Six points are equally spaced around a circle of radius 1. Three of these points are the vertices of a triangle that is neither equilateral nor isosceles. What is the area of this triangle

Answer:

27π

Step-by-step explanation:

Volume = 1/3πr²h
=1/3*(3)²9π
=27π

3 ice cream machines will make a total of 480 gallons for the same length of time and 4 ice cream machines will make a total of 640 gallons for the same length of time.

For the quantity 3 ice cream machines will make.

2 ice cream machines produce a total of 320 gallons in a certain amount of time.

If 2 ice cream machines can make a total of 320 gallons. Then,

3 ice cream machines will produce a total of x gallons in the same amount of time

x = (320 × 3)/2

x = 480 gallons

Hence, 3 ice cream machines will produce a total of 480 gallons.

For the quantity 4 ice cream machines will make.

If 2 ice cream machines can make a total of 320 gallons. Then,

4 ice cream machines will produce a total of y gallons in the same amount of time

y = (320 × 4)/2

y = 640 gallons

Hence, 4 ice cream machines will produce a total of 640 gallons.

--The given question is incomplete, the complete question is

"Two ice cream machines make a total of 320 gallons in a certain amount of time. Working together for the same length of time, how many gallons of ice cream can each number of machines make?

3 ice cream machines

4 ice cream machines"--

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Answer:

The graph passes through the point (0, 4), and for each increase of 1 in the x-values, the y-values increase by a factor of 1.5

Step-by-step explanation:

Answer:

y = -2x + 13

Step-by-step explanation:

In order to write an equation of the line in slope-intercept form, we first have to look at the equation:

y = mx + b

We know that m represents slope, and b represents the y-intecept, in other words, what y is equal to when x is equal to zero.

We already have y and x, (4,5), and the slope of the line, since the two lines are parellel, they share the same slope. So now, we have to pulg those values into the equation and solve for b:

(4,5)

y = 5

x = 4

m = -2

y = mx + b

(5) = (-2)(4) + b

5 = -8 + b

13 = b

So, equation of the line in slope-intercept form that passes through point (4, 5), is y = -2x + 13

Remeber that you can always graph to check your work!

I hope this helps! :)

The area of the trapezium is 36.3 unit squared.

The figure above is a trapezium. Therefore, the area of a trapezium can be represented as follows:

Hence,

area of a trapezium = 1 / 2 (a + b)h

where

Therefore,

area of a trapezium = 1 / 2 (a + b)h

where

Using Pythagoras's theorem,

c² = a² + b²

where

h² = 6² - 1.5²

h²= 36 - 2.25

h = √33.75

h = 5.80947501931

h = 5.81

Therefore,

area of the trapezium = 1 / 2 (4 + 8.5)5.8

area of the trapezium = 1 / 2 (12.5)5.8

area of the trapezium = 72.5 / 2

area of the trapezium = 36.3 units²

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In this problem, you have a square with side length 6 and four smaller squares with side length 2 placed in the corners. The points W, X, Y, and Z are the vertices of the smaller squares. The goal is to find the maximum possible distance between points A and P.

[asy] size(200); pair A,B,C,D,P,W,X,Y,Z; A=(0,4); B=(4,4); C=(4,0); D=(0,0); P=(2,2); W=(0,6); X=(6,6); Y=(6,0); Z=(0,0); draw(A--B--C--D--cycle); draw(W--X--Y--Z--cycle); draw(A--P--C); label("A",A,NW); label("B",B,NE); label("C",C,SE); label("D",D,SW); label("P",P,S); label("W",W,N); label("X",X,NE); label("Y",Y,S); label("Z",Z,SW); [/asy]

The point A is located on the bottom-left corner of the square with side length 6, and the point P is the center of the square $ABCD$. To find the maximum distance from A to P, we need to find the longest diagonal of the square $ABCD$.

The longest diagonal of a square is the one that goes from one corner of the square to the opposite corner. In this case, the longest diagonal goes from point A to point C, which is a distance of 6 units. Therefore, the maximum possible distance from A to P is 6 units.

The minimum number of hours you will have cruise control on is 1.83 hour(s) and the maximum is 2.83 hour(s).

To find the minimum and maximum numbers of hours you will have cruise control on, you need to calculate the distance you will travel with cruise control on and divide it by your speed.

