The comparison test, [tex]$\sum_{n=1}^{\infty} a_{n}$[/tex] diverges since [tex]$\sum_{n=M}^{\infty} b_{n}$[/tex] diverges.
Suppose that [tex]$\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$[/tex] are series with positive terms. We assume that does not exist and that [tex]$\sum_{n=1}^{\infty} b_{n}$[/tex]diverges.
To prove that [tex]$\sum_{n=1}^{\infty} a_{n}$[/tex] also diverges, we will use the comparison test. Since [tex]$\sum_{n=1}^{\infty} b_{n}$[/tex] diverges, we can find a number M such that [tex]$\sum_{n=M}^{\infty} b_{n}$[/tex] diverges. Since [tex]$\lim_{n\rightarrow \infty} \frac{a_{n}}{b_{n}}$[/tex] does not exist, we can find a number N such that for all [tex]$n \geq N$, $\frac{a_{n}}{b_{n}} > 1$[/tex].
Now, for any [tex]$n \geq \max \{M,N\}$[/tex], we have [tex]$\frac{a_{n}}{b_{n}} > 1$[/tex]. Thus, [tex]$a_{n} > b_{n}$[/tex]. By the comparison test, [tex]$\sum_{n=1}^{\infty} a_{n}$[/tex] diverges since [tex]$\sum_{n=M}^{\infty} b_{n}$[/tex] diverges.If the ratio of two series with positive terms,[tex]$\sum_{n=1}^{\infty}a_n$[/tex] and [tex]$\sum_{n=1}^{\infty}b_n$[/tex], does not converge to a limit as n approaches infinity and [tex]$\sum_{n=1}^{\infty}b_n$[/tex] diverges, then [tex]$\sum_{n=1}^{\infty}a_n$[/tex] must also diverge.
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the visualization below shows voting patterns in the united states, canada, and mexico in the united nations general assembly on a variety of issues. specifically, for a given year between 1946 and 2015, it displays the percentage of roll calls in which the country voted yes for each issue. this visualization was constructed based on a dataset where each observation is a country/year pair.1
(a) List of variables is Years, Country, Arms control and disarmament, colonialism, economic development, human rights, nuclear weapons, and materials, and Palestinian conflict.
(b) Each variable is numeric if the variable assumes number values and it will be categorical if it assumes labels.
(a) The list of variables is:
The variables that can be used in this visualization can be Years, Country, Percentage that said yes, Arms control and disarmament, colonialism, economic development, human rights, nuclear weapons, and materials, and Palestinian conflict.
(b) Category of different variables:
Years: Numerical and discrete
years are number values but the voting has not been done in the same interval so it is a numerical and discrete variable.
Country: Categorical and not ordinal
country name assumes as labels hence it is a categorical variable.
The percentage that said yes: Numerical and continuous
The percentage indicates number values so it is a numerical value.
And following is also assumed as a label so these are also categorical values:
Arms control and disarmament: categorical and not ordinal
colonialism: categorical and not ordinal
economic development: categorical and ordinal
human rights: categorical and ordinal
Nuclear weapons and materials: categorical and not ordinal.
Palestinian conflict: Categorical and not ordinal
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The complete question is:
the visualization below shows voting patterns in the united states, Canada, and Mexico in the united nations general assembly on a variety of issues. specifically, for a given year between 1946 and 2015, it displays the percentage of roll calls in which the country voted yes for each issue. this visualization was constructed based on a dataset where each observation is a country/year pair.
1) List the variables used to create this visualization.
2) Indicate wheater each variable in the study is numerical or categorical. If numerical, identify as continuous or discrete. If categorical, identify as the variable is ordinal.
A number is multiplied by 6, and that product is added to 4. The sum is equal to the product of 2 and 23. Find the number.
Answer:
The number is 7
Step-by-step explanation:
1. Create the equation needed to solve this
Lets think of a number as x. We have a number multiplied by 6 and have 4 added to it.
