The given statement "If ∑an is a convergent series, then S = limsn, where sn is the nth partial sum. " is true. This is because the sum of the series is defined as the limit of the sequence of partial sums.
Given that ∑an is a convergent series, sn is the nth partial sum, S=limsn
To prove limn→∞ an = 0
Since ∑an is convergent, we know that the sequence {an} must be a null sequence, i.e., it converges to 0. This means that for any ε>0, there exists an N such that |an|<ε for all n≥N.
Now, let's consider the partial sums sn. We know that S=limsn, which means that for any ε>0, there exists an N such that |sn−S|<ε for all n≥N.
Using the triangle inequality, we can write:
|an|=|sn−sn−1|≤|sn−S|+|sn−1−S|<2ε
Therefore, we have shown that limn→∞ |an| = 0, which implies limn→∞ an = 0, as required.
Hence, the proof is complete.
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A 20ft ladder is set up that it reaches up 16ft if Christian pulls it 2 feet farther from its base how far up the side of the house is the ladder
The ladder reaches up 20ft the side of the house.
If a 20ft ladder reaches 16ft up the side, what would be the new distance of the ladder's base from the house if it is moved 2ft farther from its initial position?The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
This theorem can be used to solve problems involving right triangles, such as finding the length of the sides or the height of an object.
In this problem, we are given the length of the ladder and the height up the side of the house that it reaches.
We can use the Pythagorean theorem to find the distance from the base of the ladder to the side of the house.
We can then use this distance and the height up the side of the house that the ladder reaches to find the length of the ladder using the Pythagorean theorem again.
Let's call the distance from the base of the ladder to the side of the house "x". We can then use the Pythagorean theorem to find the height that the ladder reaches up the side of the house.
According to the Pythagorean theorem, the length of the ladder (which is the hypotenuse of the right triangle formed by the ladder, the ground, and the side of the house) is equal to the square root of the sum of the squares of the other two sides.
So, if we let "h" be the height up the side of the house that the ladder reaches, we have:
ladder length = √(x^2 + h^2)
We know that the ladder is 20ft long and reaches up 16ft, so we can set up the equation:
20 = √(x^2 + 16^2)
Squaring both sides of the equation, we get:
400 = x^2 + 256
Subtracting 256 from both sides, we get:
144 = x^2
Taking the square root of both sides, we get:
x = 12
So the ladder is leaning against the house 12ft away from the base, and we can use the Pythagorean theorem to find the height up the side of the house that the ladder reaches:
ladder length = √(12^2 + 16^2) = √(144 + 256) = √400 = 20
Therefore, the ladder reaches up 20ft the side of the house.
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Question 3 Next ſsin"" e cos"" Evaluate the indefinite integral xdu
But there seems to be some missing information in your question. Please provide more context or details so that I can assist you accurately.
Hi! I'd be happy to help you evaluate the indefinite integral. Based on the provided terms and information, it seems like you want to evaluate the following integral:
∫x * sin(e * cos(x)) dx
To solve this integral, we can use integration by parts, which is defined as:
∫u dv = u * v - ∫v du
Let's choose u = x and dv = sin(e * cos(x)) dx. Then, we need to find du and v:
du = dx
v = ∫sin(e * cos(x)) dx
Unfortunately, the integral for v does not have a simple closed-form expression. However, you can use numerical methods or software (like Wolfram Alpha) to approximate it. Once you have an approximation for v, you can plug it back into the integration by parts formula to obtain an approximation of the original integral:
∫x * sin(e * cos(x)) dx ≈ x * v - ∫v dx
Keep in mind that this is an indefinite integral, so don't forget to add the constant of integration, C, to your final answer.
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15 Points! 15 Points 15 Points!
A student is graduating from college in 12 months but will need a loan in the amount of $10,720 for the last two semesters. The student may receive either an unsubsidized Stafford Loan or a PLUS Loan. The terms of each loan are:
Unsubsidized Stafford Loan: annual interest rate of 5. 95%, compounded monthly, and a payment grace period of six months from time of graduation
PLUS loan: annual interest rate of 6. 55%, compounded monthly, with a balance of $11,443. 63 at graduation
Which loan will have a lower balance, and by how much, at the time of repayment?
