Hence, the value of CosФ + Cos (2π+Ф) + Cos (4π+Ф) is 0.6 .
What is Cosine function ?The cosine function One of the fundamental trigonometry functions is cos x as the others being the co-secant, cotangent, secant, sine, and tangent. Let theta represent the angle that can be calculated by rotating the unit circle's arc anticlockwise from the x-axis. Cos theta is the arc endpoint of horizontal coordinate after that.
What is the trigonometry identities?trigonometric functions are used in an expression or equation, trigonometric Identities are helpful. For every value of a variable appearing on both sides of an equation, a trigonometric identity is true. Certain trigonometric functions such as sine, cosine, and tangent of one or more angles are involved geometrically in these identities.Three main are trigonometry functions are sine, cosine, and tangent, while the other three are cotangent, secant, and co-secant. All six trigonometric functions serve as the foundation for the trigonometric identities.
Given, CosФ = 0.2,
thus we can use the trigonometric identity:
cos(2 n π+Ф) = cosФ
where any integer n is used. Therefore:
Cos(2 π+ Ф) = CosФ
cos (4π+Ф) = cosФ
These values, put into the given expression
than we get,
cos Ф+ cos (2π+Ф) + cos (4π+Ф) = cos Ф+ cosФ + cosФ.
= 3 cosФ
= 3×0.2
= 0.6
So,CosФ + Cos (2π+Ф) + Cos (4π+Ф) = 0.6
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I need help completing the table and finding the velocity
We can conjecture that the value of the instantaneous velocity at t=3 is 81.6.
What is the average velocity over an interval?To find the average velocity over an interval, we use the formula:
average velocity = (change in distance)/(change in time)
Using the position function s(t) = -16t^2 + 100t, we can find the change in distance over an interval [a, b] by subtracting the value of s(a) from the value of s(b):
change in distance = s(b) - s(a)
change in distance = [-16(b^2) + 100b] - [-16(a^2) + 100a]
change in distance= -16(b^2 - a^2) + 100(b - a)
Similarly, we can find the change in time over the same interval by subtracting a from b:
change in time = b - a
Using these formulas, we can complete the table as follows:
Time Interval | Average Velocity
[2,3] | 84
[2.9,3] | 82
[2.99,3] | 81.6
[2.999, 3] | 81.6
[2.9999,3] | 81.6
Based on these results, we can make a conjecture about the value of the instantaneous velocity at t = 3.
From the table, the average velocity over smaller and smaller intervals around t=3 is approaching a value of 81.6.
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i need the answer to this question
Answer:
Area of annulus is 40.85cm² to 2d.p
Step-by-step explanation:
Area of shaded part=Area of bigger circle -Area of smaler circle
Area of annulus= πR²- πr²= π(R²-r²)
A=3.142(7²-6²)
A=3.142(49-36)
A=3.142×13
A=40.846cm²
A=40.85cm² to 2d.p
a rectangle has a perimeter of 48 in. the length and width are scaled by a factor of 2.5. what is the perimeter of the resulting rectangle?
If a rectangle has a perimeter of 48 in. the length and width are scaled by a factor of 2.5, the perimeter of the resulting rectangle is 225 inches.
Let L be the length of the original rectangle and W be the width. Then, the perimeter of the original rectangle is P = 2L + 2W = 48 inches.
If we scale the length and width by a factor of 2.5, we get a new length of 2.5L and a new width of 2.5W. The perimeter of the new rectangle would be:
P' = 2(2.5L) + 2(2.5W)
= 5L + 5W
To find the new perimeter, we need to find the new values of L and W. Since the length and width are scaled by the same factor, we can write:
2.5L = kL
2.5W = kW
where k is the scaling factor.
Since the new rectangle is scaled by a factor of 2.5, k = 2.5. Therefore:
L' = 2.5L = 2.5(12) = 30 inches
W' = 2.5W = 2.5(6) = 15 inches
The new perimeter is:
P' = 5L' + 5W'
= 5(30) + 5(15)
= 225 inches
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2
If 3 12, what is the value of x?
