Answer:
first is 18
second is 36
third is -10
fourth is -10
12. What is m∠GEA?
Answer:
90°
Step-by-step explanation:
Circumcenter of a triangle is obtained by drawing perpendicular bisectors of the sides of a triangle. Hence GE is perpendicular to AC.
Therefore, m∠GEA = 90°
17. Convert the following measures of liquid measure. a. 3,450 deciliters to cubic decimeters _______ b. 124.3 hectoliters to deciliters _______ c. 9 liters to cubic centimeters _______ d. 32.5 liters to cubic decimeters _______
Step-by-step explanation:
. 345,000 cm³.
b. 124,300 hl.
c. 9,000 cm³.
d. 32.5 dm³.
Step-by-step explanation:
To solve this problem you must apply the proccedure shown below:
a. 3,450 deciliters to cubic decimeters:
1 deciliter=100 cubic decimeters
(3,450dl)(\frac{100cm^{3}}{1dl})=345,000cm^{3}(3,450dl)(
1dl
100cm
3
)=345,000cm
3
b. 124.3 hectoliters to deciliters:
1 hectoliter=1,000 deciliters
(124,3hl)(\frac{1,000dl}{1hl})=124,300hl(124,3hl)(
1hl
1,000dl
)=124,300hl
c. 9 liters to cubic centimeters:
1 liter=1,000 cubic centimeters
(9L)(\frac{1,000cm^{3}}{1L})=9,000cm^{3}(9L)(
1L
1,000cm
3
)=9,000cm
3
d. 32.5 liters to cubic decimeters:
1 liter=1 cubic decimeter
32.5L=32.5dm^{3}32.5L=32.5dm
3
your answer follow me plzzz
Answer:
The first one is 345,000 cm³.
The second is 124,300 hl.
the Third is 9,000 cm³.
Anddd the fourth one is 32.5 dm³
:)
Starting from an airport, an airplane flies 210 miles southeast and then 210 miles south. How far, in miles, from the airport is the plane? (Round your answer to the nearest mile.)
Answer:
The plane is 388 miles far from the airport.
Step-by-step explanation:
We know that, the angle between southeast and south directions is [tex]135^\circ[/tex].
The plane travels as per the triangle as shown in the attached image.
A is the location of airport.
First it travels for 210 miles southeast from A to B and then 210 miles south from B to C.
[tex]\angle ABC = 135^\circ[/tex]
To find:
Side AC = ?
Solution:
As we can see, the [tex]\triangle ABC[/tex] is an isosceles triangle with sides AB = BC = 210 miles.
So, we can say that the angles opposite to the equal angles in a triangle are also equal. [tex]\angle A = \angle C[/tex]
And sum of all three angles of a triangle is equal to [tex]180^\circ[/tex].
[tex]\angle A+\angle B+\angle C = 180^\circ\\\Rightarrow \angle A+135^\circ+\angle A = 180^\circ\\\Rightarrow \angle A = \dfrac{1}{2} \times 45^\circ\\\Rightarrow \angle A =22.5^\circ[/tex]
Now, we can use Sine Rule:
[tex]\dfrac{a}{sinA} = \dfrac{b}{sinB}[/tex]
a, b are the sides opposite to the angles [tex]\angle A and \angle B[/tex] respectively.
[tex]\dfrac{210}{sin22.5^\circ} = \dfrac{b}{sin135^\circ}\\\Rightarrow \dfrac{210}{sin22.5^\circ} = \dfrac{b}{cos45^\circ}\\\Rightarrow b = 210\times \dfrac{1}{\sqrt2 \times 0.3826}\\\Rightarrow b = 210\times \dfrac{1}{0.54}\\\Rightarrow b \approx 388\ miles[/tex]
So, the answer is:
The plane is 388 miles far from the airport.
g Suppose that three hypothesis tests are carried out, each using significance level 0.05. What is the worst-case probability of a type I error in at least one of these tests?
