The path of a ball thrown from a cliff is modelled by the quadratic function h(t)=-4.9(t-1)^2+35.

From what height is the hall thrown?

Range of the function?

How long does it take for the ball to land?

\(a \in \mathbb{Z}_{n}\) such that \(a^{2} = [1] \in \mathbb{Z}_{n}\) and \(a \neq [1]\)

Let \(n \in \mathbb{N}\) with \(n>2\). We consider the set \( S = \{a \in \mathbb{Z}_{n} \ | \ a^{2} = [1] \in \mathbb{Z}_{n}\} \). We have to prove that \( S \neq \emptyset \).

We prove by contradiction. Suppose \( S = \emptyset \). This implies that for all \( a \in \mathbb{Z}_{n}, \ a^{2} \neq [1] \in \mathbb{Z}_{n}\). Thus, \( [1] \) is not a square in \(\mathbb{Z}_{n}\). But since \(n >2\), \([1]\) has at least two square roots in \(\mathbb{Z}_{n}\) which implies that \( S \neq \emptyset \).

Therefore, \(S \neq \emptyset\) and thus there exists \(a \in \mathbb{Z}_{n}\) such that \(a^{2} = [1] \in \mathbb{Z}_{n}\) and \(a \neq [1]\).

This proves that there exists an \(a \in \mathbb{Z}_{n}\) such that \(a^{2} = [1] \in \mathbb{Z}_{n}\) and \(a \neq [1]\).

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The given function f(x) = 5 (2.3)^x is an exponential function.

The given function is f(x) = 5 (2.3)^x.

To determine if the function is linear, quadratic, or exponential, we need to examine the form of the function.

A linear function has the form f(x) = mx + b, where m is the slope and b is the y-intercept.

A quadratic function has the form f(x) = ax^2 + bx + c, where a, b, and c are constants.

An exponential function has the form f(x) = ab^x, where a and b are constants.

The given function, f(x) = 5 (2.3)^x, is in the form of an exponential function, with a = 5 and b = 2.3. Therefore, the function is exponential.

In conclusion, the given function f(x) = 5 (2.3)^x is an exponential function.

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Mean: The mean is the average of a set of numbers. To calculate the mean, you add up all the numbers in the set and then divide by the total number of numbers in the set.

Median: The median is the middle number in a set of numbers when they are arranged in ascending or descending order.

Mode: The mode is the number that appears most frequently in a set of numbers.

Hi there! Yes, I can certainly show and explain the methods for calculating Mean, Median, and Mode. These are all measures of central tendency, which are used to describe the center of a data set.

Mean: The mean is the average of a set of numbers. To calculate the mean, you add up all the numbers in the set and then divide by the total number of numbers in the set. For example, if you have the numbers 2, 4, 6, 8, and 10, the mean would be (2 + 4 + 6 + 8 + 10) / 5 = 6.

Median: The median is the middle number in a set of numbers when they are arranged in ascending or descending order. If there is an odd number of numbers in the set, the median is the middle number. If there is an even number of numbers in the set, the median is the average of the two middle numbers. For example, if you have the numbers 2, 4, 6, 8, and 10, the median would be 6. If you have the numbers 2, 4, 6, 8, 10, and 12, the median would be (4 + 6) / 2 = 5.

Mode: The mode is the number that appears most frequently in a set of numbers. If there is more than one number that appears the same number of times, there can be more than one mode. For example, if you have the numbers 2, 4, 6, 8, 10, and 10, the mode would be 10 because it appears twice and the other numbers only appear once.

I hope this helps! Let me know if you have any other questions.

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To get the chart that you have circled in the second picture, you need to create a table using the given formulas and then create a chart using the table. Here are the steps:

1. Create a table with the following columns: Iteration, Random, Demand, Revenue, Cost, Refund, and Profit.
2. In the first row of the table, enter the given formulas in the respective columns. For example, in the first row of the Random column, enter =RAND(), in the first row of the Demand column, enter =ROUND ( NORM.INV ( RAND(), Mean, Standard Deviation),0), and so on.
3. Copy the formulas down to the 5000th row to get the values for all 5000 iterations.
4. Select the entire table and click on the Insert tab in the Excel ribbon.
5. In the Charts group, click on the type of chart that you want to create. In this case, it looks like you want to create a scatter chart.
6. In the Chart Design tab, click on the Select Data button in the Data group.
7. In the Select Data Source dialog box, click on the Add button in the Legend Entries (Series) section.
8. In the Edit Series dialog box, enter a name for the series, select the Profit column for the Series X values, and select the Demand column for the Series Y values.
9. Click on the OK button to close the Edit Series dialog box and then click on the OK button to close the Select Data Source dialog box.
10. Your chart should now be created and should look like the one that you have circled in the second picture.

