Why would there be different published values for the normal range of a particular measurement? why do these values have to be updated periodically?

Answer:  B. [tex]\dfrac{1}{98}[/tex].

Step-by-step explanation:

Given : A hat contains four red marbles, two blue marbles, seven green marbles, and one orange marble.

Total marbles = 4+2+7+1 = 14

P(Blue)= [tex]\dfrac{2}{14}=\dfrac{1}{7}[/tex]

P(Orange) = [tex]\dfrac{1}{14}[/tex]

if we randomly select two marbles , then the probability of selecting  one will be orange and one will be blue marble =  P(Blue) x P(Orange)  [both events are independent]

[tex]=\dfrac{1}{7}\times\dfrac{1}{14}=\dfrac{1}{98}[/tex]

Hence, the probability that one will be orange and one will be blue is  [tex]\dfrac{1}{98}[/tex].

Therefore , the correct answer is  B. [tex]\dfrac{1}{98}[/tex].

Answer:

9

Step-by-step explanation:

Number of tulips per row has to be a factor of the number of available tulips.

Red tulips: 45 = 3×3×5

Yellow tulips: 63 = 3×3×7

Highest no. of tulips each row is 9 (3×3 = 9)

Answer:

712/1 410/1

Step-by-step explanation:

theres a fraction for 712 and 410

Donna ate more popcorn than Jack.

A fraction is a part of whole number, and a way to split up a number into equal parts. Or, A number which is expressed as a quotient is called fraction. It can be written as the form of p : q, which is equivalent to p / q.

We have to given that;

Donna ate 7/12 box of popcorn, and Jack ate 4/10 box of popcorn the boxes of popcorn.

Now, By using benchmark fraction we can determine who ate more popcorn as,

Here, Jack ate 4/10 box of popcorn.

⇒ 4/10

⇒ 2/5

And, Donna ate 7/12 box of popcorn.

⇒ 7/12

Take LCM of 5 and 12,

⇒ LCM {5, 12} = 60

Hence, We get;

⇒ 2/5 = 24/60

⇒ 7/12 = 35/60

Clearly, We get;

⇒ 7/12 > 2/5

Hence, Donna ate more popcorn than Jack.

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In 15 coin tosses, an unbiased coin can land tails either exactly 8 times or exactly 5 times in 46,683 ways.

The question relates to the concept of probability, specifically calculating the number of outcomes in coin tosses. The outcomes in which a coin lands tails exactly 8 times or exactly 5 times can be calculated using the formula for combinations, which is nCr = n! / r!(n-r)!. For n=15 (the number of trials) and r=8 (the number of successful outcomes), the number of ways the coin can land tails 8 times is 15C8 = 15! / 8!(15-8)! = 43,680 ways. Similarly, for the coin to land tails 5 times the number of ways is 15C5 = 15! / 5!(15-5)! = 3,003 ways. Therefore, the coin can land tails either exactly 8 times or exactly 5 times in 43,680 + 3,003 = 46,683 ways.

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The correct answer is that there are [tex]\(\binom{15}{8} + \binom{15}{5}\)[/tex] ways for the coin to land tails either exactly 8 times or exactly 5 times.

To solve this problem, we will use the concept of combinations, which is denoted by the binomial coefficient [tex]\(\binom{n}{k}\)[/tex], where [tex]\(n\)[/tex] is the total number of trials and [tex]\(k\)[/tex] is the number of successful trials. The binomial coefficient calculates the number of ways to choose [tex]\(k\)[/tex] successes from [tex]\(n\)[/tex] trials.

For the first part of the question, where we want the coin to land tails exactly 8 times out of 15 tosses, we calculate the number of combinations using the binomial coefficient:

[tex]\[ \text{Number of ways for 8 tails} = \binom{15}{8} \][/tex]

For the second part of the question, where we want the coin to land tails exactly 5 times out of 15 tosses, we calculate the number of combinations similarly:

[tex]\[ \text{Number of ways for 5 tails} = \binom{15}{5} \][/tex]

To find the total number of ways for either event to occur, we add the two separate probabilities together:

[tex]\[ \text{Total number of ways} = \binom{15}{8} + \binom{15}{5} \][/tex]

Now we calculate each binomial coefficient:

[tex]\[ \binom{15}{8} = \frac{15!}{8!(15-8)!} = \frac{15!}{8!7!} \][/tex]

[tex]\[ \binom{15}{5} = \frac{15!}{5!(15-5)!} = \frac{15!}{5!10!} \][/tex]

We can simplify these expressions by canceling out common factors in the numerator and the denominator:

[tex]\[ \binom{15}{8} = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} \][/tex]

[tex]\[ \binom{15}{5} = \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1} \][/tex]

After calculating these values, we would add them together to get the final answer. However, since we are not providing numerical calculations here, we leave the answer in the form of the sum of the two binomial coefficients:

[tex]\[ \text{Total number of ways} = \binom{15}{8} + \binom{15}{5} \][/tex]

This is the number of ways the coin can land tails either exactly 8 times or exactly 5 times in 15 tosses.

Answer:Circumference = 2 \pi r

circumference = 2 \pi (1.5)

circumference = 9.42 feet

The circumference is 9.42 feet, so we need to get fencing for 10 feet (assuming the fence comes in feet, i.e. you cant buy half a foot)

1 feet = $.075

10 feet = $0.75 x 10 = $7.50

Answer:

the answer would be c)  

Step-by-step explanation:

because dy and dx are equal ex10 and that would eventually turn to 5x4 plus 10 x4  

but correct me if im incorrect.

The equation of the function is y = 4 sin( [tex]\frac{1}{2}[/tex] t - [tex]\frac{4}{3}\pi[/tex] ) - 2

Step-by-step explanation:

If the equation is y = A sin (B x + C) + D , where

∵ The amplitude is 4

∴ A = 4

∵ The period is 4π

∴  T= 4π ⇒ T = [tex]\frac{2\pi }{B}[/tex]

∵ The phase shift is [tex]-\frac{4}{3}\pi[/tex]

∴ C =  [tex]-\frac{4}{3}\pi[/tex]

∵ The vertical shift is -2

∴ D = -2

Substitute these values in the y = A sin( [tex]\frac{2\pi }{T}[/tex] t + C) + D

∴ y = 4 sin( [tex]\frac{2\pi }{4\pi }[/tex] t - [tex]\frac{4}{3}\pi[/tex] ) - 2

- Simplify it

∴ y = 4 sin( [tex]\frac{1}{2}[/tex] t - [tex]\frac{4}{3}\pi[/tex] ) - 2

The equation of the function is y = 4 sin( [tex]\frac{1}{2}[/tex] t - [tex]\frac{4}{3}\pi[/tex] ) - 2

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Answer:

-4.99

1.9435

3.25E4

1.56e-9

Step-by-step explanation:

Answer:

The probability that the selected person is a woman engineering major is 0.0396.

Step-by-step explanation:

The proportion of students at the university who are women is 0.42.

P (W) = 0.42

The proportion of students at the university who are engineering majors is 0.18.

P (E) = 0.18

The proportion of engineering majors that are women is 0.22.

P (W|E) = 0.22

The proportion of students at the university that are woman and engineering major is:

[tex]P (W|E)=\frac{P(W\cap E)}{P(E)} \\P(W\cap E)=P(W|E)\times P(E)\\= 0.18\times0.22\\=0.0396[/tex]

Thus, the probability that the selected person is a woman engineering major is 0.0396.

The probability that a randomly selected student from the university is a woman engineering major is 99/100.

To find this probability, we will use the information given about the percentages of women and engineering majors at the university, as well as the percentage of women among the engineering majors.

First, let's denote the total number of students at the university as T.

According to the information given:

- 42% of the students are women, so the number of women students is 0.42T

- 18% of the students are engineering majors, so the number of engineering students is 0.18T

- Of the engineers, 22% are women, so the number of women engineering students is 0.22 \times 0.18T.

Now, we want to find the probability that a randomly selected student is a woman engineering major. Probability is defined as the number of favorable outcomes divided by the total number of possible outcomes. In this case, the favorable outcomes are the number of women engineering majors, and the total number of possible outcomes is the total number of students.

