A pentagon is made up of an equilateral △ABC of side length 2cm on top of a square BCDE. Circumscribe a circle through points, A,D and E. The radius of the circle is
A1+√32
B5−2√3
C 2
D 1+√3

Answer:

B

So convert the number over 200

The measure of the largest angle of the triangle is 93°, while the measure of the smallest and the third angle is 31° and 56° respectively. Considering that the measure of the third angle is 25 degrees more than that of the smallest angle.

In any given triangle, the sum of its interior angles is equal to 180°.

Let the smallest angle be represented by x, so that the other angles are 3x and x + 25.

thus

x + 3x + x + 25 = 180° {collect like terms}

5x + 25° = 180°

5x + 25° - 25° = 180° - 25° {subtract 25° from both sides of the equation}

5x = 155°

x = 155°/5 {divide through by the coefficient of x}

x = 31°

thus the other angles are derived by substituting the value 31° for x as follows;

3(31) = 93°

31° + 25° = 56°

Therefore, the largest angle of the triangle 93°, the smallest angle is 31° and the measure of the third angle is 56°.

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Answer:

undefined

Step-by-step explanation:

the line has always the same x, so it is a vertical line

A vertical line has an undefined slope

Answer:

4500

Step-by-step explanation:

This can be solved in more than one way. One method is to use permutations and see which digits can appear how many times in which position in the possible list of even numbers between 1000 and 9999

However, I think approaching this from an arithmetic progression perspective is easier to explain.

An arithmetic progression or sequence is a sequence of numbers such that there is a common difference between consecutive numbers.

For example, 2, 4, 6, 8, 10... is an arithmetic sequence with a common difference.

Given the first term in the sequence, we can find the nth term using the formula:

[tex]a_n = a_0 + (n-1)\cdot d[/tex]

where
[tex]a_n[/tex] is the nth term,
[tex]a_0[/tex] is the first term
[tex]d[/tex] is the common difference

Treating the even numbers from 1000 to 9999 as an arithmetic sequence, we get the following:

[tex]a_0 = 1000[/tex]
[tex]a_n = 9998[/tex]  (since the last number has to be even
[tex]d = 2[/tex]

Let [tex]n[/tex] be the number of possible values

Plug this into the formula and solve for [tex]n[/tex]

9998 = 1000 + (n-1)2
9998 = 1000 + 2n - 2

9998 = 998 + 2n
9998-998 = 2n
9000 = 2n

Switch sides:

2n = 9000

Divide by 2 to get
n = 9000/2 or

n = 4500

Since the question specified that the arrow had to reach the height only after reaching its maximum altitude, then the only correct answer is 6.096 seconds.

A mathematical expression with the equals sign is referred to as an equation. Algebra is frequently used in equations. When performing calculations but not knowing the precise number, algebra is used.

Give,

[tex]20 = -16t^{2} + 100t + 5[/tex]

[tex]= > -16t^{2} + 100t - 15 = 0[/tex]

[tex]= > t = \frac{-100 ± \sqrt{100^{2} - 4(-15)(-15) } }{2(-16)} \\\\= > t = \frac{-100 ± \sqrt{9040} }{-32} \\\\Here, \sqrt{9040} = 4\sqrt{565} \\ \\= > t = \frac{-100 ± \ 4sqrt{565} }{-32} \\\\= > t = \frac{-25 ± \ sqrt{565} }{-8} \\\\= > t_{1} = 6.096 sec \\ = > t_{2} = 0.154 sec[/tex]

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The quadratic equation with the roots  1/α and 1/β using the given equation is equal to

As given in the question,

Given equation is:

2x²−5x−6=0

Roots of the given equations are α and β

Compare with standard quadratic equation:

ax² + bx + c = 0

a = 2 , b= -5 , c = -6

Sum of the roots are:

α + β = -b/a

        = -( -5/2)

