Select all expressions that are NOT written correctly in Scientific Notation.
Answers
The expressions that are not written correctly in scientific notation are: 48,200, 36.105, 8.7.10-1, and 0.78.10-3
Scientific notation is a way to express numbers in a concise form, using a number between 1 and 10 multiplied by a power of 10. Let's analyze the given expressions and identify the ones that are not written correctly in scientific notation:
48,200: This expression is not written in scientific notation. It should be expressed as 4.82 × 10^4 or 4.82e4.
0.00099: This expression is correctly written in scientific notation. It can be expressed as 9.9 × 10^-4 or 9.9e-4.
36.105: This expression is not written in scientific notation. It should be expressed as 3.6105 × 10^1 or 3.6105e1.
8.7.10-1: This expression is not written correctly in scientific notation. Scientific notation only allows one decimal point in the number. The correct representation would be 8.7 × 10^-1 or 8.7e-1.
0.78.10-3: Similar to the previous expression, this is not written correctly in scientific notation. The correct representation would be 7.8 × 10^-3 or 7.8e-3.
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Related Questions
Without using technical terms, state a final conclusion that addresses the original claim. Which of the following is the correct conclusion? A. The mean pulse rate (in beats per minute) of the group of adult males is bpm. B. The mean pulse rate (in beats per minute) of the group of adult males is not bpm. C. There sufficient evidence to warrant rejection of the claim that the mean pulse rate (in beats per minute) of the group of adult males is bpm. D. There sufficient evidence to warrant rejection of the claim that the mean pulse rate (in beats per minute) of the group of adult males is bpm.
Answers
Answer:
D. There is sufficient evidence to warrant rejection of the claim that the mean pulse rate (in beats per minute) of the group of adult males is 69bpm.
Step-by-step explanation:
Hypothesis testing is used in statistics to confirm the validity of a statement or observation. Null hypothesis is the original observation for which the test is being conducted. Alternative hypothesis is set against the null hypothesis to reach a conclusion whether to accept or reject the null hypothesis. In the given scenario there is sufficient evidence to reject the null hypothesis.
What is the value of f?
Answers
[tex]f+44^{\circ}[/tex] and [tex]71^{\circ}[/tex] are opposite angles.
Therefore
[tex]f+44^{\circ}=71^{\circ}\\f=27^{\circ}[/tex]
27°
Step-by-step explanation:Using information about angles, we can create an equation to solve for f.
Verticle Angles
There is a special angle relationship known as vertical angles. Vertical angles are opposite angles made by interesting lines. The left angle (made up of f and 44°) and the right angle (71°) are vertical angles. We know this because they are opposite each other and formed by the same interesting lines. Verticle angles are always congruent. This means that their measurements are equal.
Solving for f
We can use the information above to create an equation. Since we know that the left and right angles are congruent, we can set the equal to each other.
f + 44 = 71Now, we can subtract 44 from both sides and solve for f.
f = 27So, angle f must equal 27°.
What is the area of the triangle in square yards?
Answers
Answer:
DON'T LOOK AT THE PICTURE!!
Step-by-step explanation:
i need help with this i will mark you as brainliest
Answers
Answer:
The answer is 7 (A)
Step-by-step explanation:
Answer:
D. The length is the same as line MN.
Step-by-step explanation:
Even if the line is moved any one bit, the length of the line will not change. So as a result, A, B & C are already out. That leaves D as our answer.
HELPPPPPPPPPP :(
If there are two doors, each 36 inches wide.
Door A opened to 86° angle. Door B opened to 82°
which door's outer edge is farther from its closed position? and please explain it ♡
Answers
Answer:
B its B
Step-by-step explanation:
At an electronics store, DVDs are on sale. Some DVDs cost $15.00 each and some cost $12.00 each. Bart purchased 8 DVDs and spent a total
of $105.00. Write a system of equations and solve by using substitution to determine how many $12.00 DVDs Bart purchased.
Answers
$105
15*3=45
12*5=60
60+45=105
(12*5)+(15*3)=105
Can i get brainliest
The number of $15 DVDs and $12 DVDs at an electronics store will be 3 and 5, respectively.
What is the solution to the equation?The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.
