex](1 + (h/2)a)[/tex] is always less than or equal to 1. Therefore, the solution yn+1 is always less than or equal to yn, which means that the solution is stable for a < 0.

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We can sketch the graph of 2x²+4x.

We can start by dissecting the equation and identifying its main components before drawing the graph of 2x²+4x.

The formula reads as y = ax² + bx + c, where a = 2, b = 4, and c = 0. It is a quadratic function.

The upward opening of the graph is indicated by the positive coefficient of x2 (a). By applying the formula -b/2a, which in this case equals -4/4 = -1, one can determine the vertex of the parabola.

Hence, the parabola's vertex is located at (-1,0).

By setting y = 0 and solving for x, we may get the graph's x-intercepts:

0 = 2x² + 4x

0 = 2x(x + 2)

Hence the x-intercepts are at x = 0 and x = -2.

We can set x = 0 to determine the graph's y-intercept:

y = 2(0)² + 4(0) = 0

The y-intercept is therefore at (0,0).

Using this knowledge, we can draw the 2x²+4x graph as follows:

Vertex located at (-1,0).

x = 0 and x = -2 are the two x-intercepts.

Y-intercept is located at (0,0).

The graph has a "U"-shaped opening that faces upward.

The graph's basic drawing is shown below:

|

   |

   |

   |

   |   *    *

   |        |

   |        |

   |        |

_____|______________

   -2     0     2

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The distance around the track along the inside edge of each lane is 614.44 yd for lane 1, 634.88 yd for lane 2, 655.32 yd for lane 3, 675.76 yd for lane 4, 696.20 yd for lane 5, 716.64 yd for lane 6, 737.08 yd for lane 7 and 757.52 yd for lane 8.

The distance around the track along the inside edge of each lane can be calculated using the following formula:

Distance = (2 × 44 yd) + (2 × π × radius)

where the radius is equal to 44 yd + (lane number × 10 yd).

For lane 1, the radius is equal to 44 yd + (1 × 10 yd) = 54 yd. Therefore, the distance around the track along the inside edge of lane 1 is (2 × 44 yd) + (2 × π × 54 yd) = 614.44 yd.

The same calculation can be done for the other lanes, with the radius increasing by 10 yd for each lane. The radius for lane 2 is 64 yd, for lane 3 is 74 yd, for lane 4 is 84 yd, for lane 5 is 94 yd, for lane 6 is 104 yd, for lane 7 is 114 yd and for lane 8 is 124 yd.

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Answer:

Step-by-step explanation:

To find this answer you need to find the vertex of the parabola. Do this by using the formulas for h and k as follows:

[tex]h=\frac{-b}{a}[/tex]    and    [tex]k=c-\frac{b^2}{4a}[/tex], where a, b, and c come from the quadratic equation.

Filling in for h:

[tex]h=\frac{10}{2(.2)}=25[/tex]

Filling in for k:

[tex]k=650-\frac{100}{4(.2)} =525[/tex]

Thus, the coordinates for the vertex are (25, 525). The h value interprets the number of bowls that should be produced to minimize the cost of production (25) and the minimum production cost is the k value (525).

The survey questiοn is What is yοur favοrite brand οf sοda?

In mathematics, a survey is a technique fοr gathering data that invοlves pοsing a series οf questiοns tο participants in οrder tο learn mοre abοut their attitudes and behaviοrs. It is the mοst typical and affοrdable methοd οf data cοllectiοn. A sample size can change depending οn the situatiοn.

What dο yοu like and dοn’t like abοut the CEO? It is nοt a survey questiοn. In a survey questiοn, there shοuld be an οbject, nοt a  persοn.

Dο yοu οwn a car οr a truck? It is nοt a survey questiοn.  The survey is dοne οn a particular prοduct. But in the questiοn, nο band is mentiοned.

Hοw much credit card debt dο yοu οwe? It is nοt a survey questiοn. It is a persοnal questiοn.

What is yοur favοrite brand οf sοda? It is a survey questiοn. The survey is dοne οn a sοda brand.

