We can conclude that r - s is rational.
Proof: Suppose r and s are rational numbers.
We must show that r - s is rational.
To prove this, we will use the closure property of rational numbers under subtraction.
Starting point: Suppose r and s are rational numbers.
Conclusion to be shown: We must show that r - s is rational.
By definition, a rational number can be expressed as a fraction p/q, where p and q are integers and q is not equal to zero.
Let r = a/b and s = c/d, where a, b, c, and d are integers and b, d are not equal to zero.
Now, we can express r - s as (a/b) - (c/d).
By the closure property of rational numbers under subtraction, the difference of two rational numbers is also a rational number.
Therefore, we can conclude that r - s is rational.
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Given question is incomplete, the complete question is below
Outline a proof of the following statement by writing the "starting point" and the "conclusion to be shown" in a proof of the statement.
∀ real numbers r and s, if r and s are rational then r−s is rational.
That is, complete the sentences below.
Proof: Suppose ___________.
We must show that ______________.
Find the volume and total area of the right circular cone.
To find the volume and total area of the right circular cone, we will use the formulas below. Volume of the right circular cone: $$V = \frac{1}{3}πr^2h$$
Total surface area of the right circular cone:$$A = πr^2 + πrl$$, Where r is the radius, l is the slant height and h is the height of the cone.π (pi) is a mathematical constant that is approximately equal to 3.14159 and is used to calculate the circumference and area of a circle. The radius of the right circular cone is 3.5 cm and its height is 7 cm. To calculate the slant height, we will use the Pythagorean theorem which states that the square of the hypotenuse (l) is equal to the sum of the squares of the other two sides:$$l^2 = r^2 + h^2$$$$l = \sqrt{r^2 + h^2} = \sqrt{3.5^2 + 7^2} \approx 7.98\ cm$$
Volume of the right circular cone:$$V = \frac{1}{3}πr^2h = \frac{1}{3}π(3.5)^2(7) \approx 89.75\ cm^3$$. Total surface area of the right circular cone:$$A = πr^2 + πrl = π(3.5)^2 + π(3.5)(7.98) \approx 91.86\ cm^2$$. Hence, the volume of the right circular cone is approximately 89.75 cm³ and the total surface area is approximately 91.86 cm².
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What is AB?
Triangle ACB is right angle triangle. The length of AC is 12 and BC is 35.
Answer:
the answer is 35
Step-by-step explanation:
because if BC is 35 that means AB will have to be that same because it's a triangle
The required value of AB is 33 units for the given right triangle.
What is the right triangle?A right triangle is defined as a triangle in which one angle is a right angle or two sides are perpendicular.
Triangle ACB is a right-angle triangle. The length of AC is 12 and BC is 35.
Pythagoras's theorem states that in a right-angled triangle, the square of one side is equal to the sum of the squares of the other two sides.
Assume BC is the hypotenuse,
Since this is a right triangle, use the formula AB² + AC² = BC² and substitute values of AC = 12 and BC = 35.
AB² + AC² = BC²
AB² + (12)² = (35)²
AB² + 144 = 1225
AB² = 1225 -144
AB² = 1081
AB = 32.8785
Rounded to two decimal places,
AB = 33 units
Therefore, the required value of AB is 33 units.
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An auditor is determining the appropriate sample size for testing inventory valuation using MUS. The population has 2.620 inventory items valued at $12.625.000. The tolerable misstatement is $500.000 at a 10% ARIA. No misstatements are expected in the population. Calculate the preliminary sample size. (Confidence factor: 2,31)
The preliminary sample size is undefined since the projected misstatement is zero.
In determining the appropriate sample size for testing inventory valuation using MUS, the following steps are taken;
Plan the audit- Identify the tolerable misstatement. Assess inherent and control risk. Estimate population deviations. Determine the preliminary sample size. Select the sample to perform the audit procedures. Evaluate the results.Given that the population has 2,620 inventory items valued at $12,625,000 and the tolerable misstatement is $500,000 at a 10% ARIA, we can calculate the preliminary sample size using the formula;
Preliminary sample size = (Confidence Factor2 × Tolerable Misstatement)/Projected misstatement.
Considering that no misstatements are expected in the population, the projected misstatement will be zero.