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If there are 578 coyotes in 2003 with initial growth of 1.52, there will be approximately 336113 coyotes in 2028.

The number of coyotes in 2028 can be found by using the formula for exponential growth:

[tex]A = P(1 + r)^t[/tex]

Where:

A = final amount
P = initial amount
r = growth rate
t = time in years

Plugging in the given values:

[tex]A = 578(1 + 1.52)^25[/tex]

Using a calculator, we get:

[tex]A = 578(1.52)^25[/tex]
[tex]A = 578(581.68)[/tex]
[tex]A = 336112.64[/tex]

Therefore, there will be approximately 336113 coyotes in 2028.

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Answer:

To find the area of the trapezoid-shaped playground, we need to use the formula:

Area = (1/2) × (base1 + base2) × height

In the given diagram, the length of the top base is 24 feet, the length of the bottom base is 48 feet, and the height is 52 feet.

So, the area of the playground is:

Area = (1/2) × (24 + 48) × 52

= (1/2) × 72 × 52

= 1,872 square feet

Therefore, the area of the playground is 1,872 square feet.

The closest option to this answer is option B, which is 1,768 square feet. However, the correct answer is actually 1,872 square feet.

Given that Jordan cut strips of border for a design for a triangular sign to put on a bulletin board. Two of the strips were 15 inches long and the third was 30 inches long. We need to determine if the design can be made.

To determine whether the design can be made or not, we will check whether the sum of the lengths of any two sides of the triangle is greater than the length of the third side or not. Let a, b and c be the three sides of the triangle such that c is the longest side. According to the Triangle Inequality Theorem, For a triangle to be formed, the sum of the lengths of any two sides of the triangle should be greater than the length of the third side.Thus, a + b > cIf the above condition is satisfied, then the design can be made. If not, then the design cannot be made.
Let's check for the given design:
a + b > ca + b = 15 + 15 = 30 (Since two of the strips were 15 inches long)
Therefore, 30 > 30 (Since the third strip was 30 inches long)The given design satisfies the Triangle Inequality Theorem. Hence, the design can be made.

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1). W is closed under addition.  

  W is not closed under scalar multiplication

2. W is not closed under addition

   W is closed under scalar multiplication

1. Let V be the vector space of all 2x2 matrices with real entries. Consider the subset W of V consisting of matrices of the form:

| a b |

| 0 c |

where a, b, and c are real numbers. Note that W is non-empty, since the matrix |0 0| is in W.

|0 0|

Now, let A and B be any two matrices in W, so that A has the form:

| a1 b1 |

| 0 c1 |

and B has the form:

| a2 b2 |

| 0 c2 |

Then the sum of A and B has the form:

| a1+a2 b1+b2 |

| 0 c1+c2 |

which is clearly also in W. Therefore, W is closed under addition.

However, W is not closed under scalar multiplication. Let A be any matrix in W, so that A has the form:

| a b |

| 0 c |

where a, b, and c are real numbers. Then, if we multiply A by a non-real scalar k = xi (where x is a real number and i is the imaginary unit), we get:

kA = xi * | a b | = | xia xib |

| 0 c | | 0 xc |

which is not in W, since the entry in the (1,2) position is non-zero. Therefore, W is not closed under scalar multiplication.

2. Let V be the vector space of all polynomials with real coefficients. Consider the subset W of V consisting of all polynomials of degree at most 2.

Note that W is non-empty, since the polynomial p(x) = 0 is in W.

Now, let c be any real scalar, and let p(x) be any polynomial in W.

Then cp(x) is a polynomial of degree at most 2, since multiplying a polynomial of degree at most 2 by a scalar does not change its degree. Therefore, W is closed under scalar multiplication.

However, W is not closed under addition. Let p(x) and q(x) be any two polynomials in W of degree at most 2. Then the sum p(x) + q(x) may have degree greater than 2, and hence may not be in W. For example, if p(x) = x^2 + 2x + 1 and q(x) = -x^2 + x - 1, then p(x) + q(x) = 3x, which is not in W. Therefore, W is not closed under addition.