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Using the remainder theorem, the quotient of (x³ + 3x² + 5x + 3) divided by (x + 1) is Ox²+2x+3

What exactly is the remainder theorem?

The Remainder Theorem is an approach to Euclidean polynomial division. According to this theorem, dividing a polynomial P(x) by a factor (x - a), which is not an element of the polynomial, yields a smaller polynomial and a remainder.

Given Data

x³ +3x² +5x +3 / x+1 = x² +2x +3

x³ +x²

------------

0   2x² +5x

  2x² +2x

----------------

    0      3x +3

            3x+3

----------------

            0    0  

The quotient of (x³ + 3x² + 5x + 3) divided by (x + 1) is x² +2x+3 using remainder theorem.

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Therefore , the solution of the given problem of equation comes out to be the first plant makes 5x + 22 more items per day.

Mathematical formulas frequently employ same variable word to guarantee agreement between two claims. Many academic numbers are shown to be equal using mathematical expression, also known as assertions. In this case, the normalise method adds b + 6 to employ the example of y + 6 rather than splitting 12 into two parts. It is possible to determine the length of the line and the quantity of connections between each sign's constituents. The significance of a symbol usually contradicts itself.

Here,

The first plant cranks out 8x + 15 items every day, while the second cranks out 3x - 7 items every day. By deducting the daily output of the second plant from the daily output of the first plant, we can determine how many more items the first plant creates than the second plant:

=> (8x + 15) - (3x - 7) (3x - 7)

If we condense this phrase, we get:

=> 8x + 15 - 3x + 7

Combining related words gives us:

=> 5x + 22

Therefore, compared to the second plant, the first plant makes 5x + 22 more items per day.

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Answer:

12x - 8x^2 - 8.

Step-by-step explanation:

To find the sum of the two expressions, we add the corresponding coefficients of each term:

(x - 5x^2 - 12) + (4 + 11x - 3x^2)

= x + 11x - 5x^2 - 3x^2 - 12 + 4

= 12x - 8x^2 - 8

Answer: To solve this problem, we need to graph the three inequalities and find the overlapping region.

First, let's graph the inequality y ≥ x - 4. We can start by graphing the line y = x - 4, which has a y-intercept of -4 and a slope of 1.

 |

10|     + +

 |   +     +

 | +        

 |+          

 |          

 |          

 |        +

 |      +

 |    +

 |  +

0|-----------------

  0  1  2  3  4  5

Since we want the region where y is greater than or equal to x - 4, we shade the area above the line.

Next, let's graph the inequality y ≤ 10. This is a horizontal line passing through y = 10.

 |

10|     +----+

 |   +        +

 | +            

 |+            

 |              

 |              

 |              

 |              

 |              

 |              

0|-----------------

  0  1  2  3  4  5

Since we want the region where y is less than or equal to 10, we shade the area below the line.

Finally, let's graph the inequality 5 ≤ x + y. This is a line with a y-intercept of 5 and a slope of -1.

 |

10|     +----+

 |   +   |  +

 | +     |    

 |+      |    

 |       |    

 |       |    

 |       |    

 |        |  

 |        |  

 |         +

0|-----------------

  0  1  2  3  4  5

Since we want the region where x + y is greater than or equal to 5, we shade the area above the line.

Now we can find the overlapping region of the three shaded areas:

 |

10|     +----+

 |   +   |  +

 | +     |    

 |+      |    

 |       |    

 |       |    

 |      +    

 |    +      

 |  +        

 |+          

0|-----------------

  0  1  2  3  4  5

The region is a triangle with vertices at (0, 4), (1, 5), and (5, 0).

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

Area = (1/2) * base * height

The base of the triangle is the distance between the points (0, 4) and (5, 0), which is 5.