At a hardware store, DVDs are at a bargain. A few DVDs cost $15.00 each and some expenses $12.00 each. Bart bought 8 DVDs and spent a sum of $105.00.
Let 'x' be the number of $15 DVDs and 'y' be the number of $12 DVDs. Then the equations are given as,
x + y = 8 ...1
15x + 12y = 105 ...2
From equations 1 and 2, then we have
15x + 12(8 - x) = 105
15x + 96 - 12x = 105
3x = 9
x = 3
Then the value of y is given as,
y = 8 - 3
y = 5
The number of $15 DVDs and $12 DVDs at an electronics store will be 3 and 5, respectively.
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What level of measurement is used in the operationalization of extracurricular participation?
Measures Extracurricular Participation Students listed the university-based clubs they participated in during the academic year. Based on these lists, we created a variable reflecting whether they were involved in at least one activity (37% were involved). Of the activities listed, 28% were sports/ recreation (i.e., intercollegiate athletics, club sports, intramural sports, or campus recreation), 18% fraternities/sororities, 15% cultural, 13% departmental/professional, 9% campus programs, 8% special interest, 7% service, and 3% religious.
Answers
The level of measurement used in the operationalization of extracurricular participation is categorical/nominal.
In the operationalization of extracurricular participation, the measurement of students' involvement in university-based clubs is done using categorical/nominal level of measurement.
This is evident from the variable created to reflect whether students were involved in at least one activity, indicating a binary (yes/no) response. The subsequent breakdown of the activities listed into different categories, such as sports/recreation, fraternities/sororities, cultural, departmental/professional, campus programs, special interest, service, and religious, further supports the use of categorical measurement.
Each activity falls into a distinct category, and the percentages represent the proportions of students engaged in each category. Categorical/nominal measurement allows for classifying and organizing data into mutually exclusive categories, without any inherent order or numerical value associated with the category.
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[5 points) At a certain rate of compound interest 100 will increase to 200 in x years, 200 will increase to 300 in years, and 300 will increase to 1,500 in z years. If 600 will increase to 1,000 in n years, find an expression for n in terms of x, y, and z.
Answers
At a certain compound interest rate, the amount of 100 will increase to 200 in x years, 200 will increase to 300 in y years, and 300 will increase to 1,500 in z years. We need to find an expression for n, the number of years it takes for 600 to increase to 1,000.
The given information implies that the ratio of the final amount to the initial amount after a certain number of years is constant. We can express this ratio as follows:
(200 / 100) = (300 / 200) = (1500 / 300)
Simplifying this equation, we get:
2 = 3/2 = 5
Now, let's calculate the value of n. We know that 600 will increase to 1,000 in n years. We can set up a ratio using the same constant ratio calculated above:
(1000 / 600) = (300 / 200)
Simplifying the equation, we have:
5/3 = 3/2
Cross-multiplying, we get:
(5 * 2) = (3 * 3)
10 = 9
Since the equation is not valid, it means there is no value of n that satisfies the given conditions. Therefore, we cannot express n in terms of x, y, and z.
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A thermometer is taken from a room where the temperature is 21 degrees Celsius to the outdoors, where the temperature is 5 degrees Celsius. After one minute the thermometer reads 15 degrees Celsius.
(a) What will the reading on the thermometer be after 3 more minutes?
(b) When will the thermometer read 6 degrees Celsius?
degrees Celsius
Answers
a) the reading on the thermometer after 3 more minutes will be -3 degrees Celsius.
b) the thermometer will read 6 degrees Celsius after 1.5 minutes.
To solve the given problem, we can assume that the temperature change follows a linear pattern based on the given information.
(a) To find the reading on the thermometer after 3 more minutes, we need to determine the rate of temperature change per minute. From the initial reading of 21 degrees Celsius to the reading after one minute of 15 degrees Celsius, there was a temperature decrease of 6 degrees Celsius in one minute.
Therefore, the rate of temperature decrease is 6 degrees Celsius per minute. If this rate remains constant, after 3 more minutes, the thermometer will show a further temperature decrease of:
3 minutes * 6 degrees Celsius per minute = 18 degrees Celsius
Thus, the reading on the thermometer after 3 more minutes will be 15 degrees Celsius - 18 degrees Celsius = -3 degrees Celsius.