The Stafford loan will have a lower balance by $485. 06 at the time of repayment.
The PLUS loan will have a lower balance by $485. 06 at the time of repayment.
The Stafford loan will have a lower balance by $274. 54 at the time of repayment.
The PLUS loan will have a lower balance by $274. 54 at the time of repayment
The Stafford loan will have a lower balance by $272.54 at the time of repayment.
For the Stafford Loan, the principal is $10,720 and the interest rate is 5.95% compounded monthly, so the monthly interest rate is 0.0595/12 = 0.00495833.
After 12 months, the balance of the loan will be:
$10,720(1 + 0.00495833)¹² = $11,204.60
After the time of six-month grace period, the balance will be:
$11,204.60(1 + 0.00495833)⁶ = $11,689.66
For the PLUS Loan, the principal is $11,443.63 and the interest rate is 6.55% compounded monthly, so the monthly interest rate is 0.0655/12 = 0.00545833.
After 12 months, the balance of the loan will be:
$11,443.63(1 + 0.00545833)¹² = $11,962.20
Therefore, the Stafford Loan will have a lower balance at the time of repayment by:
$11,962.20 - $11,689.66 = $272.54
Hence, the Stafford loan will have a lower balance by $272.54 at the time of repayment.
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Question 1 (Essay Worth 30 points) 2. (10.07 HC) Consider the Maclaurin series g(x)=sin x = x - 3! + x х" х9 7! 9! + x2n+1 ... + Σ (-1). 2n+1 5! n=0 Part A: Find the coefficient of the 4th degree term in the Taylor polynomial for f(x) = sin(4x) centered at x = (10 points) Part B: Use a 4th degree Taylor polynomial for sin(x) centered at x = to estimate g(0.8) out to five decimal places. Explain why your answer is so close to 1. (10 points) x2n+1 263 Part C: The series { (-1)" has a partial sum S. when x = 1. What is an interval, |S - S5l = R5| for which the actual sum exists? 2n +1 315 Provide an exact answer and justify your conclusion. (10 points) n=0
Part A: The coefficient of the 4th degree term in the Taylor polynomial for f(x) = sin(4x) centered at x = 0 is -1/3! = -1/6.
Part B: Using a 4th degree Taylor polynomial for sin(x) centered at x = 0, we can write g(x) = sin(0.8) ≈ P4(0.8), where P4(0.8) is the 4th degree Taylor polynomial for sin(x) evaluated at x = 0.8.
Evaluating P4(0.8) using the formula for the Taylor series coefficients of sin(x), we get P4(0.8) = 0.8 - 0.008 + 0.00004 - 0.0000014 ≈ 0.78333. This estimate is very close to 1 because sin(0.8) is close to 1, and the Taylor series for sin(x) converges very rapidly for values of x close to 0.
Part C: The series { (-1)n / (2n + 1) } has a partial sum S when x = 1. To find an interval |S - S5| = R5| for which the actual sum exists, we can use the alternating series test. The alternating series test states that if the terms of a series alternate in sign, decrease in absolute value, and approach zero, then the series converges.
Since the terms of the series { (-1)n / (2n + 1) } alternate in sign and decrease in absolute value, we know that the series converges. To find an interval |S - S5| = R5|, we can use the remainder formula for alternating series, which states that |Rn| ≤ a_n+1, where a_n+1 is the first neglected term in the series.
Since the terms of the series decrease in absolute value, we know that a_n+1 ≤ |a_n|. Therefore, we have |R5| ≤ |a6| = 1/7!, which means that the actual sum of the series exists in the interval S - 1/7! ≤ S5 ≤ S + 1/7!. Therefore, an interval for which the actual sum exists is [S - 1/7!, S + 1/7!].
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Roger logs the number of miles he runs each week. The mean number of miles Roger ran in October was 30. 2 miles and the mean number of miles Roger ran in November was 25. 6. The mean absolute deviation for both months is 2. What is the difference between the means expressed as a multiple of the mean absolute deviation?
The difference between the means expressed as a multiple of the mean absolute deviation is 2.3.
How to find the difference between the means expressed as a multiple of the mean absolute deviation?To find the difference between the means expressed as a multiple of the mean absolute deviation, we need to calculate the absolute difference between the two means and divide it by the mean absolute deviation.
