An explicit definition can give a value for the nth item in a sequence by simply using the value of n: an = -27 * (-1/3)n-1
What is Geometric Progression ?
Geometric Progression can be defined as the sequence of numbers in which the common ratio between any consecutive numbers is same.
When you say, "each term in the sequence is obtained by multiplying -1/3 to the previous term," that is the recursive definition.
An explicit definition can give a value for the nth item in a sequence by simply using the value of n:
an = -27 * (-1/3)n-1
(more generally written as: an = a1 * (-1/3)^n-1 )
if n = 1
we get,
an = -27 *1 = -27
if n=2
we get,
an = 9
and if n=3
we get,
an = -3
similarly if n=4
than an= 1
Therefore, An explicit definition can give a value for the nth item in a sequence by simply using the value of n: an = -27 * (-1/3)n-1
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Hannah makes 5 out of 8 attempted free-throws! What is the average waiting time for Hannah to make her first basket, and what is the probability that she will make a basket on or before her last attempt within her average waiting time?
The probability that she will make a basket on or before her last attempt within her average waiting time are 1/8 and 2/8
How can we interpret probability?Probability of an event is a measurement of how likely an event can occur as an outcome of a random experiment;
Probability ranges from 0 to 1, both inclusive. Events whose probability is closer to 0 are rarer to occur than those whose probabilities are closer to 1 (relatively);
When converted to percentage, we just need to multiply its decimal representation by 100. In percentage form, the probability ranges from 0% to 100%;
Given:
Hannah makes 5 out of 8 attempted free-throws.
Now,
The average waiting time of hannah to make her first basket is 5/8
The probability of hannah making basket at her last attempt is 1/8
The probability of hannah making basket before her last attempt is 2/8
Hence, the probabilty will be 1/8 and 2/8;
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convert the following unsigned binary numbers to hexadecimal. a. 1101 0001 1010 1111 b. 001 1111 c. 1 d. 1110 1101 1011 0010
Answer:
a. D1AF
b. 1F
c. 1
d. EDB2
Step-by-step explanation:
You want these unsigned binary numbers converted to hexadecimal.
a. 1101 0001 1010 1111
b. 0001 1111
c. 0001
d. 1110 1101 1011 0010
HexEach group of 4 bits can take on any of 16 different values. Conveniently, each of those corresponds to a hexadecimal digit, as shown in the attachment.
To do the conversion, we replace each group of 4 bits by its hexadecimal equivalent.
a. D1AF
b. 1F
c. 1
d. EDB2
__
Additional comment
A couple of the given numbers are one or more bits short of a group of 4 bits. You may want to check your actual problem statement to see if the numbers here are the same. The attached table applies in any event.
The equation of the plane passing through the point (-8,3,7) and parallel to the plane 4x + 8y ~ 2z = 45, is given by 4x - 8y - 2z = -22 4x + 8y + 2z = -22 4x + 8y _ 2z = -22 4x + 8y - 2z = 22
The equation of the plane passing through the point (-8,3,7) and parallel to the given plane will be 4x + 8y - 2z = -22.
The given equation of the plane is 4x + 8y - 2z = 45.
To find the equation of the plane which passes through the point (-8,3,7) and is parallel to the given plane, we need to calculate the values of x, y and z in the given equation.
We have x = -8, y = 3 and z = 7 in the given point.
Substituting these values in the given equation, we get
4(-8) + 8(3) - 2(7) = 45
-32 + 24 - 14 = 45
-22 = 45
This is not possible.
Hence, the equation of the plane passing through the point (-8,3,7) and parallel to the given plane will be 4x + 8y - 2z = -22.
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What is the limit of the function?
Select True or False for each limit statement
Answer:
all are True
Step-by-step explanation:
You want to know if the end behavior is properly described for the polynomial functions shown in the table.
-2x³ at -∞A function of odd degree with a positive leading coefficient has a graph that has a generally positive slope (/).
