Consider the series ∑n=0∞2e−n.
a. The general formula for the sum of the first n terms is S_n=__?__. Your answer should be in terms of n.
b. The sum of a series is defined as the limit of the sequence of partial sums, which means ∑n=0∞2e−n=limn→∞=(__?__)=(__?__).
c. Select all true statements (there may be more than one correct answer):
A. The series is a telescoping series (i.e., it is like a collapsible telescope).
B. The series converges.
C. The series is a geometric series.
D. The series is a p-series.
Answers
a. The general formula for the sum of the first n terms is Sₙ = 2(1 - e⁻ⁿ)/(1 - e⁻¹)
b. The sum of a series is defined as the limit of the sequence of partial sums is converges.
c. The true statement are "The series converges. and The series is a geometric series." (option B and C).
a. To find the general formula for the sum of the first n terms, we need to add the first n terms of the series. Thus, we have:
Sₙ = 2e⁰ + 2e⁻¹ + 2e⁻² + ... + 2e⁻ⁿ
We can see that this is a geometric series with a first term a=2 and a common ratio r=e⁻¹. Therefore, we can use the formula for the sum of a geometric series to find the general formula for Sₙ:
Sₙ = a(1 - rⁿ)/(1 - r)
Substituting a=2 and r=e⁻¹, we get:
Sₙ = 2(1 - e⁻ⁿ)/(1 - e⁻¹)
the limit of the sequence of partial sums is zero, the series converges.
b. To determine whether the series converges or diverges, we can take the limit of the sequence of partial sums as n approaches infinity. Thus, we have:
∑n=0∞2e−n=limn→∞ S_n= limn→∞ 2(1 - e⁻ⁿ)/(1 - e⁻¹)
To evaluate this limit, we can use L'Hopital's rule, which states that if we have an indeterminate form of the type 0/0 or infinity/infinity, we can take the derivative of the numerator and the denominator until the form is no longer indeterminate. Applying L'Hopital's rule, we get:
lim n→∞ 2(1 - e⁻ⁿ)/(1 - e⁻¹) = limn→∞ 2e⁻ⁿ/(e⁻¹) = 0
c. We can now identify whether the series is a telescoping series, a geometric series, or a p-series.
The series is a geometric series, as we already saw in part (a).
Therefore, the correct answers are (B) and (C).
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Related Questions
Scarlett Squirrel teaches a hula dancing class to young squirrels.
14
1414 squirrels showed up to class on Monday,
10
1010 squirrels on Tuesday,
8
88 squirrels on Wednesday,
10
10squirrels on Thursday, and
12
12squirrels on Friday.
Find the mean number of squirrels.
Answers
Answer:
10.8
Step-by-step explanation:
14+10+8+10+12
_______________
5
= 54/5
= 10.8
answer with explanation
Answers
The area and circumference of a circle with diameter of 3 m is 7.0165 m² and 9.42 m² respectively
How to solve an equation?An equation is an expression that can be used to show the relationship between two or more numbers and variables using mathematical operators.
The area of a figure is the amount of space it occupies in its two dimensional state.
The diameter (d) of the circle is 3 meter. Hence:
Area = π * diameter²/4 = π * 3²/4 = 7.0165 m²
Circumference = π * diameter = π * 4 = 9.42 m²
The area and circumference are 7.0165 m² and 9.42 m²
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20 students each rolled dice 5 times each to measure the median. Here is the data in the picture attached. What can we infer from this graph by looking at the median data?
A. when rolling a dice multiple times, the median is less likely to fall between numbers 1 and 6.
B. it's not possible to tell the likelihood of where the median is going to be when measuring probability since rolling dice is completely randomized, etc.
C. some other response
Answers
We can deduce that option A is likely to be true based on the given graph of rolling dice median data.
What is the median?The median is the value in the middle of a data set, which means that 50% of the data points have a value less than or equal to the median, and 50% of the data points have a value greater than or equal to the median.
The graph illustrates that the median value of the rolls is closer to the center of the possible outcomes (numbers 3, 4, and 5) than the extremes (numbers 1 and 6).