The probability P that a randomly selected student is a woman engineering major is:

[tex]\[ P = \frac{\text{Number of women engineering majors}}{\text{Total number of students}} \][/tex]

[tex]\[ P = \frac{0.22 \times 0.18T}{T} \][/tex]

Since [tex]$T$[/tex]is in both the numerator and the denominator, it cancels out, leaving us with:

[tex]\[ P = 0.22 \times 0.18 \][/tex]

[tex]\[ P = 0.0396 \][/tex]

To express this probability as a fraction, we can write it as:

[tex]\[ P = \frac{396}{10000} \][/tex]

Simplifying this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4, we get:

[tex]\[ P = \frac{99}{2500} \][/tex]

Step-by-step explanation:

The only way the answer could be E is if the x² term under the radical is supposed to be t².

f(x) = ∫₄²ˣ √(t² − t) dt

f'(x) = √((2x)² − 2x) (2)

f'(x) = 2√(4x² − 2x)

f'(2) = 2√(4(2)² − 2(2))

f'(2) = 2√12

The missing figure is attached down

Answer:

The measure of EC is 1 foot

Step-by-step explanation:

Let us revise the cases of similarity

From the attached figure

∵ DE // BC

∴ ∠ADE ≅ ABC ⇒ corresponding angles

∴ ∠AED ≅ ACB ⇒ corresponding angles

In Δs ADE and ABC

∵ ∠ADE ≅ ABC ⇒ proved

∵ ∠AED ≅ ACB ⇒ proved

∵ ∠A is a common angle in the two triangles

∴ Δ ADE is similar to triangle ABC by AAA postulate

From the results of similarity the corresponding sides of the triangles are proportion

∴ [tex]\frac{AD}{AB}=\frac{DE}{BC}=\frac{AE}{AC}[/tex]

∵ AD = 8 feet

∵ DB = 2 feet

∴ AB = AD + DB

∴ AB = 8 + 2 = 10 feet

∵ AE = 4 feet

By using the proportion statement above  [tex]\frac{AD}{AB}=\frac{AE}{AC}[/tex]

∴ [tex]\frac{8}{10}=\frac{4}{AC}[/tex]

By using cross multiplication

∴ 8 × AC = 10 × 4

∴ 8 AC = 40

Divide both sides by 8

∴ AC = 5 feet

∵ AC = AE + EC

∴ 5 = 4 + EC

Subtract 4 from both sides

∴ 1 = EC

∴ The measure of EC is 1 foot

Answer:

8.8%

Step-by-step explanation:

Given that,

Cross-price elasticity between airline tickets and gasoline = 2.2

Price of the gasoline increases by 4​%.

Therefore,

Cross-price elasticity  = Percentage change in the quantity demanded of airline tickets ÷ Percentage change in the price of gasoline

2.2 = Percentage change in the quantity demanded of airline tickets ÷ 4%

2.2 × 4 = Percentage change in the quantity demanded of airline tickets

8.8% = Percentage change in the quantity demanded of airline tickets

The cross-price elasticity of demand, which in this case is 2.2, should theoretically indicate a 8.8% increase in airline ticket demand for a 4% increase in gasoline prices. However, if there is no increase in demand, other factors may be influencing the demand or the elasticity value may be inaccurate.

The cross-price elasticity of demand measures the percentage change in quantity demanded for a product (airline tickets in this case) that results from a percentage change in the price of another product (gasoline in this case). If cross-price elasticity is 2.2, it means that a 1% increase in the price of gasoline leads to a 2.2% increase in the demand for airline tickets. So if gasoline price increases by 4%, you'd expect the demand for airline tickets to increase by 8.8% (4% times 2.2). But, in this situation, if the quantity demanded increases by nothing, it could mean that other factors are playing a role and affecting the demand for airline tickets, or it may be that the relationship between the two goods is not as strong as initially determined.

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Answer:

the answer is g21T

Step-by-step explanation:

a3z, b6Y, C9X, d12W, e15V, f18U next step in the series is E. g24t

A mathematical series consists of a pattern in which the next term is obtained by adding the two terms in-front. An example of the Fibonacci number series is: 0, 1, 1, 2, 3, 5, 8, 13, … For instance, the third term of this series is calculated as 0+1+1=2.

1) Arithmetic Sequences.

2) Geometric Sequences.

3) Exponent Sequences.

4) Two-Stage Sequences.

5) Mixed Sequences.

6) Alternating Sequences.

7) Fibonacci Sequences.

8) A Combination of Sequences' Types.