Answer:
The worst-case probability is 0.05
Step-by-step explanation:
The given significance level ([tex]\alpha[/tex]) = 0.05
since Probability of a type I error is [tex]\alpha[/tex]
∴ P (type I error) = 0.05
0.05 will be the worst-case probability of a type I error in at least one of the tests.
Which is the graph of x – y = 1?
Answer:
Step-by-step explanation:
Hope you can see it.
2. The table below shows the activity on the credit card statement of Miss Pepper Mills for the month of April. She started the month with a balance of $342.57.
Date Activity Location Amount
04/05 Payment Payment $200.00
04/15 Charge Gas $26.37
04/22 Charge Macy's $105.42
04/25 Charge Starbuck's $4.24
a. Find the average daily balance.
b. If her card charges an 18.5% annual interest rate on her average daily balance, calculate Miss Pepper Mill’s finance charge for the month of April.
Answer:
Miss Pepper Mills
Credit Card Statement for the month of April:
a. Average Daily Balance:
Average balance = $618.66/5 = $123.73
b. Calculation of Miss Pepper Mill's Finance Charge for the month of April:
Finance charge
= 18.5% of $20.55 x 1/12
= $0.32
Step-by-step explanation:
a) Data and Calculations:
Activity on the credit card statement for April
Beginning balance = $342.57
Date Activity Location Amount Daily Balance
04/01 Balance $342.57
04/05 Payment Payment $200.00 $142.57
04/15 Charge Gas $26.37 $116.20
04/22 Charge Macy's $105.42 $10.78
04/25 Charge Starbuck's $4.24 $6.54
30 days Total = $618.66
Average balance = Total of the daily balances divided by 30 days
= $618.66/30 = $20.55
The value of the average daily balance and the finance charge for Miss pepper mills are $123.732 and $0.32 respectively.
The average daily balance :
(Sum of balances) ÷ number of purchasesSum of balances = (342.57+ (342.57-200) +(342.57 - 200 - 26.37) +(342.57-200-26.37-105.42) + (342.57-200-26.37-105.42-4.24)) = $618.66
Average daily balance :
(618.66) ÷ 5 = 123.732Finance charge for April :
Monthly rate = 18.5% ÷ 12 = 1.541666%Average balance × daily rate
Average balance = (618.66) ÷ 30 = 20.62220.622 * 1.5416666% = 0.32Therefore, the average daily balance and finance charge are $123.732 and $0.32 respectively
Learn more : 149.9448066146972668163077278862934016
Base: z(x)=cosx Period:180 Maximum:5 Minimum: -4 What are the transformation? Domain and Range? Graph?
Answer:
The transformations needed to obtain the new function are horizontal scaling, vertical scaling and vertical translation. The resultant function is [tex]z'(x) = \frac{1}{2} + \frac{9}{2} \cdot \cos \left(\frac{\pi\cdot x}{90^{\circ}} \right)[/tex].
The domain of the function is all real numbers and its range is between -4 and 5.
The graph is enclosed below as attachment.
