In triangle ABC, ∠ABC=70° and ∠ACB=50°. Points M and N lie on sides AB and AC respectively such that ∠MCB=40° and ∠NBC=50°. Find m∠NMC.
Answers
Answer:
∠NMC = 50°
Step-by-step explanation:
The interpretation of the information given in the question can be seen in the attached images below.
In ΔABC;
∠ A + ∠ B + ∠ C = 180° (sum of angles in a triangle)
∠ A + 70° + 50° = 180°
∠ A = 180° - 70° - 50°
∠ A = 180° - 120°
∠ A = 60°
In ΔAMN ; the base angle are equal , let the base angles be x and y
So; x = y (base angle of an equilateral triangle)
Then;
x + x + 60° = 180°
2x + 60° = 180°
2x = 180° - 60°
2x = 120°
x = 120°/2
x = 60°
∴ x = 60° , y = 60°
In ΔBQC
∠a + ∠e + ∠b = 180°
50° + ∠e + 40° = 180°
∠e = 180° - 50° - 40°
∠e = 180° - 90°
∠e = 90°
At point Q , ∠e = ∠f = ∠g = ∠h = 90° (angles at a point)
∠i = 50° - 40° = 10°
In ΔNQC
∠f + ∠i + ∠j = 180°
90° + 10° + ∠j = 180°
∠j = 180° - 90°-10°
∠j = 180° - 100°
∠j = 80°
From line AC , at point N , ∠y + ∠c + ∠j = 180° (sum of angles on a straight line)
60° + ∠c + ∠80° = 180°
∠c = 180° - 60°-80°
∠c = 180° - 140°
∠c = 40°
Recall that :
At point Q , ∠e = ∠f = ∠g = ∠h = 90° (angles at a point)
Then In Δ NMC ;
∠d + ∠h + ∠c = 180° (sum of angles in a triangle)
∠d + 90° + 40° = 180°
∠d = 180° - 90° -40°
∠d = 180° - 130°
∠d = 50°
Therefore, ∠NMC = ∠d = 50°
Related Questions
a sequence of transformations is described below horizontal stretch about a vertical line PQ, a translation, another horizontal stretch about PQ, a reflection over PQ.
Answer Choices:
Angle measures only
Segment lengths only
Both angle measures and segment lengths
Neither angle measures nor segments lengths
Answers
Answer:
Both angle measures and segment lengths.
Step-by-step explanation:
An angle is a shape formed by two rays that meets at a point. The angle is measured by degrees. The angle is formed by the sides of an angle which shares the common endpoint called the vertex. The line is horizontal stretch with a vertical line PQ. It will measure the angle and segments lengths.
Answer:
neither angle measures nor segment lines
WILL GIVE BRAINLEST PLEASE!!!!!!!! Jenny has some tiles in a bag. The tiles are of three different colors: purple, pink, and orange. Jenny randomly pulls a tile out of the bag, records the color, and replaces the tile in the bag. She does this 50 times. The results are recorded in the given table: Color of Tile Purple Pink Orange Number of times the tile is drawn 6 18 26 What is the experimental probability that Jenny will pull out a purple tile? fraction 6 over 50 fraction 44 over 50 fraction 6 over 44 fraction 18 over 44
Answers
Answer:
6/50
Step-by-step explanation:
There are 50 tiles
6 purple
18 pink
26 orange
P( purple) = purple/ total
= 6/50
Classify the following random variable according as either discrete or continuous. The temperature in degrees Celsius on January 1st in a certain city
A continuous
B discrete
Answers
Answer:
continuous
Step-by-step explanation:
A quantity like temperature is a continuous random variable. A continuous random variable is different from a discrete random variable because it can take on many values infinitely.
From the question, measuring the Temperature in degrees can take on many different values because there are an uncountable number of possible temperatures that could be taken.
-7y=-91 show your work
Answers
Answer:
[tex] \boxed{ \bold{\sf{y = 13}}}[/tex]Step-by-step explanation:
[tex] \sf{ - 7y = - 91}[/tex]
Divide both sides of the equation by -7
⇒[tex] \sf{ \frac{ - 7y}{ - 7} = \frac{ - 91}{ - 7} }[/tex]
Calculate
⇒[tex] \sf{y = 13}[/tex]
Hope I helped!
Best regards!!