This implies that when rolling a dice several times, the median is less likely to fall between the numbers 1 and 6.
However, because rolling the dice is a random process, there is always a degree of uncertainty in predicting the outcomes.
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Express as a product. 1+2cos a
Answers
Answer:
Therefore, 1 + 2cos a can be expressed as the product of 2 and (cos a/2 + 1).
1 + 2cos a = 2(cos a/2 + 1)
Step-by-step explanation:
We can use the trigonometric identity:
cos 2a = 1 - 2 sin^2 a
to rewrite 1 + 2cos a as:
1 + 2cos a = 1 + 2(1 - sin^2 a/2)
= 1 + 2 - 2(sin^2 a/2)
= 3 - 2(sin^2 a/2)
Now, using another trigonometric identity:
sin a = 2 sin(a/2) cos(a/2)
we can rewrite sin^2 a/2 as:
sin^2 a/2 = (1 - cos a)/2
Substituting this into the expression for 1 + 2cos a, we get:
1 + 2cos a = 3 - 2((1 - cos a)/2)
= 3 - (1 - cos a)
= 2 + cos a
Therefore, 1 + 2cos a can be expressed as the product of 2 and (cos a/2 + 1).
1 + 2cos a = 2(cos a/2 + 1)
12.
What is the product of
3 and 10?
Answers
Answer:
Any number multiplied by 10 equals to the number and a 0 at the end
Therefore 3×10=3
Cassie uses 1 quart cranberry juice, 1/2 gallon ginger ale, and 1 1/2 pint orange juice to make friut punch .How many 6- ounce servings can she make
Answers
She can make 20 6-ounce servings to make fruit punch
How to determine this
Cassie uses 1 quart cranberry juice,
1/2 gallon ginger ale,
1 1/2 pint orange juice.
1 quart of cranberry juice when converted to ounce
1 quart = 32 ounces
1/2 gallon of ginger ale when converted to ounce
1 gallon = 128 ounces
1/2 gallon = 128/2
1/2 gallon = 64 ounces
1 1/2 pint orange juice to make fruit punch
1 pint = 16 ounces
1 1/2 pint = 16 + 16/2 ounces
= 16 + 8
= 24 ounces
How many 6-ounces servings can she make
32 + 64 + 24
= 120 ounces
6-ounces that can be made
= 120 ounces/6 ounces
= 20 servings
Therefore, 20 6-ounce servings can be make for fruit punch
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A soccer field is 80 yards wide and 110 yards long. If a person walks across the field along the diagonal instead of walking the length and width of the field, how much shorter is the distance
Answers
So the person would save approximately 244 yards by walking diagonally across the field instead of walking the length and width of the field separately.
what is approximately ?
"Approximately" is a word that is used to indicate that a value, quantity, or measurement is close to the actual or precise value, but not exactly the same. It means that the stated value is an estimate or approximation, and may be slightly different from the true value.
In the given question,
Using the Pythagorean theorem, we can find the length of the diagonal of the soccer field:
diagonal² = length² + width²
diagonal² = 110² + 80²
diagonal² = 12,100 + 6,400
diagonal² = 18,500
diagonal = √18,500
diagonal ≈ 136 yards
So the person walking diagonally across the field would cover a distance of approximately 136 yards.
To compare this with the distance they would cover if they walked the length and width of the field separately, we can use the formula for the perimeter of a rectangle:
perimeter = 2(length + width)
perimeter = 2(110 + 80)
perimeter = 2(190)
perimeter = 380 yards
Therefore, if the person walked the length and width of the field separately, they would cover a total distance of 380 yards.
The difference between these two distances is:
380 - 136 = 244 yards
So the person would save approximately 244 yards by walking diagonally across the field instead of walking the length and width of the field separately.
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So the person would save approximately 244 yards by walking diagonally across the field instead of walking the length and width of the field separately.
what is approximately ?
"Approximately" is a word that is used to indicate that a value, quantity, or measurement is close to the actual or precise value, but not exactly the same. It means that the stated value is an estimate or approximation, and may be slightly different from the true value.