        = 5/2

Product of the roots

αβ = c/a

    = -6/2

    = -3

Now ,

( α + β )/ αβ = (5/2) / -3

⇒1/ β + 1/α = -5/6

⇒ 1/α + 1/β = -5/6

And ,

1/αβ = -1/3

Quadratic equation with roots 1/α and 1/β is given by:

x² - ( 1/α + 1/β)x + 1/αβ =0

⇒x² - (-5/6)x + (-1/3) = 0

⇒x² + (5/6)x -(1/3) =0

Therefore, the quadratic equation with roots 1/α and 1/β is equal to

x² + (5/6)x -(1/3) =0

The complete question is :

Suppose the roots of the equation 2x²−5x−6=0 are α and β.Find the quadratic equation with roots 1/α and 1/β.​

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The slope-intercept equation that represents the proportional relationship in this problem is given as follows:

y = 0.5x.

The slope-intercept representation of a linear function is defined as follows:

y = mx + b.

In which the coefficients are given as follows:

A proportional relationship is a linear function with an intercept of zero, hence the equation is defined as follows:

y = mx.

The variables in this problem are given as follows:

The plane consistently climbs 20 feet for every 40 feet it moves horizontally, hence the slope is obtained as follows:

m = 20/40

m = 0.5.

As the slope is given by the vertical movement divided by the horizontal movement, and hence the equation is given as follows:

y = 0.5x.

The problem asks for the slope-intercept equation that represents the proportional relationship in this problem.

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Alma should sit  7.5 feet apart from the fulcrum, In order to balance the seesaw.

What is Ratio ?

A ratio in mathematics demonstrates how many times one number is contained in another. For instance, the ratio of oranges to lemons in a dish of fruit is eight to six if there are eight oranges and six lemons.

Given, The weights of Alma and Bella are in the ratio 5:7 respectively.

∴ Alma's weight = 5/12 and Bella's weight = 7/12.

they are sitting 18 feet apart on a seesaw.

It means the distance between Alma and Bella is 18 feet.

Now, Alma should sit feet far from the fulcrum in order to balance the seesaw :

       = Distance between Alma and Bella × Weight of Alma

       = 18 feet × 5/12

       = 7.5 feet.

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Answer:

Slope M=6

Step-by-step explanation:

The time taken for the cargo to hit the ground will be 0.6 seconds and it travels in the horizontal direction for around 14.69 feet.

The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign. The point-slope form, standard form, and slope-intercept form are the three main types of linear equations. The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.

Here,

The calculations are attached in images.

To calculate t we will use that is y because at the moment of cargo will have a height 0, so we get,

= −4.9t² +1.75

-1.75=-4.9t²

t²=-1.75/-4.9

t²=√0.3571

=0.6 seconds

Now we will calculate x, then y = 0

because since the cargo will be on the ground it will no have height, so

0= -0.0081x²+1.75

-1.75 =-0.0081x²

x²=-1.75/-0.0081

x= √216.04 = 14.69 feet

It will take the cargo 0.6 seconds to hit the ground, and it will travel 14.69 feet in a horizontal direction.

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After 40 days 4467 bees will be the population after 40 days.

An exponential function is a Mathematical function in the form f (x) = ax, where “x” is a variable and “a” is a constant which is called the base of the function and it should be greater than 0.

Given here: A hive of bees contains 27 bees when it is first discovered. After 4 days, there are 45 bees. The population of bees also increase exponentially.

let the function that gives the population of bees after given days

f(x)=aˣ27

a⁴×27=45

       a= [tex]\sqrt[4]{45/27}[/tex]

        a=1.1362

Thus for x= 40 we have f(40)=1.1362⁴⁰×27

                                               = 4465.69

                                               =4467 (approx)

Hence, After 40 days 4467 bees will be the population after 40 days

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Answer:

x = 4

Step-by-step explanation:

A line with undefined slope is a vertical line.

Vertical lines have equation in the form:

x = anumber

In this case they gave you the info you need (4,6) The 4 is the x and the 6 is the y. (4,6) is in the form (x,y).

So x is 4.

x = 4

The number of pizzas that must be sold to make at least a profit of $500 is 220.

Profit is total revenue less total cost.