Step-by-step explanation:
Let be [tex]z (x) = \cos x[/tex] the base formula, where [tex]x[/tex] is measured in sexagesimal degrees. This expression must be transformed by using the following data:
[tex]T = 180^{\circ}[/tex] (Period)
[tex]z_{min} = -4[/tex] (Minimum)
[tex]z_{max} = 5[/tex] (Maximum)
The cosine function is a periodic bounded function that lies between -1 and 1, that is, twice the unit amplitude, and periodicity of [tex]2\pi[/tex] radians. In addition, the following considerations must be taken into account for transformations:
1) [tex]x[/tex] must be replaced by [tex]\frac{2\pi\cdot x}{180^{\circ}}[/tex]. (Horizontal scaling)
2) The cosine function must be multiplied by a new amplitude (Vertical scaling), which is:
[tex]\Delta z = \frac{z_{max}-z_{min}}{2}[/tex]
[tex]\Delta z = \frac{5+4}{2}[/tex]
[tex]\Delta z = \frac{9}{2}[/tex]
3) Midpoint value must be changed from zero to the midpoint between new minimum and maximum. (Vertical translation)
[tex]z_{m} = \frac{z_{min}+z_{max}}{2}[/tex]
[tex]z_{m} = \frac{1}{2}[/tex]
The new function is:
[tex]z'(x) = z_{m} + \Delta z\cdot \cos \left(\frac{2\pi\cdot x}{T} \right)[/tex]
Given that [tex]z_{m} = \frac{1}{2}[/tex], [tex]\Delta z = \frac{9}{2}[/tex] and [tex]T = 180^{\circ}[/tex], the outcome is:
[tex]z'(x) = \frac{1}{2} + \frac{9}{2} \cdot \cos \left(\frac{\pi\cdot x}{90^{\circ}} \right)[/tex]
The domain of the function is all real numbers and its range is between -4 and 5. The graph is enclosed below as attachment.
Need answers ASAP!!!!! (due today)
Answer:
15) 2.08m
Step-by-step explanation:
We kow tanA= p/b
Here, A=33°
b=3.2m
Then,
tan33°=p/3.2
0.65=p/3.2
p=0.65*3.2
p=2.08
So, The height of tree is 2.08m
14) 59.58ft
tan50°=p/b
1.19=p/50
p=59.58ft
So, The height of signpost is 59.58ft
In both of these problems, we will be using trigonometry! Remember, SOH-CAH-TOA.
14. x = 13.5950 ft
Visualization of the problem is attached below.
We want to find out the opposite side to the angle, and we know the adjacent side. Therefore, we should use the tangent function.
tan(50) = x / 50
x = tan(50) * 50
x = 13.5950 ft (round off wherever you need)
15. x = 241.0016 m
The visualization of the problem is already given. We know the same information as we need in the previous problems, an angle and an adjacent side, and we want to find the opposite side. Therefore, we should use the tangent function.
tan(33) = x / 3.2
x = tan(33) * 3.2
x = 241.0016 (round off wherever you need)
Hope this helps!! :)
I really need help! Please help, i don't understandddd
Answer:
x is 2
Step-by-step explanation:
To solve this you have to use the Pythagoram theorem A^2+B^2=C^2. So 9+16=C^2.
25=C^2
c=5
Than since u know the radius of the circle is 3, its 5-3 so x is 2.
In order to determine the average price of hotel rooms in Atlanta, a sample of 64 hotels was selected. It was determined that the average price of the rooms in the sample was $112 with a standard deviation of $16. Use a 0.05 level of significance and determine whether or not the average room price is significantly different from $108.50.
Which form of the hypotheses should be used to test whether or not the average room price is significantly different from $108.50?
H0:
a. mu is greater than or equal to $108.50
b. mu is greater than $108.50
c. mu is less than $108.50mu is less than or equal to $108.50
d. mu is equal to $108.50mu is not equal to $108.50
Ha:
a. mu is greater than or equal to $108.50
b. mu is greater than $108.50mu is less than $108.50
c. mu is less than or equal to $108.50
d. mu is equal to $108.50mu is not equal to $108.50
Answer:
H0 :
a. mu is greater than or equal to $108.50
Ha:
c. mu is less than or equal to $108.50
Step-by-step explanation:
The correct order of the steps of a hypothesis test is given following
1. Determine the null and alternative hypothesis.
2. Select a sample and compute the z - score for the sample mean.
3. Determine the probability at which you will conclude that the sample outcome is very unlikely.
4. Make a decision about the unknown population.
These steps are performed in the given sequence
In the given scenario the test is to identify whether the average room price significantly different from $108.50. We take null hypothesis as mu is greater or equal to $108.50.