The minimum distance you will travel with cruise control on is

140 miles - 30 miles = 110 miles.

The maximum distance you will travel with cruise control on is

140 miles + 30 miles = 170 miles.

To convert these distances into hours, you will divide them by your speed of 60 miles per hour.

Minimum hours: 110 miles / 60 mph = 1.83 hours

Maximum hours: 170 miles / 60 mph = 2.83 hours

Therefore, the minimum number of hours you will have cruise control on is 1.83 hour(s) and the maximum is 2.83 hour(s).

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In the triangle the line segment joining the midpoints 'S' and 'T' of the sides PQ and PR implies that ST || QR.

As given in the question,

Diagram is attached.

As per the attached diagram  we have,

Given : In triangle PQR with midpoints S and T of sides PQ and PR respectively.

To prove : ST is parallel to QR.

Proof : First construct a line through R parallel to PQ such that it meet the extended line of ST at point 'U'

In triangle PST and triangle RTU,

PQ|| RU and SU is the transversal

∠PST ≅ ∠RUT ( alternate angles )

∠PTS ≅ ∠RTU ( vertically opposite angles)

PT ≅TR ( 'T' is the midpoint of PR )

By AAS we have

ΔPST ≅ΔRTU

By congruent part of congruent triangle we have,

PS ≅ RU

⇒ SQ ≅ RU

In quadrilateral QSUR,

SQ ≅ RU

SQ || RU ( as PQ || RU by construction )

Quadrilateral with one pair of opposite side congruent and parallel implies it is a parallelogram.

Quadrilateral QSUR is a parallelogram.

⇒ SU || QR

S - T - U

⇒ ST is parallel to QR

Therefore, in the given triangle line segment formed by joining the midpoints of the other two sides is parallel to third side.

The above question is incomplete , the complete question is :

Prove theorem 8.9 : The line segment joining the midpoints of the two sides of the given triangle is parallel to the third side .

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The person is using approximately 2,333.333 gallons of gasoline.

The average is defined as the mean value which is equal to the ratio of the sum of the number of a given set of values to the total number of values present in the set.

To calculate the amount of gasoline used, you can use the formula:

gasoline used = total miles driven / mpg

Where total miles driven is 70,000 miles and mpg is 30.

Plugging these values into the formula, we get:

Gasoline used = 70,000 miles / 30 mpg = 2,333.333 gallons

So the person is using approximately 2,333.333 gallons of gasoline.

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Answer:

Step-by-step explanation:

By using the Trapezoidal sum with the four subintervals indicated by the table the sum is 52.95.

What is Trapezoidal sum?

The trapezoidal rule is a method for roughly approximating the definite integral in calculus. The area of the region that roughly fits the shape of a trapezoid under the graph of the function f(x) is how the trapezoidal rule operates.

A Riemann sum is a technique for calculating the area under a curve by adding the areas of rectangles or trapezoids drawn along the curve. Triangular Riemann Sum: A trapezoidal Riemann sum determines the area under a curve by adding the areas of trapezoids.

Let,  [tex]\frac{1}{10}\int\limits^1_00 {H(t)} \, dt[/tex]  is the average temperature of the tea, in degree Celsius, over the 10 minutes.

[tex]\frac{1}{10}\int\limits^1_0_0 {H(t)} \, dt = \frac{1}{10}(2*\frac{66+60}{2}+ 3*\frac{60 + 52}{2}+4*\frac{52+44}{2}+1*\frac{44+43}{2} \\ \frac{1}{10}\int\limits^1_0_0 {H(t)} \, dt = 52.95[/tex]  

Hence, by using the Trapezoidal sum with the four subintervals indicated by the table the sum is 52.95.

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The measure of the middle angle of  triangle abc is 52°

Let the largest angle of triangle be x, the smallest angle be z, and the third angle be y.

Given that the measure of the largest angle is 32 less than 3 times the smallest angle.

So, we get an equation,

x = 3z - 32              ............(1)

Also, the measure of the middle angle is 8 more than half the measure of the largest angle.