(b) To find when the thermometer will read 6 degrees Celsius, we need to determine the time it takes for the temperature to decrease from 15 degrees Celsius to 6 degrees Celsius.
The initial reading is 15 degrees Celsius, and the final desired reading is 6 degrees Celsius. Therefore, we need to calculate the time it takes for a temperature decrease of:
15 degrees Celsius - 6 degrees Celsius = 9 degrees Celsius
Since the rate of temperature decrease is 6 degrees Celsius per minute, we can set up the equation:
9 degrees Celsius = 6 degrees Celsius per minute * t minutes
Solving for t (the time it takes to reach 6 degrees Celsius):
t = 9 degrees Celsius / 6 degrees Celsius per minute = 1.5 minutes
Therefore, the thermometer will read 6 degrees Celsius after 1.5 minutes.
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The Internal Revenue Service estimates that 8% of all taxpayers filling out forms make mistakes.
An IRS employee randomly selects and checks 10 forms for mistakes. What is the probability that exactly one has an error?
The same IRS employee announces at lunch one day that she is going to check 25 randomly selected forms after lunch.
Calculate the mean and the standard deviation of the number of forms with mistakes she should expect to find.
The IRS employee checked 25 randomly selected forms and didn't find any mistakes. Does this provide convincing evidence to cause her supervisor to believe that she is missing mistakes? Explain.
Answers
Probability of exactly one error in 10 forms: Use binomial distribution with n = 10 and p = 0.08..Mean and standard deviation of forms with mistakes: Calculate using expected value and standard deviation formulas for a binomial distribution.
To calculate the probability that exactly one form has an error out of 10 randomly selected forms, we use the binomial probability formula: P(X = k) = (n C k) * p^k * (1-p)^(n-k), where n is the number of trials, p is the probability of success, and k is the number of successes. In this case, n = 10, p = 0.08, and k = 1. Plugging in these values, we can calculate the probability as P(X = 1) = (10 C 1) * (0.08)^1 * (1-0.08)^(10-1).
The mean (expected value) of the number of forms with mistakes can be calculated as μ = n * p, where n is the number of trials and p is the probability of success. In this case, n = 25 and p = 0.08. The standard deviation can be calculated as σ = sqrt(n * p * (1-p)), where sqrt represents the square root. Plugging in the values, we can calculate the mean and standard deviation.
Finding no mistakes in 25 randomly selected forms does not provide strong evidence to conclude that the employee is missing mistakes. The probability of not finding any mistakes in 25 forms can be calculated using the binomial distribution with parameters n = 25 and p = 0.08. If the actual proportion of mistakes is 8%, it is possible to observe a sample without mistakes due to random chance. To draw a more reliable conclusion, a larger sample size or additional evidence would be needed.
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10. Use the diagram below to find the value of x.
Answers
Answer:
x=20
Step-by-step explanation:
Hello There!
Remember the exterior angle of a triangle rule:
An exterior angle of a triangle is equal to the sum of the opposite interior angles
Knowing this, we can create an equation to solve for x
exterior angle (100) = sum of opposite interior angles (3x+2x)
100 = 2x+3x
now we solve for x
step 1 combine like terms
2x+3x=5x
now we have 100=5x
step 2 divide each side by 5
5x/5=x
100/5=20
we're left with x = 20
Answer:
With their steel hoofs, their long legs, their stag-like muscles, their thick skins, their powerful horns, they could walk the roughest ground, cross the widest deserts, climb the highest mountains, swim the widest rivers, fight off the fiercest bands of wolves, endure hunger, cold, thirst and punishment as few beasts of the earth have ever shown themselves capable of enduring.
What is the probability of 3 people sharing the same birthdays? How many different pairs of people are there when there are three humans? (Think nPr or nCr)
Answers
The probability of three people sharing the same birthdays is approximately [tex]0.0000075[/tex] or [tex]0.00075[/tex]%, and there are three different pairs of people when there are three humans.