Profit = revenue - cost (price of one pizza x number of pizzas sold) - [cost to rent the equipment + (cost of one pizza x number of pizzas made)] ≥ profit

$4.50p - ($250 + $2p) ≥ $300

$4.50p - $250 - $2p ≥ $300

$2.50p ≥ $300 + $250

$2.50p ≥ $550

Divide

p ≥ $550 / 2.50

p ≥ 220

This illustrates the concept of inequality.

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The portion of gold in three bracelets that are shown. The bracelet that has a portion of gold is 0.65. The correct option is c.

Gold is actually a very soft metal. Rings constructed of pure gold would bend and disfigure over time. Other elements are added to strengthen the ring. 100% gold is far too soft and far from as durable as it should be.

Gold is a pure element. It is found in the earth's crust when making jewellery from gold, and some other metals like copper or silver is mixed with it.

Therefore, the correct option is c. 0.65.

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The probability that all three pairwise distances between the points are less than the radius of the circle is 1/12.

What is probability?

The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true. An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty.

Since it doesn't matter where the first point is picked, it can be placed anywhere on the circle.

The following point must be within an arc of 60 degrees on either side, providing a total of 120 degrees and a chance of 1/3. The final point must be situated within 60 degrees of the first two.

If the first two points are apart by 60 degrees, the third point can be placed within an arc with a minimum area of freedom of 60 degrees and a chance of 1/6. If the first two points are the same, the third point can be placed with a maximum degree of freedom of a 120 degree arc and a 1/3 probability.

The chance changes linearly with increasing distance from the first point, up to a maximum of 60 degrees.

As a result, we may find 1/4, the average likelihood we can position the third point based on a fluctuating second point, by averaging the probabilities at either end.

Total probability is 1 × 1/3 × 1/4 = 1/12.

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The pattern of the value is given as aₙ =  2aₙ₋₁, where n is a set of natural numbers,

Given that,
We have to determine the pattern of the given series of numbers.

Geometric progression is a sequence of series whose ratio with adjacent values remains the same.

Here,
Given series,
2, 4, 8,16, 32, .... so on
From observation, it is to be determined that each proceeding value is twice the previous value.
So,
The pattern is given as,
aₙ =  2aₙ₋₁

Thus, the pattern of the value is given as aₙ =  2aₙ₋₁, where n is a set of natural numbers,

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There are 19 tragedies out of 52 plays, so the probability of selecting a tragedy is 19/52. Therefore, the odds against selecting a tragedy are 33/52

Probability is a way of selecting how likely something random is to happen. It is a mathematical idea that is used to estimate the probability of an event occurring given a collection of circumstances or results. Probability is commonly expressed as a fraction or decimal, with a value of 1 representing an event that is certain to occur and a value of 0 representing an event that is impossible to occur. In order to generate predictions and to assist in making decisions based on the possibility of various events, probability might be utilized. It is a crucial instrument in a variety of disciplines, such as physics, economics, and statistics.

How to solve?

probability of selecting a tragedy = 19/52.

The odds against selecting a tragedy = 1 - 19/52 = 33/52

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Given that, AB ≅ ED, CA ≅ CE, and AE bisects BD, the two-column proof that shows ΔABC is congruent to  ΔEDC by SSS Congruence Theorem.

The side-side congruence theorem (SSS) states that if two triangles has three pairs of sides that are congruent to each other, then both triangles are congruent.

We are given the following information that is;

AE bisects BD; AB is congruent to DE

Since AE bisects BD, thus the third pair of sides of ΔABC and ΔEDC, BC and CD, will be congruent.

This implies that both triangles has three pairs of congruent sides (AB ≅ ED, CA ≅ CE, and BC ≅ CD), meets the criterion of the SSS Congruence Theorem.

Therefore,  ΔABC ≅  ΔEDC by SSS Congruence Theorem.

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The 5 separate possibility that officer seats may be filled in 336 different ways.

7 + 4 + 6 = 17 students are running for the 5 officer positions. If at least 3 of the 5 positions must be filled by juniors, there are 6 ways to choose which 3 juniors will fill these positions.

Then there are 7 + 4 - 3 = 8 remaining students 3 juniors and 5 from the remaining sophomores and freshmen to fill the remaining 2 positions.