We get:
6x + 4
Now this sum is equal to the product of 2 and 23. We connect this to get
6x + 4 = 2(23)
We can simplify this to
6x + 4 = 46
2. Solve for x
6x + 4 = 46
Subtract 4 from both sides
6x = 42
Divide both sides by 6
x = 7
Therefore the number is 7
When Boris bought his house, he got his mortgage through a bank. The mortgage was a personal, amortized loan for 91500, at an interest rate of 3.3% , with monthly payments for a term of 40 years.
Answer: As the loan is amortized, Boris will make a series of equal payments each month for the duration of the loan, with each payment comprising of both interest and principal. The amount of interest in each payment will decrease over time, while the amount of principal in each payment will increase.
To calculate the monthly payments, we can use the formula:
M = P [ i(1 + i)^n ] / [ (1 + i)^n - 1]
Where:
M = monthly payment
P = the principal or the loan amount = 91500
i = interest rate as a decimal = 3.3%/12 = 0.00275
n = total number of payments = 40 years x 12 months/year = 480
By substituting the values in the formula:
M = 91500 [ 0.00275 (1 + 0.00275)^480 ] / [ (1 + 0.00275)^480 - 1]
M = $439.65
So the monthly payment that Boris will make is $439.65. This amount will stay the same throughout the loan term, but the proportion of interest to principal in each payment will change as the loan is amortized.
Step-by-step explanation:
the lengths of the legs of a right triangle are 7 in. and 2 in. what is the length of the hypotenuse? enter your answer rounded to the nearest tenth in the box.
Use the figure below to select the classification of each set of angles in a)-d)
The classification of the angles are;
a) ∠3 and ∠5; Alternate Angles
b) ∠1, ∠2 and ∠3; three interior angles of a triangle.
c) ∠2 and ∠3; Complementary angles
d) ∠4 and ∠5; Supplementary angles
How to classify angles?We know that the sum of angles in a triangle is 180 degrees and as such it means that ∠1, ∠2 and ∠3 are classified as three interior angles of a triangle.
We know that alternate angles are defined as angles that occur on opposite sides of the transversal line and have the same size. Thus;
∠3 and ∠5 are alternate angles.
Since ∠1 is 90 degrees, then it means that ∠2 and ∠3 are complementary angles.
Similarly, ∠4 and ∠5 are called supplementary angles because they sum up to 180 degrees.
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rogress: The movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer A motor oil retailer needs to fill 50 one quart bottles, and he has two tanks: one that contains 12 gallons of oil and one that contains 2 gallons of oil. Which will he need to fill the bottles? The 2 gallon tank is enough. He will need both tanks. The 2 gallon tank is not enough, but the 12 gallon tank is enough. Question ID: 53297 Even both tanks will not be enough.
The motor oil dealer will have to use both tanks since he needs 15 gallons of oil but will still be short by one gallons.
Find the solution ?The motor oil dealer will have to use both tanks since he needs 15 gallons of oil but will still be short by one gallons.
A motor oil store has two tanks: one holds 12 gallons of oil, and the other holds 2 gallons of oil. In order to determine which tank will require to fill the bottles, the following computation must be done:
Quarts to gallons is 4 to 1.
1/4 x 60 = X
60/4 = X
15 = X
Since the motor oil vendor requires 15 gallons of oil, he must use both tanks but will still be short by 1 gallons
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20% of a is 11. What is a 
Therefore , the solution of the given problem of percentage comes out to be a value is 55.
A percentage can be specified.In mathematics, a percentage is a number any figure that is stated as a proportion of 100. The abbreviations "pct.," "pct," and "pc" are also occasionally used. The "%" symbol, however, is frequently used to denote it. The % amount has no dimensions and is flat. Because the numerator of percentages is 100, they are effectively fractions. To show that a number is a percentage, the percent sign (%) should be put in front of it. On a test, for instance, you would receive a 75% if you gave the answer 75 out of the total questions.
Here,
Given :
=> 20 /100 * a =11
=> a = 11 *100 /20
=> a = 1100 /20
=> a = 55
Thus , a is 55 .
Therefore , the solution of the given problem of percentage comes out to be a value is 55.
