Locate the value of the constant π on a number line using rational numbers.

The exact binomial probability that 45 or more people in the survey get enough exercise is 0.0853.

The mean (also called the arithmetic mean or average) is a measure of central tendency that represents the typical or average value of a set of data. The mean is calculated by summing up all the values in the data set and dividing by the number of values.

a. To calculate the probability that 45 or more people in the survey get enough exercise using a normal approximation for p hat, we need to first find the mean and standard deviation of the sample proportion.

The mean of the sample proportion is:

μ = p = 0.20

The sample proportion's standard deviation is:

σ = √[(p(1-p))/n] = √[(0.20)(0.80)/200] = 0.034

Using the normal distribution with a continuity correction, we can find the probability of 45 or more people getting enough exercise:

z = (45 - 0.20200 + 0.5) / √(0.200.80*200) = 1.24

P(Z > 1.24) = 0.1082

Therefore, the probability that 45 or more people in the survey get enough exercise is approximately 0.1082.

b. To find the exact binomial probability that 45 or more people in the survey get enough exercise, we can use the binomial cumulative distribution function (CDF) with n = 200 and p = 0.20:

P(X ≥ 45) = 1 - P(X < 45) = 1 - binomcdf(200, 0.20, 44) = 0.0853

Therefore, the exact binomial probability that 45 or more people in the survey get enough exercise is 0.0853.

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Answer: T'=(-1,-1), U'=(-3,-1), V'=(-1,-4)

Step-by-step explanation:

Answer:

75

Step-by-step explanation:

f(4) means to use 4 in place of x.

f(x) = 3(5)^(x-2)

fill in 4 for

f(4) = 3(5)^(4-2)

do the 4-2 subtraction first

f(4) = 3(5)^2

do the 5 to the 2nd power next.

f(4) = 3(25)

last, multiply.

f(4) = 75

This is just following the typical order of operations.

The statement that "σ(n) < 2n holds true for all n of the form n = p²" has been proved.

Let p be any prime number, and let σ(n) be the sum of all positive divisors of the integer n.

As p is a prime number, and 2 is the smallest prime number, so, p\(\geq\)2

So, the positive divisors of the integer n are: 1,p,p².

As σ(n) represents the sum of all positive divisors of the integer n.

σ(n)=1+p+p²

In order to prove that σ(n) < 2n,for all n of the form n = p².

1+p+p²<2p²

p²-p-1>0

It is know that, p\(\geq\)2.

So, p²-p-1\(\geq\)1

Thus, σ(n) < 2n holds true for all n of the form n = p².

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Answer:

Hope this will help you :)

Step-by-step explanation:

Answer:(2,-3)

Step-by-step explanation:

Answer:

C. 5i

Step-by-step explanation:

Answer:

Electric car = Exponential function (decreasing cost)

Gasoline car = Exponential function (increasing cost)

Step-by-step explanation:

The given data the Electric car, we have;

Time (years)   \({}\)     Cost (thousands of dollars)

0                           \({}\)      35

5                           \({}\)      27

10                           \({}\)     21

15                           \({}\)     16

20                           \({}\)    12.5

The difference between successive term is such that each term decreases by approximately the same percentage given as follows

1st and 2nd term (35 - 27)/35 × 100 ≈ 22.8571% decrease

2nd and 3rd term (27 - 21)/27 × 100 ≈ 22.222% decrease

3rd and 4th term (21 - 16)/21 × 100 ≈ 23.81% decrease

4th and 5th term (16 - 12.5)/16 × 100 ≈ 21.875% decrease

Therefore. the electric car is best approximated by exponential function

The given data the Gasoline car, we have;

Time (years)   \({}\)     Cost (thousands of dollars)

0                           \({}\)      35

5                           \({}\)      40.2

10                           \({}\)    46.3

15                           \({}\)    53.2

20                           \({}\)    61.2

The difference between successive term is such that each term increases by approximately the same percentage given as follows

1st and 2nd term (40.2 - 35)/35 × 100 ≈ 14.857% decrease

2nd and 3rd term (46.3 - 40.2)/40.2 × 100 ≈ 15.174% decrease

3rd and 4th term (53.2 - 46.3)/46.3× 100 ≈ 14.9% decrease

4th and 5th term (61.2 - 53.2)/53.2× 100 ≈ 15.03% decrease

Therefore. the Gasoline car is best approximated by exponential function

Answer:

\(x-2y+3=0\)

Step-by-step explanation:

Equation of a circle:  \((x-h)^2+(y-k)^2=r^2\)

(where (h, k) is the centre, and r is the radius)

Given equation: \((x-3)^2+(y+2)^2=20\)

Therefore, the centre is (3, -2) and the radius is √20

To find the equation of the tangent

The tangent of a circle is perpendicular to the radius at the point.

Therefore, first find the gradient (slope) of the line that passes through the centre of the circle and the given point.

\(\textsf{let}\:(x_1,y_1)=(1,2)\)

\(\textsf{let}\:(x_2,y_2)=(3,-2)\)

\(\textsf{slope}\:(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{-2-2}{3-1}=-2\)

If two lines are perpendicular to each other, the product of their gradients will be -1.