Thus; the Preliminary sample size = (2.31 × 500,000)/0. Preliminary sample size = (2.31 × ∞) / 0. The preliminary sample size is undefined.
In conclusion, the preliminary sample size is undefined since the projected misstatement is zero.
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can somebody please double-check these for me.
just in case if the pictures blurry, I will provide the answers that I put below
my answers:
1. I got 27
2. I got 18.154
3. I got 44
4. I got 41.5
please correct me on any mistakes that I may have made
1. is a central angle, therefore, the arc will have the same measurement as the angle. Set the equation:
Arc = Central Angle
2x - 7 = 47
Isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS.
First, add 7 to both sides of the equation:
2x - 7 (+7) = 47 (+7)
2x = 47 + 7
2x = 54
Next, divide 2 from both sides of the equation:
(2x)/2 = (54)/2
x = 54/2 = 27
27 is your answer.
2. is a inscribed angle, meaning that the angle measurement will be half of the arc. Set the equation:
212 = 2(13x - 24)
Isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS.
First, divide 2 from both sides of the equation:
(212)/2 = (2(13x - 24))/2
106 = 13x - 24
Next, add 24 to both sides of the equation:
106 (+24) = 13x - 24 (+24)
106 + 24 = 13x
130 = 13x
Finally, divide 13 from both sides of the equation:
(130)/13 = (13x)/13
x = 130/13
x = 10
10 is your answer.
3. is a central angle. The arc and the angle will, therefore, have the same measurement:
137 = 3x + 5
First, subtract 5 from both sides of the equation:
137 (-5) = 3x + 5 (-5)
137 - 5 = 3x
132 = 3x
Next, divide 3 from both sides of the equation to isolate the variable, x:
(132)/3 = (3x)/3
x = 132/3 = 44
44 is your answer.
4. is a inscribed angle. The arc will be twice the measurement of the angle.
86 = 2(2x + 3)
First, isolate the variable x by dividing 2 from both sides of the equation.
(86)/2 = (2(2x + 3)/2
43 = 2x + 3
Next, subtract 3 from both sides of the equation:
43 (-3) = 2x + 3 (-3)
40 = 2x
Finally, divide 2 from both sides of the equation:
(40)/2 = (2x)/2
x = 40/2 = 20
20 is your answer.
~
Write the word sentence as an inequality.
A number y is less than 6.
Answer:
y - 6
Step-by-step explanation:
y is the number and it is less than 6 so 6 has to be negative:
y - 6
Question 6: Integration (12 marks) a. Which of the following definitions best describes the result of integrating a positive function f(x)? A The value of f(x) when == 0 B. The area between the curve of f(x) and the x-axis. C. The difference between the minimum of f(x) and the maximum of f(x). D. The gradient of f () at the point where x = 0. (1 mark) b. Which of the following is the general antiderivative of the function f(x) = 23+8x?? A 10x4 + 24x2 B. 2x° (x2 + 4) C. 2x6 + 8x4 D. 32° +2x4+C (1 mark) Which of the following statements is true for an odd function 9(2) ? 1 C. A. B. В S 0-2500 S = g(x) = 0 5 (2) = 0 Soo-a C. D (1 mark) d. By using the substitution 4x + 2 = u, show that the expression below is true. 1 +1 dx +C (4x + 2) 1600 + 8 (5 marks) e. Find the volume of the solid obtained when the area bounded by the curves below is rotated about the x-axis. Use the result shown in part (e) to assist you. Sa+gads 1 O x 1 = 0 (4 marks)
(a) The area between the curve of f(x) and the x-axis best describes the result of integrating a positive function f(x).
(b) The general antiderivative of the function f(x) = 23 + 8x is 2x³ + 4x² + C.
Hence, option (C) is the correct answer.
(c) An odd function satisfies f(-x) = -f(x). Thus, for an odd function f(x), the integral from -a to a is equal to zero because f(x) and -f(x) will have opposite signs, and the areas will cancel each other out. Hence, option (A) is the correct answer.
(d) To use the substitution u = 4x + 2, we need to find dx in terms of du.du = d/dx (4x + 2) dx= 4dxIntegrating both sides gives ∫du/4 = ∫dx/ (4x + 2). Therefore, the given expression becomes, ∫ 1/(4x + 2) dx = (1/4)∫du/u= (1/4)ln|u|+C= (1/4) ln|4x + 2| + C. Hence, (1/4) ln|4x + 2| + C is true by using the substitution 4x + 2 = u.