In mathematics, a series is the cumulative sum of a given sequence of terms. Typically, these terms are real or complex numbers, but much more generality is possible

You can find the right answer in number series by taking the difference between consecutive pairs of numbers, which form a logical series. In this example, the differences between succeeding pairs of numbers are 1,2, 4, 8, 16, and 32. So, the next difference must be 64.

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Answer:

Replacing t with 4, 32.5*4 + 5 is 135 and 28.75 * 4 + 20 is also 135. Since these two are equal, the equation is verified.

or

If you substitute 4 for t in the equation, you get a true statement. That means it would cost the Burns family $135 for 4 tickets and parking for either lower-level or middle-level seats.

Step-by-step explanation:

Answer:The price for the lower level is $32.50 a ticket, plus $5.00 for a discounted parking pass. The middle-level tickets are $28.75 each, plus $20.00 for a parking pass.

Step-by-step explanation: assigment

Answer:

  L = s^2/(30.25Cd)

Step-by-step explanation:

In accident investigation, the speed of a vehicle can be estimated using a polynomial function that relates speed (s) to the length of skid marks (L). The drag coefficient Cd will depend on the condition of the road surface and tires, but might be expected to be between 0.7 and 0.8.

If the skid marks end in a collision, the length of the marks that might have been made can be estimated using this formula, then that length added to the actual length of marks to estimate the original speed. The speed at the point of collision can be estimated by the damage caused, and/or the movement created.

In the above formula, length is in feet, and speed is in miles per hour.

In Physics, a polynomial function such as[tex]y = -gt^2 + v0*t + h0[/tex] can model an object's motion under gravity. A rational function like P = a / (1 + bQ) can model the relationship between supply and demand in Economics.

In the field of Physics, polynomial functions are often used to model physical phenomena. For example, the motion of an object under the force of gravity can be represented by a second-degree polynomial, or quadratic function. The equation[tex]y = -gt^2 + v0*t + h0[/tex]is an example of a quadratic function modeling free fall, where g represents the acceleration due to gravity, v0 is initial velocity, t represents time and h0 is initial height.

On the other hand, rational functions are used in various fields too. In Economics for instance, it can model situations like the relationship between supply and demand in market equilibrium. A simple model could be P = a / (1 + bQ) where P represents the price, Q is the quantity of good sold, and a, b are constants.

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Answer:

$151,800

Step-by-step explanation:

For the publisher, the expected revenue is $26.40 per copy. For 225,000 copies, the revenue is [tex]225000\times26.40 = 5,940,000[/tex]

The expenses incurred by the publisher are as follows:

Cost of 1 print = $3.75

Cost of 225,000 prints = [tex]225000\times3.75 = 843,750[/tex]

Editing/Design = $27,500

Publicity/Advertising/Administrative = $135,150

Author's fee = 6.5% of retail price per copy for 225,000 copies = [tex]225000\times26.40\times6.5/100= 386,100[/tex]

Total cost = 843,750 + 27,500 + 135,150 + 386,100 = $1,392,500

Let the cost of kelp for copies be k.

Then the total cost = 1,392,500 + k

If the expected profit is 4,092,100, then

Revenue = total cost + profit

5,940,000 = 1,392,500 + k + 4,092,100

5,940,000 = 5,484,600 + k

k = 5,940,000 - 5,484,600 = 455,400

Since Carlita is paying [tex]\frac{1}{3}[/tex] of k, her share of the kelp =

[tex]\frac{1}{3}\times455400 = 151800[/tex]

Carlita will pay $151,800

Answer:

1/2

Step-by-step explanation:

(1/4)^x

when x=(1/2)

(1/4)^(1/2)

this can be rewritten in this way

√(1/4)

2 is the radical

then we distribute the root

√1 / √4

and solve

1/2

Answer:

√820

√36/83

√48/81

√111

√17

Step-by-step explanation:

Wszystkie liczby, które nie są wymierne, są uważane za irracjonalne. Liczbę niewymierną można zapisać jako liczbę dziesiętną, ale nie jako ułamek. Liczba niewymierna ma niekończące się powtarzające się cyfry po prawej stronie przecinka dziesiętnego.

mam nadzieję, że to pomoże

Answer:

3,604,389

Step-by-step explanation:

Monthly payment P = [tex]\frac{a}{[(1+r)^{n} - 1] / [r(1+r)^{n}] }[/tex]

where a = total loan amount, r = periodic rate, n = number of payment periods

a = 500,000 ; r = 0.12 ; n = (5 x 12) = 60 months

P = [tex]\frac{500000}{[(1+0.12)^{60} - 1] / [0.12 (1+0.12)^{60} ] }[/tex]

P = [tex]\frac{500000}{(896.597) / (107.712)}[/tex]

p = 60,073.151

Total amount paid = 60073.151 x 60 = 3,604,389

Final answer:

Liza will have to pay a monthly amount of P2,997.75 for a loan of P500,000 at a 12% interest rate compounded monthly over 5 years. The total amount paid at the end of the term will be P179,865.

Explanation:

To determine the total sum Liza will pay for a loan of P500,000 with an interest rate of 12% compounded monthly over 5 years, we first need to calculate the monthly payment she would need to make. The problem is essentially asking to find the annuity payment for a present value, which can be calculated using the formula for present value of an ordinary annuity:

PV = PMT * [1 - (1 + r)-n] / r

Where:

PV = Present Value of the annuity (P500,000 in this case)

PMT = Monthly payment

r = Monthly interest rate (12% per year compounded monthly, so 0.12/12 per month)

n = Total number of payments (5 years × 12 months/year = 60 payments)

First, we calculate the monthly interest rate:

r = 0.12 / 12 = 0.01 or 1%

Now, setting up the equation:

500,000 = PMT * [1 - (1 + 0.01)-60] / 0.01

We find that PMT = 500,000 / 166.7916 (Using the present value factor derived from the reference information)

PMT = P2997.752

Therefore, the monthly payment Liza has to make is P2,997.75. To find the total sum paid after 5 years:

Total Sum = Monthly Payment * Number of Payments

Total Sum = P2,997.75 * 60 = P179,865

Answer:

  4·6 = 24 ≠ 12 = LCM(4, 6)

Step-by-step explanation:

Any pair of numbers with a common factor will refute Sarah's conjecture.

One such pair is 4 and 6, which have a product of 24. The LCM of 4 and 6 is 12, which is that product divided by their common factor of 2.

Sarah's statement is incorrect.

Sarah's statement that the least common multiple (LCM) of any two whole numbers is obtained by merely multiplying them together is incorrect. For example, consider the whole numbers 8 and 12. The product of these two numbers is 8  * 12 = 96. However, the LCM of 8 and 12 is actually 24, since 24 is the smallest number that is divisible by both 8 and 12. To find the LCM of two numbers, you need to list their multiples and find the smallest number that appears on both lists (in this case, the multiples of 8 are 8, 16, 24, 32, ... and the multiples of 12 are 12, 24, 36, 48, ...).

The arrival and departure of the airplanes are arranged according to a

schedule that is sequential.

The number of planes it passes going in the opposite direction is 13 planes

Reasons:

The given parameters are;

The frequency of airplane departure = Every 30 minutes

Time it takes to fly from LA to NY = 3 hours 5 minutes

Required;

The number of planes it passes going in opposite direction.

Solution:

As the airplane takes off from NY, we have that the airplane that took off 3

hours earlier from LA is 5 minutes away from the NY.

The airplane are moving in opposite directions, therefore, the relative

speed of the airplane is twice the speed of each airplane.

The time after takeoff at which the airplane from NY pass the first airplane

from LA is 5 minutes ÷ 2 = 2.5 minutes.

The time after which the airplane passes subsequent airplane is two times

faster or half the normal time which is 30 minutes ÷ 2 = 15 minutes, which

gives;

3 hours 5 minutes = 185 minutes

We have an arithmetic progression with first term, a = 2.5, common

difference = 15, and nth term, aₙ = 185, which gives;

Given that the number of planes that the airplane passes are whole

number (discrete) values, we have;

The number of planes it passes going in the opposite direction, n = 13 planes

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We want to see how many planes does a plane that flies from NY to LA sees during the travel.

The answer is 6 planes.

We know that every 30 minutes, an airplane departs from LA to NY.

The travel does take 3 hours and 5 minutes (or 185 minutes in total).

At the same time that an airplane departs from LA to NY, one departs from NY to LA, we want to see how many airplanes it will see.

