Answers
Answer: [tex]\\ \lim\limits_{k \to \infty} (1+\frac{4}{k})^k =e^4.[/tex]
Step-by-step explanation:
[tex]\displaystyle\\ \lim_{k \to \infty} (1+\frac{4}{k})^k \\x=\frac{x}{4} *4\\So,\ \lim_{k \to \infty} (1+\frac{4}{k})^\frac{k}{4}*4 \\ \lim_{k \to \infty} ((1+\frac{4}{k})^\frac{k}{4} )^4.\\Use\ the\ second\ wonderful\ limit:\\\boxed { \lim_{x \to \infty} (1+\frac{1}{x})^x=e },\\\\So,\\ \lim_{k \to \infty} (1+\frac{4}{k})^k =e^4.[/tex]
We can also apply l'Hôpital's rule by first rewriting the limit as
[tex]\displaystyle \lim_{k\to\infty} \left(1 + \frac4k\right)^k = \lim_{k\to\infty} \exp\left(\ln \left(1 + \frac4k\right)^k\right) = \exp\left(\lim_{k\to\infty} \frac{\ln\left(1+\frac4k\right)}{\frac1k}\right)[/tex]
Applying the rule gives
[tex]\displaystyle \lim_{k\to\infty} \frac{\ln\left(1+\frac4k\right)}{\frac1k} = \lim_{k\to\infty} \frac{\left(-\frac4{k^2}\right)/\left(1+\frac4k\right)}{-\frac1{k^2}} = 4 \lim_{k\to\infty} \frac1{1 + 4k} = 4[/tex]
so that the overall limit is
[tex]\displaystyle \lim_{k\to\infty} \left(1 + \frac4k\right)^k = \lim_{k\to\infty} \exp(4) = \boxed{e^4}[/tex]
Related Questions
Consider the line 4x + 2y = 8.
What is the slope of a line perpendicular to this line?
Answers
m1 x m2 = - 1
Rearrange 4x + 2y = 8.
4x + 2y = 8
-4x, then ÷ 2
y = - 4x + 8 / 2
y = - 2x + 4
m1 = - 2, find reciprocal of - 2, by placing 1 over the 2
So, - 2 will become - 1/2, but multiplying this gives us a positive 1. Thus, m2 = 1/2
m1 x m2 = - 1
-2 x 1/2 = - 1
That means slope perpendicular to the line
4x + 2y = 8 is y = 1/2x, where a half is the gradient/slope.
Ans: 1/2
Hope this helps!
Solve for x. You must show all your work to receive credit.
Answers
Step-by-step explanation:
since the 2 arc angles cover the full circle, we know their sum must be 360°.
so,
23x - 5 + 37x + 5 = 360
60x = 360
x = 6
and so the angle at V = 5×6 + 17 = 30+17 = 47°
Simplify these expressions
4. 9a+7-2a+13=
5. 21-3b+5b-7=
6.3c squared + 8c squared - 15=
Answers
Answer:
4) 7a+20
5) 14+2b
6) 25c² - 15
Step-by-step explanation:
4) 9a+7-2a+13= 9a-2a+7+13 = 7a+20
5) 21-3b+5b-7= 21-7+5b-3b = 14+2b
6) (3c)² + (8c)² - 15 = 9c² + 16c² - 15 = 25c² - 15
=7a+20
5. Given equation,21-3b+5b-7
=2b+14
6. Given equation,3c square +8c square-15
=11c^2-15
Geometry: fill in the blanks, ASAP! It’s urgent
Answers
Answer:
3. 125
4. 115
5. m<1 = 70, m<2 = 55, m<3 = 55
6. m<2 = 50, m<3 = 50, m<4 = 80, m<6 = 130
Answer each question and tell your reasoning.
a. Is 60% of 400 equal to 87?
b. Is 60% of 200 equal to 87?
c. Is 60% of 120 equal to 87?
2. 60% of x
is equal to 87. Write an equation that expresses the relationship between 60%, x
, and 87. Solve your equation.
Equation:
x=
3. Write an equation to help you find the value of each variable. Solve the equation.
60% of c
is 43.2.
Equation:
c=
38% of e
is 190.