(e) The given function can be graphed as below: [tex]\int_0^1 (x^2 + 1) dx = \frac{4}{3}[/tex] We need to use the disk method to find the volume of the solid generated by rotating the region bounded by the curves about the x-axis. We need to consider an elemental area, find its volume, and integrate it over the region of interest. We know that the volume of the disk is given by V = πr²h, where r is the radius and h is the height of the disk. Let us consider an elemental area, A of the region rotated about the x-axis. If we rotate this area through a small angle, θ, then the area of the sector generated is given by d A = πr²dθ/2π = r²dθ/2. The radius of the disk is x, and the height is given by g(x) - f(x). Thus, V = ∫[g(x) - f(x)]²πx²dx.In this case, we have g(x) = x + 1 and f(x) = x². Substituting these values, V = π∫(x + 1 - x²)² x² dx. The limits of integration are from 0 to 1.
Therefore, V = π∫[x⁴ - 2x³ + x² + 2x + 1]dx= π[x⁵/5 - x⁴/2 + x³/3 + x² + x]₀¹= π[(1/5) - (1/2) + (1/3) + 1 + 1]
The volume of the solid obtained is, V = π[(8/15) + 2] = (14π/15).
Hence, the volume of the solid obtained when the area bounded by the curves below is rotated about the x-axis is (14π/15).
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in how many ways can you divide 7 candies and 14 stickers among 4 children such that each child gets at least one candy and also gets more stickers than candies?
The total number of ways is: {Total number of ways} =[1120]
Let's say we have 7 candies and 14 stickers that need to be distributed among 4 children in such a way that each child gets at least 1 candy and more stickers than candies. So, let's divide the candies first. Then we can use the formula of stars and bars to distribute the remaining stickers.
Let's assume that the candies have already been divided into 4 groups such that each group contains at least 1 candy.
Then, the total number of ways of dividing 7 candies among 4 children is given by 3 stars and 4 bars (one less than the number of children). For example, the following diagram shows one possible distribution of the candies:
Each star represents one candy, and the bars represent the separations between the groups.
For example, the above diagram represents the distribution of candies as follows:
Child 1: 1 candy
Child 2: 2 candies
Child 3: 1 candy
Child 4: 3 candies
Now, we need to distribute the 14 stickers among the 4 children in such a way that each child gets more stickers than candies. We can do this by using the formula of stars and bars again. This time, we need to distribute 14 stickers among 4 children such that each child gets more than 1 candy. Let's represent the candies by stars and the stickers by bars, then we need to distribute 14 bars among 3 stars and 4 bars.
Here, the first three bars represent the stickers for the first child, the next two bars represent the stickers for the second child, the next bar represents the stickers for the third child, and the remaining eight bars represent the stickers for the fourth child.
So, the total number of ways of distributing 7 candies and 14 stickers among 4 children such that each child gets at least 1 candy and more stickers than candies is given by the product of the number of ways of distributing the candies and the number of ways of distributing the stickers.
Therefore, the total number of ways is: {Total number of ways} =[1120]
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Please help!! I'm super confused:(
Answer: Singing
Mode: The value that appears the most in a list
Also by the way, the frequency table is most likely just a table of every single value and how many times they appear in the list. Its super easy all you have to do is count them
1. Start Time: 3:30 P.M.
End Time: 7:00 P.M.
Elapsed Time:
Answer:
7:00 = 6:60
Step-by-step explanation:
6:60 - 3:30 = 3 hours and 30 minutes
Which sequences are geometric? Check all that apply. –2, –4, –6, –8, –10, … 16, –8, 4, –2, 1 –15, –18, –21.6, –25.92, –31.104, … 4, 10.5, 17, 23.5, 30, … 625, 125, 25, 5, 1, …
Answer:
16,-8,4,-2,1
-15,-18,-21.6,25.92,-31.104...
625,125,25,5,1
Step-by-step explanation:
Answer: 16, –8, 4, –2, 1. has a common ratio,r = (-1/2)
-15, –18, –21.6, –25.92, –31.104 has a common ratio, r = (1.2)
625, 125, 25, 5, 1 has a common ratio, r= (1/5)
Step-by-step explanation: just took the test
HELP ME!!!!!! Correct answers only!!!!!