The height of a cylinder is 9.5 cm. The diameter is 1.5 cm longer than the height. Which is closest to the volume of the cylinder?
Answer:
853.8cm^3
Step-by-step explanation:
[tex]h = 9.5cm\\d = 1.5cm + 9.5 = 10.7\\r =d/2=10.7/2=5.35\\\\V = \pi r^2 h\\V = 3.14 \times (5.35)^2 \times 9.5\\\\V =853.8 cm^3[/tex]
Graph the line y=4/3x +1
The slope would be 4/3 and the y-intercept is 1
Create a table x and y and in x there is -3/4 and 0 and for the y side is 0 and 1. The line would be in the 2 quadrant with 2 points on on the y axis 1 and the other on the x axis 0.9 and that would be the graphed description of the line. Sorry if this is hard to understand i don’t have a access to draw or insert an image.
The graph of the linear equation is on the image at the end.
How to graph the line?To do it, we need to find two points on the line, so let's evaluate it.
When x = 0
y = (4/3)*0 + 1 = 1 ----> (0, 1)
When x = 3
y = (4/3)*3 + 1 = 5 ---> (3 , 5)
Now just graph these two points and connect them with a line, that will be the graph of the linear equation.
Learn more about linear equations at:
https://brainly.com/question/1884491
#SPJ6
Solve the equation: 1/3(y - 2) - 5/6(y + 1) = 3/4(y - 3) - 2
Answer:
y=2.2
Step-by-step explanation:
1: Distribute all numbers to get rid of all parenthesis
2: Solve
Hope this helped :) LET ME KNOW IF YOU NEED THE ANSWER IN A DIFFERENT FORM I CAN GET IT FOR YOU
amanda teaches the art of quilling to 4 students. These students each teach art of quilling to 4 other students. If this process continues for 5 generation after amanda, BLANK people other than amanda will know the art of qiulling
Answer:
1024
Step-by-step explanation:
4 * 4 * 4 * 4 * 4
Find the circumference of C in terms of π
radius of c Is 5
Answer:
Given that
radius of circle =5units
So, circumference of circle=2πr
=2×π×5
=10π units
hope it helps u...
plz mark as brainliest...
Answer:
[tex]\boxed{Circumference = 10\pi \ units}[/tex]
Step-by-step explanation:
Circumference = [tex]2\pi r[/tex]
Where r = 5
=> Circumference = 2π(5)
=> Circumference = 10π units
Identify the initial amount a and the growth factor b in the exponential function. f(x) = 620 • 7.8x
Answer:
Initial amount (a)= 620
Growth factor (b)= 7.8
Step-by-step explanation:
620 is the initial amount and is multiplied by 7.8 x which is the growth factor.
1/2 of a right angle is a? answers: A. reflex angle B. obtuse angle C. acute angle D. straight angle
Answer:
C. acute angle
Step-by-step explanation:
As you know ,right angle is equal to 90 degrees so half of 90 degrees is 45 degree which is an acute angle (acute angles are the angles which are less than 90 degrees)
Hope this helps and pls mark as brainliest :)
Answer: acute
Step-by-step explanation:
An angle that is less than 90 degrees
The population, p, in thousands of a resort community is given by P(t)=700t/4t[tex]x^{2}[/tex]+9
Answer:
Step-by-step explanation:
pt=700 is basically evaluate it form the bottom to the top and u must mark me as brainly
Which graph is defined by the function given below ? y=(x-3)(x-3)
Answer:
Graph B
Step-by-step explanation:
We are looking for a graph with a double root at x=3. In such cases, the function will "touch' the x-axis at x=3. The graph that shows such behavior is Graph B.
Answer:B
X intercepts (3,0) Y intercepts (0,9)
A hypothesis test is to be performed for a population proportion. For the given sample data and null hypothesis, compute the value of the test statistic, Z.