In the given question,
Using the Pythagorean theorem, we can find the length of the diagonal of the soccer field:
diagonal² = length² + width²
diagonal² = 110² + 80²
diagonal² = 12,100 + 6,400
diagonal² = 18,500
diagonal = √18,500
diagonal ≈ 136 yards
So the person walking diagonally across the field would cover a distance of approximately 136 yards.
To compare this with the distance they would cover if they walked the length and width of the field separately, we can use the formula for the perimeter of a rectangle:
perimeter = 2(length + width)
perimeter = 2(110 + 80)
perimeter = 2(190)
perimeter = 380 yards
Therefore, if the person walked the length and width of the field separately, they would cover a total distance of 380 yards.
The difference between these two distances is:
380 - 136 = 244 yards
So the person would save approximately 244 yards by walking diagonally across the field instead of walking the length and width of the field separately.
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A table is being sold for 67% off the regular price the sale price is $536 what is the regular price
Answers
Answer:
Step-by-step explanation:
Use a simple Equation
Price = Number x Price
So Price = 536.
Since it was 67% off we multiple the price of the original by .67.
So
536 = .67x
Solve for x = 536/.67 = 800
The original Price is $800.
check by multiplying by the sale %.
248 meters per second = miles per hour
Answers
Answer:
554.76 miles per hour
Step-by-step explanation:
[tex]\frac{248 meters}{1 second}[/tex] x [tex]\frac{3600 seconds}{1 hour}[/tex] x [tex]\frac{1 mile}{1609.34 meters}[/tex]
= 554.76 miles per hour
[tex]\sf 554.76(mi/h).[/tex]
Step-by-step explanation:1. Write the initial expression.[tex]\sf \dfrac{248(m)}{s}[/tex]
2. Using conversion factors, convert from meters to miles.a) First, convert the meters to kilometers.
[tex]\sf 248(m)*\dfrac{1(km)}{1000(m)} =0.248(km)[/tex]
Here, the meters cancel, since one is on the numerator and the other one is expressed as a denominator, and we simpyl multiply the value by the fraction, resulting in a number that would be a value for kilometers.
b) Convert from kilometers to miles.
[tex]\sf 0.248(km)*\dfrac{1(mi)}{1.60934(km)} =0.1541(mi)[/tex]
3. Convert the seconds into hours.[tex]\sf 1(seconds)*\dfrac{1(hour)}{3600(seconds)} =\frac{1}{3600}(h)[/tex]
4. Rewrite the initial expression using the calculated converted values.[tex]\sf \dfrac{0.1541(mi)}{\dfrac{1}{3600} (h)} =554.76(mi/h)[/tex]
Joe took 1/2 hours to clean the bathroom. Then he took 3 1/4 hours to clean the den. How much total time did Joe take to clean the two rooms? Write your answer as a mixed number in simplest form.
Answers
Answer:
He took 3 3/4 hours to clean both rooms.
Step-by-step explanation:
1/2 = 2/4
3 1/4 + 2/4 = 3 3/4
Therefore, he took 3 3/4 hours.
multiply (−3x+4)(2x−1)
Answers
After multiplying (−3x+4) with (2x−1) we get a quadratic equation which is equal to −6x² + 11x − 4.
To multiply the expressions (−3x+4)(2x−1), we use the distributive property of multiplication:
(−3x+4)(2x−1) = −3x(2x) + 4(2x) − 3x(−1) + 4(−1)
Simplifying the above expression, we get:
= −6x² + 8x + 3x − 4
= −6x² + 11x − 4
Therefore, the product of (−3x+4)(2x−1) is −6x² + 11x − 4.
We can also use the FOIL method to multiply the two expressions:
(−3x+4)(2x−1) = −3x(2x) −3x(−1) + 4(2x) + 4(−1)
= −6x² + 3x + 8x − 4
= −6x² + 11x − 4
Either method can be used to get the same result.