O 4
08
O 18
24
Answer:
8
Step-by-step explanation:
12 ÷ 3 = 4
2 x 4 = 8
Hope I could help :)
URGENT!! ILL GIVE
BRAINLIEST! AND 100 POINTS
Answer:
1) 26
2) 118
3) 144
4) 58
5) 38
6) 96
7) 84
8) 156
9) 240
Step-by-step explanation:
Draw a Venn diagram to confirm my answers.
35% of the 240 students were interested in athletics, so .35 × 240 = 84 students were interested in athletics. Since 26 students were interested in both athletics and academic clubs, 84 - 26 = 58 students were interested in athletics only.
3/5 of the 240 students were interested in academic clubs, so 3/5 × 240 = 144 students were interested in academic clubs. Since 26 students were interested in both athletics and academic clubs,
144 - 26 = 118 students were interested in academic clubs only. From this information, you should be able to complete the two-way table.
Question 5 of 25
What are the domain and range of y= sin x? Select one choice for domain
and one for range.
A. Domain:
/+nr
B. Range: All real numbers
C. Domain: All real numbers
D. Range: -1≤1
The domain and range of this sine function y = sinx include the following:
C. Domain: All real numbers
D. Range: -1 ≤ y ≤ 1
What is a range?In Mathematics and Geometry, a range can be defined as the set of all real numbers that connects with the elements of a domain.
Furthermore, the horizontal extent of any graph of a function represents all domain values and they are always read and written from smaller to larger numerical values, and from the left-hand side of the graph to the right-hand side.
By critically observing the graph shown in the image attached above, we can reasonably and logically deduce the following domain and range:
Domain = {-∞, ∞}, x|x ∈ R, or all real numbers.
Range = {-1, 1}, y ≥ or -1 ≤ y ≤ 1.
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The doubling time of a bank account balance is 10 years. By what factor does the balance grow in 30 years?
The doubling time of a bank account balance is the amount of time it takes for the balance to double. In this case, the doubling time is 10 years, which means that the balance will double every 10 years.
Let's say the initial balance in the bank account is B. After 10 years, the balance will have doubled to 2B. After another 10 years (i.e., 20 years from the initial balance), the balance will have doubled again to 4B. Finally, after another 10 years (i.e., 30 years from the initial balance), the balance will have doubled one more time to 8B.
Therefore, after 30 years, the balance in the bank account will have grown by a factor of 8B/B, which simplifies to just 8. This means that the balance in the bank account will be 8 times the initial balance after 30 years.
To summarize, the balance in the bank account will double every 10 years due to the given doubling time. After 30 years, it will have doubled three times, resulting in a growth factor of 2 x 2 x 2 = 8, which means the balance will be 8 times the initial balance.
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The following letter cards are put in a bag.
REFLEX
A card is picked at random.
A
B
C
D +
E-
0
Which letter on the probability scale shows the probability of:
a) picking a card with an 'E' on it?
b)
picking a card that does not have an 'E' on it?
1
The probability of picking a card with an 'E' is 2/6 and not picking an E is 4/6
The probability of picking a card with an 'E' on it?From the question, we have the following parameters that can be used in our computation:
REFLEX
In the above, we have
Letters = 6
E = 2
So, we have
P(E) = 2/6
This also means that
P(Not E) = 1 - 2/6
P(Not E) = 4/6
Hence, the letters are
Probability scale shows the probability of picking a card with an 'E' on it = E'Probability scale shows the probability of picking a card with not an 'E' on it = Other lettersRead mroe about probability at
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If the distance between A (0,4) and B(3,a) is 5 units then Find the value of a.