The absolute difference between the means is:
|30.2 - 25.6| = 4.6
To express this difference as a multiple of the mean absolute deviation, we divide it by the mean absolute deviation:
4.6 / 2 = 2.3
Therefore, the difference between the means expressed as a multiple of the mean absolute deviation is 2.3.
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From the information given, find the quadrant in which the terminal point determined by t lies. input i, ii, iii,
or iv.
(a) sin(t) < 0 and cos(t) < 0, quadrant
(b) sin(t) > 0 and cos(t) < 0, quadrant
(c) sin(t) > 0 and cos(t) > 0, quadrant
(d) sin(t) < 0 and cos(t) > 0, quadrant
;
Answer:
Step-by-step explanation:
In option (a), sin(t) < 0 and cos(t) < 0, In trigonometry, the terminal point of an angle t is the point on the unit circle where the angle intersects with the circle.
The position of the terminal point determines the quadrant in which the angle lies.
To determine the quadrant, we need to look at the signs of the sine and cosine functions. In quadrant I, both sine and cosine are positive. In quadrant II, sine is positive and cosine is negative. In quadrant III, both sine and cosine are negative. In quadrant IV, sine is negative and cosine is positive.
In option (a), sin(t) < 0 and cos(t) < 0, both the sine and cosine functions are negative. This means that the terminal point lies in quadrant III.
In option (b), sin(t) > 0 and cos(t) < 0, the sine function is positive and the cosine function is negative. This means that the terminal point lies in quadrant II.
In option (c), sin(t) > 0 and cos(t) > 0, both the sine and cosine functions are positive. This means that the terminal point lies in quadrant I.
In option (d), sin(t) < 0 and cos(t) > 0, the sine function is negative and the cosine function is positive. This means that the terminal point lies in quadrant IV.
In summary, the signs of the sine and cosine functions can be used to determine the quadrant in which the terminal point lies.
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Will any ramp with one angle of 4. 8 degrees have a slope ratio of 1 : 12?
Yes, any ramp with an angle of 4.8 degrees will have a slope ratio of 1:12.
The slope ratio is the ratio of the vertical rise to the horizontal run of the ramp, and it is equivalent to the tangent of the angle of inclination of the ramp.
The tangent of 4.8 degrees is approximately 0.0084, which means that for every 1 unit of vertical rise, there is 0.0084 units of horizontal run. To convert this to a ratio, we can multiply both sides by 100 to get:
1 unit of rise : 100 x 0.0084 = 0.84 units of run
Simplifying this ratio by dividing both sides by 0.84, we get:
1 unit of rise : 1.19 units of run
which is equivalent to a slope ratio of 1:12 (since 12 = 1/0.084). Therefore, any ramp with an angle of 4.8 degrees will have a slope ratio of 1:12.
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What is your net pay after FICA has been taken out if you make $47,000?
Remember that FICA is 7.65%
Answer:
3595.5
Step-by-step explanation:
b. use the overhead rate in (a) to determine the amount of total and per-unit overhead allocated to each of the three products, rounded to the nearest dollar.
The amount of total and per-unit overhead allocated to each of the three products using overhead rate is equal to,
Total Per Unit Factory Overhead , Cost Factory Overhead Cost
Flutes $530 $1,060,000
Clarinets $795 $1,192,500
Oboes $397.5 $695,625
Total $1,722.5 $2,948,125
Budgeted factory overhead cost = $2,948,125
The single plantwide overhead rate
= Dividing the budgeted factory overhead cost by the total budgeted direct labor hours.
For this,
Flutes= 2,000×2
= 4,000 hours
Clarinets= 1,500×3
= 4,500 hours
Oboes= 1,750×1.5
= 2,625 hours
Total direct labor hours = 11,125
Substitute the value we have,
⇒ Single plantwide overhead rate = $2,948,125 / (2,000 x 2.0 + 1,500 x 3.0 + 1,750 x 1.5)
= $2,948,125 / 11,125
= $265 per direct labor hour
To allocate overhead to each product
=Multiply the overhead rate by the budgeted direct labor hours per unit for each product.