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Describe the sampling distribution of p. Assume the size of the population is 15,000. n=700, p=0.6 Question content area bottom Part 1 Choose the phrase that best describes the shape of the sampling distribution of p below. A. Approximately normal because n≤0.05N and np(1−p)<10. B. Approximately normal because n≤0.05N and np(1−p)≥10. C. Not normal because n≤0.05N and np(1−p)<10. D. Not normal because n≤0.05N and np(1−p)≥10. Part 2 Determine the mean of the sampling distribution of p. μp=enter your response here (Round to one decimal place as needed.) Part 3 Determine the standard deviation of the sampling distribution of p. σp=enter your response here (Round to three decimal places as needed.)
Answers
The correct answer is B. The sampling distribution of p is approximately normal because n is less than 5% of the population size (n ≤ 0.05N) and np(1-p) is greater than or equal to 10 (np(1-p) ≥ 10).
Part 2:
The mean of the sampling distribution of p is equal to the population proportion, which is 0.6. Therefore, μp = 0.6.
Part 3:
The standard deviation of the sampling distribution of p is given by the formula:
σp = sqrt((p(1-p))/n)
Substituting p = 0.6 and n = 700, we get:
σp = sqrt((0.6(1-0.6))/700)
σp ≈ 0.024
Therefore, the standard deviation of the sampling distribution of p is approximately 0.024.
Part 1:
The shape of the sampling distribution of p can be determined using the following criteria:
- np(1-p) ≥ 10
- n ≤ 0.05N
Here, n=700, N=15000, and p=0.6.
np(1-p) = 700 * 0.6 * (1-0.6) = 168.
Since np(1-p) ≥ 10, the first criterion is met.
n/N = 700/15000 = 0.0467, which is less than 0.05. So the second criterion is also met.
Therefore, the sampling distribution of p is approximately normal.
The correct answer is (B) Approximately normal because n≤0.05N and np(1−p)≥10.
Part 2:
The mean of the sampling distribution of p is given by the formula:
μp = p = 0.6
Therefore, the mean of the sampling distribution of p is 0.6.
The answer is μp = 0.6.
Part 3:
The standard deviation of the sampling distribution of p is given by the formula:
σp = sqrt [ p(1-p) / n ]
Substituting the values, we get:
σp = sqrt [ 0.6 * (1-0.6) / 700 ]
σp = 0.026
Therefore, the standard deviation of the sampling distribution of p is 0.026.
The answer is σp = 0.026.
I need help with my assignments
Answers
Answer :
Answer : 1.796
The explanation is in the pictures.
HELLPPPP !!!! i need help with the answer I'm stuck!
Answers
Answer:
1. f(x)= -2x^2-12x-19
2. f(x)= 2x^2+12x+17
3. f(x)= -2x^2+12x-17
4. f(x)= 2x^2-12x+17
Step-by-step explanation:
Hope this helps!
Answer: f(x)=-2x^2+12x-17 is the 3rd picture (vertex at (3,1))
f(x)=2x^2-12x+17 is the 4th picture (vertex at (3,-1))
f(x)=2x^2+12x+17 is the 2nd picture (vertex at (-3,-1))
f(x)=2x^2+12x-17 does not correlate to a picture.
f(x)=-2x^2-12x-19 is the 1st picture (vertex at (-3,-1))
4X Minus 3y Equal 21
Answers
Step-by-step explanation:
The given equation is:
4x - 3y = 21
To solve for y, we can rearrange the equation as follows:
4x - 21 = 3y
Dividing both sides by 3, we get:
y = (4x - 21) / 3
Therefore, the solution to the equation is:
y = (4x - 21) / 3
Willow owns 225 books. Of them, 5/9
are mysteries. How many of Willow's books
are mysteries?
A) 225
B) > 225
C) < 225
D) 0
(Side note:) please provide explanation if can!
Answers
(5/9) x 225 = 125
Therefore, there are 125 mystery books, which is greater than the total number of books (225).
the question is on the screenshot
Answers
When the simultaneous equation is solved, we have that; x = 2 and y = 3
What is the solution to a simultaneous equationSimultaneous equations are a set of two or more equations that are solved together to find the values of the variables that satisfy all of the equations at the same time.