Step-by-step explanation:
Using the distance formula, we know that the distance between points A and B is:
sqrt((3 - 0)^2 + (a - 4)^2) = 5
Simplifying:
sqrt(9 + (a - 4)^2) = 5
Squaring both sides:
9 + (a - 4)^2 = 25
Simplifying:
(a - 4)^2 = 16
Taking the square root of both sides (note that we can take the positive or negative square root since both will satisfy the equation):
a - 4 = ±4
Solving for a:
a - 4 = 4 or a - 4 = -4
a = 8 or a = 0
However, we need to check that these values of a actually make sense based on the given information. If a = 0, then the distance between A and B is:
sqrt((3 - 0)^2 + (0 - 4)^2) = sqrt(9 + 16) = 5
So a = 0 is a valid solution. If a = 8, then the distance between A and B is:
sqrt((3 - 0)^2 + (8 - 4)^2) = sqrt(9 + 16) = 5
So a = 8 is also a valid solution.
Therefore, the possible values of a are a = 0 or a = 8.
The value of a can be 0 or 8 if the distance between the given points is 5.
A straight line drawn between two points is the shortest distance between the two points. We can calculate the shortest distance by the distance formula.
Let A(0,4) = (x1,y1)
B(3,a) = (x2,y2)
Using distance formula,
[tex]D = \sqrt{(x2-x1)^2 + (y2-y1)^2}[/tex]
As distance = 5 units
Therefore,
[tex]\sqrt{(x2-x1)^2 + (y2-y1)^2} = 5[/tex]
[tex]\sqrt{(3-0)^2 + ((a-4)^2} = 5[/tex]
[tex]\sqrt{(3)^2 + (a^2 + 16 + - 8a)} = 5[/tex]
[tex]\(9 + 16 + a^2 - 8a = 25[/tex]
[tex]25 + a^2 -8a = 25[/tex]
[tex]a^2 - 8a = 0[/tex]
[tex]a(a-8) = 0[/tex]
a = 0 and a = 8
Therefore, the answer is 0 or 8.
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with 3 to 4 paragraphs on your thoughts on the income gap between the richest and poorest in the United States. Make sure your content is factual and address the possible causes and what we might do as a nation to correct it in some way. At least 2 sources - APA format.
find the range of h(x)=-4x^2-6x-4
The range of h(x)=-4x^2-6x-4 is (∞, -13).
How to find the range of range of h(x)=-4x^2-6x-4Using the vertex notion, find the range of the function h(x)=-4x2-6x-4.
The function's vertex is at x = -b/2a,
where a and b are the coefficients of x2 and x, respectively.
Because a = -4 and b = -6 in this situation, the vertex is at x = -(-6)/(2*(-4)) = 3/4.
To find the vertex's y-coordinate, enter x = 3/4 into the function:
h(3/4) = -4(3/4)^2 - 6(3/4) - 4 = -13
As a result, the function's vertex is at (3/4, -13).
Hence, the function's range is (∞, -13).
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Trevor bought an antique desk. The net value of the desk is equal to the resale value of the desk minus what Trevor paid to buy it. Exponential function f, shown in the table, represents the net value of the antique desk, rounded to the nearest whole dollar, x years after Trevor purchased it.
x 0 1 2 3 4 5
f(x) -37 -26 -13 0 15 32
The exponential function that represents the net value of the antique desk is: f(x) = -37 × [tex]0.7027^{x}[/tex]
Define Exponential function?The exponential function is a mathematical function denoted as "exp(x)" or "[tex]e^x[/tex]", where "e" is Euler's number (a mathematical constant approximately equal to 2.71828) and "x" is the input variable. The exponential function is a special type of mathematical function that grows or decays rapidly as the input value changes.
What is known by the term net value?"Net value" typically refers to the residual value or worth of something after all relevant expenses, debts, or liabilities have been subtracted from its total value or assets.
The table provides values of f(x) for x = 0, 1, 2, 3, 4, and 5. The corresponding values of f(x) are -37, -26, -13, 0, 15, and 32, respectively.