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fill in the blank. let where is an integer greater than___. compute the number of values of for which the sum of the squares of the digits of is at most .
Let where is an integer greater than 9. compute the number of values of for which the sum of the squares of the digits of is at most 25.
The question asks us to count the number of values of for which the sum of the squares of the digits of is at most 25. To solve this problem, we can use a loop to iterate through all possible values of starting from 9 and up. For each value of , we can then calculate the sum of the squares of the digits of by using modulo and division operations to extract each digit and square it. If the sum of the squares of the digits of is at most 25, we can add 1 to the count of values. After the loop completes, we will have the total number of values of for which the sum of the squares of the digits of is at most 25.
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What is the distance between 60 and 120
The circumference of a personal sized pizza at center court is 12.56 inches. Their large pizza is three times the diameter as the personalized sized pizza. What is the area of the large pizza?
The area of the large pizza is approximately equal to 112.984 square inches.
How to determine the area of a large pizza
Herein we need to determine the area occupied by a large pizza (A), in square inches, and based on given input by circumference of a small pizza (s), in inches. Then, we need the following formulas:
Circumference
s = π · D
Area
A = 0.25π · D²
Where D is the diameter, inches.
First, determine the diameter of the small pizza:
D = s / π
D = 12.56 / π
D ≈ 3.998 in
Second, calculate the diameter of the large pizza:
D' = 3 · (3.998 in)
D' = 11.994 in
Third, obtain the area of the large pizza:
A = 0.25π · (11.994 in)²
A ≈ 112.984 in²
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A ladder leans against the side of a house. The angle of elevation of the ladder is 74° when the bottom of the ladder is 12 ft from the side of the house. Find the length of the ladder. Round your answer to the nearest tenth.
?
12
74°
Answer: We can use trigonometry to solve this problem.
Let x be the length of the ladder, and h be the height of the ladder.
We can use the tangent function to relate the angle of elevation, the length of the ladder, and the height of the ladder:
tan(74°) = h / x
We know that x = 12 ft and we can find h.
We can cross multiply and solve for h:
x * tan(74°) = h
12 * tan(74°) = h
We can use a calculator to find the value of h, this will give us the height of the ladder.
Finally, we can use the Pythagorean theorem to find the length of the ladder:
x^2 = h^2 + 12^2
x = √(h^2 + 12^2)
Round your answer to the nearest tenth, we have x = √(h^2 + 12^2)
We can use the value of h to find the length of the ladder.
However, without the value of h it is not possible to find the length of the ladder.
Step-by-step explanation:
Suppose that the annual amount of rainfall (in million tons) accumulated in a lake follows a gamma distribution with alpha = 3 and gamma = 5.
a) Find the expected annual rainfall accumulated in this lake.
b) Find the standard deviation of the annual amount of rainfall in this lake.
c) Find P(X < 4), use R.
d) Find P(2 < X < 5), use R
a) The expected annual rainfall is 15 million tons.
b) The standard deviation is the square root is 15 million tons.
c) Pgamma(4, shape = 3, scale = 5)
d) Pgamma(5, shape = 3, scale = 5) - Pgamma(2, shape = 3, scale = 5)
Gamma Distribution Probabilitiesa) The expected annual rainfall accumulated in this lake is given by the mean of the gamma distribution, which is alpha * gamma. In this case, the expected annual rainfall is 3 * 5 = 15 million tons.
b) The standard deviation of the annual amount of rainfall in this lake is given by the square root of the variance, which is the square root of alpha * gamma². In this case, the standard deviation is the square root of 3 * 5² = 15 million tons.
c) To find P(X < 4), we can use the cumulative distribution function (CDF) of the gamma distribution. In R, we can use the pgamma() function to find the CDF of the gamma distribution at a given value.
pgamma(4, shape = 3, scale = 5)
This will return the probability that X is less than 4, which is the cumulative probability.
d) To find P(2 < X < 5), we can use the cumulative distribution function (CDF) of the gamma distribution. In R, we can use the pgamma() function to find the CDF of the gamma distribution at a given value.
pgamma(5, shape = 3, scale = 5) - pgamma(2, shape = 3, scale = 5)
This will return the probability that X is between 2 and 5, which is the cumulative probability.