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Answer:

744416

Step-by-step explanation:

Just add everything together

a) Tοtal cοst C(x) = 200x + 1500

   Revenue cοst R(x) = 300x

The definitiοn οf an equatiοn in algebra is a mathematical statement that demοnstrates the equality οf twο mathematical expressiοns. Fοr instance, the equatiοn 3x + 5 = 14 cοnsists οf the twο equatiοns 3x + 5 and 14, which are separated by the 'equal' sign.

Let x be the number οf scοοters.

It cοsts yοu $1500 fοr tοοls and equipment tο get started, and the materials cοst $200 fοr each scοοter.

Here fixed cοst = $1500 and variable cοst = $200

We knοw that cοst functiοn = variable cοst per unit + fixed cοst

Cοst functiοn :

C(x) = 200x+1500

Yοur scοοter sell fοr $300

Revenue functiοn :

R(x) = 300x

(b) The sοlutiοn means

When yοu start a business assembling initial cοst is $1500 fοr tοοls and equipment.

The material cοst will increase $200 fοr each number οf scοοter increases.

The revenue will increase $300 fοr each number οf scοοter increases.

c) Let us take number οf scοοters x=50

Then cοst C(50)=200*50+1500 = 10000+1500=$11500

Revenue Cοst R(50)=300*50=$15000

Then prοfit = Revenue- cοst = 15000-11500 = $3500.

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To solve this problem, we can use a system of equations.

Let's call the number of Php 5 coins x and the number of Php 10 coins y. The first equation will represent the total number of coins: x + y = 120. The second equation will represent the total amount of money: 5x + 10y = 950. Now we can use the elimination method to solve for one of the variables. Let's multiply the first equation by -10 and then add the two equations together:

-10x - 10y = -1200

5x + 10y = 950

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

-5x = -250

Divide both sides by -5 to get x:

x = 50. Now we can plug this value back into the first equation to solve for y:

50 + y = 120

y = 70

So there are 50 Php 5 coins and 70 Php 10 coins.

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x = 18 ...... y = 20/3

x = 30 ..... y = ?

____________

[tex]y = \dfrac{30 \cdot 20}{3 \cdot 18} = \dfrac{100}{9}[/tex]

Answer: Let x = 0.727272...

Then, 100x = 72.727272...

Subtracting x from 100x gives:

100x - x = 72.727272... - 0.727272...

99x = 72

x = 72/99

We can simplify this fraction by dividing both the numerator and denominator by their greatest common factor, which is 9. Therefore:

x = (72/9) / (99/9) = 8/11

So, a = 8 and b = 11. Therefore, the recurring decimal 0.727272... can be expressed as the fraction 8/11.

Step-by-step explanation:

The company will need 58,800 cubic inches of wax to make 420 candles.

To calculate the amount of wax required to make 420 candles,

The volume of a rectangular prism is given by the formula V = l x w x h, where l is the length, w is the width, and h is the height.

So, we have

V = 5 in x 4 in x 7 in = 140 cubic inches

To find the total amount of wax required to make 420 candles,

We have

Total amount of wax = 140 cubic inches/candle x 420 candles

= 58,800 cubic inches

Hence, the company needs 58,800 cubic inches

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Since you sailed 0.5 miles 60º north of east to reach the marina, the distance you sailed eastward is 0.5cos(60º) = 0.25 miles and the distance you sailed northward is 0.5sin(60º) = 0.433 miles.

To return to your cabin, you sailed directly south for 0.433 miles, forming the adjacent side of a right triangle, and then sailed directly west for 0.25 miles, forming the opposite side of the same right triangle.

Therefore, the distance you sailed on your return trip is the hypotenuse of the right triangle, which can be found using the Pythagorean theorem:

distance = sqrt((0.433)^2 + (0.25)^2) ≈ 0.51 miles

Rounded to the nearest hundredth, the distance you sailed on your return trip is 0.51 miles, which is closest to option (a) 0.68 miles.

A figure which is a dilation of figure E include the following: A. figure F.

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

In Geometry, a scale factor is the ratio of two corresponding length of sides or diameter in two similar geometric figures such as equilateral triangles, quadrilaterals, and other types of polygons.