Answer: 22.94 cubic meters
I'm pretty sure it is this.
I think it's 25.23 cubic meters if not then I don't know
Answer: 891ft^3
Step-by-step explanation:
сalculate the cross product. (use symbolic notation and fractions where needed.) (i+j ) × k = ___________
The cross product of
[tex](i+j) \times k[/tex]= 0.
To calculate the cross product of (i+j) and k,
Step 1: Assign unit vectors to the given vectors:
(i+j) = i + j + 0k
k = 0i + 0j + k
Step 2: Apply the cross product formula:
(i+j) × k = (i × 0i) + (i × 0j) + (i × k) + (j × 0i) + (j × 0j) + (j × k) + (0k × 0i) + (0k × 0j) + (0k × k)
Step 3: Simplify the cross product using the properties of the cross product:
(i × 0i) = (j × 0j) = (0k × 0i) = (0k × 0j) = 0
(i × k) = - (k × i)
(j × k) = - (k × j)
(k × k) = 0
Step 4: Substitute the simplified cross products into the formula:
(i+j) × k = 0 + 0 + (i × k) + 0 + 0 + (j × k) + 0 + 0 + 0
= 0 + 0 + (i × k) + 0 + 0 + (j × k) + 0 + 0 + 0
= (i × k) + (j × k)
Step 5: Calculate the cross products:
(i × k) = (0 - 0)k - (0 - 0)j = 0k - 0j = 0k
(j × k) = (0 - 0)i - (0 - 0)k = 0i - 0k = 0i
Step 6: Substitute the calculated cross products into the formula:
(i+j) × k = 0k + 0i
= 0k + 0
= 0
Therefore k=0.
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A farmer wants to seed and fence a section of land. Fencing costs $27 per yard. Grass seed costs $2 per square foot. How much does it cost to fence and seed the pasture? (No links)
Answer:
she have 29 seed for the pasture
please help :c
Angela bought a total of 3 dozen cookies for Easter. Each cookie was 0.65 (INCLUDING TAX!) Of the cookies, 3/4 of them were shaped like Easter eggs. What was the cost of the Easter egg shaped cookies?
please add the work along with the question
Answer: $17.55
Step-by-step explanation:
36 x 0.65 = $23.40 total for all cookies
($23.40/1) x (3/4) = (70.2/4)
70.2 divided by 4 = $17.55
or
$23.40 x .75 = $17.55
help i hate fractions pls
Answer:
the answer is B
Step-by-step explanation:
because you just put 100 underneath
Answer:
B, 71/100
Step-by-step explanation:
To do this, just put each fraction into a calculator, and whichever fraction results in the decimal you're looking for will be your answer.
PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!!PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!
Answer:
One solution
I actually do not think you're going to give me brainliest
simplify this number 300mm:9m
Answer:
1 : 30
Step-by-step explanation:
300mm:9m
We need to change the meters to mm
1 meter is 1000 mm
so 9 m is 9000 mm
300mm:9000mm
Divide both sides by 300
300mm/300 : 9000/300
1 : 30
please help help help help !!!!! ASAP
Factorise
3у^2 - 54y + 243
Answer:
3(y-9)^2
Step-by-step explanation:
Answer:
See in the picture mark brainliest if correct
evalute sin 60. cos 30 sin +sin 30 .cos 60 what is the value of sin(30-60) what can you conclude
Answer:
See explanations below
Step-by-step explanation:
evaluate sin 60. cos 30 sin +sin 30 .cos 60 what is the value of sin(30-60) what can you conclude
According to the trigonometry identity
Sin 30 = 1/2
Sin 60 = √3/2
Cos 30 = √3/2
Cos 60 = 1/2
sin 60. cos 30 +sin 30 .cos 60
= √3/2(√3/2) + 1/2(1/2)
= √9/4 + 1/4
= 3/4 + 1/4
= 4/4
= 1
sin(30-60) = sin30cos60 - cos30sin60
sin(30-60) =1/2(1/2) - √3/2(√3/2)
sin(30-60) = 1/4 - √9/4
sin(30-60) = 1/4 - 3/4
sin(30-60) = (1-3)/4
sin(30-60) = -2/4
sin(30-60) = -1/2
hence the former fraction gives a positive values while the later gives a negative
Martin is dilating a blue triangle to create a yellow triangle. If he used a scale factor of
Which statement is true?