Answer:
[tex] \boxed{\sf y = 13} [/tex]
Step-by-step explanation:
Solve for y:
[tex] \sf \implies - 7y = - 91[/tex]
Divide both sides of -7y = -91 by -7:
[tex] \sf \implies \frac{ - 7y}{ - 7} = \frac{ - 91}{ - 7} [/tex]
[tex] \sf \frac{ - 7}{ - 7} = 1 : [/tex]
[tex] \sf \implies y = \frac{ - 91}{ - 7} [/tex]
[tex] \sf \implies y = \frac{ \cancel{ - 7} \times 13}{ \cancel{ - 7}} [/tex]
[tex] \sf \implies y = 13[/tex]
The multiplicative inverse of – 1 in the set {-1,1}is
Answers
Answer: The multiplicative inverse of – 1 in the set {-1,1} is -1.
Step-by-step explanation:
In algebra, the multiplicative inverse of a number(x) is a number (say y) such that
[tex]x\times y=1[/tex] [product of a number and its inverse =1]
if x= -1, then
[tex]-1\times y=1\Rightarrow\ y=-1[/tex]
That means , the multiplicative inverse of -1 is -1 itself.
Hence, the multiplicative inverse of – 1 in the set {-1,1} is -1.
PLzzzz answer quick will give good rate nd say thanks
What is the slope of the line? 5(y+2)=4(x-3)
A 2/3
B 4/5
C 5/4
D 3/2
Answers
Step-by-step explanation:
Here,
[tex]5y = 4x - 12 - 10[/tex]
[tex]y = \frac{4}{5} x - \frac{22}{5} [/tex]
slope is 4÷5
Answer:
The answer is option BStep-by-step explanation:
Equation of a line is y = mx + c
where
m is the slope
c is the y intercept
5(y+2)=4(x-3)
To find the slope first expand the terms in the equation
That's
5y + 10 = 4x - 12
Write the equation in the form y = mx+ c
That's
5y = 4x - 12 - 10
5y = 4x - 22
Divide both sides by 5 to make y stand alone
y = 4/5x - 22/5
Comparing with the general equation above the slope of the line is
4/5Hope this helps you
A population of values has a normal distribution with μ= 106.9 and σ=14.5
You intend to draw a random sample of size n=20
What is the probability that a single randomly selected value is less than 109.8?
P(X < 109.8)
How do you the probability that a sample of size n= 20 is randomly selected with a mean less than 109.8?
P(M < 109.8)
Also, I have to round the answer to the 4th decimal place. How do I do that?
Answers
Step-by-step explanation:
Find the z-score.
z = (x − μ) / σ
z = (109.8 − 106.9) / 14.5
z = 0.2
Use a chart or calculator to find the probability.
P(Z < 0.2) = 0.5793
Find the mean and standard deviation of the sampling distribution.
μ = 106.9
σ = 14.5 / √20 = 3.242
Find the z-score.
z = (x − μ) / σ
z = (109.8 − 106.9) / 3.242
z = 0.894
Use a calculator to find the probability.
P(Z < 0.894) = 0.8145
Which statements about the dilation are true? Check all that apply. Triangle X prime Y prime Z prime. Point X prime is 2 units from the center of dilation C and point Z prime is 3 units from the center of dilation. Triangle X Y Z. Point X is 5 units from point C and point Z is 7.5 units from point C. The center of dilation is point C. It is a reduction. It is an enlargement. The scale factor is 2.5. The scale factor is Two-fifths.
Answers
Answer:
I only know two right answers.
A: The center of dilation is point C.
C: It is an enlargement.
E: The scale factor is 2/5.
Step-by-step explanation:
These two answers are correct because When you look in the center you see a C.
You tell if it is a reduction because the pre image is small but the image is big.
The center of dilation is point C.
It is an enlargement.
The scale factor is 2/5
The correct options are D, F, H.
What is dilation?Resizing an item uses a transformation called dilation. Dilation is used to enlarge or shorten the structures. The result of this transformation is an image with the same shape as the original. However, there is a variation in the shape's size. The initial form should be stretched or contracted during a dilatation.
Given:
The transformation of the figure is dilation.
The figure is given in the attached image.
From the diagram:
The center of dilation is point C.
It is an enlargement.
The scale factor is 2/5
Therefore, all the correct statements are given above.
To learn more about the dilation in geometry;
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Yelena needs to swim a total of 8 miles this
week. So far, she swam 5 miles. Use the
equation 5 + m=8 to find how many more
miles Yelena needs to swim.