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Due to gravity, an object falls 16t^2 feet in t seconds. You drop a rock from a bridge that is 75 feet above the water. Will the rock hit the water in 2 seconds?
Explain
If due to gravity, an object falls 16t^2 feet in t seconds. You drop a rock from a bridge that is 75 feet above the water. The rock will hit the water in 2 seconds?
How to find seconds the rock will hit the rock?Given data:
Object falls = 16t² feet in t seconds
Rock from a bridge = 75 feet above the water
So,
72 =16t²
t² = 72/16
t²= 4.5
t = √4.5
t = 2.12 seconds to hit the water
Therefore we can conclude that the rock will hit the water in 2 seconds.
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Problem Gregory forms two different numbers with the digits 1, 2, 3, 4, 5, and 6. Both numbers have 3 digits, and each digit is used only once. If he adds these two numbers, what is the greatest sum Gregory can get? Show your work.
The greatest sum that Gregory can get is given as follows:
1173.
How to obtain the greatest sum?When two three digit numbers are added, the numbers in the equivalent positions are added with each other, that is, the third digit is added with the third digit, the second with the second and the third with the third.
Hence the greatest sum would be obtained as follows:
The first digits would be of 5 and 6.The second digits would be of 3 and 4.The third digits would be of 1 and 2.Hence the sum would be given as follows:
642 + 531 = 1173.
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Use properties to rewrite the given equation. Which equations have the same solution as 2.3p – 10.1 = 6.5p – 4 – 0.01p? Select two options.
Two equations that have the same solution as the given equation are;
2.3. p - 10.1 = 6.49. p - 4
230.p - 1010 = 650.p - 400 - p
The correct options are B and C.
How to explain the equation?The given equation is; 2.3·p - 10.1 = 6.5·p - 4 - 0.01·p
Simplifying the above equation by collecting like terms gives;
2.3·p - 10.1 = 6.5·p - 0.01·p - 4 = 6.49·p - 4
2.3·p - 10.1 = 6.49·p - 4
Both sides of an equation will remain equal, when both sides are multiplied by the same amount.
Multiplying both sides of the original equation by 100 gives the same solution as follows;
100 × (2.3·p - 10.1) = 100 × (6.5·p - 4 - 0.01·p)
230·p - 1010 = 650·p - 400 - p
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Use properties to rewrite the given equation. Which equations have the same solution as 2.3p – 10.1 = 6.5p – 4 – 0.01p? Select two options. 2.3p – 10.1 = 6.4p – 4 2.3p – 10.1 = 6.49p – 4 230p – 1010 = 650p – 400 – p 23p – 101 = 65p – 40 – p 2.3p – 14.1 = 6.4p – 4
for how many positive integers $p$ does there exist a triangle with sides of length $3p-1,$ $3p,$ and $p^2 1?$
Number of integers 'p' for which there exist a triangle with the given side length 3p -1 , 3p , p² + 1 is equal to five that is 1, 2, 3, 4, 5.
Using triangular inequality of triangle :
sum of two sides > third side
a. 3p - 1 + p² + 1 > 3p
⇒p² > 0
True result.
b. 3p + p² + 1 > 3p - 1
⇒ p² > -2
True result.
c. 3p - 1 + 3p > p² + 1
⇒ p² -6p + 2 < 0
Roots are : p = [ -(-6) ± √(-6)² -4(1)(2) ] /2(1)
= ( 6 ± √28 ) 2
= 5.6 or 0.35
Integer value of p are in between 0.35 to 5.6 are :
1, 2, 3, 4, 5
Therefore, the integer value of 'p' for the given side length of the triangle is equal to five that is 1, 2, 3, 4, 5.
For how many positive integers 'p' does there exist a triangle with sides of length 3p-1 , 3p , and p² + 1 ?