Answer
A
B
The perimeter of the yellow triangle will be 12/25 times the perimeter of the blue triangle.
The perimeter of the yellow triangle will be times the perimeter of the blue triangle.
The area of the yellow triangle will be times the area of the blue triangle.
С
D
The area of the yellow triangle will be
4
25
times the area of the blue triangle.
Previous
Answer:
See Explanation
Step-by-step explanation:
The question has missing details, as the scale factor is not given. However, I will give a general explanation on how to calculate the area and perimeter of a dilated shape (triangle).
The following assumptions, apply:
(1) Scale factor of 1/2 from the blue to the yellow triangle.
(2) The dimension of the blue triangle are:
[tex]Base = x[/tex]
[tex]Sides= y\ and\ z[/tex]
[tex]Height= h[/tex]
First, calculate the dimensions of the yellow triangle.
The dimension will be the product of the scale factor and the dimensions of the blue triangle.
So, we have:
[tex]Base = \frac{1}{2} * x = \frac{1}{2}x[/tex]
[tex]Sides = \frac{1}{2}y\ and\ \frac{1}{2}z[/tex]
[tex]Height = \frac{1}{2}h[/tex]
The perimeter of the blue triangle is:
[tex]P_1 =Base + Sides[/tex]
[tex]P_1 = x + y + z[/tex]
The perimeter of the yellow triangle is:
[tex]P_2 =Base + Sides[/tex]
[tex]P_2 = \frac{1}{2}x + \frac{1}{2}y + \frac{1}{2}z[/tex]
Factorize
[tex]P_2 = \frac{1}{2}[x + y + z][/tex]
Recall that: [tex]P_1 = x + y + z[/tex]
So:
[tex]P_2 = \frac{1}{2}*P_1[/tex]
This implies that the perimeter of the yellow triangle is a product of the scale factor and the perimeter of the blue triangle.
The area of the blue triangle is:
[tex]A_1 = \frac{1}{2}* Base * Height[/tex]
[tex]A_1 = \frac{1}{2} * x* h[/tex]
[tex]A_1 = \frac{1}{2} xh[/tex]
The area of the yellow triangle is:
[tex]A_2 = \frac{1}{2} * Base * Height[/tex]
[tex]A_2 = \frac{1}{2}* (\frac{1}{2}x) * (\frac{1}{2}h)[/tex]
Rewrite as:
[tex]A_2 = \frac{1}{2}* \frac{1}{2} [\frac{1}{2}x h][/tex]
[tex]A_2 = (\frac{1}{2})^2 *[\frac{1}{2}x h][/tex]
Recall that:[tex]A_1 = \frac{1}{2} xh[/tex]
So:
[tex]A_2 = (\frac{1}{2})^2 *A_1[/tex]
This implies that the area of the yellow triangle is a product of the square of the scale factor and the area of the blue triangle.
Max is mixing oil and gas for his moped. He uses 3.75 liters of gas and 1.5 liters of oil. How many liters of gas are used per liter of oil?
Answer:
2.5 liters of gas is used with per liter of oil
Step-by-step explanation:
Max is missing oil and gas for his moped.
Amount of gas used = 3.75 liters
Amount of oil required = 1.5 liters
Ratio in which gas and oil are mixed = [tex]\frac{3.75}{1.5}[/tex]
= [tex]\frac{2.5}{1}[/tex]
That means if he uses 1 liter of oil then the gas required for the mixture = 2.5 liters
Therefore, 2.5 liters of gas is used per liter of oil.
what is the x 19 + 13x = 32
Answer:
1
Step-by-step explanation:
2. Determine the points in C for which the following functions are holomorphic: (a) f(z) = z² (b) g(z) = x² - y² + 2xy (where z = x + iy)
There are no points in C for which the function g(z) is holomorphic.
The functions given are :
f(z) = z² and g(z) = x² - y² + 2xy (where z = x + iy)
We need to determine the points in C for which the functions are holomorphic.