415 people were asked if they were satisfied with their jobs. 49% said they were. H0: p= 0.3
a. 8.446
b. 2.612
c. 0.415
d. 4.125
Answer:
The correct option is a
Step-by-step explanation:
From the question we are told that
The sample size is n = 415
The sample proportion is [tex]\r p = 0.49[/tex]
Now
The null hypothesis is [tex]H_o : p = 0.3[/tex]
The alternative hypothesis is [tex]H_a : p \ne 0.3[/tex]
The test statistics is mathematically evaluated as
[tex]t = \frac{\r p - p }{ \frac{\sqrt{ p (1- p )} }{n} }[/tex]
substituting values
[tex]t = \frac{0.49 - 0.3 }{ \sqrt{ \frac{0.3 (1- 0.3 ) }{415} }}[/tex]
[tex]t = 8.446[/tex]
If Juan drives 50 mph for 1/2 hour then 60 mph for 1 1/2 an hour, how far does he drive?
Answer:
115 miles
Step-by-step explanation:
First find the distance at 50 mph
d = 50 mph * .5 hours
= 25 miles
Then find the distance at 60 mph
d = 60 mph * 1.5 hours
= 90 miles
Add the distances together
25+90
115 miles
Answer:
he drives a 115 miles
Step-by-step explanation:
if he drives 50 mph for half an hour he drove 25 miles then if he drives 60 mph for 1 hour and 30 minutes he would of drove 90 miles. 60 + 30=90
90+25=115 so he drove 115 miles.
The radius of a right circular cone is increasing at a rate of 1.1 in/s while its height is decreasing at a rate of 2.4 in/s. At what rate is the volume of the cone changing when the radius is 109 in. and the height is 198 in.
Answer:
[tex]79591.8872 in^3/s[/tex]
Step-by-step explanation:
we know that the volume of a right circular cone is give as
[tex]V(r,h)= \frac{1}{3} \pi r^2h\\\\[/tex]
Therefore differentiating partially with respect to r and h we have
[tex]\frac{dV}{dt} = \frac{1}{3}\pi [2rh\frac{dr}{dt} +r^2\frac{dh}{dt}][/tex]
[tex]\frac{dV}{dt} = \frac{\pi}{3} [218*198*1.1+109^2*2.4][/tex]
[tex]\frac{dV}{dt} = \frac{\pi}{3} [47480.4+28514.4]\\\\\frac{dV}{dt} = \frac{\pi}{3} [75994.8]\\\\ \frac{dV}{dt} = 3.142 [25331.6]\\\\ \frac{dV}{dt} =79591.8872 in^3/s[/tex]
please Help will mark brainliest !!1!Use the linear combination method to solve the system of equations. Explain each step of your solution. 2x -3y = 13 x+2=- 4
Answer:
work is shown and pictured
Keith biked 26 miles today and 32 miles yesterday. Which equation shows m, the number of miles he biked all together?
Answer:
m = 26+32
Step-by-step explanation:
To determine the total number of miles biked, add the days together
m = 26+32
Answer:
26+32=m
Step-by-step explanation:
That is the general eqaution of this, because u must add both days worth of biked miles together.
The Aluminum Association reports that the average American uses 56.8 pounds of aluminum in a year. A random sample of 51 households is monitored for one year to determine aluminum usage. If the population standard deviation of annual usage is 12.2 pounds, what is the probability that the sample mean will be each of the following? Appendix A Statistical Tables a. More than 61 pounds b. More than 57 pounds c. Between 55 and 58 pounds d. Less than 55 pounds e. Less than 48 pounds
Answer:
(a) The probability that the sample mean will be more than 61 pounds is 0.0069.
(b) The probability that the sample mean will be more than 57 pounds is 0.4522.
(c) The probability that the sample mean will be between 55 and 58 pounds is 0.6112.
(d) The probability that the sample mean will be less than 55 pounds is 0.14686.
(e) The probability that the sample mean will be less than 48 pounds is 0.00001.