Using the given data, we can create the following exponential function in the form f(x) = a × [tex]b^{x}[/tex], where a and b are constants to be determined:
f(0) = a ×[tex]b^{0}[/tex] = -37
f(1) = a × b¹= -26
f(2) = a × b² = -13
f(3) = a × b³ = 0
f(4) = a × b⁴ = 15
f(5) = a × b⁵ = 32
To find the values of a and b, we can use a system of equations. We can divide the equations f(1) = a × b¹ = -26 by f(0) = a ×b⁰ = -37 to eliminate a:
(-26) / (-37) = b¹ / b⁰
0.7027 ≈ b
Substituting the value of b back into any of the equations, we can solve for a. Using f(0) = a ×b⁰ = -37:
-37 = a × 1
a = -37
So, the exponential function that represents the net value of the antique desk is:
f(x) = -37 × [tex]0.7027^{x}[/tex]
The values are rounded to the nearest whole dollar.
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Answer:
During the first three years Trevor owned the desk, the resale value was less than the cost. Then it exceeded the cost after year 3.
Step-by-step explanation:
See attachment
Find the surface area of
a right cylinder with a
radius of 2.5 feet and a
height of 7.5 feet.
Round your answer to
the nearest hundredth.
The surface area is about
feet.
長
square
The surface area is about 157.08 square feet. use formula for the surface area of a right cylinder to get answer .
what is surface area ?
Surface area is the measure of the total area that the surface of a three-dimensional object occupies. In other words, it is the sum of the areas of all the faces or surfaces of the object. Surface area is usually measured in square units
In the given question,
The formula for the surface area of a right cylinder is:
S = 2πr² + 2πrh
where r is the radius of the cylinder, h is its height, and π is a mathematical constant approximately equal to 3.14159.
Substituting the given values, we get:
S = 2π(2.5)² + 2π(2.5)(7.5)
S ≈ 2π(6.25) + 2π(18.75)
S ≈ 39.27 + 117.81
S ≈ 157.08
Rounding this to the nearest hundredth, we get:
The surface area is about 157.08 square feet.
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Explain the effect of China's fiscal stimulus on global aggregate demand and why it might have lost some of its force. Part 2 China's fiscal stimulus increases global aggregate demand because it _________. This fiscal stimulus might have lost some of its force because _______. A. increases world saving; the stimulus decreases China's government budget surplus and raises the world real interest rate B. increases China's potential GDP and income, which increases other countries exports to China; China's government debt is growing C. raises the value of the Chinese yuan, which increases China's purchases of imports in the long run; China is becoming more self-sufficient D. increases China's GDP and imports, which increases other countries exports to China; trade restrictions limits China's imports
Answer:
China's fiscal stimulus increases global aggregate demand because it increases China's GDP and imports, which increases other countries' exports to China. This fiscal stimulus might have lost some of its force because it decreases China's government budget surplus and raises the world real interest rate.
Step-by-step explanation:
hope this helps
HELPPP PLSSSSSSSSSS
Decide the type or transformation and state the rule
The transformation rule used for the diagram is: Shift by 3 units upwards and a shift by 5 units to the left.
What is the transformation rule that was used?
There are different methods of transformation such as:
Reflection
Dilation
Rotation
Translation
Now, from the given graph, we see that both objects remain the same size and the object on the right was first if all shifted by 3 units upwards.
Now, the transformed image in the left is the exact copy and not a mirrored copy and so there was not reflection but only another translation which was by 5 units to the left
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PLS HELP!!
hassan is finding the quotient of (2+3i) and (-4-i)
Proration means what your paying monthly
What is proration?
It's when you only pay for part of something you used.
Like if you only use half of a month of a service, you only pay for half of the monthly cost.y cost.
When is proration used?
When you sign up for a service that you don't use for a full month.
When you change your service in the middle of a billing cycle.
Why is proration used?
To make sure people only pay for what they use.
To be fair and not overcharge people for services they didn't fully use.
chatgpt
The speeds of vehicles traveling on a highway are normally distributed with an unknown population mean and standard deviation. A random sample of 13 vehicles is taken and results in a sample mean of 55 miles per hour and a sample standard deviation of 7 miles per hour. find the margin of error for a 95% confidence interval estimate for the population mean using the Student's t-distribution
The margin of error is given as 4.22 mph
How to solve for margin of errorMargin of Error = t * (sample standard deviation / √(sample size))
degrees of freedom (df), which is calculated as:
df = n - 1 = 13 - 1 = 12
95% confidence level with 12 degrees of freedom = 2.179.