Equation:
e=
Answers
Answer:
Step-by-step explanation:
400*60% can be rewriten as 400×0.6 which is 240, so a is false.
the same steps can be repeated for b and c
b) 200*0.6=120, therefore is false
c) 120*0.6=72, therefore is false
2)
[tex]60 percent * x=87\\0.6x=87\\x=\frac{87}{0.6} \\\\x=145[/tex]
Now we check the answer by using the same steps in problems a,b,c
[tex]145*0.6=87\\87=87[/tex]
3)
[tex]0.6c=43.2\\c=\frac{43.2}{0.6} \\c=72[/tex]
and finally
[tex]0.38e=190\\e=\frac{190}{0.38} \\e=500[/tex]
In a given right triangle, If 0 º < x < 90 º and cos x = √ 3 / 2, then sin x = ?
Answers
Answer:
1/2
Step-by-step explanation:
The "Pythagorean relation" between trig functions can be used to find the sine.
Pythagorean relationThe relation between sine and cosine is the identity ...
sin(x)² +cos(x)² = 1
This can be solved for sin(x) in terms of cos(x):
sin(x) = √(1 -cos(x)²)
ApplicationFor the present case, using the given cosine value, we find ...
sin(x) = √(1 -(√3/2)²) = √(1 -3/4) = √(1/4)
sin(x) = 1/2
__
Additional comment
The sine and cosine of an angle are the y and x coordinates (respectively) of the corresponding point on the unit circle. The right triangle with these legs will satisfy the Pythagorean theorem with ...
sin(x)² + cos(x)² = 1 . . . . . . where 1 is the hypotenuse (radius of unit circle)
A calculator can always be used to verify the result.
Convert 2 ounces into grams. Round your answer to the nearest whole number
Answers
Answer: 57
I'm pretty sure that's what it is
Step-by-step explanation:
Over a period of 3 hours, the outside temperature changed an average of -2.250 Fahrenheit per hour. Which statement correctly describes the change in temperature from the beginning the end of the 3-hour period?
A.
The temperature decreased by 6.75 degrees Fahrenheit.
B.
The temperature increased by 0.75 degrees Fahrenheit.
C.
The temperature increased by 6.75 degrees Fahrenheit.
D.
The temperature decreased by 0.75 degrees Fahrenheit.
Answers
Answer: A
Step-by-step explanation: The question states that the change wasn't a positive one, it was a negative one as it states that the temperature changed an average of -2.250 Fahrenheit per hour. So, we multiply that by 3 to get -6.75. This is why A is correct.
A gardener uses 1/3 of a liter of water to water 2/7 of a garden.
Answers
1 1 / 6 litres of water is needed to watering the entire garden at the given rate.
How to find the amount of water using rate?We need to find how many litres of water is needed to watering the entire garden at this rate.
Let the total garden be x parts then,
Therefore,
2 / 7 x = 1 / 3
cross multiply
2x × 3 = 7
6x = 7
divide both sides by 6
6x / 6 = 7 / 6
x = 1 1 / 6
Therefore, Thus, 1 1 / 6 litres of water is needed to watering the entire garden at the given rate.
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Select the correct answer. daniel has a list of numbers that are not in any order. he wants to find the order of the numbers with the help of a spreadsheet. which statistical function will help him do so? a. mode b. rank c. median d. average
Answers
The correct option is B.
Rank
What is Rank function?The RANK function in a Microsoft Excel spreadsheet enables users to order the various pieces of data by their numerical values. By entering an absolute value for the range of cells the user wants to rank, they can also use the rank function. If an absolute value is not specified, the program's auto-formula may cause the rank of the various cells to alter.
What is Spreadsheet?A spreadsheet is a computer program used for organizing, organizing, analyzing, and storing tabular data. Spreadsheets were created as electronic alternatives to paper accounting worksheets. Data typed into table cells serves as the program's input.