When the leading coefficient is negative, the slope is generally negative (\). In this case, the sign of the end behavior is opposite the sign of x.
x → -∞, -2x³ → ∞ . . . . . True
2x⁴ at ∞A function of even degree with a positive leading coefficient has a graph that generally opens upward (U). The sign of the end behavior will match the sign of the leading coefficient.
x → ∞, 2x⁴ → ∞ . . . . . True
-9x⁵ at ∞This function is of odd degree with a negative leading coefficient. The sign of the end behavior is opposite the sign of x.
x → ∞, -9x⁵ → -∞ . . . . . True
All the Limit statement are True
Explain about the limit of the function?
If the polynomial functions indicated in the table's final behavior are appropriately explained, you wish to know.
-2x³ at -∞
The graph of an odd-degree function with a positive leading coefficient has a generally upward slope (/).
The slope is often negative when the leading coefficient is negative. The sign of the end behavior is thus the opposite of the sign of x.
x -, -2x3 -∞ are True
2x⁴ at ∞
The graph of an even degree function with a positive leading coefficient typically opens upward (U). The final behavior's sign will coincide with the leading coefficient's.
x, 2x4 ∞ are True
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In the cuboid below all lengths are in centimetres. The cuboid has a volume of 196cm cubed. Find the value of k.
The value of k in the cuboid is 7 cm when the volume is 196cm³.
What is volume?The space taken up by any three-dimensional solid constitutes a volume, to put it simply. They can be a cube, cuboid, cone, cylinder, or sphere.
The volumes of various shapes vary. In three-dimensional geometry, we have examined a variety of solids and shapes that are defined in three dimensions, including cubes, cuboids, cylinders, cones, etc. We are going to learn to calculate the volume for each of these shapes.
The volume of cuboid = l × b × h
Here,
4 × k × k = 196cm³
k² = 196/4
k² = 49
k = √49
k = 7
Thus, The value of k in the cuboid is 7 cm when the volume is 196cm³.
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You randomly choose one of the tiles shown. Select the favorable outcome(s) of choosing a 6.
Answer:
Step-by-step explanation:
I DONT UNDERSTAND
A two member beach volleyball team decided they wanted new uniforms. The
swimsuits cost $58 and the visors cost $24 each. What is the total cost of the new
uniform for all of the members?
O $58
O $82
O $116
O $164
5x - 4y = 27; - 2x + y = 3
Answer: x=-13, y=-23
Step-by-step explanation:
y in terms of x
-2x+y=3
y=3+2x
Substitute in other equation:
5x-4(3+2x)=27
5x-12-8x=27
-3x=39
x=39/-3
x=-13
Solve for y, substitute for x:
-2(-13)+y=3
26+y=3
y=-23
elvis and costello bought 36 lemmons to make lemmondae elvid squxed 1/3 of them costello squezzed 4/9 their mom did the rest how many lemmons did the mom sqeze
Their mom squeezed 8 lemons and Elvis and Costello bought 36 lemons.
What is the equation?The equation is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are equal.
Let's represent the number of lemons Elvis squeezed "E", the number of lemons Costello squeezed "C", and the number of lemons their mom squeezed "M".
We know that:
E + C + M = 36 (because they bought 36 lemons in total)
E = (1/3) x 36 = 12 (because Elvis squeezed 1/3 of the lemons)
C = (4/9) x 36 = 16 (because Costello squeezed 4/9 of the lemons)
We can use the first equation to find the value of M:
M = 36 - E - C
Substituting in the values we know:
M = 36 - 12 - 16
M = 8
So their mom squeezed 8 lemons.
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Find the volume of the solid of revolution obtained by rotating the region bounded by the curve
y=3xâx2 and the line y=2x about the y-axis.
The region enclosed by the curve y=3x-x2 and the line y=2x can be rotated about the y-axis to determine the volume of the solid of revolution. The cylindrical shell method can be used to determine this volume.
The cylindrical shell method can be used to determine the volume of the solid of revolution created by rotating the area enclosed by the curve y=3x-x2 and the line y=2x about the y-axis.
We must first determine the region's boundaries.
At x=2 and y=4, y=2x.
When y=3x-x2, x=0, and y=0.
As a result, the region's boundaries are x=0 to x=2 and y=0 to y=4.