The number of planes that it will see is given by the number of times that 30 minutes (the interval between planes going from LA to NY) enters in 185 minutes (the total time of the flight).

This gives:

185/30 = 6.2

But we can't have a 0.2 of a plane, so we round the result to the nearest whole number, 6.

This means that on the flight from NY to LA, you can expect to see 6 planes going from LA to NY.

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The average rate of change for the function is 2

Step-by-step explanation:

The average rate of change of a function (also equal to the slope of the function) over a certain interval [tex][x_1,x_2][/tex] is given by

[tex]m=\frac{f(x_2)-f(x_1)}{x_2-x_1}[/tex]

Where f(x) is the value of the function at x.

In this problem, the function is

[tex]f(x)=2x+2[/tex]

And the interval is −1 ≤ x ≤ 1, so we have:

[tex]x_1 = -1\\f(x_1)=f(-1)=2(-1)+2=0[/tex]

And

[tex]x_2=1\\f(x_2)=f(1)=2(1)+2=4[/tex]

Therefore, the average rate of change of the function is

[tex]m=\frac{+4-0}{1-(-1)}=\frac{4}{2}=2[/tex]

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Answer:

(a) 2 feet.

(b) 2 feet.

Step-by-step explanation:

We have been given that the velocity function [tex]v(t)=\frac{1}{\sqrt{t}}[/tex] in feet per second, is given for a particle moving along a straight line.

(a) We are asked to find the displacement over the interval [tex]1\leq t\leq 4[/tex].

Since velocity is derivative of position function , so to find the displacement (position shift) from the velocity function, we need to integrate the velocity function.

[tex]\int\limits^b_a {v(t)} \, dt[/tex]

[tex]\int\limits^4_1 {\frac{1}{\sqrt{t}}} \, dt[/tex]

[tex]\int\limits^4_1 {\frac{1}{t^{\frac{1}{2}}} \, dt[/tex]

[tex]\int\limits^4_1 t^{-\frac{1}{2}} \, dt[/tex]

Using power rule, we will get:

[tex]\left[\frac{t^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}}\right] ^4_1[/tex]

[tex]\left[\frac{t^{\frac{1}{2}}}{\frac{1}{2}}}\right] ^4_1[/tex]

[tex]\left[2t^{\frac{1}{2}}\right] ^4_1[/tex]  

[tex]2(4)^{\frac{1}{2}}-2(1)^{\frac{1}{2}}=2(2)-2=4-2=2[/tex]

Therefore, the total displacement on the interval  [tex]1\leq t\leq 4[/tex] would be 2 feet.

(b). For distance we need to integrate the absolute value of the velocity function.

[tex]\int\limits^b_a |{v(t)|} \, dt[/tex]

[tex]\int\limits^4_1 |{\frac{1}{\sqrt{t}}}| \, dt[/tex]

Since square root is not defined for negative numbers, so our integral would be [tex]\int\limits^4_1 {\frac{1}{\sqrt{t}}} \, dt[/tex].

We already figured out that the value of [tex]\int\limits^4_1 {\frac{1}{\sqrt{t}}} \, dt[/tex] is 2 feet, therefore, the total distance over the interval [tex]1\leq t\leq 4[/tex] would be 2 feet.

Answer:

Step-by-step explanation:

Answer:

Step-by-step explanation:

Let us assume that we have n different dots on a paper.  We are to connect pairwise by a line.  We have to find out how many lines can be formed.

Let us prove by induction.

If there is one dot then we have no line = 1(1-1) =0

Thus n(n-1) is true for 1 dot

Let us assume that for n dots we have n(n-1) lines

Add one more point now total points are n+1.

Already the existing n points are connected by a line.

So the extra point has to be connected to each of n point

i.e. n lines should be added from the new point to the n points and again n lines from the points to the new point(Assuming lines are different if initial and final point are different)

So 2n lines would be added

So total number of lines for n+1 points

[tex]= n(n-1) +n+n= n^2-n+2n \\= n^2+n\\=n(n+1)[/tex]

Thus true for n+1 if true for n.  Already true for n =1

So proved by induction for all natural numbers n.

Using mathematical induction, we prove the number of lines connecting pairs of dots for n dots is n(n-1)/2. We confirm the base case for n=1 and then assume the proposition holds for k, leading to it holding for k+1, thus proving it for all natural numbers n.