Based on the image (see attachment), we can logically deduce that only figure E and figure F share the same (common) line of symmetry.

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The function that represents exponential growth is C. Of(x) = 0.3(1.05)^(x), where x is the input variable.

Exponential growth occurs when a quantity increases at a constant percentage rate over time, which is exactly what happens in the function C. As x increases, the term (1.05)^(x) grows larger and larger, leading to a corresponding increase in the value of the function. The coefficient 0.3 scales the rate of growth so that it starts at a manageable level.

Functions A, B, and D do not represent exponential growth. Function A decreases over time as the base is less than 1, function B represents a linear growth, and function D represents linear growth as well.

The function with Exponential growth is  f(x) = 0.3 [tex](1.05)^x[/tex].

An exponential function is a nonlinear function with the formula y = [tex]a(b)^x[/tex], where a = 0 and b = 1.

The function is an exponential growth function for a > 0 and b > 1. The function is an exponential decay function when a > 0 and 0 b 1.

First function, f(x) = 134 [tex](0.75)^x[/tex]

Here the value of b < 1 then it Exponential Decay.

Now,  f(x) = 513 + 0.2x is linear function.

Now,  f(x) = 0.3 [tex](1.05)^x[/tex]

Here the value of b > 1 then it Exponential Growth.

Now,  f(x) = 15+ 1.6x is linear function.

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The probability of rolling a prime number every time

The probability of rolling a prime number on a single die is 3/6 or 1/2, since there are three prime numbers (2, 3, 5) out of six possible outcomes.

To find the probability of rolling a prime number ten times in a row, we need to multiply the probabilities together. This is because each roll is independent of the others, so the probability of rolling a prime number on each roll is the same.

So the probability of rolling a prime number ten times in a row is:

(1/2) × (1/2) × (1/2) × (1/2) × (1/2) × (1/2) × (1/2) × (1/2) × (1/2) × (1/2) = 1/1024

Therefore, the probability of rolling a prime number every time a die is rolled ten times is 1/1024.

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The sum of the polynomials is 6x^(2)-3x-6.

To add the polynomials using a vertical format, we will first write each polynomial in a column, lining up like terms vertically. Then, we will add the coefficients of each like term together to find the sum.

Here is the solution in a vertical format:

So, the sum of the polynomials is 6x^(2)-3x-6.

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Answer:

Zoe will save 30% of the original cost of the sweater, which is $40. To calculate the amount saved, we must multiply 30% by the original cost of the sweater, which gives us 0.30 x $40 = $12. Therefore, Zoe will save $12 on the sweater with the 30% discount

Step-by-step explanation:

Answer:

She will save $12.00

Step-by-step explanation:

.3 x 40 = 12

30% as a decimal is .3.

The cost of the sweater on sale would be

.7 x 40 = 28

If we take 30% off, we are leaving 70% on.

28 + 12 = 40 the original cost.

Helping in the name of Jesus.

The center of the circular track with the equation x² - 18x + y² - 22x = -177 is (20, 0). See below for other solutions

The equation of a circle that passes through the origin is represented as

x² + y² = r²

Where

r = radius

For circle 13, we have

Point = (0, 6)

So, the radius is

0² + 6² = r²

r = 6

This gives

x² + y² = 6²

For the point (√11, 5), we have

(√11)² + 5² = 6²

36 = 36 --- true

The point (√11, 5) is on the circle

For circle 14, we have

Point = (-7, 0)

So, the radius is

-7² + 0² = r²

r = 7

This gives

x² + y² = 7²

For the point (√14, 6), we have

(√14)² + 6² = 7²

50 = 49 --- falsee

The point (√14, 6) is not on the circle

Andy's error is that he did not square 12 in (√23)² + (11)² ≠ 12

The correct solution is

(√23)² + (11)² = 12²

144 = 144

The point (√23, 11) is on the circle

The equation of a circle is represented as

(x - a)² + (y - b)² = r²

For circle 16, we have

Center = (-1, 5)

Radius, r = 4

So, we have

Equation: (x + 1)² + (y - 5)² = 4²

For circle 17, we have

Center = (2, 0)

Point = (-2, 3)

So, we have

Equation: (x - 2)² + (y - 0)² = (-2 - 2)² + (3 - 0)²

Equation: (x - 2)² + y² = 25

Given that

x² - 18x + y² - 22x = -177

This gives

x² - 40x + y² = -177

Factorize

(x - 20)² + y² = -177 + 400

Evaluate

(x - 20)² + y² = 223

From the above, we have

Center = (20, 0)

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After 5 hours of draining, the depth of the water in Mara's swimming pool will be 11/4 feet

The depth of the water variations via- 3/ 4 foot each hour, this means that the depth decreases by employing 3/4 foot every hour.