Substitute the value we have,
Flutes,
$265 x 2.0 = $530 total overhead cost, $265 per unit
Clarinets,
$265 x 3.0 = $795 total overhead cost, $265 per unit
Oboes,
$265 x 1.5 = $397.5 total overhead cost, $265 per unit
And
Total factory overhead cost allocated = Estimated manufacturing overhead rate× Actual amount of allocation base
For,
Flutes
= 4,000× 265
= $1,060,000
Clarinets
= 4,500×265
= $1,192,500
Oboes
= 2,625×265
= $695,625
This implies,
The total and per-unit overhead allocated to each product, rounded to the nearest dollar is,
Total Per Unit Factory Overhead , Cost Factory Overhead Cost
Flutes $530 $1,060,000
Clarinets $795 $1,192,500
Oboes $397.5 $695,625
Total $1,722.5 $2,948,125
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The above question is incomplete, the complete question is:
Bach Instruments Inc. makes three musical instruments: flutes, clarinets, and oboes. The budgeted factory overhead cost is $2,948,125. Overhead is allocated to the three products on the basis of direct labor hours. The products have the following budgeted production volume and direct labor hours per unit:
Budgeted Production Volume Direct Labor Hours Per Unit
Flutes 2,000 units 2.0
Clarinets 1,500 3.0
Oboes 1,750 1.5
a. Determine the single plantwide overhead rate.
$ per direct labor hour
b. Use the overhead rate in (a) to determine the amount of total and per-unit overhead allocated to each of the three products, rounded to the nearest dollar.
Total Per Unit
Factory Overhead Cost Factory Overhead Cost
Flutes $ $
Clarinets
Oboes
Total $
a restaurant meal cost $90 and there was a 6% meals tax on the price of the meal hey a)how much was the mail tax b) what was the total price of the meal with a meal tax
A restaurant meal cost $90 and there was a 6% meals tax on the price of the meal. The meal tax was $5.40 and the total price of the meal with the meal tax was $95.40.
Find the meal tax, you need to calculate 6% of the $90 meal cost.
Convert the percentage to a decimal by dividing by 100, so 6% = 0.06.
Multiply the meal cost by the decimal. In this case, $90 × 0.06 = $5.40.
So, the meal tax was $5.40.
Find the total price of the meal with the meal tax, you need to add the meal tax to the original cost of the meal.
Add the meal tax to the original cost, so $90 + $5.40 = $95.40.
The total price of the meal with the meal tax was $95.40.
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David has 3
1
2
cups of blueberries. He uses
1
4
of a cup of blueberries to make a breakfast smoothie. He uses
1
2
of the remaining blueberries to make blueberry pancakes. How many cups of blueberries does he use for the pancakes?
Write your answer as a whole number, fraction, or mixed number. Simplify any fractions
The fraction, David used 13/8 cups of blueberries for the pancakes.
Let's solve it step-by-step using the given information.
1. David has 3 1/2 cups of blueberries.
2. He uses 1/4 cup for a breakfast smoothie.
3. He uses 1/2 of the remaining blueberries for pancakes.
Step 1: Calculate the remaining blueberries after making the smoothie.
3 1/2 - 1/4 = (7/2) - (1/4)
To subtract the fractions, they need a common denominator, which in this case is 4.
(7/2) * (2/2) - (1/4) = (14/4) - (1/4) = 13/4 cups
Step 2: Calculate the amount of blueberries used for the pancakes.
David uses 1/2 of the remaining blueberries for the pancakes, so:
(13/4) * (1/2) = 13/8 cups
David uses 13/8 cups of blueberries for the pancakes.
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f(x)=x^2+6x+8
Rewrite the function into vertex form
y=(x+3)^2-1
have a good day :)
Answer: F(x) = (x + 3)^2 - 1
Step-by-step explanation:
Max has eight circular chips that are all the same size and shape in a bag.
(3 chips are square, and 5 are stars)
Max reaches into the bag and removes one circular chip. What is the theoretical probability that the circular chip has a star on it? Write your answer as a fraction, decimal, and percent
The probability of drawing a star-shaped chip is 5/8.
The theoretical probability of drawing a star-shaped circular chip from the bag is 5/8 or 0.625 or 62.5%. Out of the total of eight circular chips, five are stars, and three are squares.