Step-by-step explanation:
We are given that the Aluminum Association reports that the average American uses 56.8 pounds of aluminum in a year.
A random sample of 51 households is monitored for one year to determine aluminum usage. Also, the population standard deviation of annual usage is 12.2 pounds.
Let [tex]\bar X[/tex] = sample mean
The z-score probability distribution for the sample mean is given by;
Z = [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] ~ N(0,1)
where, [tex]\mu[/tex] = average aluminum used by American = 56.8 pounds
[tex]\sigma[/tex] = population standard deviation = 12.2 pounds
n = sample of households = 51
(a) The probability that the sample mean will be more than 61 pounds is given by = P([tex]\bar X[/tex] > 61 pounds)
P([tex]\bar X[/tex] > 61 pounds) = P( [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] > [tex]\frac{61-56.8}{\frac{12.2}{\sqrt{51} } }[/tex] ) = P(Z > 2.46) = 1 - P(Z [tex]\leq[/tex] 2.46)
= 1 - 0.9931 = 0.0069
The above probability is calculated by looking at the value of x = 2.46 in the z table which has an area of 0.9931.
(b) The probability that the sample mean will be more than 57 pounds is given by = P([tex]\bar X[/tex] > 57 pounds)
P([tex]\bar X[/tex] > 57 pounds) = P( [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] > [tex]\frac{57-56.8}{\frac{12.2}{\sqrt{51} } }[/tex] ) = P(Z > 0.12) = 1 - P(Z [tex]\leq[/tex] 0.12)
= 1 - 0.5478 = 0.4522
The above probability is calculated by looking at the value of x = 0.12 in the z table which has an area of 0.5478.
(c) The probability that the sample mean will be between 55 and 58 pounds is given by = P(55 pounds < [tex]\bar X[/tex] < 58 pounds)
P(55 pounds < [tex]\bar X[/tex] < 58 pounds) = P([tex]\bar X[/tex] < 58 pounds) - P([tex]\bar X[/tex] [tex]\leq[/tex] 55 pounds)
P([tex]\bar X[/tex] < 58 pounds) = P( [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]\frac{58-56.8}{\frac{12.2}{\sqrt{51} } }[/tex] ) = P(Z < 0.70) = 0.75804
P([tex]\bar X[/tex] [tex]\leq[/tex] 55 pounds) = P( [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] [tex]\leq[/tex] [tex]\frac{55-56.8}{\frac{12.2}{\sqrt{51} } }[/tex] ) = P(Z [tex]\leq[/tex] -1.05) = 1 - P(Z < 1.05)
= 1 - 0.85314 = 0.14686
The above probability is calculated by looking at the value of x = 0.70 and x = 1.05 in the z table which has an area of 0.75804 and 0.85314.
Therefore, P(55 pounds < [tex]\bar X[/tex] < 58 pounds) = 0.75804 - 0.14686 = 0.6112.
(d) The probability that the sample mean will be less than 55 pounds is given by = P([tex]\bar X[/tex] < 55 pounds)
P([tex]\bar X[/tex] < 55 pounds) = P( [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]\frac{55-56.8}{\frac{12.2}{\sqrt{51} } }[/tex] ) = P(Z < -1.05) = 1 - P(Z [tex]\leq[/tex] 1.05)
= 1 - 0.85314 = 0.14686
The above probability is calculated by looking at the value of x = 1.05 in the z table which has an area of 0.85314.
(e) The probability that the sample mean will be less than 48 pounds is given by = P([tex]\bar X[/tex] < 48 pounds)
P([tex]\bar X[/tex] < 48 pounds) = P( [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]\frac{48-56.8}{\frac{12.2}{\sqrt{51} } }[/tex] ) = P(Z < -5.15) = 1 - P(Z [tex]\leq[/tex] 5.15)
= 1 - 0.99999 = 0.00001
The above probability is calculated by looking at the value of x = 5.15 in the z table which has an area of 0.99999.