We have the equations;
9x - 7y = -3 --- (1)
-3x + 5y = 9 ---- (2)
Multiply (1) by 1 and (2) by (3)
9x - 7y = -3 ---- (3)
-9x + 15y = 27 ----- (4)
Add (3) and (4)
8y = 24
y = 3
Substitute y = 3 into (1)
9x - 7(3) = -3
9x - 21 = -3
9x = -3 + 21
9x = 18
x = 2
Thus the solution is x = 2 and y = 3
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Using a table find the range of a function for the given domain f(x)equals to 2x^2-7 with domain X equals to {-4,-2,0,3}
Answers
The range of function for the given domain are {25, 8, -7, 11}
Given that, a function f(x) = 2x²-7, we need to find the range of the function, for the given domain, x = {-4, -2, 0, 3},
So,
Domain = All the input values.
Range = All the output values.
So, to find the range we will put the value of x which is the domain and the value obtained of f(x) will be the range,
f(x) = 2x²-7
For, x = -4,
f(-4) = 16×2 - 7 = 25
f(-4) = 25
f(-2) = 4 × 2 - 7 = 8
f(-2) = 8
f(0) = -7
f(3) = 9 × 2 - 7
f(3) = 11
Hence the range of function for the given domain are {25, 8, -7, 11}
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The size of fish is very important to commercial fishing. A study conducted in 2012 found the length of Atlantic
cod caught in nets in Karlskrona to have a mean of 49.9 cm and a standard deviation of 3.74 cm (Ovegard, Berndt
& Lunneryd, 2012). Assume the length of fish is normally distributed.
b) What is the length in cm of the longest 15% of Atlantic cod in this area?
Answers
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by
[tex]\text{Z}=\dfrac{\text{X}-\mu}{\sigma}[/tex]
After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X.
In this problem, we have that:
A study conducted in 2012 found the length of Atlantic cod caught in nets in Karlskrona to have a mean of 49.9 cm and a standard deviation of 3.74 cm, so [tex]\mu=49.9,\sigma=3.74[/tex].
What is the length in cm of the longest 15% of Atlantic cod in this area?
We have to find the value of X for the value of Z that has a p-value of 0.85.
Looking at the z-score table, we have that Z = 1.04 has a p-value of 0.8508. So, we have to find the value of X when .
So
[tex]\text{Z}=\dfrac{\text{X}-\mu}{\sigma}[/tex]
[tex]1.04=\dfrac{\text{X}-49.9}{3.74}[/tex]
[tex]\text{X}-49.9=3.8896[/tex]
[tex]\text{X}=53.7896[/tex]
The length of the longest 15% of Atlantic cod in this area is 53.79 cm, rounded to 2 decimal places.
Question 1
Company X tried selling widgets at various prices to see how much profit they would make. The following table shows the widget selling price, x, and the total profit earned at that price, y. Write a quadratic regression equation for this set of data, rounding all coefficients to the nearest hundredth. Using this equation, find the profit, to the nearest dollar, for a selling price of 7.75 dollars.
Question 2
A tennis ball is dropped from a certain height. Its height in feet is given by h(t)=-16t^2+14 where t represents the time in seconds after launch. What is the ball’s initial height?
Answers
The quadratic regression equation is y = -140.61x² + 2,954.23x - 7,512.85 and the ball's initial height is 14 feet.
Writing the quadratic regression equationGiven the set of data
Using a graphing tool, we have
C = -7,512.85B = 2,954.23A = -140.61A quadratic regression equation is represented as
y = Ax² + Bx + C
So, we have
y = -140.61x² + 2,954.23x - 7,512.85
When the selling price is 7.75, we have
y = -140.61(7.75)² + 2,954.23(7.75) + -7,512.85
Evaluate
y = 6937
The initial heightThe ball's initial height is the height when t = 0, which means we need to evaluate the function h(0).
h(0) = -16(0)^2 + 14
h(0) = 0 + 14
h(0) = 14
Therefore, the ball's initial height is 14 feet.
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Need help will give brainliest and 5 stars!
For this graph, for each vertical asymptote, write down the two limits that describe the graph of the function near the asymptote.