Therefore, the probability of drawing a star-shaped chip is the ratio of the number of star-shaped chips to the total number of chips in the bag, which is 5/8.
To understand this conceptually, we can think of probability as a fraction where the numerator is the number of favorable outcomes (in this case, drawing a star-shaped chip) and the denominator is the total number of possible outcomes (all the circular chips in the bag).
Thus, the theoretical probability of drawing a star-shaped chip is 5/8 because there are five star-shaped chips out of the total eight circular chips in the bag.
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I seriously need help with this please anyone.
1. Complete the Pythagorean triple. (24,143, ___)
2. Given the Pythagorean triple (5,12,13) find x and y
3. Given x=10 and y=6 find associated Pythagorean triple
4. Is the following a possible Pythagorean triple? (17,23,35)
Answer:
no it is not possible
Step-by-step explanation:
a)cos 165 in terms of sine and cosine of acute angle
Cos 165 in terms of sine and cosine of acute angle would give cos(165) = -(1 + √3) / (2√2).
How to find the cosine ?To find the cosine of 165 degrees in terms of sine and cosine of an acute angle, we can use the cosine angle addition formula:
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
Since 120 degrees is in the second quadrant, the cosine is negative, and the sine is positive:
cos(120) = -cos(60) and sin(120) = sin(60)
cos(165) = -cos(60)cos(45) - sin(60)sin(45)
Now we can plug in the values of the trigonometric functions:
cos(165) = - (1/2) x (1/√2) - (√3/2) x (1/√2)
cos(165) = -(1 + √3) / (2√2)
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Nolan ordered a set of beads. He received 86 beads in all. 43 of the beads were orange. What percentage of the beads were orange?
50% of the beads Nolan received were orange.
To find the percentage of beads that were orange, we need to divide the number of orange beads by the total number of beads and then multiply by 100.
Percentage of orange beads = (Number of orange beads / Total number of beads) * 100
In this case, Nolan received a total of 86 beads, and 43 of them were orange.
Percentage of orange beads = (43 / 86) * 100
Calculating this expression:
Percentage of orange beads = 0.5 * 100
Percentage of orange beads = 50%
Therefore, Out of total number of beads Nolan received, 50% were orange.
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Help with problem in photo! Find the perimeter!
The perimeter of the shape is 53.7 units
What is perimeter ?Perimeter is a math concept that measures the total length around the outside of a shape.
A theorem of circle geometry states that the tangent from a point on a circle are equal.
Therefore the base sides is calculated as
9.9 + 3.2
= 13.1
since the perimeter is the addition of all the sides then;
P = 13.1 + 21.9 + 18.7
P = 53.7
therefore the perimeter of the triangle is 53.7
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Three times a year a camera shop has a sale on packages of batteries. In the second month of the year the packages are 3 for $4.49. In April they are 5 for $7.39 and in the last month of the year, they are 4 for $5.88. List the months in order from the smallest price per package to the largest price per package.
Answer:
The order from smallest to largest price per package is: December, April, February.
Step-by-step explanation:
For the sale in February:
- Price per package = $4.49 ÷ 3 = $1.50 per package
For the sale in April:
- Price per package = $7.39 ÷ 5 = $1.48 per package
For the sale in December:
- Price per package = $5.88 ÷ 4 = $1.47 per package
A point is dilated by a scale factor of 1/3 centered about the origin resulting in the new coordinates (-6,3). what are the coordinates of the point prior to the dilation
The coordinates of the point prior to the dilation are (-2,-1) when the Scale factor is 1/3 and the new coordinates are (-6,3).
To find the coordinates of the point prior to the dilation, we need to use the formula for dilation:
(x’, y’) = (k x, ky)
where
(x’, y’) = the new coordinates
(x, y) = original coordinates
k = scale factor
Given data:
Scale factor = 1/3
New coordinates = (-6, 3)
By substuting the values in the equation we get:
(-6, 3) = (k x, ky)
Solving for x and y:
k x = -6
ky = 3
Dividing the ky equation by the k x equation we get:
y/x = 3/-6
y/x = -1/2
From the above equation, we can assume that x = 2 and y = -1.