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The ordered pair that is a solution of the equation y = | - 3x + 3 | is:
(r, 24)
(7,0)
(7, - 18)
( 7,18)
Answers
Answer:
The answer would be (7, 18)
Step-by-step explanation:
The answers simple,
in the equation y=|-3x+3|
x and y would be said like this (x,y). just like the answer choices
now in the first option, x is R, so i couldn't understand that. so I'm gonna skip that.
in the second one, (7,0) would no work as
0=|-3(7) + 3|
0=|-21+3|
0=|-18|
and since "|" is absolute value, meaning that instead of the number being negative or positive the number will just be translated into how much distance is from the number and 0, or just make the number positive.
lets continue,
0=18, so (7,0)
the second one, (7,-18)
-18=|-3(7)+3|
-18=|-18|
again, its absolute value so
-18=18, so it still doesn't work
the last one, (7,18)
18=|-3(7)+3|
18=|18|
18=18
so (7,18) would work.
the price of a video game was reduced from 60 to 45. by what percentage was the price of the video game reduced
Answers
The percentage reduction of the price of the video game from 60 to 45 was 25%.
What is the percentage reduction?The percentage reduction refers to the fractional reduction in the price of the video game.
This implies that a percentage is a fraction of a value. Percentages are calculated over 100.
A percentage is, therefore, a number or ratio expressed as a fraction of 100, often denoted using the "%" sign.
Data and Calculations:The old price of the video game = 60
The new price of the video game = 45
Reduction in price = 15 (60 - 45)
Percentage reduction = 25% (15/60 x 100)
Thus, the percentage reduction of the price of the video game from 60 to 45 was 25%.
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Find the values of the unknown angles marked with letters.
a=
b=
c=
Answers
Answer:
a = 97° , b = c = 146°
Step-by-step explanation:
a and 83° are same- side interior angles and sum to 180°
a + 83° = 180° ( subtract 83° from both sides )
a = 97°
34° and b are same- side interior angles and sum to 180°
b + 34° = 180° ( subtract 34° from both sides )
b = 146°
b and c are corresponding angles and are congruent , so
c = b = 146°
Answer:
< a = 97 degrees.
< b = 146 degrees.
< c = 146 degrees.
Step-by-step explanation:
We have 2 lines crossing 2 parallel lines.
Internal angles between the 2 parallel lines add up to 180 degrees, so
34 + <b = 180
<b = 180 - 34 = 146 degrees
Also
<a + 83 = 180 degrees (also internal angles)
---> <a = 97 degrees
< c and < b are corresponding angles so are congruent, therefore
<c = < b = 146 degrees.
Akira and her puppy are both dressing up as superheroes for Halloween this year. Akira needs 1 yard of cloth for her cape. She also needs 1 foot of cloth for her puppy's cape. How many feet of cloth should she buy for their capes?
Answers
The feet of cloth that she should buy for the capes is 4 feet.
How many feet should she buy?
The mathematical operations that would be used to determine the answer are addition and multiplication.
Addition is the process of summing two or more numbers together. Multiplication is the process of determining the product of two or more numbers.
The first step is to convert yard to feet
According to the unit of conversion : 1 yard = 3 feet
1 x 3 = 3 feet
The second step is to add the feet needed for Akira's cape with the foot needed for the puppy's cape
Total feet needed = 3 + 1 = 4 feet
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What is the solution to the system of equations?
Answers
[tex](\frac{3}{2},\frac{5}{2})[/tex]
Step-by-step explanation:The first equation can be rearranged by subtracting y from both sides. This gives you x = 4 - y.
The second equation is x = y - 1
Therefore, as both equations are equal to x, you can say that they are equal to each other, so 4 - y = y - 1.
If you add 1 to each side, you get 5 - y = y
If you add y to each side, you get 5 = 2y
Divide both sides by 2 to get y = 5/2
Now you can substitute 5/2 for y. For example, the second equation becomes x = 5/2 - 1
x is therefore 3/2
This can be written as [tex](\frac{3}{2},\frac{5}{2})[/tex], which is the final answer.
Use part 1 of the fundamental theorem of calculus to find the derivative of the function. g(x) = x t3 t5 dt 0
Answers
Using the fundamental theorem of calculas the derivative of function g(x)=[tex]xt^{3}+t^{5}[/tex] at x=0 is [tex]t^{3}[/tex].
Given a function g(x)=[tex]xt^{3}+t^{5}[/tex].
We are required to find the derivative of the function g(x) at x=0.
Function is relationship between two or more variables expressed in equal to form. The values entered in a function are part of domain and the values which we get from the function after entering of values are part of codomain of function. Differentiation is the sensitivity to change of the function value with respect to a change in its variables.
g(x)=[tex]xt^{3}+t^{5}[/tex]
Differentiating with respect to x.
d g(x)/dx=[tex]t^{3}[/tex]+0 [Differentiation of x is 1 and differentiation of constant is 0]
=[tex]t^{3}[/tex]
Hence using the fundamental theorem of calculas the derivative of function g(x)= [tex]xt^{3}+t^{5}[/tex] at x=0 is [tex]t^{3}[/tex].