The height of the triangle is the distance between the point (1, 5) and the line 5 = x + y. We can find the equation of the line perpendicular to 5 = x + y and passing through (1, 5). This line has a slope of 1 and passes through (1, 5), so its equation is y = x + 4. We can find the intersection of this line and the line 5 = x + y by solving the system of equations:

y = x + 4

y = 5 - x

Substituting y = x + 4 into the second equation, we get:

x + 4 = 5 - x

Solving for x, we get:

x = 1

Step-by-step explanation:

The average rate of change is the slope of the line between the points

     ( -4, f(-4) ) and ( 4, f(4) )

f(-4) = 16 + 40 + 5 = 61

f(4) = 16 - 40 + 5 = 19

The slope (AKA average rate of change) is then

    [tex]m =\dfrac{19-61}{4-(-4)}=\dfrac{-42}{8} = -\dfrac{21}{4}[/tex]

Answer:

A. 63

Step-by-step explanation:

in triangle TRS, it is an isosceles triangle with TS congruent to RS. So, angle T congruent to angle r. So, the angle is 63.

Answer:

The fraction that looks the same even if you turn it upside down is 6/9 or six-ninths.

Step-by-step explanation:

If a person tosses a coin 9 times, there are 84 ways in which a person can get 6 heads in 9 coin tosses.

When a person tosses a coin 9 times, there are two possible outcomes for each toss - either heads or tails. Hence, there are a total of 2^9 = 512 possible outcomes for 9 coin tosses.

To find the number of ways in which a person can get 6 heads, we need to consider the number of ways in which 6 heads can occur in the 9 coin tosses, while the remaining 3 tosses can result in tails. The number of ways in which 6 heads can occur in 9 tosses is given by the binomial coefficient C(9,6), which is equal to 84.

Hence, there are 84 ways in which a person can get 6 heads in 9 coin tosses. This can be calculated using the formula for binomial coefficients:

C(9,6) = 9!/(6!3!) = (987)/(321) = 84

Alternatively, we can also calculate this using a combination of multiplication and addition. We can choose any 6 out of the 9 coin tosses to result in heads, which can be done in C(9,6) ways.

For each of these ways, the remaining 3 coin tosses will result in tails, which can occur in only one way. Hence, the total number of ways in which a person can get 6 heads in 9 coin tosses is given by:

Number of ways = C(9,6) * 1 = 84 * 1 = 84

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a: The cutοff time fοr the fastest 5% οf athletes in the men's grοup is apprοximately 3447.68 secοnds.

b: The cutοff time fοr the slοwest 10% οf athletes in the grοup apprοximately 4253.68 secοnds.

The nοrmal distributiοn is a cοntinuοus prοbability distributiοn that is widely used in statistics tο describe real-wοrld phenοmena that tend tο cluster arοund a central value.

a) Tο find the cutοff time fοr the fastest 5% οf athletes in the men's grοup, we need tο find the value οf x such that P(X < x) = 0.05, where X is a nοrmally distributed randοm variable with mean μ = 4370 and standard deviatiοn σ = 585.

Using a standard nοrmal distributiοn table οr a calculatοr, we can find that the z-scοre cοrrespοnding tο the 5th percentile is apprοximately -1.645. Therefοre, we can write:

(x - μ) / σ = -1.645

Substituting the given values, we get:

(x - 4370) / 585 = -1.645

Sοlving fοr x, we get:

x = 4370 + (-1.645) * 585

x ≈ 3447.68 secοnds

Hence, the cutοff time fοr the fastest 5% οf athletes in the men's grοup is apprοximately 3447.68 secοnds.

b) Tο find the cutοff time fοr the slοwest 10% οf athletes in the wοmen's grοup, we need tο find the value οf x such that P(X > x) = 0.1, where X is a nοrmally distributed randοm variable with mean μ = 5296 and standard deviatiοn σ = 826.

Using a standard nοrmal distributiοn table οr a calculatοr, we can find that the z-scοre cοrrespοnding tο the 10th percentile is apprοximately -1.28. Therefοre, we can write:

(x - μ) / σ = -1.28

Substituting the given values, we get:

(x - 5296) / 826 = -1.28

Sοlving fοr x, we get:

x = 5296 + (-1.28) * 826

x ≈ 4253.68 secοnds

Hence, the cutοff time fοr the slοwest 10% οf athletes in the grοup apprοximately 4253.68 secοnds.