Therefore, the coordinates of the point prior to the dilation are (-2,-1).
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HELP! BRAINLIST! THIS IS DUE IT IS END OF UNIT ASSESSMENT
The surface area of the cone is approximately 793.10 cm².
What is cone?
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex.
What is area of cone?
The surface area of a cone is given by the formula:
Surface Area = πr² + πrℓ
where r is the radius of the circular base and ℓ is the slant height of the cone.
According to given information:To find the surface area of a cone, we need to find the area of its base and the area of its lateral surface and then add them together.
The formula for the lateral surface area of a cone is given by:
L = πrℓ
where r is the radius of the base and ℓ is the slant height of the cone.
The formula for the area of the base of a cone is given by:
B = πr²
where r is the radius of the base.
Given the slant height ℓ = 19 and the radius r = 9, we can find the lateral surface area of the cone as follows:
L = πrℓ
= π(9)(19)
≈ 538.63 cm²
Next, we can find the area of the base of the cone as follows:
B = πr²
= π(9)²
≈ 254.47 cm²
Therefore, the surface area of the cone is:
A = B + L
≈ 793.10 cm²
So, the surface area of the cone is approximately 793.10 cm².
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A mechanic had412 gallons of motor oil at the start of the day. At the end of the day, only 5 pints remained
The mechanic used 412 - 411.375 = 0.625 gallons of motor oil during the day.
A mechanic had 412 gallons of motor oil at the start of the day and ended up with only 5 pints of oil remaining.To solve this problem, we need to convert both measurements to the same unit.
1 gallon = 8 pints (since there are 8 pints in a gallon)
So the mechanic started with:
412 gallons * 8 pints/gallon = 3,296 pints
And ended with:
5 pints
To find how much motor oil the mechanic used during the day, we can subtract the ending amount from the starting amount:
3,296 pints - 5 pints = 3,291 pints
To convert this back to gallons, we divide by 8:
3,291 pints / 8 pints/gallon = 411.375 gallons (rounded to three decimal places)
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Find the perimeter of the polygon with the vertices G(2, 4), H(2,-3), J(-2,-3), and K(-2, 4).
The perimeter is ___ units.
Check the picture below.
Calcula los siguientes lÃmites página. 115 ejercicio
a) lim n = +[infinity] infinito 6-4n²
----------
2(n)²
b) lim n = +[infinity] infinito 4n²+3n-2
--------------
2n ³ -4n
c) lim n = +[infinity] infinito 2n ³ -4n
---------------
4n
d) lim x = +[infinity] infinito -8x4 +2
------------
2x² +4
a) Para calcular este límite, podemos dividir tanto el numerador como el denominador por n² y luego aplicar la regla de L'Hôpital:
lim n → ∞ [(6 - 4n²)/(2n²)]
= lim n → ∞ [6/(2n²) - (4n²)/(2n²)]
= lim n → ∞ [3/n² - 2]
= -2
Por lo tanto, el límite es -2.
b) Podemos dividir tanto el numerador como el denominador por n³ para simplificar el límite:
lim n → ∞ [(4n² + 3n - 2)/(2n³ - 4n)]
= lim n → ∞ [(4/n + 3/n² - 2/n³)/(2/n² - 4/n²)]
= lim n → ∞ [(4 + 3/n - 2/n²)/(2 - 4/n)]
= lim n → ∞ [(4n + 3 - 2n²)/(2n² - 4)]
= lim n → ∞ [-2n²/(2n² - 4)]
= -1
Por lo tanto, el límite es -1.
c) Podemos dividir tanto el numerador como el denominador por n³ para simplificar el límite:
lim n → ∞ [(2n³ - 4n)/(4n)]
= lim n → ∞ [(2n² - 4)/(4)]
= lim n → ∞ [(n² - 2)/2]
= +∞
Por lo tanto, el límite es +∞.
d) Podemos dividir tanto el numerador como el denominador por x⁴ para simplificar el límite:
lim x → ∞ [-8x⁴ + 2]/[2x² + 4]
= lim x → ∞ [-8 + 2/x⁴]/[2/x² + 4/x⁴]
= -4/1
= -4
Por lo tanto, el límite es -4.