The function given in the question is incomplete. The right function will be g(x)=[tex]xt^{3}+t^{5}[/tex].
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.
And that the terms of this series may be arranged without changing the value of the series, determine the sum of the reciprocals of the squares of the odd positive integers
Answers
The terms of this series may be arranged without changing the value of the series. The sum of the reciprocals of the squares of the odd positive integers is [tex]\pi ^{2} /8[/tex].
In mathematics, a sequence is the cumulative sum of a given collection of terms. Usually, those phrases are actual or complicated numbers, but plenty of extra generalities are feasible.
A series is described as an arrangement of numbers in a specific order. then again, a chain is described as the sum of the factors of a sequence.
In mathematics, a series is, more or less speaking, a description of the operation of including infinitely many quantities, one after the alternative, to a given beginning quantity. The look at of series is a primary part of calculus and its generalization, mathematical analysis.
k=1
1/(1)2+1/(2)2+1/(3)2+1/(4)2+1/(5)2+1/(6)2+1/(7)2+.
up to ∞ terms = 2/6
[1/(1)2+1/(3)2+1/(5)2+1/(7)2+]+[1/(2)²+1/(4)²+1/(6)²+
..∞0] = T²/6
→ [1/(1)² + 1/(3)² + 1/(5)2+1/(7)2+......00] + [1/4 (1)² + 1/4(2)²+
1/4(3)²+....0] =²/6
[1/(1)²+1/(3)²+1/(5)2+1/(7)2+.......)] + 1/4[1/(1)² + 1/(2)²+
1/(3)²+....x] = 2/6
⇒ [1/(1)² + 1/(3)² + 1/(5)²+1/(7)²+..] + 1/4 [π²/6] = 2/6
⇒ [1/(1)² + 1/(3)² + 1/(5)²+1/(7)²+] = (1-1/4)/6
⇒ [1/(1)²+1/(3)2+1/(5)2+1/(7)2+..∞ = 3/4 x π²/6
=
↑
[1/(1)2+1/(3)2+1/(5)2+1/(7)²+] = 2/8
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Which function describes this graph?
Answers
The graph in the question is of the function f(x)=[tex]log _{0.2}x[/tex] which is option a.
Given a graph which is downward sloping and we are required to find the function whose the graph is.
Function is like a relationship between two or more variables expressed in equal to form. It may be a linear function,exponential function, logarithmic function or quadratic function.
In the given graph the values of y becomes negative when the curve crosses then x axis. For all x>1 the value of log x is greater than 0. So the value of [tex]log_{2}x,log_{4}x,log_{20}x[/tex] should be greater than 0 for x greater than 1. But in graph it is negative. That's why the graph will be of y=[tex]log_{0.2}x[/tex].
Hence the given graph is of the function f(x)=[tex]log _{0.2}x[/tex] which is option a.
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Which solution finds the value of x in the triangle below?
A right triangle is shown. The hypotenuse has a length of 8. Another side has a length of x. The angle between the hypotenuse and the other side is 60 degrees.
Secant 60 degrees = StartFraction 8 Over x EndFraction. 2 = StartFraction 8 Over x EndFraction. 2 x = 8. x = 4.
Cosecant 60 degrees = StartFraction 8 Over x EndFraction. StartFraction 2 StartRoot 3 EndRoot Over 3 EndFraction = StartFraction 8 Over x EndFraction. x times 2 StartRoot 3 EndRoot = 24. x = StartFraction 24 Over 2 StartRoot 3 EndRoot EndFraction. x = StartFraction 12 Over StartRoot 3 EndRoot EndFraction times StartFraction StartRoot 3 EndRoot Over StartRoot 3 EndRoot EndFraction. x = StartFraction 12 StartRoot 3 EndRoot Over 3 EndFraction. x = 4 StartRoot 3 EndRoot.