The following formula determines the solid's volume:
Volume equals b a 2 y r2 d y
where x is a constant and r is the cylindrical shell's radius.
As a result, the solid of revolution's volume is given by:
Volume = 4.002y x 2.2dy
= 2π ∫4 0 (3x2-x3)dx
= 2π [x3-x4/4]
4 0
= 2π (16/4)
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Write an equivalent expression by distributing the "-" sign outside the parentheses:
-(-3.4c + 1)
Answer: 3.4c - 1
Step-by-step explanation:
multiply each part within the parentheses by the negative sign
so -3.4c x -1 = 3.4c
and 1 x -1 = -1
add those two together and we get 3.4c - 1
Identify the type of sampling used in this example.
A researcher randomly selected 50 of the nation's middle schools and interviewed all of the teachers at each school. a. random
b. cluster
c. stratified
d. systematic
The type of sampling used in this example is a. random sampling
One of the most straightforward methods for gathering information from the entire population is random sampling. Every member of the subset has an equal chance of being chosen as part of the sampling process when using random sampling. For instance, if an organization employs 300 people overall, a sample group of 30 workers will be chosen to participate in the survey. In this instance, the population is the entire workforce of the organization, and the sample is 30 employees. Since all of the employees who were picked to participate in the survey were chosen at random, every worker has an equal chance of getting chosen.
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How do I solve this math problem
The estimate of the integral by parts ∫₀³f(x)g'(x) = 62.1
What is integration?Integration is the process of reverse differentiation. It is the process of finding the anti-derivative of a function.
How to estimate the value of the integral?Since we have the integral [tex]\int\limits^3_0 {f(x)g'(x)} \, dx[/tex] and f(x) = x³. To evaluate the integral within the limits, we use integration by parts.
So, for any integral by parts
[tex]\int\limits^a_b {u(x)v'(x)} \, dx = [u(x)v(x)]^{a} _{b} - \int\limits^a_b {v(x)u'(x)} \, dx[/tex]
So, for the given integral [tex]\int\limits^3_0 {f(x)g'(x)} \, dx[/tex], we have that
[tex]\int\limits^3_0 {f(x)g'(x)} \, dx = [f(x)g(x)]^{3} _{0} - \int\limits^3_0 {g(x)f'(x)} \, dx[/tex]
So, we have that
[tex]\int\limits^3_0 {f(x)g'(x)} \, dx = [f(x)g(x)]^{3} _{0} - \int\limits^3_0 {g(x)f'(x)} \, dx\\= [f(x)g(x) - f(0)g(0)] - [{g(3)f'(3) - g(0)f'(0)}][/tex]
Now, we have that f(x) = x³. So,f'(x) = 3x².
So,
f(3) = 3³ = 27 and f'(3) = 3(3)² = 27.Also, form the table
g(0) = 2.3 and g(3) = 4.9Substituting these into the equation, we have that
[tex]\int\limits^3_0 {f(x)g'(x)} \, dx = [f(3)g(3) - f(0)g(0)] - [{g(3)f'(3) - g(0)f'(0)}]\\= [27 \times 4.9 - 0 \times 2.3] - [{4.9 \times 27 - 2.3\times 27}]\\= (132.3 - 0) - (132.3 - 62.1)\\= 132.3 - 132.3 + 62.1\\= 0 + 62.1\\= 62.1[/tex]
So, integral [tex]\int\limits^3_0 {f(x)g'(x)} \, dx = 62.1[/tex]
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Question 2 of 16
In a survey of the students' lunch preferences at a school, 523 said they prefer chicken, 295 said they prefer pasta, and 126 said they prefer salad.
How many more students prefer chicken than students combined who prefer pasta and salad?
Answer:
102
Step-by-step explanation:
523 said they prefer chicken,
295 said they prefer pasta,
and 126 said they prefer salad.
523-295-126 =102
TRAVEL Marion is travelling 500 miles by car. She drives 250 miles at an average rate of x miles per hour the first day and the remaining 250 miles at an average rate of x−5
miles per hour. Select the expression that represents the amount of time Marion spent traveling the 500 miles. (Hint: Write out the units to determine how to solve the problem.)