(a) To check whether f(z) = z² is holomorphic or not, we will verify the Cauchy-Riemann equations (CRE) which are:
u x = v y and v x = - u y
Let us assume that f(z) = u(x, y) + iv(x, y)
Substituting in f(z) = z², we have f(z) = (x + iy)²= x² + 2ixy - y²
Now comparing with u(x, y) + iv(x, y), we get :
u(x, y) = x² - y² and v(x, y) = 2xy
Now applying the CRE, we get :
u x = 2xv
y = 2xu
y = - 2yv
x = 2y
We can see that both the CRE are satisfied.
Hence, f(z) = z² is holomorphic for all values of z in C.
(b) Similarly, for g(z) = x² - y² + 2xy (where z = x + iy), we have g(z) = u(x, y) + iv(x, y)
Substituting in g(z) = x² - y² + 2xy, we have g(z) = x² - y² + 2ixy
Now comparing with u(x, y) + iv(x, y), we get :
u(x, y) = x² - y² and v(x, y) = 2xy
Now applying the CRE, we get :
u x = 2xv
y = 2xu
y = - 2yv
x = 2x
Since the CRE are not satisfied, g(z) = x² - y² + 2xy (where z = x + iy) is not holomorphic at any point in C.
Therefore, we can say that there are no points in C for which the function g(z) is holomorphic.
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in cell b8, find the value from the appropriate probability table to construct a 90onfidence interval. Shipment Time to Deliver (Days)
1 7.0
2 12.0
3 4.0
4 2.0
5 6.0
6 4.0
7 2.0
8 4.0
9 4.0
10 5.0
11 11.0
12 9.0
13 7.0
14 2.0
15 2.0
16 4.0
17 9.0
18 5.0
19 9.0
20 3.0
21 6.0
22 2.0
23 6.0
24 5.0
25 6.0
26 4.0
27 5.0
28 3.0
29 4.0
30 6.0
31 9.0
32 2.0
33 5.0
34 6.0
35 7.0
36 2.0
37 6.0
38 9.0
39 5.0
40 10.0
41 5.0
42 6.0
43 10.0
44 3.0
45 12.0
46 9.0
47 6.0
48 4.0
49 3.0
50 7.0
51 2.0
52 7.0
53 3.0
54 2.0
55 7.0
56 3.0
57 5.0
58 7.0
59 4.0
60 6.0
61 4.0
62 4.0
63 7.0
64 8.0
65 4.0
66 7.0
67 9.0
68 6.0
69 7.0
70 11.0
71 9.0
72 4.0
73 8.0
74 10.0
75 6.0
76 7.0
77 4.0
78 5.0
79 8.0
80 8.0
81 5.0
82 9.0
83 7.0
84 6.0
85 14.0
86 9.0
87 3.0
88 4.0
This formula calculates the value corresponding to a 90% confidence level using the T.INV function. The first argument, 0.95, represents 1 minus the desired confidence level (0.05 for a 90% confidence level). The second argument, `COUNT(A2:A89)-1`, calculates the degrees of freedom by subtracting 1 from the count of data points.
To find the value from the appropriate probability table to construct a 90% confidence interval in cell B8, you can use the T.INV function in Excel. Here's how you can do it:
1. In cell B8, enter the formula:
```
=T.INV(0.95, COUNT(A2:A89)-1)
```
This formula calculates the value corresponding to a 90% confidence level using the T.INV function. The first argument, 0.95, represents 1 minus the desired confidence level (0.05 for a 90% confidence level). The second argument, `COUNT(A2:A89)-1`, calculates the degrees of freedom by subtracting 1 from the count of data points.
Make sure to adjust the cell references in the formula based on the actual location of your data in the spreadsheet.
Note: The T.INV function returns the value from the Student's t-distribution, which is used for small sample sizes or when the population standard deviation is unknown.
If you have a large sample size (usually considered more than 30), you can use the Z.INV function instead to calculate the value from the standard normal distribution.
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Who can help , desperately need
Answer:
1/4 is x and 1 is b
Step-by-step explanation:
Determine volume of a cylindre r2 + y2 = 4 inside a sphere r2 + y2 +22 = 16
The volume of the cylinder inside the given sphere is 8 cubic units.
How to determine the volume of the cylinder inside the given sphere?To determine the volume of the cylinder inside the given sphere, we need to find the limits of integration and set up the integral.