Still, also after one hour of draining, the intensity of the water might be

If the primary depth of the water was five bases.

5-(3/4) = 41/4 ft

After hours, the depth might be

-(3/4) = 31/2 fr

After three hours

-(3/4) = 23/4 ft

After 4 hours

-(3/4) = 2 ft

After five hours

2-(3/4) = 11/4 bases

Thus, after 5 hours of draining, the depth of the water in Mara's swimming pool will be 11/4 feet

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Reflect point A over the line to get point A', -1,2 is A'.

What is a graph?

A graph is a structure that fundamentally consists of a set of items where some pairs of the objects are "connected" in some way. This definition comes from discrete mathematics, more especially graph theory. The items are represented by mathematical abstractions called vertices, and each pair of connected vertices is known as an edge.

A graph is frequently depicted in a diagram by a collection of dots or circles for the vertices and lines or curves for the edges. Among the topics covered by discrete mathematics are graphs.

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a) To solve for the unknowns using Gaussian elimination, follow these steps:
1. Multiply the first equation by -1 and add it to the second equation to get x2 = 7 + x3.
2. Multiply the first equation by -3 and add it to the third equation to get x3 = 3.
3. Substitute x3 = 3 into the first equation to get x1 = 2.

b) To solve for the unknowns using Gauss-Jordan elimination, follow these steps:
1. Multiply the first equation by -1 and add it to the second equation to get x2 = 7 + x3.
2. Multiply the first equation by -3 and add it to the third equation to get x3 = 3.
3. Substitute x3 = 3 into the second equation to get x2 = 4.
4. Substitute x3 = 3 and x2 = 4 into the first equation to get x1 = 2.

c) To solve for the unknowns using LU decomposition, follow these steps:
1. Compute the LU decomposition of the coefficient matrix A.
2. Solve Ly = b using forward substitution to get y = [1, 4, 3]^T.
3. Solve Ux = y using backward substitution to get x = [2, 4, 3]^T.

d) To solve for the unknowns using matrix-inverse based solution, follow these steps:
1. Compute the inverse of the coefficient matrix A.
2. Multiply the inverse of A with b to get x = [2, 4, 3]^T.

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Using synthetic division, the value of h(-4) is - 24.

1. Set up the synthetic division grid with the divisor (-4) in the top left corner and the coefficients of the polynomial in the top row.
-4 | 2   0   -25   4

2. Bring down the first coefficient to the bottom row.
-4 | 2   0   -25   4
   |                      
     2

3. Multiply the divisor (-4) by the first number in the bottom row (2) and put the result (-8) in the second column of the top row.
-4 | 2   0   -25   4
   |     -8            
     2

4. Add the numbers in the second column (0 and -8) and put the result (-8) in the second column of the bottom row.
-4 | 2   0   -25   4
   |     -8            
     2  -8

5. Repeat steps 3 and 4 for the remaining columns.
-4 | 2   0   -25   4
   |     -8     32  -28    
     2  -8      7   -24

6. The last number in the bottom row (-24) is the remainder and the value of h(-4). The other numbers in the bottom row (2, -8, 7) are the coefficients of the quotient polynomial.

Therefore, h(-4) = 24.

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Ans - The pilot is 1868.08 mi from her starting position, To find the distance the pilot is from her starting position, we can use the law of cosines. The law of cosines states that for any triangle with sides a, b, and c, and angle C opposite side c:

[tex]c^2 = a^2 + b^2 - 2ab*cos(C)[/tex]

In this case, the pilot's original path is one side of the triangle (a), her new path after the course correction is another side of the triangle (b), and the distance she is from her starting position is the third side of the triangle (c). The angle between the two paths is 20 degrees (C).