Multiple choice question.
A)
x(x−5)/500x−1250
B)
62,500/x(x−5)
C)
500x−1250/x(x−5)
D)
500/2x−5
Answer:
C) (500x - 1250)/x(x - 5)----------------------------------
We know that the equation for the time is:
time = distance / speedFind the time at travel in the first day:
250/xFind the time at travel in the second day:
250/(x - 5)To find the total time add app the two expressions we got above:
250/x + 250/(x - 5) = 250[1/x + 1/(x - 5)] = 250(x - 5 + x)/x(x - 5) = 250(2x - 5)/x(x - 5) = (500x - 1250)/x(x - 5)There matching choice is C.
The area of a square with a side 2.4m in length
Answer:
Below
Step-by-step explanation:
Squares have equal side lengths
area = side X side
= 2.4 m X 2.4 M = 5.76 m^2
premise 1: if 901,223 is divisible by 6, then it is divisible by 3 premise 2: conclusion: 901,223 is not divisible by 6 supply the missing premise that would make this a valid argument. state what valid argument form you used.
Premise 3: 901,223 is not divisible by 6.
Valid argument form used: Modus Tollens.
Step 1: Divide 901,223 by 6.
901,223 ÷ 6 = 150,203.83333
Step 2: Round down the answer.
150,203.83333 rounded down = 150,203
Step 3: Check if the answer is divisible by 3.
150,203 ÷ 3 = 50,067.3333
Step 4: Round down the answer again.
50,067.3333 rounded down = 50,067
Step 5: Check if the answer is divisible by 3.
50,067 ÷ 3 = 16,689
Step 6: Round down the answer again.
16,689 rounded down = 16,689
Step 7: Check if the answer is divisible by 3.
16,689 ÷ 3 = 5,563
Step 8: Round down the answer again.
5,563 rounded down = 5,563
Step 9: Check if the answer is divisible by 3.
5,563 ÷ 3 = 1,854.3333
Step 10: Round down the answer again.
1,854.3333 rounded down = 1,854
Step 11: Check if the answer is divisible by 3.
1,854 ÷ 3 = 618
Step 12: Check if the answer is divisible by 3.
618 ÷ 3 = 206
Step 13: Check if the answer is divisible by 3.
206 ÷ 3 = 68.66666666
Step 14: Round down the answer again.
68.66666666 rounded down = 68
Step 15: Check if the answer is divisible by 3.
68 ÷ 3 = 22.66666666
Step 16: Round down the answer again.
22.66666666 rounded down = 22
Step 17: Check if the answer is divisible by 3.
22 ÷ 3 = 7.333333
Step 18: Round down the answer again.
7.333333 rounded down = 7
Step 19: Check if the answer is divisible by 3.
7 ÷ 3 = 2.333333
Step 20: Round down the answer again.
2.333333 rounded down = 2
Step 21: Check if the answer is divisible by 3.
2 ÷ 3 = 0.666666
Conclusion: 901,223 is not divisible by 6
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the expression \root(4)(10mn) and (10mn)^((1)/(4))are equivalent or not equivalent
The expression \root(4)(10mn) and (10mn)^((1)/(4)) are equivalent
What are index forms?Index forms are described as that number written in the form of an exponential expression.
It is also described as a single number raised to another number.
The exponent of an exponential expression explains how many times to multiply the base by itself to simply the expression.
Standard forms and scientific notation are other names for index forms.
It is important to note that the fourth root of a number or variable is the biquadratic root.
From the information given, we have that;
[tex]\sqrt[4]{10mn}[/tex]
[tex](10mn)^1^/^4[/tex]
Note the following;
√2 = 2^1/2
∛2 = 2^1/3
Hence, the expressions are equivalent
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Identify the slope and one point on the line, given the equation
The slope of the line of the equation y - 12 = 3/4(x + 7), is: 3/4.
How to Identify the Slope of a Line in Point-Slope Form?The point-slope form of a linear equation is given as y - y1 = m(x - x1), where m is the slope of the line, (x1, y1) is a point on the line, and y and x are the variables.
