Given the function f(x) =3x^2-6x-9 is the point (1,-12) on the graph of f?

Given The Function F(x) =3x^2-6x-9 Is The Point (1,-12) On The Graph Of F?

Answers

Answer 1

The equation x² + y - 36 = 0 is symmetry about the y - axis.

We have to given that;

The equation of graph is,

⇒ x² + y - 36 = 0

Now, We get;

⇒ x² + y - 36 = 0

⇒ y = - x² + 36

By plotting the equation in graph we get;

The equation x² + y - 36 = 0 is symmetry about the y - axis.

Thus, The equation x² + y - 36 = 0 is symmetry about the y - axis.

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Given The Function F(x) =3x^2-6x-9 Is The Point (1,-12) On The Graph Of F?

Related Questions

Drag and drop to complete the proof below:

Given: DE←→
is tangent to circle C, at point F

Prove: ∠FEC and ∠ECF are complementary

Answers

To complete the proof below, we have;

<EFC is right angle; definition of a right angle

m<EFC = 90 degrees; definition of a tangent line

m<EFC + m<FEC + m<ECF = 180; Triangle sum theorem

90° + m<FEC + m<ECF = 180; Substitution property of equality

m<FEC + m<ECF = 90; Subtraction property of equality

<FEC + <ECF are complementary angles; definition of complementary angles

How to match the statements

To match the statement with their corresponding equations, we need to know the following;

The sum of the angles in a triangle is 180 degreesThe measure of the angle at right angle is 190 degreesComplementary angles are pair of angles that sum up to 90 degrees

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with each heartbeat, blood pressure increases as the heart contracts, then decreases as the heart rests between beats. The maximum blood pressure is called
the systolic pressure and the minimum blood pressure is called diastolic pressure. When a doctor records an individual's blood pressure such as "120 over 80" it
is understood as "systolic over diastolic". Suppose that the blood pressure for a certain individual is approximated by p (t)-80+30 sin (120xt) where p is the
blood pressure in mmHg (millimeters of mercury) and is the time in minutes after recording begins.
(a) Find the period of the function and interpret the results.
(b) Find the maximum and minimum values and interpret this as a blood pressure reading.
(c) Find the times at which the blood pressure is at its maximum.
Part: 0/3
Part 1 of 3
(a) Find the period of the function and interpret the results.
The period is minutes and represents the time for one complete heartbeat.
This implies that the heart rate is beats per minute. (Write your answers as simplified fractions, if necessary.)

Answers

The period of the function is 1/60 minutes

As a result, blood pressure changes between each heartbeat as a consequence of these oscilllations happening every second.

Max value = 110 mmHg

the minimum = 50 mmHg.

This suggests that systolic pressure remains at 110 mmHg and diastolic pressure is still maintained at 50 mmHg.

blood pressure peaks occur at times t = (1/2 + 2 * k) / 120 seconds.

How to find the period

(a) The given function is:

p(t) = 80 + 30 * sin(120 * pi * t)

This is equivalent to sinusoidal function in the form of:

p(t) = A + B * sin(C * t)

Where:

A is the baseline value,

B is the amplitude, and

C determines the frequency of the function.

Information given in the problem

A = 80, B = 30, and C = 120 * pi.

The period of a sinusoidal function is given by:

Period = 2 * pi / C

Period = 2 * pi / (120 * pi) = 1/60 minutes

The period of the function is 1/60 minutes, which means that the blood pressure oscillates every 1/60 minutes or 1 second. As a result, blood pressure changes between each heartbeat as a consequence of these oscilllations happening every second.

(b) maximum and minimum values of a sinusoidal function

Max value = A + B

Min value = A - B

Substituting the values of A and B:

Max value = 80 + 30 = 110 mmHg

Min value = 80 - 30 = 50 mmHg

Max value = A + B giving values of 110 mmHg

the minimum is A - B delivering 50 mmHg.

This suggests that systolic pressure remains at 110 mmHg and diastolic pressure is still maintained at 50 mmHg.

(c)  the time at which the blood pressure is at its maximum, we solve for t when the sinusoidal function is at its peak.

sin(120 * pi * t) = 1

Taking the inverse sine of both sides:

120 * pi * t = pi/2 + 2 * pi * k (where k is an integer)

Solving for t:

t = (1/2 + 2 x k)/120 (for k = 0, 1, 2, ......)

implying that blood pressure peaks occur at times t = (1/2 + 2 * k) / 120 seconds.