Ryan buys some jumpers to sell on a stall.
He spends £130 buying 40 jumpers.
He sells 80% of the jumpers for £12 each.
He then puts the rest of the jumpers on a Buy one get one half price offer.
He manages to sell half the remaining jumpers using this offer.
How much profit does Ryan make?
Answer: £326
Step-by-step explanation:
Step 1: Calculate the cost per jumper
To find out how much Ryan spent on each jumper, we divide the total cost by the number of jumpers.
[tex]\frac{130}{40} = 3.25[/tex]
This gives us a cost of £3.25 per jumper.
Step 2: Calculate the revenue from selling 80% of the jumpers
Ryan sells 80% of the 40 jumpers, so:
[tex]\text{0.8 x 40 = 32}[/tex]
So he sold 32 Jumpers.
He sells each jumper for £12:
[tex]\text{32 x 12 = 384}[/tex]
So his revenue from selling these jumpers is £384
Step 3: Calculate the revenue from selling the remaining jumpers on the Buy one get one half price offer
Ryan has 8 jumpers left after selling 80% of them. He puts these on a Buy one get one half price offer, which means that for every jumper sold at full price, he sells another one at half price.
This means that he sells 4 jumpers at full price (£12 each) and 4 jumpers at half price (£6 each).
His revenue from selling these jumpers is:
[tex]\text{(4 x 12) + (4 x 6) = 72}[/tex]
Step 4: Calculate the total revenue
Ryan's total revenue is the sum of the revenue from selling 80% of the jumpers and the revenue from selling the remaining jumpers on the Buy one get one half price offer.
This is:
[tex]\text{384 + 72 = 456}[/tex]
So Ryan's total revenue is £456
Step 5: Calculate the total cost
Ryan's total cost is the amount he spent on buying the jumpers, which is £130.
Step 6: Calculate the profit
Ryan's profit is the difference between his total revenue and his total cost:
[tex]\text{456 - 130 = 326}[/tex]
Therefore, Ryan makes a profit of £326.
4/625 x 625/9 cross cancellation
Answer:
Step-by-step explanation:
4/625 x 625/9 = 4 x 1 / 5 x 5 x 5 x 1 = 4/625. The cross cancellation did not change the result.
Mr linden drives from his home to his office every day. if he drives at an average speed of 70 km/h for 45 min, what is the distance of the journey from his home to his office?
If he drives at an average speed of 70 km/h for 45 min, the distance of the journey from Mr. Linden's home to his office is 52.5 km.
To find the distance of Mr. Linden's journey from his home to his office, we can use the formula:
Distance = Speed x Time
Since Mr. Linden drives at an average speed of 70 km/h for 45 minutes, we first need to convert the time to hours:
45 minutes = 0.75 hours
Now, we can plug in the values we have into the formula:
Distance = 70 km/h x 0.75 hours
Distance = 52.5 km
Therefore, the distance of the journey from Mr. Linden's home to his office is 52.5 km.
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How would you do a point circle problem like this without arctan?
To do this, we can use the Pythagorean theorem and trigonometric ratios instead.
1. Determine the coordinates of the given point, let's call it P(x, y), and the center of the circle, let's call it O(h, k). Also, note the radius, r.
2. Calculate the distance between point P and the center O using the Pythagorean theorem: d^2 = (x-h)^2 + (y-k)^2, where d is the distance.
3. Set d equal to the radius of the circle: r^2 = (x-h)^2 + (y-k)^2.
4. Now, let's find the angle θ between the x-axis and the line OP without using arctan. To do this, we'll use the sine and cosine ratios:
sin(θ) = (y-k) / r and cos(θ) = (x-h) / r
5. To eliminate the need for arctan, we can use the Pythagorean identity sin^2(θ) + cos^2(θ) = 1. Substitute the sine and cosine ratios we found earlier:
((y-k) / r)^2 + ((x-h) / r)^2 = 1
6. Simplify the equation by multiplying both sides by r^2:
(y-k)^2 + (x-h)^2 = r^2
You'll notice that this equation is the same as the one we found in step 3, confirming that the point P lies on the circle. You've now solved the point circle problem without using arctan, by employing the Pythagorean theorem and trigonometric ratios instead.