To identify the slope of a line in point-slope form, you can simply look at the coefficient of x, which is the value of m. In the equation y - y1 = m(x - x1), m is the slope of the line.
This value tells you the rate of change or steepness of the line. A positive slope value implies that the line is increasing and a negative value implies that the line is decreasing.
Given the equation of a line in point-slope form as y - 12 = 3/4(x + 7). Therefore, the slope (m) can be identified as 3/4.
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i currently average a 7 over the first 3 test of the year. what do i need to average over the 4 test in order to bring my overall to an 8?
Answer:
11
Step-by-step explanation:
You have an average of 7 on 3 tests, and you want to know the score required of the 4th test to make the average be 8.
AverageThe average is the sum, divided by the number.
If you have an average of 7 on the first 3 tests, the sum of those scores is ...
average = sum/number
7 = sum/3
21 = sum
If you want an average of 8 on 4 tests, the sum of the scores must be ...
8 = sum/4
32 = sum
In order to increase the sum of scores from 21 to 32, you must have a score of ...
32 -21 = 11
You need to score 11 on the 4th test to bring the average to an 8.
__
Alternate solution
The difference between the scores you have and the average you want is ... 3(7 -8) = -3.
In order to bring your average up to 8, you need to have a total of 3 more points than that average on your 4th test: 8 + 3 = 11. (This brings the total of differences from average to zero: (-1) +(-1) +(-1) +(3) = 0.)
Your 4th test must have a score of 11 to bring your average on 4 tests up to 8.
While trying to decipher the madness that is Calc 2, you and your classmates Donald and Joe are discussing a problem. Donald has come up with the following work for the problem.
∫3xsin(x3)dx=x3 du=3x2 dx
=x/x∫3xsin(x3)dx
=x/1∫3x2sin(x3)dx
=x/1∫sin(u)du=x1⋅−cos(4)+C
=∫xcos(x)3)+C
Joe, on the other hand strongly believes Donald is wrong. A heated argument develops about who is correct and why. In your judgement, is Donald's work correct or not? Is there a way to double: check? Be sure to support your thinking.
Donald's work seems to be inaccurate. He substitutes u = x3 in the first step, however he does not correctly adjust the bounds of integration. He divides by x in the subsequent phase, which results in a division by zero at x = 0.
Additionally, it appears like Donald made a mistake while integrating sin(u)du in the final step. It should have been -cos(u) + C rather than x*cos(u) + C. One could try an alternative approach or use a calculator or computer to assess the integral to verify the work. Another option would be to look up the right answer in a book or internet resources.
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Mabel was asked whether the following equation is an identity: ( 4 x − 3 ) ( x − 2 ) 2 = 4 x 3 − 19 x 2 + 4 x − 12 (4x−3)(x−2) 2 =4x 3 −19x 2 +4x−12left parenthesis, 4, x, minus, 3, right parenthesis, left parenthesis, x, minus, 2, right parenthesis, squared, equals, 4, x, cubed, minus, 19, x, squared, plus, 4, x, minus, 12 She performed the following steps: ( 4 x − 3 ) ( x − 2 ) 2 ↪ Step 1 = ( 4 x − 3 ) ( x − 2 ) ( x − 2 ) ↪ Step 2 = ( 4 x − 3 ) ( x 2 − 2 x − 2 x + 4 ) ↪ Step 3 = ( 4 x − 3 ) ( x 2 − 4 x + 4 ) ↪ Step 4 = 4 x 3 − 16 x 2 + 16 x − 3 x 2 − 12 x − 12 ↪ Step 5 = 4 x 3 − 19 x 2 + 4 x − 12 Step 1 Step 2 Step 3 Step 4 Step 5 =(4x−3)(x−2) 2 =(4x−3)(x−2)(x−2) =(4x−3)(x 2 −2x−2x+4) =(4x−3)(x 2 −4x+4) =4x 3 −16x 2 +16x−3x 2 −12x−12 =4x 3 −19x 2 +4x−12 For this reason, Mabel stated that the equation is a true identity. Is Mabel correct? If not, in which step did she make a mistake?