Secant 60 degrees = StartFraction 8 Over x EndFraction. One-half = StartFraction 8 Over x Endfraction. x = 16
Cosecant 60 degrees = StartFraction 8 Over x EndFraction. StartFraction StartRoot 3 EndRoot Over 2 EndFraction = StartFraction 8 Over x EndFraction. x times StartRoot 3 EndRoot = 16. x = StartFraction 16 Over StartRoot 3 EndRoot EndFraction. x = StartFraction 16 Over StartRoot 3 EndRoot EndFraction times StartFraction StartRoot 3 EndRoot Over StartRoot 3 EndRoot EndFraction. x = StartFraction 16 StartRoot 3 EndRoot Over 3 EndFraction.
Answers
The value of x in the triangle is 4
How to determine the solutions?The complete question is added as an attachment
From the question, the given parameters are as follows:
Hypotenuse = 8
Adjacent = x
Angle = 60
The cosine of the angle is then calculated as:
cos(angle) = Adjacent/Hypotenuse
Substitute the known values in the above equation
cos(60) = x/8
Multiply both sides by 8
x = 8 * cos(60)
Evaluate the product
x = 4
Hence, the value of x in the triangle is 4
So, the complete parameters are:
Hypotenuse = 8
Adjacent = x
Angle = 60
x = 4
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Answer:
Secant 60 degrees = StartFraction 8 Over x EndFraction. 2 = StartFraction 8 Over x EndFraction. 2 x = 8. x = 4.
Step-by-step explanation:
The answer above is correct.
Out of total 47 observation arranged in ascending order, the 12th and 13th observation are 25 and 30 respectively. What is the value of Quartile 1
Answers
Answer: i believe the answer is 26
Step by step explanation
Total no. of observation = (12+13) = 25
Median = Average of 18/2 observation + (18/2+1)
=1/2 (12th observation + 13 observation)
1/2 × (25 +27) = 1/2 × 52
=26
A train arrived at the station. If you count cars starting from the first one, the dining car is th 7th in the row. If you count cars starting from the last one, then the dining car is the 8th in a row. How many cars are there in this train?
Answers
Answer:
15
Step-by-step explanation:
from the first then it is 0+7=7
and from the last it is 0+8=8
then the number of cars available in the station is 8+7=15
A written contract is not legally binding until _____. a. money is exchanged b. it is signed by all parties involved c. all requirements of the contract have been fulfilled d. it is signed by all parties involved and notarized by an authorized notary official please select the best answer from the choices provided a b c d
Answers
Answer: Answer is B, Mark me brainliest please
Step-by-step explanation:
Simplify the expression. c–6 • 2c7
Answers
The answer is 2c.
Applying the Law of Exponents,
c⁻⁶ x 2c⁷2c⁷⁻⁶2cIn a certain exam of grade ten, 75% students got high score in mathematics, 65%
students got high score in English. If 6% of them did not get high score in both
mathematics and English, then calculate:
i. the percent of students who got high score in both the subjects.
ii. the total number of students who got high score either in mathematics or
in English if 300 students had attended the exam.
Mathematics, grade 10
Answers
(i) The percentage of students who got high scores in both the subjects English and Mathematics is 46%.
(ii) The total number of students who got high scores either in Mathematics or in English if 300 students had attended the exam exists 138.
What is probability?The probability exists in the analysis of the possibilities of happening of an outcome, which exists acquired by the ratio between favorable cases and possible cases.
The number of students who got high scores in Mathematics was 75%.
The number of students who got high scores in English was 65%.
(i) The percentage of students who got high scores in both the subjects
100% - 6% = 94%
(75% + 65%) - 94%
= 140% - 94%
= 46%
Therefore, the percentage of students who got high scores in both the subjects English and Mathematics is 46%.
(ii) The total number of students who got high scores either in Mathematics or in English if 300 students had attended the exam
= 300 [tex]*[/tex] 46%
= 300 [tex]*[/tex] (46 / 100)
= 300 [tex]*[/tex] 0.46
= 138.
Therefore, the total number of students who got high scores either in Mathematics or in English if 300 students had attended the exam exists 138.
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Marika is training for a track race. She start sprinting by 100 yards. She gradually increases her distance, adding 4 yards a day for 21 days. the explicit formula that models this situation is an=100+(n-1)4. how far does seh sprint on day 21
Answers
Answer:
180 yards
Step-by-step explanation:
The number of yards she sprints on day 21 can be found by evaluating the formula with n=21.