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Some easy math i totally cant solve it

Answers

Answer:

correct answer is D

Step-by-step explanation:

Computer systems analysts
Database administrators
Computer software engineers,
systems software
Gaming and sports book writers
and runners
Environmental science and
protection technicians, including
health
Manicurists and pedicurists
Physical therapists
Physician assistants
504
650
119 154
350 449
18 24
36
78
47
100
220
29.0
28.6
28.2
28.0
28.0
27,6
27.1
146
34
99
5
10
22
47
18
Bachelor's degree
Bachelor's degree
Bachelor's degree
Short-term on-thi
Associate degree
Postsecondary vocational award
Master's degree
Master's degree
173
66
83
27.0
Based on the above table, what is the fastest growing listed occupation for those without a college degree?
a. Gaming and sports book writers and runners
b. Dental assistants
c. Marriage and family therapists
d. Manicurists and pedicurists

Answers

Based on the above table , the fastest growing listed occupation for those without a college degree is Gaming and sports book writers and runners.

Therefore option A is correct.

What is a college degree?

The college degree is described as a qualification awarded to students after completing the requirements for a specific course of study of which the two most common types of degrees are the Bachelor of Arts and Bachelor of Science.

With reference to the table, the occupations that do not require a college degree are:

Gaming and sports book writers and runnersEnvironmental science and protection technicians, including healthManicurists and pedicurists

We were able to conclude that Based on the above table , the fastest growing listed occupation for those without a college degree is Gaming and sports book writers and runners with a rate of 24.6%.

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Every number has only one quality: its distance from zero.
O A. True
OB. False

Answers

Answer:

True

Step-by-step explanation:

Answer:

False

Step-by-step explanation:

Every number has multiple qualities beyond its distance from zero. Some of the qualities of a number include its sign (positive or negative), its magnitude (Absolute Value), its parity (whether it is even or odd), and its relationship to other numbers (greater than, less than, or equal to). Additionally, numbers can have properties such as being prime or composite, rational or irrational, and real or complex. Therefore, a number is characterized by more than just its distance from zero.

107/103 X 101/100 =
What is 107 over 103 times 101 over 100

Answers

107 over 103 times 101 over 100 would result in 10807/10300.

Mathematical operations

In order to multiply the fractions 107/103 and 101/100, we can simply multiply their numerators together and their denominators together:

(107/103) x (101/100) = (107 x 101) / (103 x 100)

Now we can simplify this fraction by finding the greatest common factor of the numerator and denominator and dividing both by it:

(107 x 101) / (103 x 100) = 10807 / 10300

Therefore, 107/103 times 101/100 simplifies to 10807/10300.

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On a coordinate plane, point A is (negative 2, 3), point B is (2, 4), and point C is (0, negative 1). The points are connected with lines. Use the graph to find the coordinates of each vertex in triangle ABC. is the coordinate of Point A. is the coordinate of Point B. is the coordinate of Point C.

Answers

The coordinate of Point A is (-2, 3). The coordinate of Point B is (2, 4). The coordinate of Point C is (0, -1).

Can u mark my answer as the Brainlyest if it work Ty

A manufacturer cuts a piece of metal for a microscope. The resulting piece of metal can be
represented in a coordinate plane by a triangle with vertices A (8,8), B(5, 3), and C(11,3)
. One unit in the coordinate plane represents one millimeter.
Prove that AABC is isosceles.
Find the exact length of each side.
AB =
BC =
AC =

Answers

We can actually see here that the triangle ABC is an isosceles triangle. This is because two sides of the triangle are equal.

What is an isosceles triangle?

An isosceles triangle is actually a geometric shape having three sides, two of which are equal in length. Consequently, two of the angles that are across from those sides are also equal.

In order to prove that the triangle is an isosceles triangle, we will find the following:

Distance between  vertices A and B:

AB = √((8-5)² + (8-3)²) = √(3² + 5²) = √34 = 5.83mm

Then, we also find the distance between vertices B and C:

BC = √((11-5)² + (3-3)²) = 6mm

Finally, we find the distance between vertices A and C:

AC = √((11-8)² + (3-8)²) = √(3² + 5²) = √34  = 5.83mm

Thus, we see here that AB and AC are actually equal which makes the triangle an isosceles.

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Gate posts are parallel and m<4 =47, what is m<1

Answers

The value of angle 1 given the value of angle 4 as 47° will be 133°

How to explain parallel lines

Parallel lines are two lines on a two-dimensional plane that, no matter how far they are extended, never collide. They always keep the same gap between them.