First, we need to find the length of sides a and b. The pilot flew for 1.5 hrs at a speed of 685 mi/h on her original path, so:

a = 1.5 hrs * 685 mi/h = 1027.5 mi

She then flew for 2 hrs at the same speed on her new path, so:

b = 2 hrs * 685 mi/h = 1370 mi

Now we can plug these values into the law of cosines to find the distance the pilot is from her starting position (c):

[tex]c^2 = 1027.5^2 + 1370^2 - 2*1027.5*1370*cos(20)[/tex]
[tex]c^2 = 3,488,806.25[/tex]
[tex]c = sqrt(3,488,806.25)[/tex]
[tex]c = 1868.08 mi[/tex]

Therefore, the pilot is 1868.08 mi from her starting position.

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The distance between the points C=(-7,6) and E=(-2,2) is 6.4 units.

To calculate the distance between two points, we use the distance formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

In this case, the points are C=(-7,6) and E=(-2,2), so we can plug in the values into the formula:

d = √((-2 - (-7))^2 + (2 - 6)^2)

d = √((5)^2 + (-4)^2)

d = √(25 + 16)

d = √(41)

d = 6.4

Therefore, the distance between the points C=(-7,6) and E=(-2,2) is 6.4 units.

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Answer: 1.1 The diameter of the tin on the picture is approximately 90 mm.

1.2 The circumference of the base of the tin is approximately 565 mm (2πr = 2π(45) = 565).

Step-by-step explanation: math is stressful fr

We can write the diameter and circumferance of base as -

D = 2√(750ρ/πh)

C = 2π√(750ρ/πh)

In mathematics, volume is the space taken by an object. Volume is a measure of three-dimensional space. It is often quantified numerically using SI derived units or by various imperial or US customary units. The definition of length is interrelated with volume.

here, we have,

Given is to find the diameter and height of the tin can.

Assume the density of coffee as {ρ}. We can write the volume of the tin can as -

Volume = mass x density

Volume = 750ρ

We can write -

πr²h = 750ρ

r = √(750ρ/πh)

D = 2r

D = 2√(750ρ/πh)

Now, we can write the circumferance as -

C = 2πr

C = 2π√(750ρ/πh)

Therefore, we can write the diameter and circumferance of base as -

D = 2√(750ρ/πh)

C = 2π√(750ρ/πh)

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The answer of the given question based on the Irons is trying to lay out the bases for a game of kickball such that the infield is a square the answer is  first base and third base should be about 35.4 feet apart.

The Pythagorean theorem is  fundamental concept in geometry that relates to three sides of right triangle. It states that in right triangle, the square of  length of  hypotenuse (the side opposite the right angle) is equal to  sum of the squares of  lengths of other two sides.

We can use the Pythagorean theorem to find this distance.

Let's call the distance between home base and first base "a". We know that the distance between home base and third base is also "a", because the infield is a square. We also know that the distance between home base and second base (which is the hypotenuse of the right triangle formed by home base, first base, and second base) is 25 feet.

Using the Pythagorean theorem, we can solve for "a":

a² + 25² = a² + a²

2a² = 25²

a² = (25²)/2

a = sqrt((25²)/2) ≈ 17.7 feet

So the distance between first base and third base should be approximately 35.4 feet (2a). Rounding to the nearest tenth of a foot, this is 35.4 feet. Therefore, first base and third base should be about 35.4 feet apart.

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Therefore , the solution of the given problem of unitary method comes out to be a 2-liter soda costs $4.98 and a platter of nachos costs $11.99.

The measurements taken from this femtosecond section must be multiplied by two in order to complete the task using the unitary variable technique. In essence, the characterised by a group and the hue groups are both removed from the unit approach when a desired object is present. For example, 40 pens with a expression price will indeed cost Inr ($1.01). It's possible that one country will have total influence over the approach taken to accomplish this. Almost every living creature has a distinctive quality.

Here,

Let's use "N" for the price of a platter of nachos and "S" for the price of a 2-liter soda.

The first aspect of the issue reveals that:

=> 5N + 2S = 67.87

The second component of the issue reveals the following to us:

=>2N + S = 27.94

=> S = 27.94 - 2N

When we use this expression in place of S in the first equation, we obtain:

=> 5N + 2(27.94 - 2N) = 67.87

When we simplify and account for N, we obtain:

=> 5N + 55.88 - 4N = 67.87\sN = 11.99

We can determine S by substituting this number for N in the second equation:

=> 2(11.99) + S = 27.94\sS = 4.98

As a result, a 2-liter soda costs $4.98 and a platter of nachos costs $11.99.

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