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The surface area of building with the skyview is found as 331.625 sq. yd.

A solid shape having six square faces is called a cube. Because every square face shares a comparable side length, each face is the same size. A cube has 8 vertices and 12 edges. An intersection of three cube edges is referred to as a vertex.

Total area = TSA of cuboid + CSA of hemisphere - area of circle

Total area = 2(lb + bh + hl) + 2πr² - πr²

Total area = 2(6*10 + 10*6 + 6*6) + 2*3.14*2.5² - 3.14*2.5²

On simplification:

Total area = 331.625

Thus, surface area of the building with the skyview is found as 331.625 sq. yd.

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Answer:

the answer is (A) 21-1/2.

Step-by-step explanation:

First, we need to calculate the rate of descent in feet per mile:

750 ft / 3 miles = 250 ft/mile

Next, we can set up a proportion to solve for the distance traveled:

(distance traveled) / (total altitude change) = (distance traveled) / (altitude change due to descent) + (altitude at which descent begins)

Let m be the distance traveled:

m / (32000 ft - 16000 ft) = m / (250 ft/mile * x miles) + 16000 ft

where x is the number of miles traveled when the pilot is 16000 ft above the ground.

Simplifying:

m / 16000 ft = m / (250 ft/mile * x miles) + 1

Multiplying both sides by 16000 ft:

m = m / (250 ft/mile * x miles) * 16000 ft + 16000 ft * 16000 ft

Multiplying both sides by (250 ft/mile * x miles):

m * (250 ft/mile * x miles) = m * 16000 ft + 16000 ft * (250 ft/mile * x miles)

Simplifying:

250 * x * m = 16000 * m + 4000 * x * m

Dividing both sides by m:

250 * x = 16000 + 4000 * x

Subtracting 4000 * x from both sides:

-3750 * x = -16000

Dividing both sides by -3750:

x = 4.266666... miles

Rounding to the nearest half mile gives us:

x ≈ 4.5 miles

Therefore, the answer is (A) 21-1/2.

The total weight of the 4 small jars of coffee is W = 400 grams

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be represented as A

Now , the value of A is

Substituting the values in the equation , we get

Let the weight of large jar be x

Let the weight of small jar be y

5 large jars of coffee have a total weight of 1250 grams.

So , 5x = 1250

Divide by 5 on both sides , we get

x = 250 grams

And , 2 large jars of coffee and 7 small jars of coffee have a total weight of 1200 grams.

So , 2x + 7y = 1200

2 ( 250 ) + 7y = 1200

Subtracting 500 on both sides , we get

7y = 1200 - 500

7y = 700

Divide by 7 on both sides , we get

y = 100 grams

On simplifying the equation , we get

So , the weight of 4 small jars is 4y = 400 grams

Hence , the weight of 4 small jars is y = 400 grams

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Answer:

25 I am not sure

Step-by-step explanation:

The proportion of at-home games that were wins is 0.33, or 33%.

The proportion of at-home games that were wins can be found by dividing the number of at-home wins by the number of at-home games. This can be represented as a fraction:

At-home wins / At-home games

Using the information given in the question, we can plug in the values for at-home wins and at-home games:

0.20 / 0.60

Simplifying the fraction gives us:

1/3

Converting this to a decimal gives us:

0.33

Therefore, the proportion of at-home games that were wins is 0.33, or 33%.

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When x is the subject of the formula a/b = 2x/x+5, the value of x = 5a/(2b - a).

The alphabetic letter that conveys a numerical value or a number is known as a variable in mathematics. A variable is used to represent an unknown quantity in algebraic equations.

Any alphabet from a to z can be used for these variables. Most frequently, the variables "a," "b," "c," "x," "y," and "z" are utilised in equations. By performing mathematical operations on variables as if they were express numbers, one is able to handle a variety of problems in a single computation. A quadratic recipe is a common example that demonstrates how to explain each quadratic condition by simply substituting the numerical estimates of the condition's coefficients for the variables that correspond to it.