The probability of three people sharing the same birthday depends on the assumptions made about the distribution of birthdays. Assuming that birthdays are uniformly distributed throughout the year and that leap years are not considered, there are [tex]365[/tex] possible birthdays for each person. The first person can have any birthday, and the probability that the second person shares the same birthday is [tex]$\frac{1}{365}$[/tex]. Similarly, the probability that the third person shares the same birthday as the first two is also [tex]$\frac{1}{365}$[/tex]. Multiplying these probabilities together, we get [tex]$\left(\frac{1}{365}\right) \times \left(\frac{1}{365}\right) = \frac{1}{133,225}$[/tex], approximately [tex]0.0000075[/tex] or [tex]0.00075\%[/tex].When there are three humans, the number of different pairs of people can be calculated using the combination formula, also known as [tex]$\binom{n}{r}$[/tex]. In this case, [tex]$n$[/tex] represents the total number of people ([tex]3[/tex]), and [tex]$r$[/tex] represents the number of people chosen at a time ([tex]2[/tex] for pairs). Applying the formula, we have [tex]$\binom{3}{2} = 3$[/tex]. Therefore, there are three different pairs of people when there are three humans: ([tex]1,2[/tex]), ([tex]1,3[/tex]), and ([tex]2,3[/tex]).In conclusion, the probability of three people sharing the same birthdays is extremely low (approximately [tex]0.0000075 \ or \ 0.00075\%[/tex]), and when there are three humans, there exist three different pairs of people.
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Consider the following sentence and scheme of abbreviation: Only lunatics drive on the autobahn. Lx: x is a lunatic Dx: x drives on the autobahn Which of the following would be accurate translations of the above sentence into the language of PL? a. (x)(DxLx) b. (Ex)(DxLx) C. (x)(LxDx) d. (Ex)(LxDx)
Answers
Option b. (Ex)(DxLx) is the accurate translation of the given sentence into the language of PL.
The accurate translation of the sentence "Only lunatics drive on the autobahn" into the language of PL (Propositional Logic) can be represented as:
b. (Ex)(DxLx)
Let's break down the translation scheme:
Lx: x is a lunatic (Lunatics)
Dx: x drives on the autobahn (Drive on the autobahn)
The sentence states "Only lunatics drive on the autobahn," which implies that there are some individuals who drive on the autobahn and they are lunatics. This can be expressed using existential quantification (Ex) to indicate that there exists at least one individual who satisfies the condition. Therefore, option b. (Ex)(DxLx) accurately translates the given sentence.
Option a. (x)(DxLx) would mean "For all x, x drives on the autobahn implies x is a lunatic." This does not capture the meaning of the original sentence because it suggests that everyone who drives on the autobahn is a lunatic, which is not what the sentence implies.
Option c. (x)(LxDx) would mean "For all x, x is a lunatic implies x drives on the autobahn." Again, this does not capture the intended meaning of the original sentence as it suggests that all lunatics drive on the autobahn.
Option d. (Ex)(LxDx) would mean "There exists an x such that x is a lunatic and x drives on the autobahn." This translation does not capture the sense of the original sentence, which states that only lunatics drive on the autobahn, not that there exists at least one lunatic who drives on the autobahn.
Therefore, option b. (Ex)(DxLx) is the accurate translation of the given sentence into the language of PL.
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Ops gets $140 per day and laborers get $100 per day. If 35 people were hired and payroll was $4820, how many ops were employed and how many laborers?
Answers
Answer:
2 laborers and 33 ops
Step-by-step explanation:
Given
Let:
L = Laborers
P = Ops
From the statements, we have:
[tex]L + P = 35[/tex] -- total people hired; and
[tex]140P + 100L = 4820[/tex] ---number of hirees
Required
Solve for L and P
Make L the subject in: [tex]L + P = 35[/tex]
[tex]L = 35 - P[/tex]
Substitute [tex]L = 35 - P[/tex] in [tex]140P + 100L = 4820[/tex]
[tex]140P + 100(35 - P) = 4820[/tex]
[tex]140P + 3500 - 100P = 4820[/tex]
Collect like terms
[tex]140P - 100P = 4820 - 3500[/tex]
[tex]40P = 1320[/tex]
Solve for P
[tex]P = 1320/40[/tex]
[tex]P = 33[/tex]
Substitute [tex]P = 33[/tex] in [tex]L = 35 - P[/tex]
[tex]L = 35 - 33[/tex]
[tex]L = 2[/tex]
A car worth $14,000 depreciates at a rate of 4% per month. How long until it is worth $10,000?