Margin of Error = t * (s / √n)
Margin of Error = 2.179 * (7 / √13)
Margin of Error ≈ 2.179 * (7 / 3.606)
Margin of Error ≈ 2.179 * 1.939
Margin of Error ≈ 4.22 mph
The margin of error is given as 4.22 mph
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What is the volume of the rectangular prism?
A rectangular prism where the area of the base is 12 square centimeters and the height is 7 centimeters.
84 cm3
84 cm2
19 cm3
19 cm2
84cm³
V= 12 x 7 = 84
volume is always cubed btw
There are approximately 7.48 liquid gallons in a cubic foot. If a cylindrical water tank holds 1,500 liquid gallons and has a radius of 3.4 feet, what is the approximate height of the water tank? Approximate using π = 3.14 and round to the nearest tenth.
In the cylinder , height of the water tank is 5.5 feet.
What is volume?
The space taken up by any three-dimensional solid constitutes a volume, to put it simply. A cube, cuboid, cone, cylinder, or sphere can be one of these solids. Cubic units are used to measure the volume of solids. The volume will be given in cubic metres, for instance, if the dimensions are given in metres.
Given that,
A cubic foot contains roughly 7.48 liquid gallons.
We have to find what is roughly the height of a cylindrical water tank with a capacity of 1,500 liquid gallons and a radius of 3.4 feet.
We know that,
7.48 liquid gallons are contained in one cubic foot.
The cylindrical water tank's radius is 3.4 feet.
1,500 liquid gallons are the maximum capacity.
The cylindrical water tank's volume is calculated to hold 1500 liquid gallons ,
= [tex]\frac{1500}{7.48}[/tex]
= 200.53 ft³
Volume of the cylinder= πr²h
=> 200.53= 3.14 ×3.4×3.4×h
=> 200.53 = 3.14 × 11.56 × h
=> 200.53 = 36.29 × h
=> h= 5.5
=> h= 5.5 feet (Approximately)
Therefore, height of the water tank is 5.5 feet.
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An adult patient has come into the burn unit during your shift. You take an inventory of the location of their burns in the chart below.
Part of Body Burned?
(1 = yes, 0 = no)
Head 1
Left Arm 1
Right Arm 1
Upper Front Torso 0
Upper Back Torso 0
Lower Front Torso 1
Lower Back Torso 1
Upper Left Leg 0
Upper Right Leg 0
Lower Left Leg 0
Lower Right Leg 1
Given that their weight/mass is 50.9 kg, determine how much fluid they should receive per hour in the first 8 hours of their care. Answer to the nearest hundredth of a liter (two decimal places). ( Please help me by showing work, thank you!)
Answer:
To determine the fluid replacement rate for a burn patient in the first 8 hours, we can use the Parkland formula:
Fluid (in liters) = 4 mL × body weight in kg × % total body surface area (TBSA) burned
First, we need to calculate the TBSA burned. We can use the Rule of Nines to estimate this:
Head: 9%
Left Arm: 9%
Right Arm: 9%
Upper Front Torso: 0%
Upper Back Torso: 0%
Lower Front Torso: 18%
Lower Back Torso: 18%
Upper Left Leg: 0%
Upper Right Leg: 0%
Lower Left Leg: 0%
Lower Right Leg: 9%
Total TBSA burned = 9 + 9 + 9 + 18 + 18 + 9 = 72%
Now we can plug in the values into the Parkland formula:
Fluid (in liters) = 4 mL × 50.9 kg × 72% = 14,694.72 mL
To convert mL to liters, we divide by 1000:
Fluid (in liters) = 14,694.72 mL ÷ 1000 = 14.69 L
This is the total amount of fluid the patient needs in the first 8 hours. To determine the hourly rate, we divide by 8:
Hourly fluid rate = 14.69 L ÷ 8 = 1.84 L/hour
Therefore, the patient should receive approximately 1.84 liters of fluid per hour in the first 8 hours of their care.
express 2x^2-10x+8 in the form of a(x+b)^2+c where abc are constants and use your answer to state the minimum value of y=2x^2-10x+8
For the given quadratic equation, the value of constant abc is 45/8.