Answers
The limits on the vertical asymptotes of the function are given as follows:
lim x→6- f(x) = ∞
lim x→6+ f(x) = - ∞
lim x→2- f(x) = ∞
lim x→2+ f(x) = - ∞
lim x→6- f(x) = - ∞
lim x→6+ f(x) = ∞
What are a function's vertical asymptotes?The values of x that are outside of the domain—in a fraction, the zeroes in the denominator—are the vertical asymptotes.
When a graph reaches its vertical asymptotes, it just approaches the point rather than crossing it.
The limits on the vertical asymptotes of the function are given as follows:
lim x→6- f(x) = ∞
lim x→6+ f(x) = - ∞
lim x→2- f(x) = ∞
lim x→2+ f(x) = - ∞
lim x→6- f(x) = - ∞
lim x→6+ f(x) = ∞
The vertical asymptotes are then provided as x = -6, x = -2, and x = 2.
Each asymptote has limits to the left (shown by the minus sign) and the right (shown by the plus sign).
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Suppose Patrick Goldsmith deposited $1000 in an account that earned simple interest at an annual rate of 10% and left it there for 5 years. At the end of the 5 years, Patrick deposited the entire amount from that account into a new account that earned 10% compounded quarterly. He left the money in this account for 5 years. How much did he have after the 10 years? (Round your answer to the nearest cent.)
Answers
Answer:
Step-by-step explanation:
After 5 years at a simple interest rate of 10%, Patrick would have earned $500 in interest ($1000 x 10% x 5 years). So at the end of the 5 years, he would have $1500 in the account.
If this entire amount is deposited into a new account that earns 10% compounded quarterly, we need to determine the quarterly interest rate first.
The quarterly interest rate is (1 + 0.10/4)^4 - 1 = 0.025 (rounded to three decimal places).
After 10 years (or 40 quarters) at a quarterly interest rate of 0.025, the compounded amount is:
$1500 x (1 + 0.025)^40 = $4045.56
Therefore, Patrick would have $4045.56 in the account after 10 years when rounding to the nearest cent.
Hope that Helps :)
What is the probability of flipping a coin 12 times and getting heads 5 times?
Round your answer to the nearest tenth of a percent.
OA. 12.1%
OB. 3.1%
OC. 19.3%
OD. 5.4%
Answers
Answer:
The correct answer is D. 5.4% after rounding my answer off to the nearest tenth of a percent.
Directions: Read the scenario below and solve. Show your
work.
If given x > 0 and y> 0, in which quadrant or axis will each
ordered pair described below lie? Explain your answer.
a. (-x, y)
b. (-x, 0)
c. (x, -y)
d. (0. y)
Answers
The quadrant or axis for each of the given ordered pairs is:
a. (-x, y) - second quadrant
b. (-x, 0) - X-axis
c. (x, -y) - fourth quadrant
d. (0, y) - Y-axis
How to determine the quadrantsa. (-x, y):
As the x-coordinate is negative (-x), and the y-coordinate is positive (y), the ordered pair (-x, y) will lie in the second quadrant.
b. (-x, 0):
The x-coordinate being negative (-x), and the y-coordinate 0 renders that the ordered pair (-x, 0) lies on the x-axis. This horizontal axis records an unchanging value for y: zero.
c. (x, -y):
The x-coordinate here is a positive value (x), and the y coordinate a negative one (-y); it implies that the ordered pair (x, -y) lies within the fourth quadrant.
d. (0, y):
In this case, the x-coordinate is 0 and the y-coordinate is positive (y); therefore, the ordered pair (0, y) has its place on the y-axis. The y-axis is the vertical line on which x retains a permanent numerical value of zero.
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How to draw a quilt of isosceles triangular
Answers
The creation of an isosceles triangle quilt can only be achieved through a number of discreet steps:
The StepsInitially, select the bespoke size of both individual triangles and the full-scale length and breadth of the snug.
Subsequently, draft a lattice on either a sheet of paper or digital drafting software, with each square serving as a single triangular unit.
Commencing your task from the bottom row, resonate along the equidistant bases while drawing with precision to ensure an immaculate line-up.