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two planes take off from the same airstrip. the first plane flies west for 150 miles and then flies 30 degrees south of west for 220 miles. the second plane flies east for 220 miles and then flies x degrees south of east for 150 miles. if x < 30, which plane is farther from the airstrip after the second leg? justify your answer.
The first plane is farther from the airstrip after the second leg
Which plane is farther from the airstrip after the second leg?The first plane
Initial distance = 150 miles west
Additional distance = 220 miles 30 degrees SW
This means that
Additional distance = 220 * cos(30) miles SW
Additional distance = 190.53 miles
Total distance = 150 + 190.53 = 340.53
The second plane
Initial distance = 220 miles east
Additional distance = 150 miles x degrees SE
This means that
Additional distance = 150 * cos(x) miles SW
Total distance = 220 + 150 cos(x)
Given that
x < 30
We have
Second plane < First plane
220 + 150 cos(x) < 340.53
Evaluate
cos(x) < 0.8035
Take the arc cos of both sides
x < 36
x < 36 is not the same as x < 30
Hence, the first plane is farther
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A 83-inch candle burns down in 5 hours. Assuming the candles are the same
thickness and make (that is, directly proportional), how long would it take a 15-inch
candle to burn down?
Step-by-step explanation:
83 in burn in 5 hours.
so, we have a 83/5 speed ratio.
directly proportional means
y = kx
with k being constant for all values of x.
83 = k×5 = 5k
k = 83/5
15 in burn in x hours.
therefore,
15 = 83/5 × x
75 = 83x
x = 75/83 hours
this can be also calculated like this :
because of the direct proportionality, 15/x must have the same ratio as the original 83/5.
83/5 = 15/x
83x/5 = 15
83x = 75
x = 75/83 hours = 0.903614458... hours
1. Subtract
4 1/2
- 1 7/8
2. 20% of what amount is $6.50
The value obtained when we subtract the given fractions is 21/8 and the 20% of 32.5 gives the amount $6.50.
What are percentages?We must divide the value by the entire value to find the percentage, and then multiply the resulting number by 100.
Given the following expression:
4 (1/2) - 1 (7/8)
Convert the mixed fractions into simple fractions:
9/2 -15/8
Take the LCM of the denominator:
36/ 8 - 15/8
21/8
2. 20% of x = 6.50
20/100 (x) = 6.50
20x = 650
x = 650/20
x = 32.5
Hence, the value obtained when we subtract the given fractions is 21/8 and the 20% of 32.5 gives the amount $6.50.
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Order the correlation coefficients from least to greatest -0.50,0.60,,98,-0.10,0.68
answer: In this case, -0.50, -0.10, 0.60, 0.68, 0.98 are all correlation coefficients and -0.50 is the least and 0.98 is the greatest among the given correlation coefficients.
but why? The correlation coefficients can be ordered from least to greatest as:
-0.50, -0.10, 0.60, 0.68, 0.98
A correlation coefficient is a measure of the strength and direction of a linear relationship between two variables. The value of the coefficient ranges from -1 to 1, where:
A coefficient of -1 represents a perfect negative correlation, meaning that as one variable increases, the other variable decreases.
A coefficient of 0 represents no correlation, meaning that there is no linear relationship between the two variables.
A coefficient of 1 represents a perfect positive correlation, meaning that as one variable increases, the other variable also increases.
a set of data has a mean that is much larger than the median. which of the following statements is most consistent with this information?
This suggests that the data set has a skewed distribution with a long tail of larger values.
This suggests that the data set has a skewed distribution, meaning that the data is not evenly distributed. The mean being much larger than the median indicates that there are a few larger values in the data set that are pulling the mean up and creating a long tail of larger values. This makes the distribution look lopsided, with a majority of the values being clustered around the median, and a few larger values that are skewing the mean higher.
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Show that the following number is rational by writing it as a ratio of two integers. 52.4726172617261... Enter a fraction.
The following number is rational by writing it as a ratio of two integers. 52.4726172617261... = 52497370/999900.