Therefore, the gradient of the tangent is:  

\(m_t \cdot-2=-1\implies m_t=\dfrac12\)

Now use the point-slope formula of a linear equation to find the equation of the tangent:

\(\implies y-y_1=m_t(x-x_1)\)

\(\implies y-2=\dfrac12(x-1)\)

\(\implies 2y-4=x-1\)

\(\implies x-2y+3=0\)

Answer:

28p + 14q + 21r

Step-by-step explanation:

7(4p + 2q + 3r) ← multiply each term in the parenthesis by the 7 outside

= 28p + 14q + 21r

Answer:

70%

Step-by-step explanation:

1. The difference in the distance of Jupiter from the sun than Mercury will be 4.816.

2. The difference in the distance of Neptune from the sun than Mars will be

3. The greatest distance between Earth and Uranus is Uranus.

4. The greatest distance between Venus and Saturn is Saturn

From the information, the average distance from the sun to each planets have been given. The difference in the distance of Jupiter from the sun than Mercury will be:

= 5.203 - 0.387

= 4.816

The difference in the distance of Jupiter from the sun than Mars will be:

= 30.07 - 1.524

= 28.546

The greatest distance between Earth and Uranus is Uranus and the greatest distance between Venus and Saturn is that of Saturn.

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Using DeMoivre’s theorem, the answer in regular form would be (5 + 5√3i)⁷ = -5000000 + 8660254.03i

The De Moivre's Theorem is used to simplify the computation of powers and roots of complex numbers and is used in together with polar form.

Convert the complex number to polar form. The polar form of a complex number is z = r(cos θ + isin θ),

r = |z|  magnitude of z

it becomes

r = √((5)² + (5√3)²) = 10

θ = arg(z) is the argument of z.

θ = atan2(b, a) = atan2(5√3, 5) = π/3

(5 + 5√3i) = 10 × (cos π/3 + i sin π/3)

De Moivre's theorem to raise the complex number to the 7th power

(5 + 5√3i)⁷

= 10⁷× (cos 7π/3 + i sin 7π/3)

= 10⁷ × (cos 2π/3 + i sin 2π/3)

Convert this back to rectangular form:

Real part = r cos θ = 10⁷× cos (2π/3) = -5000000

Imaginary part = r sin θ = 10⁷ × sin (2π/3) = 5000000√3 = 8660254.03i

∴  (5 + 5√3i)⁷ = -5000000 + 8660254.03i

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Answer:10^7 (1/2 - √3/2 i)

Step-by-step explanation:

To use DeMoivre's theorem, we first need to write the number in polar form. Let's find the magnitude and argument of the number:

Magnitude:

|5 + 5√(3i)| = √(5^2 + (5√3)^2) = √(25 + 75) = √100 = 10

Argument:

arg(5 + 5√(3i)) = tan^(-1)(√3) = π/3

So the number can be written in polar form as:

5 + 5√(3i) = 10(cos(π/3) + i sin(π/3))

Now we can use DeMoivre's theorem:

(5 + 5√(3i))^7 = 10^7 (cos(7π/3) + i sin(7π/3))

To simplify, we need to find the cosine and sine of 7π/3:

cos(7π/3) = cos(π/3) = 1/2

sin(7π/3) = -sin(π/3) = -√3/2

Explanation:

So the final answer in rectangular form is:

10^7 (1/2 - √3/2 i)

The inequalities that match the given graph are:

x ≤ 0 and y ≤ 0 (option a) for the third quadrant.

From the graph, we can determine the quadrants of the coordinate plane where x and y are negative or positive.

If x ≤ 0 and y ≤ 0, then we are in the third quadrant of the coordinate plane, where both x and y are negative.

If x ≥ 0 and y ≥ 0, then we are in the first quadrant of the coordinate plane, where both x and y are positive.

If x ≥ 0 and y ≤ 0, then we are in the fourth quadrant of the coordinate plane, where x is positive and y is negative.

If x ≤ 0 and y ≥ 0, then we are in the second quadrant of the coordinate plane, where x is negative and y is positive.

Therefore, the inequalities that match the given graph are:

x ≤ 0 and y ≤ 0 (option a) for the third quadrant.

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Answer:

A(0,-3) B (5,0) C(0,-7)D(5,-4)

Step-by-step explanation:

Answer:

100 mL

Step-by-step explanation:

0.5 x 1000 =500mg

250mg/5ml= 5mg per 1ml

500mg/x mL = 5mg/1ml

cross multiply

500=5mg XmL

=100mL

△ABC and △DEF are congruent

AC is congruent to DF because, AC = 9 in, DF = 9 in.

is congruent to EF is corresponding to BC

The triangles are congruent by the SAS Triangle Congruence Theorem onl;y when x = 4

Answer:

14

Step-by-step explanation:

Answer:

A, x=14

Step-by-step explanation:

We know that <HJL is a right triangle, therefore 90 degrees.