Parallel lines have the same slope, which means they have the same angle of inclination or steepness. In other words, they rise and fall at the same pace along their length.

The value will be:

= 180° - 47°

= 133°

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Find the missing angle
A
B
C
D

Answers

The solution is the missing angle is 74.

Here, we have,

The sum of all the angles of a triangle (of all types) is equal to 180°.

The sum of the length of the two sides of a triangle is greater than the length of the third side.

In the same way, the difference between the two sides of a triangle is less than the length of the third side.

here, we have,

from the given diagram, we get,

the given angles are 55 and 51

let,  the missing angle be x

we know that,

The sum of all the angles of a triangle (of all types) is equal to 180°.

so, x+55+51 = 180

or, x + 106 = 180

or, x = 74

Hence, The solution is the missing angle is 74.

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A civil patrol unit of 15 members includes 4 officers. In how many ways can 3 members be selected for a search and rescue mission such that at least 1 officer is included?

Answers

Step-by-step explanation:

There are 4 ways to select an officer

   multiply this by the number of ways to select  2 others from the 15

4  *   15 C 2 = 4 *  15!/(2!13!)  = 420 ways to select

two 1−quart bottles
two 1−quart bottles

two 1−quart bottles and two 1−pint bottles
two 1−quart bottles and two 1−pint bottles

one 1−quart bottle and eight 8−ounce fluid glasses
one 1−quart bottle and eight 8−ounce fluid glasses
Select the objects that hold the same amount of liquid as a 96−fluid-ounce jug. Select all that apply.

three 1−quart bottles
three 1−quart bottles

two 8−ounce fluid glasses and two 1−pint bottles
two 8−ounce fluid glasses and two 1−pint bottles

Answers

To determine which objects hold the same amount of liquid as a 96-fluid-ounce jug, we need to convert each of the given measurements to fluid ounces and then add them up. Here are the conversions:

Two 1-quart bottles = 64 fluid ounces (2 x 32)

Two 1-pint bottles = 32 fluid ounces (2 x 16)

One 1-quart bottle and eight 8-ounce fluid glasses = 96 fluid ounces (32 + 8 x 8)

So, the objects that hold the same amount of liquid as a 96-fluid-ounce jug are:

One 1-quart bottle and eight 8-ounce fluid glasses

Three 1-quart bottles

Therefore, the correct answer is:

one 1−quart bottle and eight 8−ounce fluid glasses

three 1−quart bottles

I'm very confused on this question.

Answers

The value of angle x is determined as 107⁰.

What is the x?

The value of angle x is calculated by applying principle of intersecting chord theorem.

The intersecting chord theorem states that half of the difference between the arc of two intersecting chords is equal to the tangent angle.

Based on the angle of intersecting chord theorem, we will have the following equation.

m∠MNP = ¹/₂( arc MLP - arc MP )

x = ¹/₂ [(360 -73) - (73)]

x = ¹/₂ (287 - 73 )

x = ¹/₂ (214)

x = 107⁰

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sketch the graph of each function.
22. g(x)= -2x³-8x2 +18x+72

Answers

The graph of the cubic equation is in the equation of the end.

How to sketch the graph of the function?

To do it, we need to find some points that are solutions of the equation. Then we can graph these points on a coordinate axis and then connect these points with a curve proper of a cubic relation.

when x = 0

g(0) =  -2*0³-8*0² +18*0+72 = 72

So we have the point (0, 72)

when x = 1

g(1) =  -2*1³-8*1² +18*1+72

      = -2 - 8 + 18 + 72 = 80

So we have the point (1, 80)

when x = -1

g(-1) =  -2*-1³-8*-1² +18*-1+72

      =    2 - 8 - 18 + 72 = 48

(-1, 48)

And so on, when you have enough points, you can connect them. The graph that should you get is one like the graph in the image at the end.

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how do you find the net of a rectangular prism

Answers

Answer:

The formula looks like this:

Surface Area = 2 l w + 2 l h + 2 h w ,

where SA = surface area, l = length, w = width, and h = height. In the rectangular prism net above, l = 8 inches, w = 5 inches, and h = 3 inches. Simply put these numbers into the formula and solve for surface area.

NO LINKS!! URGENT HELP PLEASE!!!