Then, for the succeeding section, diverge by slightly altering the position of the point of each triangle in accordance with the intermediary gap that exists between two triangles below it.
Continue repeating this pattern while constructing rows, enveloping further ground until fully complete coverage has been ensured.
Last but not least, rigorously maintain accuracy by exploiting straight edges strategically laid beside precise measuring instruments.
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. To determine how funny the jokes were, the researchers asked a group of 86 undergraduates to rate the jokes on a scale from 1 (very unfunny) to 21 (very funny). Participants rated a “lawyer joke” as one of the funniest jokes, with a rating of 14.48 ± 4.38 (M ± SD).
Assuming that these data are normally distributed,
1. What was the rating that marks the cutoff for the top 10% of participant ratings for this joke?
Answers
If the "Lawyer's-Joke" is rates as the funniest, then the rating which is the top 10% of participant rating for this joke is 20 marks.
A "Normal-Distribution" is defined as a probability distribution that is symmetric, bell-shaped, and characterized by its mean and standard deviation.
First, we standardize the rating of 14.48 using the formula:
⇒ z = (x - μ) / σ;
where x is = rating of 14.48, μ is = mean rating, and σ is = standard deviation,
⇒ z = (14.48 - 14.48) / 4.38,
⇒ z = 0,
Next, we take the "z-score" that corresponds to the top 10% of the standard normal distribution. We know that the "z-score" is approximately 1.28,
Substituting the value in formula,
We get,
⇒ z = (x - μ) / σ ⇒ 1.28 = (x - 14.48)/4.38,
⇒ x - 14.48 = 1.28 × 4.38,
⇒ x = 20.97 ≈ 20 marks.
Therefore, the rating that marks the cutoff for the top 10% of participant ratings for this joke is approximately 20 on the scale of 1 to 21.
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Suppose that 10 years ago you bought a home for $110,000, paying 10% as a down payment, and financing the rest at 9% interest for 30 years.
a. Let's consider your existing mortgage:
i. How much money did you pay as your down payment?
ii. How much money was your mortgage (loan) for?
iii. What is your current monthly payment?
iv. How much total interest will you pay over the life of the loan?
b. This year, you check your loan balance. Only part of your payments has been going to pay down the loan; the rest has been going towards interest. You see that you still have $88,536 left to pay on your loan. Your house is now valued at $150,000.
i. How much of the loan have you paid off? (i.e., how much have you reduced the loan balance by? Keep in mind that interest is charged each month - it's not part of the loan balance.)
ii. How much money have you paid to the loan company so far?
iii. How much interest have you paid so far?
iv. How much equity do you have in your home (equity is value minus remaining debt)
c. Since interest rates have dropped, you consider refinancing your mortgage at a lower 6% rate.
i. If you took out a new 30-year mortgage at 6% for your remaining loan balance, what would your new monthly payments be?
ii. How much interest will you pay over the life of the new loan?
d. Notice that if you refinance, you are going to be making payments on your home for another 30 years. In addition to the 10 years you've already been paying, that's a total of 40 years.
i. How much will you save each month because of the lower monthly payment?
ii. How much total interest will you be paying
Answers
a.
i. 10% of $110,000 = $11,000 down payment
ii. Loan (mortgage) = $110,000 - $11,000 = $99,000
iii. Monthly payment = P * r * (1 + r)n / ((1 + r)n - 1), where P represents the principal amount, r represents the monthly interest rate, and n represents the number of monthly installments.
First, we must determine the quantity of monthly payments. Because it is a 30-year loan, the monthly installments are 30 * 12 = 360.
The monthly interest rate of 9% divided by 12 equals 0.0075.
When we plug in the values, we get:
$793.07 monthly payment
iv. Loan total interest paid = (monthly payment * number of instalments) - principal amount = ($793.07 * 360) - $99,000 = $184,465.20
How much money have you paid to the loan company so far?b
i. Reduction in loan balance = $99,000 - $88,536 = $10,464
ii. To date, the total amount paid to the loan firm is (monthly payment * a number of payments) = ($793.07 * 120) = $95,168.40.