The question involves calculating the number of lines that can be drawn to connect each pair of dots on a paper. To find this number for n dots, we can use mathematical induction.

Let's denote P(n) as the proposition that for n dots, the number of lines required to connect each pair of dots is n(n-1)/2, which simplifies to the sum of the first n-1 natural numbers.

Base Case

For n=1, there are no lines needed since there is only one dot, and no connections to be made: P(1) holds because 1(1-1)/2 = 0.

Inductive Step

We assume that P(k) is true for some natural number k, meaning that k dots require k(k-1)/2 lines. Now, we consider n=k+1 dots. The new dot must be connected to each of the k existing dots, adding k new lines. Therefore, the total number of lines for k+1 dots is k(k-1)/2 + k, which simplifies to (k+1)k/2, and is exactly P(k+1).

By the principle of mathematical induction, since we have proven the base case and the inductive step, we conclude that P(n) holds for all natural numbers n.

The probability that a person who preferred a picnic responded by email is 2/3.

To find the probability that a person who preferred a picnic responded by email, we need to compare the number of email responses to the total number of responses. Let's assume the number of phone responses is x. Since the number of email responses is twice the number of phone responses, the number of email responses is 2x.

The total number of responses would be the sum of the email and phone responses, which is 2x + x = 3x.

The probability that a person who preferred a picnic responded by email would be the number of email responses divided by the total number of responses, which is 2x/3x = 2/3.

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To find the probability that a person selected at random who preferred a picnic responded by email, you take the 5% who responded by email and divide it by the total response percentage (30%). This results in a probability of 1/6, or approximately 16.67%.

We are given that 20% who preferred a picnic responded by phone and that the number of email responses was twice the number of phone responses. This means that 5% responded by email because twice 5% is 10%, and 10% plus the original 20% for phone responses gives us the 30% of people who responded in total. To find the probability that a randomly selected person who preferred a picnic responded by email, we can use relative frequency probability calculation.

The probability (P) can be expressed by the following equation:

P (Email | Picnic) = Number of people who preferred a picnic and responded by email / Total number of people who preferred a picnic

Given that the number of email responses is twice the number of phone responses, and knowing the percentage that corresponds to each mode of response:

To calculate the probability that the person responded by email, we take the proportion:

P (Email | Picnic) = 5% / 30%

After calculating this, we get: P (Email | Picnic) = 1/6 or approximately 0.1667, which is about 16.67%.

Answer:

Volume of cylinder [tex]=2797.7[/tex]

Step-by-step explanation:

Let [tex]a[/tex] be diameter and [tex]b[/tex] the height of the cylinder

[tex]Then\ radius(r)=\frac{diamater\ (a)}{2}\\\\r=\frac{18}{2}\\\\r=9[/tex]

Volume of cylinder [tex]=\pi\times(radius)^2\times(height)[/tex]

[tex]=\pi\times(9)^2\times11\ \ \ \ \ \ (\pi=3.14)\\\\=3.14\times9\times9\times11\\\\=2797.74\\\\\approx2797.7[/tex]

Answer: The initial number of shoes in the store is 300.

Step-by-step explanation:

Given : Kelly is a salesperson at a shoe store, where she must sell a pre-set number of pairs of shoes each month.

At the end of each work day the number of pairs of shoes that she has left to sell that month is given by the equation

[tex]S=300-15x[/tex] , where S =  Number of pair of shoes Kelly still needs to sell.

x =  Number of days she has worked that month.

When x= 0 , we get S= 300

i.e. When she started working in that month , she has 300 pairs of shoes to sell.

Therefore , Number 300 means that the initial number of shoes in the store is 300

Answer:

3

Step-by-step explanation:

You'll need to use the Angle Bisector Theorem.

CU / ZU = BC / BZ

From the first part of the problem, you used Pythagorean theorem to find that BZ = 10.

Let's say that CU = x.  That means ZU = 8 − x.  Plugging in values:

x / (8 − x) = 6 / 10

Cross multiply:

10x = 6(8 − x)

Solve:

10x = 48 − 6x

16x = 48

x = 3

Answer:

Step-by-step explanation:

2,2+1,2+1+3,2+1+3+5,...

2,3,6,11,18,27