Use a flowchart to prove the triangles below are congruent:

Answers

Answer:

- angle T is congruent to angle U (given)

- NS is congruent to SH (given)

- angle TSN is congruent to USH (vertical angles are congruent)

So triangle TNS is congruent to triangle UHS by AAS.

Need help I have 20 min! x=In^20 Write in exponential form.

Answers

The given equation in exponential form will be [tex]20 = x^e[/tex]

Given is an equation, x = ㏑ 20,

So, we know that,

㏑(x) = [tex]log_e(x)[/tex]

And,

logₐ (x) = b

x = bᵃ

Therefore,

x = ㏑ 20

will be converted into,

[tex]20 = x^e[/tex]

Hence, the given equation in exponential form will be [tex]20 = x^e[/tex]

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solve for x line LK is 59; line LM no measurement; line MK no measurement; angle L is 90 degrees and angle K is 58 degrees

Answers

The value of x which I suppose is the angle M is calculated to be 32°

Calculation of angles in a triangle

In a right-angled triangle KLM, it is a known fact that the sum of the three angles in any triangle is 180°.

Since angle L is a right angle and has a measure of 90°, we can find angle M by subtracting the sum of angles K and L from 180°.

Thus,

angle M = 180° - angle K - angle L

= 180° - 58° - 90°

= 32°

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A teacher gives 6 students some cards to play a game

Answers

If teacher has total "52 cards", and teacher gives each student "1 card" until all 52 cards are distributed, then number of students which got exactly 9 cards are (b) 4 students.

There are 6 students and 52 cards. The teacher gives each student one card until all 52 cards are given out in game. We want to know how many students receive exactly 9 cards.

To solve the problem, we divide the total number of cards (52) by the number of students (6) to find the average number of cards per student.

⇒ 52 cards / 6 students = 8.67 cards per student;

Since we can't give a student a fraction of a card, we need to round down to 8 cards,

If we give each student 8 cards, that will total of 8 cards × 6 students = 48 cards. which leaves 4 cards left over that we can't give out evenly.

This means that four-students will receive 9 cards each and two students will receive 8 cards each.

Therefore, the correct option is (b).

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The given question is incomplete, the complete question is

A teacher gives 6 students some cards to play a game, She has 52 cards in total, The teacher gives each 1 card until all 52 cards are given.

How many students gets exactly 9 cards?

(a) 2

(b) 4

(c) 5

(d) 6.

Triangle ABC shown below has m∠B=36∘, a=5, and c=13. Find the area of the triangle.

Answers

Answer: 19.1 is correct

Step-by-step explanation:

where a and c are the lengths of two sides of the triangle and C is the angle between them. To use this formula, we need to find side b and angle A.

We can use the Law of Cosines to find side b:

at+a+fair+Daniel+and+Claire+went+on+a+ride+that+has+two+separate+circular+tracks.+Daniel+rode+in+a+purple+car+that+travels+a+total+distance+of+265+feet+around+the+track+.+Ciara+rode+in+a+yellow+car+that+travels+a+total+distance+of+170+feet+around+the+track.+They+drew+drew+a+sketch+of+the+ride.+What+is+the+difference+ofthe+radii+of+the+two+circle+tracks

Answers

Note that the difference in the radii of the two circular tracks is about 13.68 ft

How did we get that?

recall that the circumference of a circle is denoted by the expression

C = 2πr

In this case, C = Circumference
and r   radius

Since we have two tracks

Let ra = radius of the purple track and

rb = radius of the yello track

So

265 = 2πra

170 = 2πrb

making ra and rb subject of the expression we have


ra = 256/(2π)  =40.7436654315 ≈ 40.74

rb = 170 / (2π) = 27.0563403256 ≈ 27.06

Hence, the difference is

40.74 - 27.06 = 13.68ft

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Full Question:

at a fair Daniel and Claire went on a ride that has two separate circular tracks. Daniel rode in a purple car that travels a total distance of 265 feet around the track . Ciara rode in a yellow car that travels a total distance of 170 feet around the track. They drew drew a sketch of the ride. What is the difference ofthe radii of the two circle tracks

A soda can has a radius of 1 inch and a height of 5 inches and a density of 3.2 g/mL. What is the mass?

Answers

The volume of the soda can is given by the formula V = pi*r^2*h, where r is the radius and h is the height. Substituting the given values, we get V = 3.1416*(1 inch)^2*(5 inches) = 15.708 cubic inches.