We multiplied by 120 (rather than 10*12) because we're computing total payments made so far, not the total payments made over the life of the loan.
iii. Total interest paid thus far = total money given to the loan business minus principal amount = $95,168.40 - $99,000 = -$3,831.60
The negative figure shows that the interest paid thus far has been less than the principal amount. This is because, in the early phases of the loan, the majority of the monthly payment goes towards interest, with only a little portion going toward principal reduction.
iv. Home equity = value of the home - remaining debt
= $150,000 - $88,536
= $61,464
c.
i. New monthly payment = P * r * (1 + r)n / ((1 + r)n - 1), where P represents the remaining loan value, r represents the new monthly interest rate (6% / 12 = 0.005), and n represents the number of monthly payments remaining (360 - 120 = 240).
When we plug in the values, we get:
$63.63 is the new monthly payment.
Total interest paid on the new loan during the life of the loan = (new monthly payment * number of payments) - remaining loan balance = ($63.63 * 240) - $88,536 = $32,028.40
d.
Monthly savings = old monthly payment minus new monthly payment = $793.07 - $63.63= $729.44
Total interest paid on the new loan during the life of the loan = (new monthly payment * number of payments) - remaining loan balance = ($63.63 * 360) - $88,536 = $37,847.80
Because the new loan is being paid off over a longer period of time, the total interest paid during the life of the new loan is more than the remaining interest on the previous loan.
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Parking lots A and B charge a set fee. A charges 36.50 for 5 hours. B charges 38.00 for 5 hours. Lot A charges 49.50 for 7 hours and lot B charges 50.00 for 7 hours. How many hours are needed to park where it would cost the same?
Answers
Answer:
if you park for 12 hours in both parking lots A and B, it will cost the same amount, which is $64.50.
Step-by-step explanation:
Let's assume that the cost of parking for x hours in both lots is the same, then we can set up the following equations:
For 5 hours of parking:
36.50 = 38.00
This is not true, so we know that the cost for parking in both lots will not be the same for 5 hours.
For 7 hours of parking:
49.50 = 50.00
This is also not true, so we know that the cost for parking in both lots will not be the same for 7 hours either.
Let's try setting up the equation for x hours of parking:
Fee for parking in Lot A = Fee for parking in Lot B
36.50 + (x-5)*7 = 38.00 + (x-5)*7
Simplifying the equation, we get:
36.50 + 7x - 35 = 38.00 + 7x - 35
Solving for x, we get:
x = 12
Therefore, if you park for 12 hours in both parking lots A and B, it will cost the same amount, which is $64.50.
3. Set A has a standard deviation of 6. Set B has a standard deviation of 9. Based on
this information, which of the following statements must be true?
A) The range for set A and set B is the same.
B) Set A is larger than set B.
3
C) The mean of set B is 1.5 greater than the mean of set A.
D) The data in set A varies less than the data in set B.
Answers
The statement that is true about standard deviation is
Option D is the correct answer.
We have,
Standard deviation is a measure of how spread out the data is in a set.
A smaller standard deviation indicates that the data points are closer to the mean, while a larger standard deviation indicates that the data points are more spread out from the mean.
Since the standard deviation of set A is 6 and the standard deviation of set B is 9, we can conclude that the data in set A varies less than the data in set B.
So,
The data in set A varies less than the data in set B.
Thus,
The statement that is true about standard deviation is
The data in set A varies less than the data in set B.
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The statement that is true about standard deviation is
Option D is the correct answer.
We have,
Standard deviation is a measure of how spread out the data is in a set.
A smaller standard deviation indicates that the data points are closer to the mean, while a larger standard deviation indicates that the data points are more spread out from the mean.
Since the standard deviation of set A is 6 and the standard deviation of set B is 9, we can conclude that the data in set A varies less than the data in set B.
So,
The data in set A varies less than the data in set B.
Thus,
The statement that is true about standard deviation is
The data in set A varies less than the data in set B.
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Given this equation what is the value of x at the indicated point
Answers
Answer:
y = 4
Step-by-step explanation:
[tex]y = -3*-2 - 2\\y = 4[/tex]