<HJK + < KJL = <HJL

<HJK + 50 = 90

<HJK = 40

We replace HJK with 3x-2

3x-2 =40

3x=42

x= 14, A

I hope this helped! :)

Answer:

Step-by-step explanation:

With reference < H

perpendicular (p) = 12

base (b) = 5

so now

tangent of < H

= p / b

= 12 / 5

hope it helps :)

Answer:

yes the square root of 144 is 12

Step-by-step explanation:

12 x 12 = 144 and 144 divided by 12 = 12

The expression that simplifies to 100 by arranging digits from 1 to 7 is \(26\times 3\ + \ 7\times 4 \ -5 \ -1\)

The given digits include;

1, 2, 3, 4, 5, 6, 7

To form an expression that simplifies to 100:

This expression is formed as follows;

\(26\times 3\ + \ 7\times 4 \ -5 \ -1\)

The above expression can be simplified to 100 as follows;

\((26\times 3)\ + \ (7\times 4) \ -(5) \ -(1) \\\\= (78) + (28) - (6)\\\\= 106 - 6\\\\= 100\)

Thus, the expression that simplifies to 100 by arranging digits from 1 to 7 is \(26\times 3\ + \ 7\times 4 \ -5 \ -1\)

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Answer:

6 years is the correct answer.

Step-by-step explanation:

Given that

Principal, P =  £8000

Rate of interest, R = 3% compounding annually

Amount, A >  £9500

To find: Time, T = ?

We know that formula for Amount when interest in compounding:

\(A = P \times (1+\dfrac{R}{100})^T\)

Putting all the values:

\(A = 8000 \times (1+\dfrac{3}{100})^T\)

As per question statement, A >  £9500

\(\Rightarrow 8000 \times (1+\dfrac{3}{100})^T > 9500\\\Rightarrow (1+0.03)^T > \dfrac{9500}{8000}\\\Rightarrow (1.03)^T > 1.19\)

Putting values of T, we find that at T = 6

\(1.03^6 = 1.194 > 1.19\)

\(\therefore\) Correct answer is T = 6 years

In 6 years, the amount will be more than £9500.

Answer:

1.)Angle Addition Postulate

2.)Alternate interior Angle Theorem

3.)m/_1+m/_2+m/_

Step-by-step explanation:

I took the test

Answer:

x=19, y=\(19\sqrt{3}\)

Step-by-step explanation:

This triangle is a 30-60-90 triangle.

This means the hypotenuse (38) is double the short leg (x)

x=19

Then use the pythagorean theorem

a^2+b^2=c^2

19^2+y^2=38^2

361+y^2=1444

Subtract 361 from both sides

y^2=1083

put it in square roots and simplify

\(\sqrt{1083} =19\sqrt{3}\)

y=\(19\sqrt{3}\)

To solve this problem:

  ⇒ need to use a special right triangle theorem:

Let's consider the information given:

   ⇒ one angle is 30 degrees

      ⇒ one angle marked with a little square signifies that it is 90

          degrees

          ⇒ (all the angles added up are 180 degrees) so the last angle is                  

               60 degrees

Therefore we have a '30-60-90' triangle which states:

           ⇒ half the length of the hypotenuse (longest side of the triangle)

              ⇒ x = 38/2 = 19

          ⇒ is the square root of 3 divided by 2 of the hypotenuse

              ⇒ y = \(\frac{\sqrt{3} }{2} *38=19\sqrt{3}\)

Therefore:

    x = 19

    y = \(19\sqrt{3}\)

Hope that helps!

The probability of getting a black card and a numbered card is 9/26.

To calculate the probability of getting a black card (E1), we need to determine the number of black cards in a standard deck of 52 cards.

There are 26 black cards in total, which consist of 13 spades (black) and 13 clubs (black).

Therefore, the probability of drawing a black card (P(E1)) is:

P(E1) = Number of favorable outcomes / Total number of possible outcomes

P(E1) = 26 / 52

Simplifying this fraction, we get:

P(E1) = 1/2

So the probability of drawing a black card is 1/2.

To calculate the probability of drawing a numbered card (E2), we need to determine the number of numbered cards (2 through 10) in a standard deck.

Each suit (spades, hearts, diamonds, clubs) contains one card for each numbered value from 2 to 10, totaling 9 numbered cards per suit.

Therefore, the probability of drawing a numbered card (P(E2)) is:

P(E2) = Number of favorable outcomes / Total number of possible outcomes

P(E2) = 36 / 52

Simplifying this fraction, we get:

P(E2) = 9/13

So the probability of drawing a numbered card is 9/13.

To calculate the probability of both events occurring together (getting a black card and a numbered card), we multiply the individual probabilities:

P(E1 ∩ E2) = P(E1) × P(E2)

P(E1 ∩ E2) = (1/2) × (9/13)

Simplifying this fraction, we get:

P(E1 ∩ E2) = 9/26

Therefore, the probability of getting a black card and a numbered card is 9/26.

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Answer:

B

Step-by-step explanation:

\(x\geq -2\) means that \(x\) can be all values that are greater than -2, and the line under the inequality sign adds that \(x\) can be equal to it as well.

Since B represents all values of \(x\) that are greater than -2 along with -2 itself due to the closed circle, it is the correct answer.

Answer:

it is c i took the test i hope this helps