Yes, Mabel is correct. She correctly followed all the steps and arrived at the correct answer.
A polynomial identityYes, Mabel is correct. All of the steps she took were mathematically valid and the final equation is the same as the original equation, so her conclusion that it is an identity is correct. An identity is an equation that is true for all values of the variables. This equation, ( 4 x − 3 ) ( x − 2 ) 2 = 4 x 3 − 19 x 2 + 4 x − 12, is an algebraic identity, meaning that it is always true for any value of x. The steps Mabel took were mathematically correct. In Step 1, she correctly multiplied out the parentheses. In Step 2, she correctly factored out the (x-2). In Step 3, she correctly applied the distributive property to expand the equation. In Step 4, she correctly multiplied out the parentheses. In Step 5, she correctly simplified the equation. The equation, ( 4 x − 3 ) ( x − 2 ) 2 = 4 x 3 − 19 x 2 + 4 x − 12, is an identity because it is true for any value of x. Mabel was correct in her conclusion that the equation is an identity.To learn more about A polynomial identity refer to:
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Many people say they want to be rich. What are some legitimate ways you can become financially wealthy as an adult? Your own words!
Answer:
see below
Step-by-step explanation:
1. study hard and go to a good college and then find a good job.
2. be intelligent on spending and save money to invest in the financial markets
3. buy a few life insurance policies with living benefits.
4. find a niche on social media and create contents to attract followers, and then invest the earnings in stock market or bond market.
5. invest in real estate
6. start a small business and then grow into franchise.
Hannah makes 5 out of 8 attempted free-throws! What is the average waiting time for Hannah to make her first basket, and what is the probability that she will make a basket on or before her last attempt within her average waiting time?
42% of people are likely to see a movie today, or probability of 0.42.
Explain about the probability?A probability is a number that expresses the possibility or likelihood that a specific event will take place. Probabilities can be stated as proportions with a range of 0 to 1, or as percentages with a range of 0% to 100%.
By dividing the number of preferable possibilities by the total number of potential outcomes, probability, which measures the likelihood that an event will occur, is obtained. A coin flip is the most basic illustration. There are only two outcomes that can occur when you flip a coin: either heads or tails.
Probability is a measure of the likelihood that an event will occur or the likelihood that something will happen.
The likelihood that a movie is being viewed right now is:
0.5 of 0.6(if it rains today) (if it rains today).
0.3 of 1 - 0.6 = 0.4(if it does not rain today) (if it does not rain today).
therefore adding them:
p =0.5 (0.6) +0.3(0.4)
=0.3 +0.12 = 0.42
42% of people are likely to see a movie today, or 0.42.
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Elena wants to add (2.3×105)+(3.6×106) and writes (2.3×105)+(3.6×106)=5.9×106. Find Elena's mistake and fix it.
What was her mistake? I havent seen any comments saying anything about it.
The mistake of Elena is that the powers of the terms were not the same and the correct result is 3.83 ×10^6.
What was the mistake?We have to note that the addition of the exponents is not just like the addition of the ordinary numbers that we know. The operation of the addition of the exponents would require an extra care to ensure that the powers of the two terms to be added is exactly the same.
In the work of Elena; (2.3×10^5)+(3.6×10^6)=5.9×10^6. This is wrong because the powers of the terms on the right hand side are not the same. As such the correct approach would be;
0.23 × 10^6 + 3.6×10^6 = 3.83 ×10^6
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What is the answer to this question?
The unknow angle in the isosceles triangle is as follows:
m∠C = 80 degrees.
How to find the angles of an isosceles triangle?An isosceles triangle is a triangle that has two of its sides equal to each other and also have two if its angles equal to each other.
The base angles of the isosceles triangles are equal and the legs of the triangle are equal.
Therefore, in ΔABC
AB = BC
Hence,
m∠A ≅ m∠C
The angle A and angle C are the base angles. This makes them congruent to each other.