So, we get an equation,

y = 8 + (x/2)          .......(2)

From equation (1),

z = (x + 32)/3           ..........(3)

We know that the sum of all angles of triangle is 180 degrees.

x + y + z = 180

Substitute the values of y and z in above equation.

x + 8 + (x/2) + (x + 32)/3 = 180

6x + 48 + 3x + 2(x + 32) = 1080

9x + 48 + 2x + 64 = 1080

11x + 112 = 1080

11x = 1080 - 112

11x = 968

x = 88

Substituting above value of x in equation (2)

y =  8 + (88/2)

y = 8 + 44

y = 52 degrees

Therefore, the middle angle of triangle measures 52 degrees.

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The correct answer is C)  i. $146.69 ii. $551.80 iii. $662.16.

What is social security tax ?

Social Security tax is a payroll tax that is imposed on both employees and employers to fund the Social Security program in the United States. The tax is used to provide benefits to retired and disabled workers, as well as to the families of deceased workers. The tax is imposed on wages and salaries earned by employees.

A. i. $146.69 ii. $551.80 iii. $1,029.76

B. i. $1,466.92 ii. $5,518.00 iii. $10,297.58

C. i. $146.69 ii. $551.80 iii. $662.16

D. i. $1,466.92 ii. $5,518.00 iii. $6,621.60

The Social Security tax rate is 6.2% for the employee and 6.2% for the employer, for a total of 12.4%. However, the employee portion of the tax is only applied to the first $142,800 of earnings in 2021. Any earnings above this amount are not subject to the employee portion of the Social Security tax.

The correct answer is C)  i. $146.69 ii. $551.80 iii. $662.16.

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Answer:

its D

Step-by-step explanation:

because of PEMDAS

Answer:

The formula for compound interest is: A = P(1 + r/n)^(nt)

Where:

A = the future value of the investment (the amount Alex has at the end of 7 years)

P = the initial investment (£4000)

r = the annual interest rate (x/100)

n = the number of times interest is compounded per year (let's assume it is compounded annually)

t = the number of years invested (7)

Plugging in the given values and solving for x:

5263.73 = 4000(1 + x/100)^7

So x = ((5263.73/4000)^(1/7))-1*100

x = 6.33% (approximately)

So, the interest rate is 6.33%.

The equation of a line passing through (10,10) that is perpendicular to the line with equation 5x + 8y = -9 is y - 10 = (8/5)(x - 10) or y = (8/5)x - 6.

Get the slope of the line 5x + 8y = -9 by rewriting it in slope-intercept form. Subtract 5x from both sides and divide by 8.

y = (-5/8)x - (9/8)

The slope of this line is -5/8. The negative reciprocal of -5/8 is 8/5. Therefore, the slope of the line that is perpendicular to 5x + 8y = -9 is 8/5.

Using point-slope form, determine the equation of the line.

y - y₁ = m(x - x₁)

y - 10 = (8/5)(x - 10)

or using the slope-intercept form,

y = mx + b

y = (8/5)x + b

10 = (8/5)10 + b

b = -6

y = (8/5)x - 6

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68% of the data falls between one standard deviation from the mean which is 13.2 and 19.6

An equation is an expression composed of variables and numbers linked together by mathematical operations.

The empirical rule states that 68% of the data falls within one standard deviation, 95% within two standard deviation from the mean and 99.7% falls within three standard deviation from the mean.

Given a mean of 16.4 and a standard deviation of 3.2.

68% of the data falls one standard deviation from the mean = 16.4 ± 3.2 = (13.2, 19.6)

68% of the data falls between 13.2 and 19.6

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After evaluating the double integral, the result, 0.27, is determined by the stated statement.

In mathematics, an integral is either a number representing the region beneath a function's graph over a certain interval or maybe a new function, the inverse of which comprises the original result (indefinite integral). The value produced after integrates or adding the words of a function that is split into an unending number of terms is referred to as an integral value in general.

The double integral is an example.

[tex]\iint_D\left(x^2+2 y\right) d A[/tex] where d is enclosed by the lines y = x & y = x³.

In the region D, x varies from 0 to 1, and y varies from y = x³ to y = x

[tex]\therefore \iint_D\left(x^2+2 y\right) d A=\int_{x=0}^1 \int_{y=x^3}^x\left(x^2+2 y\right) d y d x[/tex]

[tex]=\int_{x=0}^1\left[x^2 y+y^2\right]_{x^3}^x d x[/tex]

[tex]=\int_{x=0}^1\left[x^2\left(x-x^3\right)+\left[x^2-\left(x^3\right)^2\right]\right] d x=\int_{x=0}^1\left[x^3-x^5+x^2-x^6\right] d x[/tex]

[tex]\left.=\frac{x^4}{4}-\frac{x^6}{6}+\frac{x^3}{3}-\frac{x^7}{7}\right]_0^1[/tex]

= [tex]\frac{1}{4}-\frac{1}{6}+\frac{1}{3}-\frac{1}{7}[/tex]

= 23/84

= 0.27

[tex]\therefore \iint_D\left(x^2+2 y\right) d A=0.27[/tex]

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1800 * ? * 1 = 198

let x be the ?