There are 8 ways to choose the first of these 2 positions and 7 ways to choose the second, so there are a total of 6 × 8 × 7 = 336 ways to choose the officers.

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given the equation y = 3x, complete the table of values.

x = -2

y = 3x

y = 3(-2)

y = -6

______________________

x = -1

y = 3x

y = 3(-1)

y = -3

______________________

x = 0

y = 3x

y = 3(0)

y = 0

______________________

x = 1

y = 3x

y = 3(1)

y = 3

______________________

x = 2

y = 3x

y = 3(2)

y = 6

______________________

As described in the analytic geometry, the points are(1.1, -5 )

The area of algebra known as analytical geometry uses an ordered pair of numbers known as coordinates to determine the position of a point on a plane. It is utilized to model many plane objects, including points, lines, and others. The terms Coordinate geometry and Cartesian geometry are both used to refer to analytical geometry.

We have to locate the point where the line is through the x line and y line coordinates

Therefore the point is  (1.1, -5 )

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The probability of both Nic and Tim winning the raffle at the same time is 1/100.

Probability is a mathematical measurement of a likelihood of an event to happen. Given that both Nic and Tim each purchase one raffle ticket out of 10 tickets and that there will be two winners selected, the odds of either Nic or Tim winning the raffle out of 10 ticket is 1/10.

Since the odds of either Nic or Tim winning the raffle ticket is 1 out of 10 or 1/10, we can then calculate the mathematical probability of two events occurring at the same time with a multiplication rule formula as follows:

1/10×1/10=1/100

Hence, the odds of both Nic and Tim winning at the same time is 1 out of 100 or 1/100.

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a) is 0.386 and of b) is 0.017

What is Probability?

One use of probability theory is calculating the odds that experiments will succeed or fail. Probabilities can be used to determine, for example, the possibility of flipping a coin and obtaining heads or tails, or the likelihood of making a research error. Understanding the formula for estimating probabilities in equiprobable sample spaces, as well as the probabilities of the complementary event, etc., as well as the likelihood of two occurrences joining together, is essential to understanding this area of mathematics.

Given that:

no of seniors = 3

no of juniors = 15

a) probability that a group consist of a senior and two juniors:

                      [tex]=\frac{^{3} C_{1} *^{15} C_{2 }}{^{18} C_{3}}\\\\=\frac{3 * 105}{316} \\\\=0.386[/tex]

b) if the name of a student given then there is only one way to select it

For example = we have to select 1 senior which name is already given means there is only one way to select it, and 2 junior in which one name is given then it can only selected by one way, but still one junior student has to be selected out of remaining 14 student which can be selected by [tex](^{14}C_{1} =14)[/tex] different ways.

                         [tex]=\frac{1 *1*{^{14} C_{1}}}{^{18} C_{3}}} \\\\= \frac{14}{316} \\\\= 0.017[/tex]

Hence, The answer of a) is 0.386 and of b) is 0.017

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The number of cars were are, 550.

What is Vehicle?

A device, typically with wheels and an engine, used to move people or things, particularly on land: Yesterday evening, a truck driver was killed when his car flipped over. Buses, trucks, and other road vehicles are included. A farm vehicle is a tractor.

Wagons, bicycles, motorized (cars, trucks, buses, mobility scooters for the disabled), railed (trains, trams), amphibious (screw-propelled, hovercraft), watercraft, and aircraft are all examples of vehicles (airplanes, helicopters, aerostats)

The ratio of buses to cars entering the lot was 2 to 11.  

650 vehicles entered the lot altogether.

Base on the given information,

650 x 11 ÷ (2 + 11)

Simplifying,

7150 ÷ 13

⇒  550

Hence, the number of cars were are, 550.

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The linear equation associated with slope is 7, and (1, 8) is on the line is y = 7x + 1.

A straight line on the coordinate plane is represented by a linear function.

A linear function always has the same and constant b.

The formula for a linear function is f(x) = ax + b, where a and b are real values.

A linear equation or function is given as ;

y = mx + b

Now,

At x = 0 ⇒ y = b so it will be a y-intercept.