Therefore,
m∠C = 80 degrees
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Evaluate geometric series sigma1^12 10×4^n-1
The value of the sigma notation is 55924050
How to evaluate the sigma notationFrom the question, we have the following parameters that can be used in our computation:
sigma1^12 10×4^n-1
Express properly
So, we have the following representation
[tex]\sum^{12}_{1} 10*4^{n-1[/tex]
The sum of the geometric series can be calculated using the formula:
S = a(1 - r^n)/(1 - r)
From the sigma notation, we have the following parameters
a = 10,
r = 4
n = 12.
Substituting these values, we get:
S = 10 * (1 - (4)^12) / (1 - 4)
Evaluate the expression
S = 55924050
Hence, the sum is 55924050
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a demographer uses birth records to keep track of the ages of all the people born in a small town. they keep a log of all the ages in years. this variable is best described as...
The variable a log of all the ages in years is best described as quantitative and continuous.
As we know the variables are classified as qualitative and quantitave.
The quantitative variables are further classified as: discrete and continuous.
A discrete variable is a variable that can only take specific numeric values but values have a clear quantitative interpretation.
This variable takes the values that are countable and have a finite number of values.
In case of continuous quantitative variable, it can take any value in an interval.
The values of continuous quantitative variable are not countable and they have an infinite number of possibilities.
As we know the age is a discrete variable as it is commonly expressed as an integer.
Here, a demographer they keep a log of all the ages in years.
So, variable is continuous variable.
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find the maclauvins series of f(x) = x ln(2x+1) what in the rc of the obtained seties
The Maclaurin series is xln(2x + 1) - x^2 + (2x^3)/3 - (2x^4)/4 + ... and the arc is (-1/2, 1/2).
How to determine the Maclaurin seriesFrom the question, we have the following parameters that can be used in our computation:
f(x) = x ln(2x+1)
The Maclaurin series of f(x) = x ln(2x + 1) can be found by using the power series expansion of the logarithm function and the definition of a derivative.
Starting with the power series expansion of ln(1 + x):
ln(1 + x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 + ...
We can then substitute x = 2x + 1 - 1 to obtain the power series expansion of ln(2x + 1):
ln(2x + 1) = (2x + 1 - 1) - (2x + 1 - 1)^2/2 + (2x + 1 - 1)^3/3 - (2x + 1 - 1)^4/4 + ...
= 2x - x^2 + (2x^3)/3 - (2x^4)/4 + ...
Next, we use the definition of a derivative to find the Maclaurin series of f(x) = x ln(2x + 1):
f'(x) = d/dx (x ln(2x + 1)) = ln(2x + 1) + x * d/dx ln(2x + 1)
= ln(2x + 1) + x * (2 - 2x + 2x^2 - 2x^3 + ...)
= ln(2x + 1) + 2x - 2x^2 + 2x^3 - 2x^4 + ...
Therefore, the Maclaurin series of f(x) = x ln(2x + 1) is:
f(x) = ∫ f'(x) dx = ∫ (ln(2x + 1) + 2x - 2x^2 + 2x^3 - 2x^4 + ...) dx
= xln(2x + 1) - x^2 + (2x^3)/3 - (2x^4)/4 + ...
How to determine the arc of the seriesThe arc of a Maclaurin series refers to the range of values for x for which the series converges to the actual value of the function.
For the Maclaurin series above, the range of convergence depends on the function ln(2x + 1) which has a singularity at x = -1/2.
This implies that the Maclaurin series converges for all x such that |x| < 1/2.
Rewrite as
(-1/2, 1/2)
So, the arc of the series is (-1/2, 1/2).
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Calculate the volume of the Surface Area of pyramid 12cm height and 12cm base and 14cm weights
The volume of the pyramid is, 72 cm³.
What is Multiplication?To multiply means to add a number to itself a particular number of times. Multiplication can be viewed as a process of repeated addition.
We have to given that;
Height of pyramid = 12 cm
Base of pyramid = 12 cm
Weight of pyramid = 14 cm
Now, We know that;
⇒ Volume of pyramid = 1/2 × Base × Height
Substitute all the values, we get;
⇒ Volume of pyramid = 1/2 × 12 × 12
⇒ Volume of pyramid = 6 × 12
⇒ Volume of pyramid = 72 cm³
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