Answers
Answer:
3 miles
Step-by-step explanation:
5 + m=8
Subtract 5 from each side
5-5 + m=8-5
m = 3
She needs to swim 3 more miles
Answer:
Yelena needs to swim 3 more miles
Step-by-step explanation:
You need to solve for the variable "m", which represents the miles. Based on the information, Yelena swam 5 miles and she needs to swim 8. Solve:
[tex]5+m=8[/tex]
To find the value of m, you need to isolate it on one side of the equation. To do this, you need to get the 8 and 5 on the same side of the equal operation. For this, you need to use reverse operations. This undoes the value from one side and does the same on the other, keeping the equation balanced. Since we have a "positive 5", we take the opposite, which would be a "negative 5". So subtract 5 from both sides of the equation:
[tex]5-5+m=8-5[/tex]
Simplify. The 5's cancel each other out, leaving 0. 8-5 is 3:
[tex]m=3[/tex]
The total miles left that Yelena needs to swim is 3 miles.
:Done
Suppose 232subjects are treated with a drug that is used to treat pain and 50of them developed nausea. Use a 0.01significance level to test the claim that more than 20%of users develop nausea. Identify the null and alternative hypotheses for this test.
A. Upper H0?: p equals 0.20
Upper H1?: p not equals 0.20
B. Upper H0?: p equals 0.20
Upper H1?: p greater than 0.20
C. Upper H0?: p greater than 0.20
Upper H1?: p equals 0.20
D. Upper H0?: p equals 0.20
Upper H1?: p less than 0.20
Identify the test statistic for this hypothesis test. Identify the P-value for this hypothesis test.
Identify the conclusion for this hypothesis test.
A. Reject Upper H 0. There is sufficient evidence to warrant support of the claim that more than 20?% of users develop nausea.
B. Fail to reject Upper H 0. There is sufficient evidence to warrant support of the claim that more than 20?% of users develop nausea.
C. Reject Upper H 0. There is not sufficient evidence to warrant support of the claim that more than 20?% of users develop nausea.
D. Fail to reject Upper H 0. There is not sufficient evidence to warrant support of the claim that more than 20?% of users develop nausea.
Answers
Answer:
A
The correct option is B
B
[tex]t = 0.6093[/tex]
C
[tex]p-value = 0.27116[/tex]
D
The correct option is D
Step-by-step explanation:
From the question we are told that
The sample size is [tex]n = 232[/tex]
The number that developed nausea is X = 50
The population proportion is p = 0.20
The null hypothesis is [tex]H_o : p = 0.20[/tex]
The alternative hypothesis is [tex]H_a : p > 0.20[/tex]
Generally the sample proportion is mathematically represented as
[tex]\r p = \frac{50}{232}[/tex]
[tex]\r p = 0.216[/tex]
Generally the test statistics is mathematically represented as
=> [tex]t = \frac{\r p - p }{ \sqrt{ \frac{p(1- p )}{n} } }[/tex]
=> [tex]t = \frac{ 0.216 - 0.20 }{ \sqrt{ \frac{ 0.20 (1- 0.20 )}{ 232} } }[/tex]
=> [tex]t = 0.6093[/tex]
The p-value obtained from the z-table is
[tex]p-value = P(Z > 0.6093) = 0.27116[/tex]
Given that the [tex]p-value > \alpha[/tex] then we fail to reject the null hypothesis
Please help! picture above plus, part B: write the quadratic expression in the numerator and the dominator in factored form. Part C: cancel the common factor of the numerator and the denominator to write the expression in simplified form.
Answers
Answer:
work is shown and pictured
Answer:
Hi, there!!!
The answer would be 2(2x-1)/x(x-4).
See explanation in picture.
Hope it helps...
g A modal class in a histogram is the class that includes a. the largest number of observations. b. the smallest observation in the data set. c. the largest observation in the data set. d. the smallest number of observations.
Answers
Answer:
a. the largest number of observations.
Step-by-step explanation:
The mode is the variable in a data that has the highest frequency. Which implies that it occurs more than other variables.
When plotting a histogram, the modal class is one that has the greatest number of observations. Showing that the variables comprised in the class has more occurrence than others. Therefore, the required answer to the question is option a.
Determine whether the statement is true or false. If it is false, rewrite it as a true statement. A sampling distribution is normal only if the population is normal. Choose the correct answer below. A. The statement is false. A sampling distribution is normal only if n30. B. The statement is false. A sampling distribution is normal if either n30 or the population is normal. C. The statement is true. D. The statement is false. A sampling distribution is never normal.