Let's analyze the equations:
Cylinder equation:[tex]r^2 + y^2 = 4[/tex]
Sphere equation: [tex]r^2 + y^2 + 2^2 = 16[/tex]
From the equations, we can see that the cylinder is centered at the origin (0, 0) with a radius of 2 and an infinite height along the y-axis. The sphere is centered at the origin as well, with a radius of 4.
To find the limits of integration, we need to determine where the cylinder intersects the sphere. By substituting the cylinder equation into the sphere equation, we can solve for the values of r and y:
[tex](2^2) + y^2 + 2^2 = 16\\4 + y^2 + 4 = 16\\y^2 = 8[/tex]
y = ±√8
We can see that the cylinder intersects the sphere at y = √8 and y = -√8. Since the cylinder has infinite height, the limits of integration for y will be from -√8 to √8.
Now we can set up the integral to calculate the volume of the cylinder:
V = ∫∫∫ dV
= [tex]\int_0^ 2 \int_{\sqrt -8} ^ {\sqrt 8}\int _{\sqrt-(16 - r^2 - y^2)} ^{\sqrt (16 - r^2 - y^2)} dz dy dr[/tex]
Since the integrand is equal to 1, we can simplify the integral to:
V = [tex]\int_0 ^ 2 \int _{-\sqrt8} ^ {\sqrt8} 2\sqrt{(16 - r^2 - y^2)}[/tex] dy dr
Evaluating this integral will give us the volume of the cylinder inside the sphere.
To evaluate the integral and calculate the volume, we can integrate the given expression with respect to y first and then with respect to r.
[tex]\int_0 ^ 2 \int _{-\sqrt8} ^ {\sqrt8} 2\sqrt{(16 - r^2 - y^2)}[/tex]
Let's begin by integrating with respect to y:
[tex]\int_{-\sqrt8} ^ {\sqrt8} 2\sqrt(16 - r^2 - y^2) dy[/tex]
We can simplify the integrand using the trigonometric substitution y = √8sinθ:
dy = √8cosθ dθ
y = √8sinθ
Replacing y and dy in the integral:
[tex]\int _{-\pi /2} ^{\pi/2} 2\sqrt(16 - r^2 - (\sqrt 8sin\theta)^2) \sqrt 8cos\theta d\theta[/tex]
= 16[tex]\int _{-\pi /2} ^ {\pi /2} \sqrt(1 - (r/4)^2 - sin^2\theta)[/tex]cosθ dθ
To simplify the integral further, we can use the trigonometric identity [tex]sin^2\theta + cos^2\theta = 1:[/tex]
16[tex]\int _{-\pi /2} ^ {\pi /2} \sqrt(1 - (r/4)^2 - sin^2\theta)[/tex]cosθ dθ
= 16 [tex]\int _{-\pi /2} ^ {\pi /2} \sqrt(r^2/16)[1 - cos^2\theta][/tex]cosθ dθ
= 4r[tex]\int _{-\pi/2} ^ {\pi/2}[/tex] sinθ cosθ dθ
= 4r [tex][ -cos^2\theta/2[/tex] ]| [-π/2 to π/2 ]
= 4r [ [tex]-cos^2(\pi/2)/2 + cos^2(-\pi/2)/2[/tex] ]
= 4r [ -1/2 + 1/2 ]
= 4r
Now, we can integrate with respect to r:
[tex]\int_0 ^ 2[/tex] 4r dr
= 2[tex]r^2[/tex]| [0 to 2]
= 2[tex](2^2 - 0^2)[/tex]
= 2(4)
= 8
Therefore, the volume of the cylinder is 8 cubic units.
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Has anyone done the Alg1B Portfolio - Unit 6 for connections academy
Answer:
I have
Step-by-step explanation:
PLZ HELP ME, ASAP I WILL CROWN BRAINLIEST!!!!
Drag the correct steps into order to solve the equation 5x5 + 5 = 10 for x
Answer:
x = 1
Step-by-step explanation:
5x^5 + 5 = 10
Start the solution by subtracting 5 from both sides:
5x^5 = 5
Dividing both sides by 5 yields x^5 = 1.
Taking the 5th root of both sides, we get x = 1
Can someone plz help me I beg u
Answer:
28.26
Step-by-step explanation:
The formula for finding the circumference is C=2pi(radius) and the radius is half of the diameter, which in our case is 4.5. So 2*3.14*4.5=28.26