In mathematics, any number that can be written as p/q where q ≠0 is considered a rational number. Additionally, every fraction that has an integer denominator and numerator and a denominator that is not zero falls into the category of rational numbers. The outcome of dividing a rational number, or fraction, will be a decimal number, either a terminating decimal or a repeating decimal.
The following number is rational by writing it as a ratio of two integers. 52.4726172617261...
let 52.4726172617261...=x
100x=5247.26172617.....
100x=5247+0.2617+0.00002617+....
S0, 0.2617+0.00002617+... is a geometric series,
[tex]S=\frac{a}{(r-1)}\\\\a=0.2617,r=0.0001\\\\s=\frac{0.2617}{0.9999}\\\\S=\frac{2617}{9999}\\\\now,\\100x=5247+\frac{2617}{9999}\\100x=\frac{52,467,370}{9999}\\\\x=\frac{52467370}{999900}[/tex]
So, 52.472617...can be written by x.
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ach ant has 6 legs. How many legs do 3 ants have?
Ants 1 2 3
Legs 6 ?
Answer:
Step-by-step explanation:
the answer is 18 I think?
Answer:
18
Step-by-step explanation:
6 (legs) x 3 (ants) = 18 legs
It can also be represented as 6+6+6 (18)
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when people order books from a popular online source, they are shipped in boxes. suppose that the mean weight of the boxes is 1.5 pounds with a
The standard deviation of the total weight of the boxes that are shipped from this source is 3.02 pounds.
What is Standard Deviation?The term "standard deviation" (or "σ") refers to a measurement of the data's dispersion from the mean. A low standard deviation implies that the data are grouped around the mean, whereas a large standard deviation shows that the data are more dispersed.
Given:
Case A:
Standard Deviation= 0.3 pounds
mean weight= 1.5 pounds
Case B:
Standard Deviation= 0.1 pounds
mean weight= 0.5 pounds
Case C:
Standard Deviation= 3 pounds
mean weight= 12 pounds
let A is the Weight of boxes, B is the weight of packing and C is the weight of boxes.
ver(total)= ver A + ver B + ver C
Below is the Formula used:
SD = √variance
So, ver (total) = 0.3² + 0.1²+ 3²
ver (total) = 9.1 SD
ver (total) =√9.1
ver (total) = 3.02
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The complete Question is:
Suppose that the mean weight of the boxes is 1.5 pounds with a standard deviation of 0.3 pound, the mean weight of the packing material is 0.5 pound with a standard deviation of 0.1 pound, and the mean weight of the books shipped is 12 pounds with a standard deviation of 3 pounds. Assuming that the weights are independent, what is the standard deviation of the total weight of the boxes that are shipped from this source?
Triangle ABC is congruent to triangle XYZ, as shown below. 4. If AEFG = AHIJ, which of the following statements must be true?
From the given options, option B is the only statement that is not true is ∠C ≅ ∠E for congruent
What is the congruence theorem?
Triangle congruence theorem or triangle congruence criteria help in proving if a triangle is congruent or not. The word congruent means exactly equal in shape and size no matter if we turn it, flip it or rotate it.
If Δ ABC is congruent to Δ DEF, then its corresponding parts must be congruent.
segment AB ≅ segment DE
segment BC ≅ segment EF
segment AC ≅ segment DF
∠A ≅ ∠D
∠B ≅ ∠E
∠C ≅ ∠F
Therefore, from the given options, option B is the only statement that is not true is ∠C ≅ ∠E.
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"Your question is incomplete, probably the complete question/missing part is:"
If triangle ABC is congruent to triangle DEF, which statement is not true?
A) Segment AB ≅ segment DE
B) ∠C ≅ ∠E
C) Segment BC ≅ segment EF
D) ∠A ≅ ∠D
From the given options, option B is the only statement - 1
Please help with geometry questions 2, 4, 9 and 10!! Thank you
The answers to all the parts is mentioned above.
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given are the geometrical figures as shown.