Answers
The statement is false. A sampling distribution is normal if either n > 30 or the population is normal.
==========================================
Explanation:
If the underlying population is normally distributed, then so is the sample distribution (such as the distribution of sample means, aka xbar distribution).
Even if the population isn't normally distributed, the xbar distribution is approximately normal if n > 30 due to the central limit theorem. Some textbooks may use a higher value than 30, but after some threshold is met is when the xbar distribution is effectively "normal".
Choice A is close, but is missing the part about the population being normal. If we know the population is normal, then n > 30 doesn't have to be required.
23. f(x) is vertically shrank by a factor of 1/3. How will you represent f(x) after transformation?
A. f(3x)
B. 3f(x)
C. 13f(x)
D. f(13x)
Answers
Answer:
Step-by-step explanation:
vertical stretching / shrinking has the following transformation.
f(x) -> a * f(x)
when a > 1, it is stretching
when 0< a < 1, it is shrinking.
when -1 < a < 0, it is shringking + reflection about the x-axis
when a < -1, it is stretching + reflection about the x axis.
Here it is simple shrinking, so 0 < a < 1.
I expect the answer choice to show (1/3) f(x).
However, if the question plays with the words
"shrink by a factor of 1/3" to actually mean a "stretching by a factor of three", then B is the answer (stretch by a factor of three).
A plot of land has vertices as follows, where each coordinate is a measurement in feet. Find the perimeter of the plot of land. (1,7),(7,7),(7,1),(1,1) please help and explain how to do this type of thing because i am lost
Answers
Answer:
Perimeter of ABCD = 36 ft
Step-by-step explanation:
Given:
A (1,7)
B (7,7)
C (7,1)
D (1,1)
Find:
Perimeter of ABCD
Computation:
Distance between two point = √(x1-x2)² + (y1-y2)²
So,
AB = √(1-7)²+(7-7)²
AB = 6 ft
BC = √(7-7)²+(7-1)²
BC = 6 ft
CD = √(7-1)²+(1-1)²
CD = 6 ft
DA = √(1-1)²+(1-7)²
DA = 6 ft
Perimeter of ABCD = AB + BC + CD + DA
Perimeter of ABCD = 6 + 6 + 6 +6
Perimeter of ABCD = 36 ft
The perimeter of the plot is the sum of side length of the plot of land.
The perimeter of the plot is 24 feet.
Represent the vertices as follows:
[tex]W = (1,7)[/tex]
[tex]X = (7,7)[/tex]
[tex]Y = (7,1)[/tex]
[tex]Z = (1,1)[/tex]
First, we calculate the side length using the following distance formula:
[tex]d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}[/tex]
So, we have:
[tex]WX = \sqrt{(1- 7)^2 + (7- 7)^2} = \sqrt{36} = 6[/tex]
[tex]XY = \sqrt{(7- 7)^2 + (7- 1)^2} = \sqrt{36} = 6[/tex]
[tex]YZ = \sqrt{(7- 1)^2 + (1- 1)^2} = \sqrt{36} = 6[/tex]
[tex]ZW = \sqrt{(1- 1)^2 + (1- 7)^2} = \sqrt{36} = 6[/tex]
The perimeter (P) is then calculated as follows:
[tex]P = WX + XY + YZ + ZW[/tex]
So, we have:
[tex]P = 6 + 6 + 6 + 6[/tex]
[tex]P = 24[/tex]
Hence, the perimeter of the plot of land is 24 feet.
Read more about perimeters at:
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Help plz! Jim is climbing a mountain that has a base 150 feet above sea level. If he climbs 233 feet then descends into a cave 64 feet, how far above sea level is Jim
Answers
Jim is 319 ft above sea level.
Answer:
150+233-64=319
Jim is 319 ft above sea level.
Step-by-step explanation:
A manager from a certain well known department store found out the money their customers carry into the store is normally distributed with a mean of $258 dollars and a standard deviation of $35. In a sample of 76 Americans who walked into that store find the probability that a random customer will have more than $260 in his or her wallet
Answers
Answer:
0.30924
Approximately ≈ 0.3092
Step-by-step explanation:
To solve for this question, we use the formula:
z = (x - μ)/σ
where x is the raw score
μ is the sample mean
σ is the sample standard deviation.