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The value of the limit lim x->2 | (x-2)(x+2)/(x^2-4) is
The value of the limit lim x->2 | (x-2)(x+2)/(x^2-4) is undefined. This is because as x approaches 2, the denominator (x^2-4) approaches 0, which means that the fraction as a whole is undefined. Therefore, there is no value that the limit can approach.
The value of the limit lim x->2 | (x-2)(x+2)/(x^2-4) is:
Step 1: Recognize that the given expression can be simplified. Notice that the denominator, x^2 - 4, is a difference of squares, so it can be factored as (x-2)(x+2).
Step 2: Simplify the expression by canceling the common factors in the numerator and the denominator: (x-2)(x+2) / (x-2)(x+2) simplifies to 1, because the factors (x-2)(x+2) cancel each other out.
Step 3: Now that the expression is simplified, substitute x = 2 to find the value of the limit: lim x->2 | 1 = 1.
Your answer: The value of the limit lim x->2 | (x-2)(x+2)/(x^2-4) is 1.
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Find the area of the quadrilateral with the given coordinates A(-2, 4),
B(2, 1), C(-1, -3), D(-5, 0)
The quadrilateral formed by the vertices A(-2, 4), B(2, 1), C(-1, -3), and D(-5, 0) has an area of 21/2 square units.
What is the area of the quadrilateral with vertices A(-2, 4), B(2, 1), C(-1, -3), and D(-5, 0)?To find the area of the quadrilateral with the given coordinates A(-2, 4), B(2, 1), C(-1, -3), D(-5, 0), we can use the formula for the area of a quadrilateral in the coordinate plane:
Area = |(1/2)(x1y2 + x2y3 + x3y4 + x4y1 - x2y1 - x3y2 - x4y3 - x1y4)|
where (x1, y1), (x2, y2), (x3, y3), and (x4, y4) are the coordinates of the vertices of the quadrilateral.
Substituting the given coordinates, we get:
Area = |(1/2)(-2×1 + 2×(-3) + (-1)×0 + (-5)×4 - 2×4 - (-1)×1 - (-5)×(-3) - (-2)×0)|Area = |(-1 - 6 + 0 - (-20) - 8 + 1 + 15)|/2Area = 21/2Therefore, the area of the quadrilateral with the given coordinates is 21/2 square units.
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Determine the maximum rate of change of f at the given point P and the direction in which it occurs (a) f(x,y) = sin(xy), P(1,0) (b) f(x,y,z) = P(8,1.3)
The maximum rate of change occurs in the direction of this unit vector.
(a) To find the maximum rate of change of f at point P(1,0), we need to find the gradient of f at that point and then find its magnitude. The direction of maximum increase is given by the unit vector in the direction of the gradient.
The gradient of f is:
∇f(x,y) = <y cos(xy), x cos(xy)>
At point P(1,0), we have:
∇f(1,0) = <0, cos(0)> = <0, 1>
The magnitude of the gradient is:
||∇f(1,0)|| = sqrt([tex]0^2[/tex] +[tex]1^2[/tex]) = 1
Therefore, the maximum rate of change of f at point P is 1, and it occurs in the direction of the unit vector in the direction of the gradient:
u = <0, 1>/1 = <0, 1>
So the maximum rate of change occurs in the y-direction.
(b) To find the maximum rate of change of f at point P(8,1.3), we need to find the gradient of f at that point and then find its magnitude. The direction of maximum increase is given by the unit vector in the direction of the gradient.
The gradient of f is:
∇f(x,y,z) = <2x, 2y, 2z>
At point P(8,1.3), we have:
∇f(8,1.3) = <16, 2.6, 2(1.3)> = <16, 2.6, 2.6>
The magnitude of the gradient is:
||∇f(8,1.3)|| = sqrt[tex](16^2 + 2.6^2 + 2.6^2)[/tex]= sqrt(275.56) ≈ 16.6
Therefore, the maximum rate of change of f at point P is approximately 16.6, and it occurs in the direction of the unit vector in the direction of the gradient:
u = <16, 2.6, 2.6>/16.6 ≈ <0.963, 0.157, 0.157>
So the maximum rate of change occurs in the direction of this unit vector.
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