Solutiona21 = 100 +(21 -1)4 = 100 +(20)(4) = 100 +80 . . . . . use 21 in place of n
a21 = 180
Marika sprints 180 yards on day 21.
35 POINTS!!! PLEASE HELP !!!!!!!!!!!!!!
( I already know the answer isn't -2, -1 so D is out of the question
A function is shown in the table.
x g(x)
−2 2
−1 0
0 2
1 8
Which of the following is a true statement for this function? (5 points)
Group of answer choices
The function is decreasing from x = 0 to x = 1.
The function is decreasing from x = −1 to x = 0.
The function is increasing from x = 0 to x = 1.
The function is increasing from x = −2 to x = −1.
Answers
Answer:
The function is increasing from x = 0 to x = 1.
Step-by-step explanation:
A function is increasing when the y-value increases as the x-value increases.
A function is decreasing when the y-value decreases as the x-value increases.
From x = -2 to x = -1 the function is decreasing as the y-value decreases as the x-value increases:
x-value -2 to -1 → increasey-value 2 to 0 → decreaseFrom x = -1 to x = 0 the function is increasing as the y-value increases as the x-value increase:
x-value -1 to 0 → increasey-value 0 to 2 → increaseFrom x = 0 to x = 1 the function is increasing as the y-value increases as the x-value increase:
x-value 0 to 1 → increasey-value 2 to 8 → increasehow do i get the method on how to solve this x+5(-2)-1
Answers
Answer:
I forgot the name but I can show you how to do it lol
Step-by-step explanation:
x+5(-2)-1
x-10-1 we multiplied 5 and -2
x-11
x=11
how I did the last one is there is a non-existent 0 with an equal sign like my teacher says so we added 11 to both sides.
What is the interquartile range of the set of data this box-and-whisker plot represents?
Answers
Answer:
3
Step-by-step explanation:
The interquartile range is where the 2nd and 5th dots are, on the opposite ends of the square, from 15-18.
Answer:
3
Step-by-step explanation:
The interquartile range (IQR) is the difference between the upper quartile (Q₃) and the lower quartile (Q₁).
In a box plot:
The left edge of the box represents the lower quartile.The right edge of the box represents the upper quartile.From inspection of the given box plot:
lower quartile (Q₁) = 15upper quartile (Q₃) = 18Therefore:
IQR = Q₃ - Q₁
= 18 - 15
= 3
So the interquartile range of the given box plot is 3.
Trigonometry Question
Answers
AB^2 + BC^2 = AC^2
6^2 + BC^2 = 10^2
36 + BC^2 = 100
BC^2 = 64
BC = 8
If AC = 10, CD and AD = 5
Tan = opp/adj
Tan = 5/8
Triangle BCD is a 45-45-90 triangle
CD = BD = 5
Sec = inverse cos
sec = hyp/adj
Sec = 8/5
PLEASE HELP IM STUCK PLS
Answers
The solution to the equation is the values that make both equations true. More simply put, it is where the two graphs intersect (cross). If we find the point where the two graphs both cross, we see the value is where x is -1 and y is 1, so the point is (-1,1).
We can also check this by plugging the values into the equations.
y = 2x + 3
1 = 2 * -1 + 3
1 = 1
y = -x
1 = -(-1)
1 = 1
Since both equations are true, the solution (-1,1) is correct.
Answer: (-1, 1)
Step-by-step explanation:
Method 1: GraphingSince the graph is already given, we could directly refer to the intersecting point of the system of equations.
According to the graph, [ y = 2x + 3 ] and [ y = -x ] intersect at (-1, 1).
Therefore the solution is [tex]\Large\boxed{(-1,1)}[/tex]
Method 2: SubstitutionGiven equation
1) y = 2x + 3
2) y = -x
Substitute the y value in the 1) equation by the 2)
(-x) = 2x + 3
Add x on both sides
-x + x = 2x + 3 + x
0 = 3x + 3
Subtract 3 on both sides
0 - 3 = 3x + 3 - 3
-3 = 3x
Divide 3 on both sides
-3 / 3 = 3x / 3
x = -1
Substitute the x value into one of the equations to find the y value
y = -x
y = - (-1)
y = 1
Therefore, the solution is [tex]\Large\boxed{(-1,1)}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