The mass of the soda can is given by the formula M = V*d, where d is the density of the soda can. Substituting the given values, we get M = 15.708 cubic inches * 3.2 g/mL = 50.226 g.

Therefore, the mass of the soda can is approximately 50.226 grams.

Sean A y B dos sucesos tales que:
P(A)= 0. 4;
P(Bc)= 0. 7 y P(AUB)= 0. 6, donde Bc
es el suceso contrario de B. Determinar si son independientes A y B

Answers

Para determinar si A y B son independientes, se debe cumplir que P(A∩B) = P(A)P(B).

Primero, se debe encontrar P(B) utilizando la probabilidad complementaria:
P(B) = 1 - P(Bc) = 1 - 0.7 = 0.3.

Luego, se puede encontrar P(A∩B) utilizando la fórmula de la unión:
P(A∩B) = P(A) + P(B) - P(AUB) = 0.4 + 0.3 - 0.6 = 0.1.

Finalmente, se puede verificar si A y B son independientes:
P(A)P(B) = 0.4 x 0.3 = 0.12.

Como P(A∩B) ≠ P(A)P(B), se concluye que A y B no son independientes

Abbie needs to cover square pyramids with fabric. The dimensions of each pyramid are shown in the table. If the fabric costs $0.001 per square millimeter, what is the surface area and cost to cover each pyramid, rounded to the nearest whole cent?

PLEASE HELP !!!

Answers

The surface area of each square pyramid, we used the formula Surface Area = Base Area + 4 x (1/2 x Base x Height of Face) is 1049.2, 1285.8 and 2100mm². The cost to cover each pyramid was $1.05, $1.29 and $2.10

To calculate the surface area of each pyramid, we need to use the formula

Surface Area = Base x Height of face + 2 x ((Base/2) x Slant Height)

For the small pyramid, we have

Surface Area = 20 x 25 + 2 x ((20/2) x 26.46)

Surface Area = 500 + 2 x 10 x 26.46

Surface Area = 1049.2 mm²

Cost to cover the small pyramid

Cost = Surface Area x 0.001

Cost = 1049.2 x 0.001

Cost = $1.05

For the medium pyramid, we have

Surface Area = 20 x 40 + 2 x ((20/2) x 42.43)

Surface Area = 800 + 2 x 10 x 42.43

Surface Area = 1285.8 mm²

Cost to cover the medium pyramid

Cost = Surface Area x 0.001

Cost = 1285.8 x 0.001

Cost = $1.29

For the large pyramid, we have

Surface Area = 30 x 40 + 2 x ((30/2) x 50)

Surface Area = 1200 + 2 x 15 x 50

Surface Area = 2100 mm²

Cost to cover the large pyramid

Cost = Surface Area x 0.001

Cost = 2100 x 0.001

Cost = $2.10

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A slide at a water park with a constant slope sends riders traveling a distance of 45 feet to the pool at the bottom of the slide. If the depth of the pool is 12 feet, and the angle of depression from the top of the slide is 45 ° what is the vertical distance from the top of the slide to the bottom of the pool? Round answers to the nearest tenth.

Answers

The required vertical distance from the top of the slide to the bottom of the pool is 43.8 feet.

We can use trigonometry to solve this problem. Let's call the vertical distance from the top of the slide to the bottom of the pool "h" (in feet). We can use the angle of depression (45°) to find the horizontal distance from the top of the slide to the bottom of the pool. Since the slide has a constant slope, the angle between the slide and the ground is also 45°. It implies that the vertical height is equal to the horizontal projection of the slide.

Apply Pythagorean theorem,
45² = h² + h²
h = 45/√2
h = 31.8

Now, as mentioned that depth of the pool is 12 feet so, the total vertical height is given as,
= h + 12
= 31.8 + 12
= 43.8 feet

Thus, the required vertical distance from the top of the slide to the bottom of the pool is 43.8 feet.

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can someone solve and graph it

f(x)=|x+1|+2​

Answers

Answer:

d y                              x+1

-----   = (l x+1 l + 2) x -----------------------

d x                            (l x+1 l + 2) x lx+1l

simplify -6 times the square root 7 X 4 times the square root 5

Answers

Answer:

-24√35

Step-by-step explanation:

Mark Brainliest!!