From the question,
x is the raw score = 260
μ is the sample mean = population standard deviation = 258
σ is the sample standard deviation
= σ/√N
N = 76 samples
σ = Population standard deviation
= 35/√76
= 4.0146919966
Hence,
z = (x - μ)/σ
= 260 - 258/ 4.0146919966
= 0.4981702212
Approximately = 0.498
We find the Probability using z score table for normal distribution
P(x = z) = P( x = 260)
= P( z = 0.498)
= 0.69076
The probability that a random customer will have more than $260 in his or her wallet is calculated as:
P(x>Z) = 1 - P( z = 0.498)
P(x>Z) = 1 - 0.69076
P(x>Z) = 0.30924
Approximately ≈ 0.3092
1 = prt is an example of
O
a variable
an expression
a constant
a formula
Answers
Complete Question
I = prt is an example of
• a variable
• an expression
• a constant
• a formula
Answer:
A formula
Step-by-step explanation:
I = prt is an example of a formula called Simple Interest.
Simple Interest can be defined as the formula that is used to calculate the interest that is accumulated on a particular amount of money which was saved in a financial institution or loaned out to a person at a given interest rate for a particular period of time.
The formula for Simple Interest is Expressed as:
I = PRT
Where :
I = Simple Interest
P = Principal = Amount saved, or loaned out
R = Interest rate that is given in percentage form
T = Time that has elapsed in Years.
Answer:
D. A Formula
Step-by-step explanation:
A formula is an equation that uses variables to state a rule.
Hope I was able to help you!
what is PI numbers?
Answers
Answer:
These are the first 100 digits of pi: 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 7067
Step-by-step explanation:
Pi goes on continuously forever, so this is a reduced version, by including the first 100 digits.
create an equation with a solution closest to 0 using digits 1 to 9
Answers
Complete Questions:
Create an equation with a solution closest to 0 using digits 1 to 9
_x + _ = _x + _
Answer:
See Explanation
Step-by-step explanation:
Given
_x + _ = _x + _
Required
Fill in the gap using 1 to 9 to give a result close to 0
First, you have to determine what kind of numbers that are close to 0;
In this case, I'll work with -0.4 to 0.4 because the number in this range approximate to 0;
Next, is to fill in the gaps using trial by error method
5x + 2 = 2x + 3
Checking the above expression
Collect Like Terms
[tex]5x - 2x = 3 - 2[/tex]
[tex]3x = 1[/tex]
Divide equation by 2
[tex]x = 0.33[/tex] (Approximated)
Another trial is
6x + 8 = 2x + 7
Checking the above expression
Collect Like Terms
[tex]6x - 2x = 7 - 8[/tex]
[tex]4x = -1[/tex]
Divide equation by 4
[tex]x = -0.25[/tex] (Approximated)
I'll stop here but note that, there are more expressions that can fill in the gaps
URGENT, PLEASE HELP! (3/5) -50 POINTS- !please no wrong answers for the points.! A) [tex]y = \frac{9}{2} x + \frac{1}{2}[/tex] B) [tex]y = - \frac{1}{2} x + \frac{7}{2}[/tex] C) [tex]y = -4x + 9[/tex] D) [tex]y = 4x + 15[/tex]
Answers
Answer:
B y = -1/2x + 7/2
Step-by-step explanation:
We know that it has a negative slope since the points go from the top left to the bottom right
We can eliminate A and D
The y intercept is where it crosses the y axis
It should cross somewhere between 2 and 4
C has a y intercept of 9 which is too big
Lets verify with a point
x = -4
y = -4(-4)+9 = 16+9 = 25 (-4,25) not even close to being near the points on the graph
checking B
y = -1/2 (-4) +7/2
= 2 + 7/2 = 11/2 = 5.5 it seems reasonable
Answer:
[tex]\Large \boxed{y=-\frac{1}{2} x+\frac{7}{2} }[/tex]
Step-by-step explanation:
Using a graph,
we can see that the line y = -1/2x + 7/2 best fits for the data.
PLS HELP ASAP:Find all the missing elements:
Answers
Answer:
B = 34.2°
C = 105.8°
c = 12.0 units
Step-by-step Explanation:
Given:
A = 40°
a = 8
b = 7
Required:
Find B, C, and c.
SOLUTION:
Using the Law of Sines, find <B:
[tex] \frac{sin(A)}{a} = \frac{sin(B)}{b} [/tex]
[tex] \frac{sin(40)}{8} = \frac{sin(B)}{7} [/tex]
Multiply both sides by 7
[tex] \frac{sin(40)}{8}*7 = \frac{sin(B)}{7}*7 [/tex]
[tex] \frac{sin(40)*7}{8} = sin(B) [/tex]
[tex] 0.5624 = sin(B) [/tex]
[tex] B = sin^{-1}(0.5624) [/tex]
[tex] B = 34.2 [/tex] (to nearest tenth).