Answers
Select the correct answer.
Which value in this data set is an outlier?
4,5, 1, 7, 4, 5, 8, 9, 6, 5, 4, 9,7
O A. 1
OB.
N
O C.
3
D. 9
Answers
Answer:
answer is 1
Step-by-step explanation:
Kaylee deposited $1,450 in an account that earns 2.596 interest compounded annually. Which function represents the situation, where tis
the time in years?
fit) = 1450(2.5)
f(t) = 1450(1.25)
FO) = 1450(.025)
f(t) = 1450(1,025)
Answers
Answer:
[tex]f(t) = 1450(1.025)^{t}[/tex]
Step-by-step explanation:
Given
[tex]P =1450[/tex] -- principal
[tex]r = 2.5\%[/tex] --- rate
[tex]n = 1[/tex] --- compounded once a year
Required
Determine the function for compound interest
Compound interest f(t) is calculated as:
[tex]f(t) =P(1 + r/n)^{nt[/tex]
So, we have:
[tex]f(t) = 1450(1 + 2.5\%/1)^{1 * t}[/tex]
[tex]f(t) = 1450(1 + 2.5\%)^{t}[/tex]
[tex]f(t) = 1450(1 + 0.025)^{t}[/tex]
[tex]f(t) = 1450(1.025)^{t}[/tex]
The average car decreases in value by about 15% per year. If a car's original value is $28,000, which function best represents its value, y, after t years?
A. y=28,000(1+15)^t
B. y=28,000(1+0.15)^t
C. y=28,000(1-15)^t
D. y=28,000(1-0.15)^t
Answers
Answer:
D
Step-by-step explanation:
D
Daily question: what is 5 to the 3rd power?
Answers
Answer:
125
Step-by-step explanation:
5 to the 3rd power means 5³
5³ = 5x5x5
= 125
16. Suppose that 7₁, 72 are linear dependent in a vector space V. Show that V₁ + V2, V₂ - V₁ are also linearly dependent.
Answers
By supposing that 7₁, 72 are linearly dependent in a vector space V. The coefficient of 7₁ is 1 and the coefficient of 72 is 0, while in the second linear combination, the coefficient of 7₁ is 0 and the coefficient of 72 is 1. Therefore, we can conclude that V₁ + V2, V₂ - V₁ is also linearly dependent.
To show that V₁ + V2, V₂ - V₁ is also linearly dependent, we will begin by using the given linear combination of vectors to determine whether they are linearly dependent or independent. Suppose that 7₁, and 72 are linearly dependent in a vector space V.
Let us recall that a set of vectors is linearly dependent if it can be represented as a linear combination of other vectors in the vector space. This implies that if 7₁, 72 are linearly dependent, then there exist scalars α and β, not all zero, such that α7₁ + β72 = 0. To show that V₁ + V2, V₂ - V₁ is also linearly dependent, we need to use the definitions of vector addition and subtraction to writing each of these vectors as a linear combination of 7₁ and 72. Let's begin with V₁ + V2.
Using the definition of vector addition, we have
V₁ + V2 = 1 · 7₁ + 1 · 72 = 7₁ + 72.
Similarly, using the definition of vector subtraction, we have
V₂ - V₁ = -1 · 7₁ + 1 · 72 = -7₁ + 72.
Now we can write V₁ + V2, V₂ - V₁ as linear combinations of 7₁ and 72:
V₁ + V2 = 1 · (7₁ + 72) + 0 · (-7₁ + 72)V₂ - V₁
= 0 · (7₁ + 72) + 1 · (-7₁ + 72)
Notice that the coefficients in each linear combination are not all zero. We can say that V₁ + V2, V₂ - V₁ is linearly dependent.
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Which scientist discovered DNA after experimenting with white blood cells?
What are the building blocks of proteins?
amino acids
ribosomes
tRNA molecules
chromosomes
Answers
Find the measure of C to the nearest tenth of a degree
Answers
You have 35 plants. You want to split them equally between 4 sections of your garden. You use as many as you can in each section. How many plants do you have left?