Which compares the end behavior of the functions f and g?
f(x) = −17x − 9 g(x) = − 78
7
8
x + 20
A. For function f, as x → ∞
x




, f(x) → ∞
f
(
x
)




. Likewise, for function g, as x → ∞
x




, g(x) → ∞
g
(
x
)




.
B. For function f, as x → ∞

x




, f(x) → −∞
f
(
x
)





. Likewise, for function g, as x → ∞
x




, g(x) → −∞
g
(
x
)





.
C. For function f, as x → ∞
x




, f(x) → −∞
f
(
x
)



-

. However, for function g, as x → ∞
x




, g(x) → ∞
g
(
x
)




.
D. For function f, as x → ∞

x




, f(x) → ∞

f
(
x
)




. However, for function g, as x → ∞
x




, g(x) → −∞
g
(
x
)





.

Answers

Answer:

The correct option that compares the end behavior of the functions f and g is D.

For function f, as x → ∞, f(x) → -∞, which means that the function approaches negative infinity as x approaches infinity. This is because the leading term of the function is -17x, which approaches negative infinity as x approaches infinity.

For function g, as x → ∞, g(x) → -∞, which means that the function also approaches negative infinity as x approaches infinity. This is because the leading term of the function is -78/87x, which approaches negative infinity as x approaches infinity.

Therefore, both functions have the same end behavior, which is approaching negative infinity as x approaches infinity.

Convert 135° from degrees to radians.

Answers

Answer:

Answer:Radians = Degrees × 3.14/180

Answer:Radians = Degrees × 3.14/180Step-by-step explanation:

Answer:Radians = Degrees × 3.14/180Step-by-step explanation:145× (3.14/180)

Answer:Radians = Degrees × 3.14/180Step-by-step explanation:145× (3.14/180)3.14/180

Answer:Radians = Degrees × 3.14/180Step-by-step explanation:145× (3.14/180)3.14/180= 0.0174

Answer:Radians = Degrees × 3.14/180Step-by-step explanation:145× (3.14/180)3.14/180= 0.0174135×0.0174

Answer:Radians = Degrees × 3.14/180Step-by-step explanation:145× (3.14/180)3.14/180= 0.0174135×0.0174 = 2.355

The volume in a set of wine bottles is known to follow a normal distribution with mean μ ounces and standard deviation of σ = 0.5 ounces. You take a sample of the bottles and measure their volumes. How many bottles do you need to sample to construct a 90% confidence interval for μ with a margin of error at most 0.1 ounces?

Answers

We need to sample at least 18 wine bottles to construct a 90% confidence interval for the mean volume of the population, with a margin of error of at most 0.1 ounces.

To determine the sample size required for a given confidence interval and margin of error, we need to use the following formula:

[tex]n = (Z^2 * \sigma^2) / E^2[/tex]

Where:

n = sample size

Z = the z-score associated with the desired confidence interval (for 90% confidence, Z = 1.645)

σ = the standard deviation of the population (0.5 ounces in this case)

E = the desired margin of error (0.1 ounces in this case)

Plugging in the values, we get:

[tex]n = (1.645^2 * 0.5^2) / 0.1^2[/tex]

n = 17.1

Thus, we need to sample at least 18 wine bottles to create a 90% confidence interval for the population's mean volume, with a margin of error of no more than 0.1 ounces.

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We need to sample at least 69 bottles to construct a 90% confidence interval for the mean volume of the wine bottles with a margin of error at most 0.1 ounces.

We have,

To construct a confidence interval for the mean volume of the wine bottles, we need to use the formula:

Confidence interval = sample mean ± margin of error

Where,

Margin of error = z (σ/√(n))

Here, we want to construct a 90% confidence interval with a margin of error at most 0.1 ounces.

This means that the margin of error should be less than or equal to 0.1, and the confidence level is 90%, so the critical value of z is 1.645

(using a standard normal table or calculator).

We are given that the standard deviation is σ = 0.5 ounces.

We don't know the sample mean or the sample size (number of bottles), so we'll use the worst-case scenario to find the sample size that gives the largest margin of error.

The worst-case scenario is when the sample mean is equal to the population mean (μ) plus the margin of error (0.1 ounces), and the sample size is as small as possible (which gives the largest margin of error).

So, we have:

0.1 = 1.645 x (0.5/sqrt(n))

Solving for n, we get:

n = (1.645 x 0.5/0.1)²

n = 68.225

Rounding up to the nearest whole number, we get:

n = 69

Therefore,

We need to sample at least 69 bottles to construct a 90% confidence interval for the mean volume of the wine bottles with a margin of error at most 0.1 ounces.

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