Find <C:
C = 180 - (34.2+40°) (sum of angles in a triangle)
C = 180 - 74.2 = 105.8°
Using the Law of Sines, find c.
[tex] \frac{c}{sin(C)} = \frac{b}{sin(B)} [/tex]
[tex]\frac{c}{sin(105.8)} = \frac{7}{sin(34.2)}[/tex]
Multiply both sides by sin(105.8)
[tex]\frac{c}{sin(105.8)}*sin(105.8) = \frac{7}{sin(34.2)}*sin(105.8)[/tex]
[tex] c = \frac{7*sin(105.8)}{sin(34.2)} [/tex]
[tex] c = 12.0 [/tex]
A research center claims that % of adults in a certain country would travel into space on a commercial flight if they could afford it. In a random sample of adults in that country, % say that they would travel into space on a commercial flight if they could afford it. At , is there enough evidence to reject the research
Answers
Complete Question
A research center claims that 30% of adults in a certain country would travel into space on a commercial flight if they could afford it. In a random sample of 700 adults in that country, 34% say that they would travel into space on a commercial flight if they could afford it. At , is there enough evidence to reject the research center's claim
Answer:
Yes there is sufficient evidence to reject the research center's claim.
Step-by-step explanation:
From the question we are told that
The population proportion is p = 0.30
The sample proportion is [tex]\r p = 0.34[/tex]
The sample size is n = 700
The null hypothesis is [tex]H_o : p = 0.30[/tex]
The alternative hypothesis is [tex]H_a : p \ne 0.30[/tex]
Here we are going to be making use of level of significance = 0.05 to carry out this test
Now we will obtain the critical value of [tex]Z_{\alpha }[/tex] from the normal distribution table , the value is [tex]Z_{\alpha } = 1.645[/tex]
Generally the test statistics is mathematically represented as
[tex]t = \frac{ \r p - p }{ \sqrt{ \frac{ p (1-p)}{n} } }[/tex]
substituting values
[tex]t = \frac{ 0.34 - 0.30 }{ \sqrt{ \frac{ 0.30 (1-0.30 )}{ 700} } }[/tex]
[tex]t = 2.31[/tex]
Looking at the values of t and [tex]Z_{\alpha }[/tex] we see that [tex]t > Z_{\alpha }[/tex] hence the null hypothesis is rejected
Thus we can conclude that there is sufficient evidence to reject the research center's claim.
Use Green’s theorem to evaluate line integral along curve C ∮_c〖( 3ydx+2xdy )〗, C : The boundary of 0≤x≤π,0≤y≤sin x
Answers
Answer:
[tex]\displaystyle \oint_C {3y \, dx + 2x \, dy} = \boxed{\bold{2}}[/tex]
General Formulas and Concepts:
Calculus
Differentiation
DerivativesDerivative NotationDerivative Property [Multiplied Constant]:
[tex]\displaystyle (cu)' = cu'[/tex]
Derivative Rule [Basic Power Rule]:
Integration
IntegralsIntegration Rule [Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]
Integration Property [Multiplied Constant]:
[tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
Multivariable Calculus
Partial Derivatives
Vector Calculus
Circulation Density:
[tex]\displaystyle F = M \hat{\i} + N \hat{\j} \rightarrow \text{curl} \ \bold{F} \cdot \bold{k} = \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}[/tex]
Green's Theorem [Circulation Curl/Tangential Form]:
[tex]\displaystyle \oint_C {F \cdot T} \, ds = \oint_C {M \, dx + N \, dy} = \iint_R {\bigg( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \bigg)} \, dx \, dy[/tex]
Step-by-step explanation:
Step 1: Define
Identify given.