{ 9 } -
We can write the proportional relationship {since both triangles are similar} as -
5/8 = x/10
{x} = 50/8
{x] = 6.25
{ 10 } -
We can write the proportional relationship {since both triangles are similar} as -
x/20 = 10/16
{x} = 200/16
{x} = 12.5
TY = 20 - 12.5
TY = 7.5
Therefore, the answers to all the parts is mentioned above.
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Algebra Tiles SOMEONE HELP ME QUICK I NEED HELP I WILL GIVE BRAINLIEST
Answer:
all
Step-by-step explanation:
all of them
If f ( x ) = 2 x 2 and g ( x ) = x - 4 2 x , determine the values of the following operations:
( f g ) ( 3 )
( f - g ) ( 4 )
f g ( 2 )
( f + g ) ( 1 )
The largest value of all four operations is
The largest value of all four operations is 54
How do you determine the largest number?In order to determine the largest value of all the four operations, one needs to plug in the values into the different operations given and then generate their values. By so doing, the following deductions are made:
(f * g)(3) = 2 * (3^2) * (3 - 4 * (2 * 3)) = 54
(f - g)(4) = 2 * (4^2) - (4 - 4 * (2 * 4)) = 40
f(g(2)) = 2 * (2^2) = 8
(f + g)(1) = 2 * (1^2) + (1 - 4 * (2 * 1)) = -3
So, the largest value of all four operations is 54. Therefore, the correct answer is as given above
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Angela observes that a drone is flying horizontally at a constant height of 5m above the ground. If the drone is flying in the opposite direction from Angela's location at a speed of 1m/s determine the rate at which the angle of elevation (theta) decreases at the moment when (theta)=pi/6
The rate of change of the angle of elevation is approximately - 2.597 degrees per second (-0.045 radians per second). (Correct choice: C)
How to determine the rate of change of the angle of elevation
In this problem we find the case of a drone that travels horizontally, at constant height (h), in meters. The rate of change of the angle of elevation (θ'), in degrees per second, can be found from trigonometric functions and derivatives:
tan θ = y / x
Where:
x - Horizontal distance, in meters. y - Vertical distance, in meters.By derivatives:
sec² θ · θ' = - y · x⁻² · x'
θ' = - (y · x⁻² · x') / sec² θ
Where x' is the speed of the drone, in meters per second.
If we know that y = 5 m, θ = π / 6 and x' = 5 m / s, then the rate of change of the angle of elevation is:
x = y / tan θ
x = (5 m) / tan (π / 6)
x ≈ 2.887 m
θ' = - [(5 m) · (2.887 m)⁻² · (5 m / s)] / sec² (π / 6)
θ' ≈ - 2.597 degrees per second (- 0.045 radians per second)
Note - Please notice that negative sign indicates that angle is decreasing.
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Each year, a physicist measures the remaining radioactivity in a sample. she finds that it reduces by 18% each year. if there was 1.52 g of radioactive material left after 4 years}
a. Find the initial quality of radioactive material
b. Hoy many more years will it take for the amount of radioactive material to reduce to 0.2 g?
a. The initial quantity of radioactive material is 0.68 g.
b. It will take an additional 7.15 years for the amount of radioactive material to reduce to 0.2 g.
a. In order to find the initial quality of radioactive material, we will use the formula:
Initial Quality = Final Quantity * (1 - Reduction Rate)^Years
We know that -
the final Quantity is 1.52 g, the reduction rate is 18% (or 0.18), and the number of years is 4.
Initial Quantity = 1.52 g * [tex](1 - 0.18)^4[/tex]
Initial Quantity = 1.52 g * 0.452
Initial Quantity = 0.68 g
Hence, the initial quantity of the radioactive material is 0.68 g.
b. In order to find how many more years it will take for the amount of radioactive material to reduce to 0.2 g, we will use the formula:
Years = log(Final Quality / Initial Quality) / log(1 - Reduction Rate)
We know that -
the final quality is 0.2 g, the initial quality is 0.68 g, and the reduction rate is 18% (or 0.18).
Years = log(0.2 / 0.0.68) / log(1 - 0.18)
Years = 7.15
Therefore, it will take an additional 7.15 years for the amount of radioactive material to reduce to 0.2 g.
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