The given formula is:

a/b = 2x/x+5

To make x the subject of the formula we have to isolate the value of x.

Using cross multiplication we have:

a(x + 5) = b(2x)

ax + 5a = 2bx

5a = 2bx - ax

5a = x (2b - a)

5a/(2b - a) = x

Hence, when x is the subject of the formula a/b = 2x/x+5, the value of x = 5a/(2b - a).

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Answer:

We can substitute m = 2.5 and n = 5 into the expression:

m + (7*9)/n = 2.5 + (7*9)/5

We can simplify the second term:

(7*9)/5 = 63/5

Substituting back into the expression:

m + (7*9)/n = 2.5 + 63/5

We can find a common denominator and add the terms:

2.5 + 63/5 = 12.5/5 + 63/5 = 75/5 = 15

Therefore, the value of the expression is 15, which corresponds to option (b) as the correct answer.

Answer:The ratio of the area of the small circle to that of the bigger circle is 25:36.

Step-by-step explanation:

The simplified fοrm οf the expressiοn [tex]2x^3 + 3y^3 + 5x^3 + 4y is 7x^3 + 3y^3 + 4y.[/tex]

Mathematical statements knοwn as expressiοns in mathematics are thοse with at least twο terms cοnnected by a separatοr and cοntaining either numbers, variables, οr bοth. It is pοssible tο add, subtract, multiply, οr divide using the mathematical οperatοrs.

Fοr instance, the expressiοn "x + y" is οne where "x" and "y" are terms with a separatοr added between them. There are twο different types οf expressiοns in mathematics: numerical and algebraic. Numerical expressiοns οnly cοntain numbers, while algebraic expressiοns alsο include variables.

Tο simplify the expressiοn [tex]2x^3 + 3y^3 + 5x^3 + 4y,[/tex] we can cοmbine the like terms:

[tex]2x^3 + 5x^3 + 3y^3 + 4y[/tex]

[tex]= (2 + 5)x^3 + (3)y^3 + (4)y[/tex]

[tex]= 7x^3 + 3y^3 + 4y[/tex]

Therefοre, the simplified fοrm οf the expressiοn

[tex]2x^3 + 3y^3 + 5x^3 + 4y is 7x^3 + 3y^3 + 4y[/tex].

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By using the concept of compound interest, the total amount is $4063.27.

We have,

The amount that was deposited during high school graduation = $1000

The amount that was deposited during college graduation after 4 years = $2000

The total number of years elapsed = 5 years

The interest rate per annum (p.a.) = 4%

The simple Interest (SI) formula is given by,

SI = P × r × t

where, P = Principal amount

r = Rate of interest

t = Time period

Compound Interest (CI) formula is given by,

CI = P (1 + (r/n))^(nt)

Where P = Principal amount

r = Rate of interest

t = Time period

n = Number of times interest is compounded

For the total amount using compound interest.

Total amount (A) = Principal + Compound interest

A = P + CI

Implying it to our data, we get the following.

For $1000 that was deposited during high school graduation,

For 5 years, n = 1, t = 5 years, r = 4% p.a.

CI = 1000[1+(4/100)]^(1×5)

CI = 1000[1+(1/25)]^5

CI = 1000×(26/25)^5

CI = 1000×1.21665

CI = $1216.65

For $2000 that was deposited during college graduation after 4 years,

The total number of years elapsed = 5 years + 4 years = 9 years

n = 1, t = 9 years, r = 4% p.a.

CI = 2000[1+(4/100)]^(1×9)

CI = 2000[1+(1/25)]^9

CI = 2000×(26/25)^9

CI = 2000×1.4233

CI = $2846.62

Therefore, the total amount (A) is $1216.65 + $2846.62 ⇒ $4063.27.

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To find the derivative of the function, we can use the chain rule and the power rule of differentiation.

What is chain rule?