Need help with this, I don’t need an explanation.
the answer is x=-2
and y=12
Need help finding the length
Answer:
27
Step-by-step explanation:
First, we need to find x. We are given the perimeter, which is 2l + 2w, so from there, we have an equation of 2(4x-1) + 2(3x+2) = 100. By working through it, we get that x = 7. We're asked to find WX, so plug 7 into 4x - 1 and get 27.
Answer:
27 unitsStep-by-step explanation:
Perimeter of rectangle is 2(l) + 2(w).
The perimeter is given 100 units.
2(4x-1) + 2(3x+2) = 100
Solve for x.
8x-2+6x+4=100
14x+2=100
14x=98
x=7
Plug x as 7 for the side WX.
4(7) - 1
28-1
= 27
Find two numbers in a given ratio such that the difference of their squares is to the sum of the numbers in a given ratio.Ratios, respectively, are 3 to 1 and 6 to 1.
According to the given situation, the computation of two number in a given ratio is shown below:-
We assume the numbers is x and y
Given that
[tex]\frac{x}{y} = \frac{3}{1}[/tex]
x = 3y
and
[tex]\frac{x^2-y^2}{x + y} = \frac{6}{1} \\\\\frac{(x + y) (x - y)}{(x + y)} = 6[/tex]
With the help of above formula we will put the value and be able to find the values of x and y
x - y = 6
3y - y = 6
2y = 6
y = 3
x = 3y = 9
x = 9, y = 3
Therefore the correct answer is x = 9 where as y = 3
What is the angle between the given vector and the positive direction of the x-axis? (Round your answer to the nearest degree.) i + 3 j
Answer:
72°
Step-by-step explanation:
Given two vectors a and b, The vector i+3j will form a right angled triangle with the x-axis (i.e the horizontal axis).
The opposite side of the triangle on the Cartesian plane will be 3units along the y axis while the adjacent will be 1 unit along the x axis.
The angle between thus two vectors is expressed as tan (theta) = opp/adj
tantheta = 3/1
theta = tan^-1(3)
Theta = 71.57° ≈ 72° to nearest degree
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate Lim Sn to obtain the value of the series or state that the series diverges.
n→[infinity]
[infinity]
Σ (4/√k+5 ) - 4/ √ k+6)
k=1
Looks like the series is
[tex]\displaystyle\sum_{k=1}^\infty\left(\frac4{\sqrt{k+5}}-\frac4{\sqrt{k+6}}\right)[/tex]
This series has n-th partial sum
[tex]S_n=\displaystyle\sum_{k=1}^n\bullet[/tex]
(where [tex]\bullet[/tex] is used as a placeholder for the summand)
[tex]S_n=\displaystyle\left(\frac4{\sqrt6}-\frac4{\sqrt7}\right)+\left(\frac4{\sqrt7}-\frac4{\sqrt8}\right)+\cdots+\left(\frac4{\sqrt{n+4}}-\frac4{\sqrt{n+5}}\right)+\left(\frac4{\sqrt{n+5}}-\frac4{\sqrt{n+6}}\right)[/tex]
In each grouped term, the last term is annihilated by the first term of the next group; that is, for instance,
[tex]\displaystyle\left(\frac4{\sqrt6}-\frac4{\sqrt7}\right)+\left(\frac4{\sqrt7}-\frac4{\sqrt8}\right)=\frac4{\sqrt6}-\frac4{\sqrt8}[/tex]
Ultimately, all the middle terms will vanish and we're left with
[tex]S_n=\dfrac4{\sqrt6}-\dfrac4{\sqrt{n+6}}[/tex]
As [tex]n\to\infty[/tex], the last term converges to 0 and we're left with
[tex]\displaystyle\sum_{k=1}^\infty\bullet=\lim_{n\to\infty}S_n=\frac4{\sqrt6}=\boxed{2\sqrt{\dfrac23}}[/tex]