[tex]\displaystyle \oint_C {3y \, dx + 2x \, dy}[/tex]
[tex]\displaystyle \text{Region:} \ \left \{ {{0 \leq x \leq \pi} \atop {0 \leq y \leq \sin x}} \right.[/tex]
Step 2: Integrate Pt. 1
Define vector functions M and N:[tex]\displaystyle M = 3y , \ N = 2x[/tex][Circulation Density] Differentiate [Derivative Rules and Properties]:
[tex]\displaystyle \frac{\partial M}{\partial y} = 3 , \ \frac{\partial N}{\partial x} = 2[/tex][Green's Theorem] Substitute in Circulation Density:
[tex]\displaystyle \oint_C {3y \, dx + 2x \, dy} = \iint_R {2 - 3} \, dx \, dy[/tex]Simplify:
[tex]\displaystyle \oint_C {3y \, dx + 2x \, dy} = - \iint_R {} \, dx \, dy[/tex][Integrals] Substitute in region R:
[tex]\displaystyle \oint_C {3y \, dx + 2x \, dy} = - \int\limits^{\pi}_0 \int\limits^{\sin x}_0 {} \, dy \, dx[/tex]
Step 3: Integrate Pt. 2
We can evaluate the Green's Theorem double integral we found using basic integration techniques listed above:
[tex]\displaystyle \begin{aligned}\oint_C {3y \, dx + 2x \, dy} & = - \int\limits^{\pi}_0 \int\limits^{\sin x}_0 {} \, dy \, dx \\& = - \int\limits^{\pi}_0 {y \bigg| \limits^{y = \sin x}_{y = 0}} \, dx \\& = - \int\limits^{\pi}_0 {\sin x} \, dx \\& = \cos x \bigg| \limits^{x = \pi}_{x = 0} \\& = \boxed{\bold{2}}\end{aligned}[/tex]
∴ we have evaluated the line integral using Green's Theorem.
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Learn more about multivariable calculus: https://brainly.com/question/14502499
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Topic: Multivariable Calculus
Unit: Green's Theorem and Surfaces
A simple random sample of 28 Lego sets is obtained and the number of pieces in each set was counted.The sample has a standard deviation of 12.65. Use a 0.05 significance level to test the claim that the number of pieces in a set has a standard deviation different from 11.53.
Answers
Answer:
Step-by-step explanation:
Given that:
A simple random sample n = 28
sample standard deviation S = 12.65
standard deviation [tex]\sigma[/tex] = 11.53
Level of significance ∝ = 0.05
The objective is to test the claim that the number of pieces in a set has a standard deviation different from 11.53.
The null hypothesis and the alternative hypothesis can be computed as follows:
Null hypothesis:
[tex]H_0: \sigma^2 = \sigma_0^2[/tex]
Alternative hypothesis:
[tex]H_1: \sigma^2 \neq \sigma_0^2[/tex]
The test statistics can be determined by using the following formula in order to test if the claim is statistically significant or not.
[tex]X_0^2 = \dfrac{(n-1)S^2}{\sigma_0^2}[/tex]
[tex]X_0^2 = \dfrac{(28-1)(12.65)^2}{(11.53)^2}[/tex]
[tex]X_0^2 = \dfrac{(27)(160.0225)}{132.9409}[/tex]
[tex]X_0^2 = \dfrac{4320.6075}{132.9409}[/tex]
[tex]X_0^2 = 32.5002125[/tex]
[tex]X^2_{1- \alpha/2 , df} = X^2_{1- 0.05/2 , n-1}[/tex]
[tex]X^2_{1- \alpha/2 , df} = X^2_{1- 0.025 , 28-1}[/tex]
From the chi-square probabilities table at 0.975 and degree of freedom 27;
[tex]X^2_{0.975 , 27}[/tex] = 14.573
[tex]X^2_{\alpha/2 , df} = X^2_{ 0.05/2 , n-1}[/tex]
[tex]X^2_{\alpha/2 , df} = X^2_{0.025 , 28-1}[/tex]
From the chi-square probabilities table at 0.975 and degree of freedom 27;
[tex]X^2_{0.025 , 27}=[/tex] 43.195
Decision Rule: To reject the null hypothesis if [tex]X^2_0 \ > \ X^2_{\alpha/2 , df} \ \ \ or \ \ \ X^2_0 \ < \ X^2_{1- \alpha/2 , df}[/tex] ; otherwise , do not reject the null hypothesis:
The rejection region is [tex]X^2_0 \ > 43.195 \ \ \ or \ \ \ X^2_0 \ < \ 14.573[/tex]
Conclusion:
We fail to reject the null hypothesis since test statistic value 32.5002125 lies between 14.573 and 43.195.
A scientist needs 120mL of a 20% acid solution for an experiment. The lab has available a 10% solution and a 25% solution. How many milliliters of the 10% solution and how many milliliters of the 25% solution should the scientist mix to make the 20% solution?