The chain rule is a formula used to find the derivative of a composite function. If y = f(g(x)), then as per chain rule the instantaneous rate of change of function ‘f’ relative to ‘g’ and ‘g’ relative to x  results in an instantaneous rate of change of ‘f’ with respect to ‘x’

Let u = x² - 3, then we can rewrite the function as:

f(x) = [tex](u^{7})^\frac{1}{2}[/tex]

Using the chain rule and the power rule, we have:

f'(x) = (1/2) x (u^7)^(-1/2) x 7u^6 x 2x

Simplifying this expression, we get:

f'(x) = 7x(u^6) / (2(u^7)^(1/2))

Now, we can substitute x = -2 into this expression to find f'(-2):

f'(-2) = 7(-2)((-2)^2 - 3)^6 / (2(((-2)^2 - 3)^7)^(1/2))

Simplifying this expression, we get:

f'(-2) = -168/(2sqrt(19)^7) = -12.77 (rounded to two decimal places)

Therefore, f'(-2) = -12.77.

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After addressing the issue at hand, we can state that As a result, decimal midfielders make up around 36.36% of the players on the pitch

The decimal number system is frequently used to express both integer and non-integer quantities. Non-integer values have been added to the Hindu-Arabic numeral system. The technique used to represent numbers in the decimal system is known as decimal notation. A decimal number consists of both a whole number and a fractional number. The numerical value of complete and partially whole amounts is expressed using decimal numbers, which are in between integers. The full number and the fractional part of a decimal number are separated by a decimal point. The decimal point is the little dot that appears between whole numbers and fractions. An example of a decimal number is 25.5. In this case, 25 is the total number, and 5 is the minimum.

The following formula can be used to determine the percentage of midfielders on the field:

Overall number of players on the field / Total number of midfielders

In this situation, the total number of midfielders is 4, and the total number of players on the pitch is 11.

As a result, the percentage of midfielders on the field is:

4 / 11

This can be expressed in decimal or percentage form as follows:

0.3636 (rounded to four decimal places) (rounded to four decimal places)

36.36% (rounded to two decimal places) (rounded to two decimal places)

As a result, midfielders make up around 36.36% of the players on the pitch.

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According to the information, the graphics would remain as seen in the attached images. In them, graph A has a greater dispersion than graph B because it integrates a greater number of values.

A dot plot is a term for a type of graph used to display data by locating points on a number line. This graph is used to graphically represent certain trends or groupings of data.

According to the above, if we want to graph the information in the statement we must include at least 5 points in each graph. Additionally, we must put at least 7 points in the central value of the graph. Finally, we must have a greater dispersion of data in graph A than in graph B.

According to the above, the graphics would remain as shown in the image.

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The percentage value that represents what percent of 64 is 4? is (b) 6 1/4%

A percentage is a way to express a fraction or portion of a whole as a number out of 100.

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

What percent of 64 is 4?

Represent the percentage with x

So, we have

x percent of 64 is 4

Express as product expression

x% * 64 = 4

Multiply through by 100

x * 64 = 400

Divide by 64

x = 6.25 or 6 1/4

Hence, the percentage is 6 1/4%

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Therefore, the ladder touches the house at a height of 28 feet above the ground.

A triangle is a geometric shape that consists of three straight sides and three angles. It is a polygon with three sides. The sides of a triangle are connected by its vertices or corners. The triangle is one of the simplest and most fundamental shapes in geometry, and it has many important properties and applications. Triangles have many practical applications in everyday life and in various fields, such as architecture, engineering, and physics. The study of triangles and their properties is an important part of mathematics and geometry.

Here,

We can use the Pythagorean theorem to solve this problem. The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the legs (the two shorter sides) is equal to the square of the length of the hypotenuse (the longest side, which is opposite the right angle). In this problem, the ladder, the side of the house, and the ground form a right triangle. The ladder is the hypotenuse, the distance from the house to the ladder is one leg, and the height we want to find is the other leg.

Let x be the height above the ground where the ladder touches the house. Then, using the Pythagorean theorem, we have:

x² + 21² = 35²

Simplifying and solving for x, we get:

x² + 441 = 1225

x² = 784

x = 28

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