We need to rewrite the given quadratic equation in the form of a(x+b)²+c, where a, b, and c are constants, and then find the product abc. To do this, we'll complete the square.
Given quadratic equation: 2x² -10x+8
Follow these steps to determine the product:
1: Divide the entire equation by the leading coefficient 2:
x² -5x+4
2: Add and subtract the square of half the linear coefficient inside the parentheses. Half of 5 is 5/2, and (5/2)² is 25/4:
x² - 5x + 25/4 - 25/4 + 4
3: Combine the terms and rewrite the equation in the form a(x+b)²+c:
(1)(x - 5/2)² - 9/4
4: Now, we can see that a = 1, b = -5/2, and c = -9/4. So the product abc is:
abc = (1)(-5/2)(-9/4) = (45/8)
Therefore, the value of abc is 45/8.
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The point (8, −15) was reflected over an axis to (−8, −15). Which axis was it reflected over? Explain.
Answer: it is reflected by y-axis
Step-by-step explanation: as we know when point (x,y) reflected by y axis the value of y remains the same and the sign of x will be changed since the image is rotated horizontaly
Answer:
y-axis
Step-by-step explanation:
When a point is reflected over the y-axis, the x-coordinate of the point becomes the additive inverse of the original number.
Here, (8, -15) became (-8, -15). The only change in value of the coordinates is the value of the x-coordinate, 8, which became its additive inverse, -8.
Answer: y-axis
Answer: it is reflected by y-axis
Step-by-step explanation: as we know when point (x,y) reflected by y axis the value of y remains the same and the sign of x will be changed since the image is rotated horizontaly
Answer:
y-axis
Step-by-step explanation:
When a point is reflected over the y-axis, the x-coordinate of the point becomes the additive inverse of the original number.
Here, (8, -15) became (-8, -15). The only change in value of the coordinates is the value of the x-coordinate, 8, which became its additive inverse, -8.
Answer: y-axis
Claire works at the Sweet Shop, where candy is bagged and sold by its weight. Claire's last 4 sales of gourmet gummy worms are shown in the table.
- The cost of the gummy worms is proportional to the amount, in pounds, of gummy -worms purchased.
- This situation can be represented by an equation in the form y = kx, where k is the constant of proportionality.
What is the constant of proportionality?
A. $8.30
B. $8.40
C. $14.70
D. $4.73
Answer:
B
Step-by-step explanation:
If you take any set of the values and divide the cost (y) by the gummy worms (x) you get 8.4
(x+5)/(x+8)=1+(6/(x+1))
Answer:
(x+10)/(x+8).
Step-by-step explanation:
(x+5)/(x+8) = (x+5)/(x+8) + (x+5)/(x+1)
= (x+5) + (x+5)/(x+1)
= (x+10)
Find the exact values of x and y.
The exact values of x and y in the triangles are calculated and listed below
Finding the exact values of x and yFigure 7
Here, we have
y = √(6² + 1²) --- pythagorean theorem
This gives
y = √37
And, we have
x = √(10² - 6²) --- pythagorean theorem
x = 8
Figure 8
Here, we have
x = 1/2 * 10 --- midsegment of equilateral triangle
This gives
x = 5
And, we have
y = √(10² - 5²) --- pythagorean theorem
y = 5√5
Figure 10
Here, we have
2x = √(6² + 8²) --- pythagorean theorem
2x = 10
x = 5
So, we have
x = y = 5
Figure 11
Here, we have
y = 5
And we have
x = √(13² - 5²) --- pythagorean theorem
x = 12
Figure 13
By similar triangles, we have
x/5 = (x + y)/10
This gives
x = y
So, we have
x = y = 3
Figure 14
By similar triangles, we have
x = 6/3 * 4
x = 8
And then, we have
y = √(6² + 8²) --- pythagorean theorem
y = 10
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An engineer designed a valve that will regulate water pressure on an automobile engine. The engineer designed the valve such that it would produce a mean pressure of 4.4 pounds/square inch. It is believed that the valve performs above the specifications. The valve was tested on 110 engines and the mean pressure was 4.6 pounds/square inch. Assume the standard deviation is known to be 0.8. A level of significance of 0.01 will be used. Determine the decision rule. Enter the decision rule.