Answers
Answer:
40 mL of 10% acid
80 mL of 25% acid
Step-by-step explanation:
x = volume of 10% acid solution
y = volume of 25% acid solution
Total volume is:
x + y = 120
Total amount of acid is:
0.10 x + 0.25 y = 0.20 (120)
Solve by substitution.
0.10 x + 0.25 (120 − x) = 0.20 (120)
0.10 x + 30 − 0.25 x = 24
0.15 x = 6
x = 40
y = 80
Commute times in the U.S. are heavily skewed to the right. We select a random sample of 45 people from the 2000 U.S. Census who reported a non-zero commute time. In this sample the mean commute time is 25.2 minutes with a standard deviation of 19.1 minutes. Required:a. Can we conclude from this data that the mean commute time in the U.S. is less than half an hour?b. Conduct a hypothesis test at the 5% level of significance. c. What is the p-value for this hypothesis test?
Answers
Answer:
The mean commute time in the U.S. is less than half an hour.
Step-by-step explanation:
In this case we need to test whether the mean commute time in the U.S. is less than half an hour.
The information provided is:
[tex]n=45\\\bar x=25.5\\s=19.1\\\alpha =0.05[/tex]
(a)
The hypothesis for the test can be defined as follows:
H₀: The mean commute time in the U.S. is not less than half an hour, i.e. μ ≥ 30.
Hₐ: The mean commute time in the U.S. is less than half an hour, i.e. μ < 30.
(b)
As the population standard deviation is not known we will use a t-test for single mean.
Compute the test statistic value as follows:
[tex]t=\frac{\bar x-\mu}{s/\sqrt{n}}=\frac{25.2-30}{19.1/\sqrt{45}}=-1.58[/tex]
Thus, the test statistic value is -1.58.
(c)
Compute the p-value of the test as follows:
[tex]p-value=P(t_{(n-1)}<-1.58)=P(t_{(45-1)}<-1.58)=0.061[/tex]
*Use a t-table.
The p-value of the test is 0.061.
Decision rule:
If the p-value of the test is less than the significance level then the null hypothesis will be rejected and vice-versa.
p-value = 0.061> α = 0.05
The null hypothesis will not be rejected at 5% level of significance.
Thus, concluding that the mean commute time in the U.S. is less than half an hour.
Rania graphs the relationship between temperature (in °C) and elevation (in m) in 9 different cities
shown below)
Answers
Answer: 7
Step-by-step explanation:
Answer :
It Is 7 On Khan Academy
◊ YusuCr ◊
:)
14. Find the distance between (7,217pi/180 ) and (5,-23pi/36 ) on the polar plane.
Answers
Answer: the distance is 3.49 units
Step-by-step explanation:
There are some ways to find the exact distance, i will calculate the distance in rectangular coordinates.
When we have a point (R, θ) in polar coordinates, we can transform it into rectangular coordinates as:
x = R*cos(θ)
y = R*sin(θ)
Then we have:
(7,217pi/180 )
R = 7
θ = (217/180)*pi
x = 7*cos( (217/180)*pi) = -5.59
y = 7*sin( (217/180)*pi) = -4.21
So this point is (-5.59, -4.21) in rectangular coordinates.
And the other point is (5,-23pi/36 )
R = 5
θ = -(23/36)*pi
x = 5*cos( -(23/36)*pi ) = -2.11
y = 5*sin( -(23/36)*pi ) = -4.53
So this point is (-2.11, - 4.53)
Then the point distance between those points is:
D = I (-2.11, -4.53) - (-5.59, -4.21) I
D = I (-2.11 + 5.59, -4.53 + 4.21) I
D = I (3.48, -0.32) I = √( (3.48)^2 + (-0.32)^2) = 3.49
What is the expression
Answers
Answer:
3
Step-by-step explanation:
z - 2x
--------
y
Let x = 3 y = -4 and z =-6
-6 - 2(3)
--------
-4
-6 -6
---------
-4
-12
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-4
3
Answer:
3
Step-by-step explanation:
To solve this, we need to plug in each of the numbers to the equation.
x = 3, y = - 4, z = - 6
[tex]\frac{z-2x}{y} = \frac{-6-2(3)}{-4}[/tex]
Let's solve the parenthesis first. - 2 * 3 = - 6.
[tex]\frac{-6-6}{-4}[/tex]
We then subtract -6 - 6.
[tex]\frac{-12}{-4}[/tex]
Then, we divide (cancel out the negatives).
[tex]-12 / -4 =3[/tex]
Our final answer is 3. Hope this helps!