Answers
answer:
3 plants are left !
step-by-step explanation:
hi there!
first lets use our times table to see which number gets use closest to 35 when it is multiplied by 4
4, 8, 12, 16, 20, 24, 28, 32,
4 * 8
= 32
then you can just subtract 35 by 32 to get you final answer of
35 - 32
= 3 plants are left over
( i hope i did the problem right since the wording of it was kinda hard to understand it )
________________________________________________________
| | | | |
| 1 | 2 | 3 | 4 |
| | | | |
________________________________________________________
| | | | |
| 5 | 6 | 7 | 8 |
| | | | |
________________________________________________________
| | | | |
| 9 | 10 | 11 | 12 |
| | | | |
________________________________________________________
| | | | |
| 13 | 14 | 15 | 16 |
| | | | |
________________________________________________________
| | | | |
| 17 | 18 | 19 | 20 |
| | | | |
________________________________________________________
| | | | |
| 21 | 22 | 23 | 24 |
| | | | |
________________________________________________________
| | | | |
| 25 | 26 | 27 | 28 |
| | | | |
________________________________________________________
| | | | |
| 29 | 30 | 31 | 32 |
| | | | |
________________________________________________________
i made a table with each number being the plant! just in case your a visual learner like me!
but i hope this help you and if i made anything confusing or if i made a mistake please let me know! but other then that i hope you have a good rest of your day :)
There are 3 plants left.
What is Division method?
Division method is used to distributing a group of things into equal parts. Division is just opposite of multiplications.
For example, dividing 20 by 2 means splitting 20 into 2 equal groups of 10.
Given that;
Total number of plants = 35
And, You want to split them equally between 4 sections of your garden.
Now,
To find the left plants after splitting into 4 sections, we can divide and find the remainder as;
The number of left plants = 35 ÷ 4
4 ) 35 ( 8
- 32
-------
3
Clearly, The remainder = 3
Thus, There are 3 plants left.
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Answer number two please please
Answers
Answer: 11
Step-by-step explanation:
Sooooo all I did was take the formula for the area of a triangle ( A= 1/2(b)(h) 0 and plug In the values. So 'b' would be 4 and 'h' would be 5.5. I assumed this rectangle had equal widths and equal heights.
what are all possible values for x in the equation x^3=375?
Answers
Answer:
Select all possible values for x in the equation.
x cubed=375.
5*the cubed root of 3
the cubed root of 375
75*the cubed root of 5
125*the cubed root of 3
I am trying to do a practice test to prepare for my real test tomorrow and I don't understand the question. Can anyone help explain it plz any help would be great.
Step-by-step explanation:
1. What type of angle am I?m<1=91 degrees, m<2=89 degrees
Complementary
supplementary
Other:
Answers
Answer:
Step-by-step explanation:
If they are added, they are supplementary. If anything else is done to them, there is no answer.
89 + 91 = 180 Supplementary angles add to 180.
Point P is located at (−2, 7), and point R is located at (1, 0). Find the y value for the point Q that is located two thirds the distance from point P to point R.
4.9
4.7
2.5
2.3
PLS HELP
Answers
Answer:
Step-by-step explanation:
2.3
3. Michael surveyed a random sample of students in his school about
the number of sports they play. There are 300 students in Michael's
school. Use the results of the survey to estimate the number of
students in Michael's school who play exactly one sport. Explain
your answer. PLEASE HELP ILL ALSO MARK IT THE MOST BRAINLY ANSWER I didn’t spell that right but oh well
Answers
Answer:
Micheal surveyed 60 students.
Answer: Look at the explanation please.
Step-by-step explanation:
Please note that I am using what is in the Mathematics textbook.
So we need to know for the students who play Exactly 1 sport.
So you have to make an equation. It will look like this...
15/60 = x/300
Note that we get the 15 from the table of the number of students. We get 60 from adding 13+15+32 which is all the students Michael has surveyed. Lastly we get 300 from the question There are 300 students in his school.
x = 75 when you do the math. Cross multiply and divide.
Hope this helps :)
A beam of light in air strikes a slab of crown glass (n = 1.52) and is partially reflected and partially refracted. Find the angle of incidence if the angle of reflection is twice the angle of refraction.
Answers
The angle of incidence for a beam of light in air striking a slab of crown glass, where the angle of reflection is twice the angle of refraction, can be determined using the laws of reflection and refraction. The angle of incidence is approximately 39.2 degrees.
we can apply the laws of reflection and refraction to find the relationship between the angles. Let's denote the angle of incidence as θ, the angle of reflection as θ_r, and the angle of refraction as θ_t.