The decision rule is Reject the null hypothesis if the calculated z-value is greater than 2.33.
To determine the decision rule, we need to perform a hypothesis test to see if the sample mean of 4.6 pounds/square inch is significantly different from the population mean of 4.4 pounds/square inch.
Null hypothesis: The population mean pressure is equal to 4.4 pounds/square inch.
Alternative hypothesis: The population mean pressure is greater than 4.4 pounds/square inch.
We will use a one-tailed z-test with a level of significance of 0.01. Since the standard deviation is known, we can use the z-test formula:
z = (x - μ) / (σ / √n)
where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Plugging in the values, we get:
z = (4.6 - 4.4) / (0.8 / √110) = 2.75
The critical value for a one-tailed test with a level of significance of 0.01 is 2.33 (obtained from a standard normal distribution table). Since our calculated z-value of 2.75 is greater than the critical value of 2.33, we reject the null hypothesis.
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What are the zeros in this function?
The zeros of the function are:
-¹/₂ ± √((3)/2) i, 1/3 and - 2
What are the zeros of the polynomial?We want to find the Zeros of the polynomial:
f(x) = 3x⁴ + 8x³ + 6x² + 3x - 2
By the rational root theorem, we know that any rational zeros of f(x) must be expressible in the form p/q for integers p,q with p a divisor of the constant term −2 and q a divisor of the coefficient 3 of the leading term.
Therefore, the only possible rational zeros are:
±¹/₃, ±²/₃, ±1, ±2
Let's check x = 1/3 to get:
f(¹/₃) = 3(¹/₃)⁴ + 8(¹/₃)³ + 6(¹/₃)² + 3(¹/₃) - 2 = 0
f(-2) = 3(-2)⁴ + 8(-2)³ + 6(-2)² + 3(-2) - 2 = 0
Thus, x = 1/3 and - 2 are zeros of the function and (3x - 1) and (x + 2) are factors.
Using these factors and using the polynomial to divide them, we have that the remaining factor is: x² + x + 1
The zeros of x² + x + 1 are the Complex cube roots of 1, since
(x - 1)(x² + x + 1) = x³ - 1
using the quadratic formula, we have:
x = -¹/₂ ± √((3)/2) i
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Identify the horizontal shift (h) and the vertical shift (k). Then find the amplitude (|a|), if applicable, the frequency (b), and the period (P). y=tan x, y=cot x
Answer:
For y = tan(x), there is no horizontal or vertical shift (h = 0, k = 0). The amplitude is not applicable in this case. The frequency is b = 1 and the period is P = π.
For y = cot(x), there is no horizontal or vertical shift (h = 0, k = 0). The amplitude is not applicable in this case. The frequency is b = 1 and the period is P = π.
Use the Quadratic Formula to solve 2x2 + 6x = –3. Which of the following gives the solutions to the nearest hundredth?
A. 2.37 and 0.63
B. 2.37 and –0.63
C. –2.37 and 0.63
D. –2.37 and –0.63
Answer:
First, we need to put the equation in standard form, which is ax^2 + bx + c = 0. So, we have 2x^2 + 6x + 3 = 0.
Using the Quadratic Formula, x = (-b ± sqrt(b^2 - 4ac)) / 2a, we can plug in the values for a, b, and c:
x = (-6 ± sqrt(6^2 - 4(2)(3))) / 2(2)
x = (-6 ± sqrt(36 - 24)) / 4
x = (-6 ± sqrt(12)) / 4
Simplifying further, we have:
x = (-6 ± 2sqrt(3)) / 4
x = (-3 ± sqrt(3)) / 2
To the nearest hundredth, the solutions are:
A. 2.37 and 0.63