By shifting 1800*1 into right hand side

x= 198/1800

x= 11/100= 0.11

Final answer = 0.11

Answer: A. Yes

B. Yes

C. No

Step-by-step explanation:

A. 2z = 40 and z = 20, 20 is a solution of the equation because when you substitute z = 20 into the equation 2z = 40, the equation becomes true.

B. n + 5 = 20 and n = 15, 15 is a solution of the equation because when you substitute n = 15 into the equation n + 5 = 20, the equation becomes true.

C. v = 12 and v = 16, 12 is not a solution of the equation because v is assigned to 12 and 16 both, but the equation only has one solution, a variable can't be assigned to two different values.

In general, when solving equations, we check if the value of the variable makes the equation true. if it does it's a solution, if not it's not a solution.

If the volume of the square based Pyramid is 125 m³ and the base is of 5m then the height of the pyramid is 15 m .

The Volume of the Square based pyramid is = 125 m³ ;

the measure of the side of base of pyramid is = 5m ;

So , the Area of the base of pyramid is = [tex]5 \times 5[/tex] = 25 m² ;

We know that the formula for the Volume of the pyramid is = [tex]\frac{1}{3} \times Area\; Of \;Square \times Height[/tex] ;

Substituting the values  ;

we get ;

⇒ [tex]125= \frac{1}{3} \times 25 \times Height[/tex] ;

On solving further ,

we get ;

⇒ [tex]\frac{125\times 3}{25}[/tex]

⇒ 15 m .

Therefore , The Height of the Pyramid is 15 m.

The given question is incomplete , the complete question is

This square-based oblique pyramid has a volume of 125 m³ and the length of the side of the base is 5m . What is the height of the pyramid ?

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The approximate value of  [tex](1.999)^5[/tex] is 31.92.

A linear approximations in maths is the use of a linear function to estimate a generic function. They are frequently used to create first order methods for solving or approximating equation solution in the method of finite difference.

To estimate [tex](1.999)^5[/tex]

we’ll find the linearization of f(x) = [tex]x^5[/tex]

at a = 2. Since

f′(x) = 5[tex]x^4[/tex]

, f(2) = 32, and

f′(2) = 80,

we have L(x) = 32+80(x−2). Thus,

[tex]x^5[/tex] ≈ 32 + 80(x − 2) when x is near 2,

so (1.999)5 ≈ 32 + 80(1.999 − 2) =

32 − 0.080 = 31.92.

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It will take the rescue craft 0.44 hours or 26.4 minutes to reach the sailboat.

The time required for the rescue craft to reach the sailboat can be calculated by using the formula for relative velocity:

Relative Velocity = Velocity of the Rescue Craft - Velocity of the Sailboat

= 29 mph - 4 mph = 25 mph

Therefore, the time required for the rescue craft to reach the sailboat is given by:

time= Distance/Relative Velocity

= 11 miles/25 mph

= 0.44 hours

Therefore, it will take the rescue craft 0.44 hours or 26.4 minutes to reach the sailboat.

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The coefficient of x⁷ in the given expression is 16.

Expressions are mathematical statements which consist of two or more terms and terms are connected to each other using mathematical operators like addition, multiplication, subtraction and so on.

Given is the expression [x³ + (4 / x²)]⁴.

[x³ + (4 / x²)]⁴ =  (x³ + 4x⁻²)⁴

                      = (x³ + 4x⁻²)² (x³ + 4x⁻²)²

                      = (x⁶ + 8x + 16x⁻⁴) (x⁶ + 8x + 16x⁻⁴)  

                      = x¹² + 8x⁷ + 16x² + 8x⁷ + 64x² + 128x⁻³ + 16x² + 128x⁻³ +

                         256x⁻⁸

Rearranging with like terms, we get,

                      = x¹² + 16x⁷ + 96x² + 256x⁻³ + 256x⁻⁸

Hence the coefficient of x⁷ is 16.