According to the law of reflection, the angle of reflection is equal to the angle of incidence: θ_r = θ.
According to Snell's law of refraction, the relationship between the angles of incidence and refraction is given by: n_1 × sin(θ) = n_2 × sin(θ_t), where n_1 and n_2 are the refractive indices of the two media.
In this case, the light travels from air (with a refractive index of approximately 1) to crown glass (with a refractive index of 1.52). Substituting the values, we have: sin(θ) = (1.52 / 1) × sin(θ_t).
Since the angle of reflection is twice the angle of refraction, we can write: θ = 2θ_t.
Substituting this relation into the previous equation, we get: sin(2θ_t) = (1.52 / 1) × sin(θ_t).
Using the double-angle trigonometric identity, sin(2θ_t) = 2sin(θ_t)cos(θ_t), we have: 2sin(θ_t)cos(θ_t) = 1.52sin(θ_t).
Dividing both sides by sin(θ_t), we obtain: 2cos(θ_t) = 1.52.
Solving for cos(θ_t), we have: cos(θ_t) = 1.52 / 2.
Taking the inverse cosine, we find: θ_t = cos^(-1)(1.52 / 2) ≈ 26.8 degrees.
Finally, substituting this value into θ = 2θ_t, we get: θ ≈ 2 × 26.8 degrees ≈ 53.6 degrees.
Hence, the angle of incidence is approximately 39.2 degrees.
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Calculate the following limits using the limit laws and limx→2f(x)=−3, limx→2g(x)=4, limx→2h(x)=7 (a) limx→2(f(x)−2g(x))= (b) limx→2(h(x)2)= (c) limx→2h(x)⋅g(x)f(x)=
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The value of limits after using limit laws is [tex]$\lim_{x \to 2} \frac{h(x) \cdot g(x)}{f(x)} = -\frac{28}{3}$.[/tex]
What are Limit Laws?
Limit laws, also known as limit properties or limit theorems, are a set of rules and principles that allow us to simplify and evaluate limits of functions. These laws provide a systematic approach to finding the limit of a more complex expression by breaking it down into simpler parts.
Given:
[tex]\lim_{x \to 2} f(x) &= -3 \\\lim_{x \to 2} g(x) &= 4 \\\lim_{x \to 2} h(x) &= 7\end{align*}\textbf{(a) Calculate} $\lim_{x \to 2} (f(x) - 2g(x))$:[/tex]
Using the limit laws, we can split the expression and apply the limit laws individually:
[tex]\lim_{x \to 2} (f(x) - 2g(x)) &= \lim_{x \to 2} f(x) - \lim_{x \to 2} (2g(x)) \\&= \lim_{x \to 2} f(x) - 2 \lim_{x \to 2} g(x) \\&= (-3) - 2(4) \\&= -3 - 8 \\&= -11[/tex]
Therefore,[tex]$\lim_{x \to 2} (f(x) - 2g(x)) = -11$.[/tex]
[tex]\textbf{(b) Calculate} $\lim_{x \to 2} (h(x))^2$:[/tex]
Again, using the limit laws, we can apply the limit to the expression:
[tex]\lim_{x \to 2} (h(x))^2 &= \left(\lim_{x \to 2} h(x)\right)^2 \\&= (7)^2 \\&= 49[/tex]
Therefore,
[tex]\lim_{x \to 2} (h(x))^2 = 49$.\textbf{\\\\(c) Calculate} $\lim_{x \to 2} \frac{h(x) \cdot g(x)}{f(x)}$:[/tex]
Applying the limit laws, we can evaluate the limit as follows:
[tex]\lim_{x \to 2} \frac{h(x) \cdot g(x)}{f(x)} &= \frac{\lim_{x \to 2} h(x) \cdot \lim_{x \to 2} g(x)}{\lim_{x \to 2} f(x)} \\\\&= \frac{7 \cdot 4}{-3}\\ \\&= \frac{28}{-3}[/tex]
Therefore,[tex]$\lim_{x \to 2} \frac{h(x) \cdot g(x)}{f(x)} = -\frac{28}{3}$.[/tex]
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