Put m = 7 , x = 1 and y = 8

8 = 7(1) + b

b = 1

Thus, y = 7x + 1

Hence "The linear equation associated with slope is 7, and (1, 8) is on the line is y = 7x + 1".

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The ∠MPN is supplementary to ∠NPI.

What is the supplementary angles?

Supplementary angles are those angles that sum up to 180 degrees. For example, angle 130° and angle 50° are supplementary angles because the sum of 130° and 50° is equal to 180°.

Supplementary angles sum to 180°.  

(Note that supplementary angles don't have to be next to each other).

Given ∠NPI = 55.6°.

Therefore, to find the measure of the angle that is supplementary to ∠NPI, subtract the measure of ∠NPI from 180°:

⇒ 180° - ∠NPI = 180° - 55.6° = 124.4°.

The angle that measures 124.4° is ∠MPN.  

Therefore, The ∠MPN is supplementary to ∠NPI.

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Note : Given question is incomplete, the complete question is:

Which angle is supplementary to ∠NPI?

five angles all with common vertex of P, with angle NPI measuring 55.6 degrees, angle IPK measuring 44.2 degrees, angle KPL measuring 80.2 degrees, angle LPM measuring 55.6 degrees, and angle MPN measuring 124.4 degrees

∠IPK

∠NPL

∠MPN

∠LPM

Answer:  Perpendicular

Reason:

y = -5 is horizontal

x = 3 is vertical

A horizontal line is always perpendicular to a vertical line. The two lines form a 90 degree angle.

a= μ-3.16*σ , b= μ+3.16*σ if each fish caught is equally likely to be any one of these 4 distinct types.

Chebyshev's inequality is a probability theory that ensures that, over a vast range of probability distributions, no more than a particular proportion of values would be present within a selected limits or range as from mean. In other words, only a certain fish caught will be discovered within a given range of the distribution's mean.

The formula for which no more than a particular number of values can exceed is 1/K2; in other words, 1/K2 of a distribution's values can be more than or equal to K standard deviations away from the distribution's mean. Furthermore, it asserts that 1-(1/K2) of a distribution's values must be within, but not include, K standard deviations of the distribution's mean.

How to solve?

from Chebyshev's inequality for Y

P(| Y - μ|≤ k*σ ) ≥ 1-1/k²

where

Y =  the number of fish that need be caught to obtain at least one of each type

μ = expected value of Y

σ = standard deviation of Y

P(| Y - μ|≤ k*σ ) = probability that Y is within k standard deviations from the mean

k= parameter

thus for

P(| Y - μ|≤ k*σ ) ≥ 1-1/k²

P{a≤Y≤b} ≥ 0.90 →  1-1/k² = 0.90 → k = 3.16

then

P(μ-k*σ≤ Y ≤ μ+k*σ ) ≥ 0.90

using one-sided Chebyshev inequality (Cantelli's inequality)

P(Y- μ≥ λ) ≥ 1- σ²/(σ²+λ²)

P{Y≥b} ≥ 0.90  →  1- σ²/(σ²+λ²)=  1- 1/(1+(λ/σ)²)=0.90 → 3= λ/σ → λ= 3*σ

then for

P(Y≥ μ+3*σ ) ≥ 0.90

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The number of ways there to pick five people for a committee is if there are six men and eight women, and the selection must include at least one man and one woman 1940.

What is the combination?

Combinations are mathematical operations that count the number of possible configurations for a set of items where the order of the selection is irrelevant. In combinations, you can select the items in any order.

There are 6 different men and 8 different women.

The committee must include at least 1 man and 1 woman.

Thus let us first see the possible cases where this condition does not satisfy.

We select 5 men and no women = (6C5)(8C0) = (6)(1) =6

We select 5 women and no men = (8C5)(6C0) = 56

Total no.of possibilities of selecting 5 people in any way = 14C5    = 2002

So the no.of possibilities with at least 1 man and 1 woman = 2002 - 6 - 56 = 1940.

Hence, the number of ways there to pick five people for a committee is if there are six men and eight women, and the selection must include at least one man and one woman 1940.

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