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Utilizing the Sieve of Eratosthenes method is simple. In order to encircle the remaining numbers, we must cancel all multiples of each prime number starting with 2 (including the number 1, which is neither a prime nor a composite).

The Sieve of Eratosthenes can be used to locate all prime numbers that are less than 100, as seen in the example below. In ten rows, start by writing the numbers 1 through 100.

Therefore, based on the preceding table, we can conclude that the prime numbers 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97 are. There are ten prime numbers between 50 and 100 in all. The required prime numbers are those that are surrounded.

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For a quadratic function [tex]y=ax^2+bx+c[/tex], we know the vertex of the graph is a maximum if and only if [tex]a < 0[/tex], and a minimum if and only if [tex]a > 0[/tex].

It is easy to see that for a function of the form [tex]y=a(x-h)^2+k[/tex], the vertex of the graph is a maximum if and only if [tex]a < 0[/tex], and a minimum if and only if [tex]a > 0[/tex].

Therefore, the vertex of the graph of [tex]f(x)[/tex] is a maximum, and the vertex of the graph of [tex]g(x)[/tex] is a minimum.

Answer: 14.825

Step-by-step explanation:

First , do 2 * 5.3 because it's 2a, you'll have to do 2 * a. This equals 10.6.

Second, you'll need to do 5.3 * 1.25. Why? Because in the problem you need to do a * b, which is that. It equals 6.62500.

Lastly, put it all together! 10.6 + 6.62500 - 2.4 equals 14.825.

a) According to the central limit theorem, the mean is $142 and the standard deviation is $0.7488.

b) Nearly normal according to the central limit theorem.

c) 0.0901 = 9.01% probability that the average cable service paid by a sample of cable service customers exceeds $170 in the central limit theorem.

Normal probability distribution:

The normal probability distribution, discovered by Abraham De Moivre in 1733, approximates the binomial probability distribution when a particular experiment has a very large number of trials. In 1774 Laplace studied the mathematical properties of the normal probability distribution. By historical error, the discovery of the normal distribution was attributed to Gauss, first mentioned in his 1809 paper. In the 19th century, many scientists discovered that measurement errors in a particular experiment followed a pattern (usual error curve). ), which was very well approximated by this probability distribution.

Central Limit Theorem:

The central limit theorem is a statistical theory that for large sample sizes with finite variance, samples are normally distributed and the mean of the sample is approximately equal to the mean of the entire population.

According to the Question:

The average monthly charge is $142 with a standard deviation of $29.

This is μ = 142 and σ = 29

Part a:

The mean and standard deviation of the sampling distribution of x to show your research and support your argument:

1500 samples (over 30).

According to the Central Limit Theorem

The mean is 142 $

The standard deviation is s = [tex]\frac{29}{\sqrt{1500} }[/tex] = 0.7488

Part b:

Shape of sampling distribution of X.

Due to the large sample size, the shape of the sample distribution is almost normal. According to the central limit theorem, which states that the sampling distribution converges to the normal distribution as the sample size increases. Mean ~ N(142, 0.749²)

Part C:

The probability that the average cable service paid by a sample of cable service customers exceeds $170.

Therefore, by the central limit theorem, the p-value of Z when X= 170

                         Z =  x - μ/ σ

By the Central Limit Theorem:

   Z = 170 - 142 / 0.7488

      = 28/ 0.7488

      = 37.34 exceeds 0.9099

Using the Z probability calculator :

P(Z > 1.335) = 0.09093

Complete Question:

The amount people pay for cable service varies quite a bit but the mean monthly fee is $142 and the standard deviation is $29. the distribution is not normal many people pay about $76 for basic cable and about $160 for premium service but some pay much more. a sample survey is designed to ask a simple random sample of 1,500 cable service customers how much they pay. let x hat be the mean amount paid.

Part a: what are the mean an standard deviation of the sample distribution of x hat show your work and justify your reasoning.

Part b: what is the shape of the sampling distribution of x hat justify your answer.

Part C: what is the probability that the average cable service paid by the sample of cable service customers will exceed $170? show your work.

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Answer:

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Let the cost of a 1 kg of apples be x.

Then pear costs x - 1.12 and banana costs x + 2.35.

The total cost is 25.02: