The ODE is separable.
[tex]\dfrac{dy}{dx} = xy \iff \dfrac{dy}y = x\,dx[/tex]
Integrate both sides to get
[tex]\displaystyle \int\frac{dy}y = \int x\,dx[/tex]
[tex]\boxed{\ln|y| = \dfrac12 x^2 + C}[/tex]
But notice that replacing the constant [tex]C[/tex] with [tex]-C[/tex] doesn't affect the solution, since its derivative would recover the same ODE as before.
[tex]\ln|y| = \dfrac12 x^2 - C \implies \dfrac1y \dfrac{dy}{dx} = x \implies \dfrac{dy}{dx} = xy[/tex]
so either of the first two answers are technically correct.
Find the area between the following curves.
y=x^3-x^2+x+8 ; y=5x^2-7x+8
Find where the two curves meet.
[tex]x^3 - x^2 + x + 8 = 5x^2 - 7x + 8 \\\\ \implies x^3 - 6x^2 + 8x = 0 \\\\ \implies x (x-2) (x - 4)= 0 \implies x=0, x=2, x=4[/tex]
The area between the curves is
[tex]\displaystyle \int_0^4 \left|\left(x^3-x^2+x+8\right) - \left(5x^2 - 7x + 8\right)\right| \, dx = \int_0^4 \left|x(x-2)(x-4)\right| \, dx[/tex]
When [tex]x[/tex] is between 0 and 2, [tex]x(x-2)(x-4)[/tex] is positive; when [tex]x[/tex] is between 2 and 4, [tex]x(x-2)(x-4)[/tex] is negative. So we split the integral at [tex]x=2[/tex] to get
[tex]\displaystyle \int_0^2 x(x-2)(x-4) \, dx - \int_2^4 x(x-2)(x-4)\,dx[/tex]
In the second integral, substitute [tex]y=x-2[/tex] to get
[tex]\displaystyle \int_0^2 x(x-2)(x-4) \, dx - \int_0^2 (y+2)y(y-2)\,dy[/tex]
[tex]\displaystyle \int_0^2 x(x-2) \bigg((x-4) - (x+2)\bigg) \, dx[/tex]
[tex]\displaystyle -6 \int_0^2 x(x-2) \, dx[/tex]
[tex]\displaystyle 6 \int_0^2 \left(2x - x^2\right) \, dx[/tex]
[tex]\displaystyle 6 \left(x^2 - \frac13x^3\right)\bigg|_0^2 = \boxed{8}[/tex]
2. Brynn borrowed $25,000 at 1% per month from a family friend to start her entrepreneurial venture on December 2, 2011. If she paid back the loan on June 16, 2012, how much simple interest did she pay?
She paid a simple interest of $3,125.00 on the borrowing of $25,000
What is simple interest?
The simple interest on a loan or an investment can be determined as the principal multiplied by interest multiplied by the number of periods.
This is quite different from compound interest where the interest earned previously would earn interest in the future alongside the principal
I=PRT
I=interest on loan=unknown
P=amount borrowed=$25,000
R=interest rate=1% per month
T=12.5 months( from December 2 2011 to December 1 2012 makes one year and from December 1-16 gives 15 days, which is 0.5 of one month)
I=$25,000*1%*12.5
I=$3,125.00
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The Mogul Runners ski club planned a trip to Park City. Of the total number of club members, 11 signed up to go. If this is 25% of the club, how many members does the ski club have?
Answer: 44
Step-by-step explanation: Since 11 members are equal to 25% of the club, 11 members is 1/4 of the club because 25% = 1/4. There are four-fourths in a whole so 11x4 = 44 members. The ski club has 44 members.
For a certain company, the cost function for producing x items is C(x)=50x+100 and the revenue function for selling x items is R(x)=−0.5(x−100)2+5,000 . The maximum capacity of the company is 120 items.
The profit function P(x) is the revenue function R(x) (how much it takes in) minus the cost function C(x) (how much it spends). In economic models, one typically assumes that a company wants to maximize its profit, or at least make a profit!
Answers to some of the questions are given below so that you can check your work.
Assuming that the company sells all that it produces, what is the profit function?
P(x)=
Assuming that the company sells all that it produces, The profit function is: P(x)=-x^/2+50x -100 or P(x)=-0.5x²+50x -100.
Profit functionProfit = R-C = -(x-100)^2/2 +5000 - 50x -100 =
-(x^2 -200x +10000)/2 -50x +4900 =
-x^2/2 +100x -5000 +5000 -50x +4900
Collect like terms
-x^/2+50x -100
Hence,
Profit function = P(x)=-0.5x²+50x -100
Maximum profit generating output can be determine by taking the derivative and setting it equal to zero
R-C = -x^2/2 + 50x -100
R'-C' = -x + 50 =0
P(0)x = 50
Maximum profit can be determine by substituting x=50 into the original profit equation, R-C
P(50) =-(1/2)(50)^2 +50(50) -100
P(50)= -1250 + 2500 -100
P(50) = $1150 profit
P(120) =-(1/2)(120)^2 +50(120) -100
P(120)= -3600 + 6000 -100
P(120) = $2300 profit
Therefore the profit function is: P(x)=-x^/2+50x -100 or P(x)=-0.5x²+50x -100.
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Combined, Tanya and Sanjay have 40 pens. If Tanya has four times as many pens as Sanjay, how many pens does Sanjay have?
Answer:Sanjay has 8
Step-by-step explanation: Because 8x4 equals 32 so thats how much tanya has a 32+8 equals 40. This is what I did I knew 10x4 is 40 so that couldn't be it so then I tried 8 because nine x four is 36 which would be to high so I tried 8 and it was perfect hope this helps.
If Combined, Tanya and Sanjay have 40 pens. If Tanya has four times as many pens as Sanjay, then Sanjay have 8 pens.
What is Equation?Two or more expressions with an equal signs is called as Equation.
Given that Tanya and Sanjay have 40 pens. If Tanya has four times as many pens as Sanjay, We need to find how many pens sanjay has.
Let sanjay has x number of pens.
Tanya has four times as many pens as Sanjay
So 4 times means four multiple of sanjay pens.
Tanya has four times as many pens as Sanjay can be write as 4x.
Combined of tanya and sanjay has 40 pens.
Combined means adding.
x+4x=40
Add the like terms
5x=40
Divide both sides by 5.
x=8
So Sanjay has eight pens and Tanya has 4(8)=32 pens.
Hence Sanjay have eight pens.
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create a pattern of 6 numbers by multiplying a number by 5 to get the next number.starts with 4
Answer:
45 405 415 425 435 445
Step-by-step explanation:
this are the pattern of 6 numbers multiplying by number 5 to get the number starts with 4
Answer:
4,20,100,500,1500,8500.
Step-by-step explanation:
Complete the statement below with the appropriate term. According to John Locke, is how we acquire basic knowledge. Choose the best answer.
Which of the following explains why, in many of the problems, historical data is used?
Information changes rapidly.
It is an opportunity to study history at the same time.
Better math problems can be made from historical data.
Up-to-date information is difficult to find.
According to John Locke, Experience is how we acquire basic knowledge.
What explains why, in many of the problems, historical data is used is; Information changes rapidly.
What is the theory of John Locke?1) John Locke's theory simply means that all knowledge comes exclusively through experience. He argues that at birth the mind is a tabula rasa, that humans fill with ideas as they experience the world through the five senses.
Thus, the missing word in the question is "Experience".
2) The reason why historical data is used in may problems is because Information changes rapidly. The information being circulated currently may be different from that circulated 100 years ago.
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Which of the following inequalities represents the number line below?
Answer:
x ≤ 4
Step-by-step explanation:
the solid dot at 4 indicates that x can equal 4
the arrow points to the left indicating values less than 4 , then
x ≤ 4
The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 39 ounces and a standard deviation of 5 ounces.
Use the Standard Deviation Rule, also known as the Empirical Rule.
Suggestion: sketch the distribution in order to answer these questions.
The answer to these questions are
Option A: This is in the attachmentOption B : 24 and 54Option C: 97.59%Option D: 84.13%
How to find the point where the distribution lies at 99.7%. The data is 3 sd from mean
Hence
39 - 3*(5) = 24
39 + 3*(5) = 54
The widget lies between 24 and 54
c. P(29.0 < x < 54.0)
= 29 - 39 / 5 and 54 - 39 / 5
= -2.0 and 3.0
We have to find P(Z < 3.0) - P(Z < -2.0)
= 0.9987 - 0.0228
= 97.59%
d. x = 44
= 44 - 39/ 5
= 1
We are to find P(z < 1.0) = 84.13%
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40 points
The table shows the heights of students in a group.
Student Height (in inches)
A 45
B 48
C 49
D 40
E 53
What is the mean height of the students in the group?
47 inches
49 inches
51 inches
53 inches
Answer:
Step-by-step explanation:
answer: 47
add the heights then divide by 5
Consider the equation V=6h where V is the volume (in cubic centimeters) of a box with a variable height h in centimeters and a fixed base of area 6cm2.
The volume (in cubic centimeters) of a box) given the fixed base area of 6cm² and height of 6 cm is 36 cm³.
VolumeV = 6h
Where,
V = volume (in cubic centimeters) of a boxh = height in centimeters andIf the height = 6 cm
Fixed base area = 6 cm²
V = 6h
= 6 cm² × 6 cm
V = 36 cm³
Therefore, the volume of the box given the fixed base area of 6cm² and height of 6 cm is 36 cm³.
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PLEASE HELP ME QUICK
In the diagram, points D and E are marked by drawing arcs of equal size centered at B such that the arcs intersect BA and BC
Then, intersecting arcs of equal size are drawn centered at points D and E. Point P is located at the intersection of these arcs.
Based on this construction, m∠ABP is ANSWER ° and m∠ABC is ANSWER°
Answer:
m∠ABP is 32°,m∠ABC is 64°.Step-by-step explanation:
According to the construction we have:
BD = BEPD = PEBP - is common side of triangles BPD and BPEIt gives us:
ΔPBD ≅ ΔPBEThen, corresponding angles of congruent triangles are congruent:
∠DBP ≅ ∠EBPSo,
∠ABP ≅ ∠CBP ⇒ m∠ABP = 32°Then,
m∠ABC = m∠ABP + m∠CBP = 32° + 32° = 64°Devon is young and has just graduated college with her associate’s degree. She plans to start saving money for retirement as soon as she starts her new job. By taking 10% of her monthly gross income, Devon is able
to contribute (a) $__________ each month to a retirement plan. The account is expected to earn interest with an APR of 5.25% compounded monthly. Round answers to two decimal places.
a. How much money will be in Devon’s retirement account if she continues to make the same monthly
.
investment for 40 years?
b. Overall, Devon contributed how much of her own money into the retirement account?
c. What percent of the final balance in Devon’s retirement account was interest?
You may not ask for help from others or give help to others on specific project questions.
Part 1: Continued
2. Later in life, Devon convinces her friend Taylor to start saving for retirement as well. Even though Taylor is the same age as Devon, she hasn’t started saving for retirement yet. With only 20 years left until they both plan to retire, Taylor decides she needs to invest at least twice as much as Devon each month to try to catch up to Devon’s retirement plan. Taylor opens a similar retirement account to Devon’s that also has an APR
5.25% compounded monthly. She will invest (b) $__________ per month. Round answers to two decimal places.
a. How much money will be in Taylor’s retirement account if she continues to make the same monthly
investment for 20 years?
b. By the time she retires, Taylor will have contributed how much of her own money overall?
c. What percent of the final balance in Taylor’s retirement account will be interest?
3. Write complete sentences that identify which person had the higher amount and by how much. Round answers to two decimal places.
a. Who had more money in their retirement account when they planned to retire? How much more?
b. Who contributed more to their retirement account? How much more?
Part 1: Devon is able to contribute (a) $450 each month to a retirement plan.
a. The amount in Devon's retirement account if she makes the same monthly investment for 40 years is $733,253.26.
b. Overall, Devon's total contribution of her own money into the retirement account is $216,000.
c. The percent of the final balance in Devon's retirement account that was interest was 70.5%.
Part 2: Taylor will invest (b) $1,733.00 per month.
a. The amount that will be in Taylor's retirement account if she makes the same monthly investment for 20 years is $733,253.26.
b. By the time that Taylor retires, she will have contributed $415,920 of her own money into the account.
c. The percent of the final balance in Taylor's retirement account that will be interest is 43.3%.
3. Both Devon and Taylor had the same amount, $733,253.26, in their retirement accounts when they planned to retire.
a. None had more money at their retirement.
b. Taylor contributed more to her retirement account by $199,920 ($414,920 - $216,000).
Data and Calculations:Devon's Investments:Gross monthly income = $4,500
APR = 5.25% compounded monthly
N (# of periods) = 480 months (12 x 40 years)
I/Y (Interest per year) = 5.25%
PV (Present Value) = $0
PMT (Periodic Payment) = $450 ($4,500 x 10%)
Results:
FV = $733,253.26 ($517,253.26 + $216,000)
Sum of all periodic payments = $216,000 ($450 x 480)
Total Interest = $517,253.26
Percentage of interest out of the FV = 70.5% ($517,253.26/$733,253.26 x 100)
Taylor's Investment:N (# of periods) = 240 months (12 x 20 years)
I/Y (Interest per year) = 5.25%
PV (Present Value) = $0
FV (Future Value) = $733,253.26
Results:
PMT = $1,733.00
Sum of all periodic payments = $415,920 ($1,733 x 240
Total Interest = $317,333.26
Percentage of interest out of the FV = 43.3% ($317,333.26/$733,253.26 x 100)
Thus, based on the time value of money of investments, it is more profitable to start investing early.
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How to Convert 11010112 to base 10?
Answer:
So this is all you need to do!
Hope it helped you:)
Directions: Find the missing side of each triangle. Round your answers to the nearest tenth if necessary.
Answer:
1) 13m, 2) 10ft, 3) 11.4ft, 4) 12.7m
Step-by-step explanation:
Since all of these triangles are right triangles, we can use the Pythagorean Theorem: a^2+b^2 = c^2
1) 5^2+12^2 = c^2 = 25+144 = c^2 = 169 = c^2 = 13 = c
2) 6^2+8^2 = c^2 = 36+64 = c^2 = 100 = c^2 = 10 = c
3) 5^2+10.3^2 = c^2 = 25+106.09 = c^2 = 131.09 = c^2 = 11.4494541 = 11.4 = c
4) 8.6^2+9.4^2 = c^2 = 73.96+88.36 = c^2 = 162.32 = c^2 = 12.7404866 = 12.7 = c
if a rectangular piece of metal has 27.75 square inches what is the length and width?
The maximum area of a rectangular piece of metal with perimeter of 27.75 in² has length of 6.9375 in and width of 6.9375 in.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Let x represent the length and y represent the width. hence:
Perimeter = 2(x + y)
27.75 = 2(x + y)
y = 13.875 - x
Area (A) = xy
A = x(13.875 - x)
A = 13.875x - x²
Maximum area is at A' = 0, hence:
A' = 13.875 - 2x
13.875 - 2x = 0
x = 6.9375
y = 6.9375
The maximum area of a rectangular piece of metal with perimeter of 27.75 in² has length of 6.9375 in and width of 6.9375 in.
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Zachary's weight is 130% of Noah's weight. If Noah weighs 75 pounds, what does Zachary weigh?
Answer:
Zachary weights 95.7 pounds
IF THE ODDS AGAINST A PARTICULAR CANDIDATE WINNING AN ELECTION ARE 9
TO 5, WHAT IS THE PROBABILITY THAT THE CANDIDATE WILL WIN?
The probability that the candidate will win is P = 0.357
How to find the probability?
We know that the odds against the particular candidate are 9 to 5.
The the odds for the candidate are 5 to 9, this means that in 5 cases the candidate will win, and in 9 cases the candidate will lose.
Then the candidate wins in 5 out of 14 cases, then the probability that the particular candidate wins the elections is given by the quotient:
P = 5/14 = 0.357
The probability that the candidate will win is P = 0.357
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Find the area of the trapezoid 19 12.6 29.2 14.5
Answer:
303.66 in²
Option 4 is the correct answer
Write an equation of the line that passes through (3,-2) and is parallel to the line y = 2x + 4.
Answer: [tex]y+2=2(x-3)[/tex]
Step-by-step explanation:
Parallel lines have the same slope, so substituting into point-slope form, the equation is
[tex]y+2=2(x-3)[/tex]
Claire buys a condo for $350,000. The value of the condo appreciates on average 5% per year where she
bought the condo
If you know the rest of the problems on the page then that would help;)
1. The best way that we can represent the condo that Claire bought is by option B. [tex]350000(1.05)^{t}[/tex]
How to determine the functionTo determine the function, we would have to do so by making use of the given equation that we have here.
This is given as
Ct = [tex]A(1+r)^t[/tex]
Where a is the principal
r is the rate of interest and
t is the time
2. we have 70 = 30(1.03)^t
70/30 = (1.03)^t
We have to take the log of the function above in order to find the value of t
t = 28.665
This is approximated to 29
Hence the answer here is option A after 29 weeks.
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We will start with a string which is pulled tight enough to vibrate at 288 Hertz when
plucked. Let the length of the string be 1 unit. We can find additional nice sounding
notes by using a string of a smaller length with the same amount of tension. To find
the string lengths, we need to use fractions whose numerators are powers of 2 and
whose denominators are powers of 3 (which are larger than ½, but smaller than 1).
The first few fractions are given below. Determine the remaining fractions:
The series of fractions is 1, 2/3, 8/9, 16/27, 64/81, 128/243, 512/729, 2048/2187, 4096/6516, 16384/19683, 32768/59049, 131072/177147.
The string lengths are fractions whose numerators are powers of 2 and denominators are powers of 3, and the fractions are larger than 1/2 but smaller than 1.
Some powers of 2 are:
2⁰ = 1, 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32, 2⁶ = 64, 2⁷ = 128, 2⁸ = 256, 2⁹ = 512, 2¹⁰ = 1024, 2¹¹ = 2048, 2¹² = 4096, 2¹³ = 8192, 2¹⁴ = 16384, 2¹⁵ = 32768, 2¹⁶ = 65536, and 2¹⁷ = 131072.
Some powers of 3 are:
3⁰ = 1, 3¹ = 3, 3² = 9, 3³ = 27, 3⁴ = 81, 3⁵ = 243, 3⁶ = 729, 3⁷ = 2187, 2⁸ = 6561, 3⁹ = 19683, 3¹⁰ = 59049, and 3¹¹ = 177147.
The fractions, for which the numerator is a power of 2, the denominator is a power of 3, and the value is between 1/2 and 1 are:
2/3, 8/9, 16/27, 64/81, 128/243, 512/729, 2048/2187, 4096/6516, 16384/19683, 32768/59049, 131072/177147.
Thus, the series of fractions is 1, 2/3, 8/9, 16/27, 64/81, 128/243, 512/729, 2048/2187, 4096/6516, 16384/19683, 32768/59049, 131072/177147.
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Information about a play took up 1/3 of the pages in the program. Information about the actors took up 1/4 of the remaining pages. Information about the director, the producer, and the designer took up 2 pages, the same number of pages devoted to the actors. How many pages were there in the program? help pls
Answer: 4
Step-by-step explanation:
x - (1/3)x - (1/6)x = 2
(6x - 2x - x)/6 = 2
3x/6 = 2
x=4
A backyard pool has a concrete walkway around it that is 5' wide on all sides the total area of the pool and the walkway is 950' at the length of the pool is 8' longer than the width find the dimensions of the pool
The dimensions of the pool would give a quadratic equation x² + 28x - 770 = 0
How to determine the dimension
For the pool, we have that;
Let the width = x feet
Length = (x+8) feet
The pool alongside the walkway gives;
Width = x + 5 + 5 = (x + 10) feet
Length = x + 8 + 5 + 5 = (x + 18) feet
Total area of the pool with walkway = 950 square feet
Note that formula for area is given as
Area = length * width = 950
Equate the length and width
(x+18) × (x + 10) = 950
Using the FOIL method, we have;
(x × x )+ (x × 10) + (18 × x) + (18 × 10) = 950
x² + 10x + 18x + 180 -950 = 0
collect like terms
x² + 28x - 770 = 0
Thus, the dimensions of the pool would give a quadratic equation x² + 28x - 770 = 0
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can someone help me with this, it's urgent!
1. D
2. B
3. E
4. C
5. B
Concept:From the three examples at the top of the page, the sum of the bottom two parts of the triangle equals to six times the value of the part at the top.
For example, the first one: 20, 10, 5 (Tip)
20 + 10 = 305 × 6 = 30For example, the second one: 4, 14, 3 (Tip)
4 + 14 = 183 × 6 = 18Solve:Question 1
Given information
Tip = 15
Bottom = ?, 51
Determine the sum
15 × 6 = 90
Find the other bottom part
90 - 51 = 39
[tex]\Large\boxed{D.~39}[/tex]
Question 2
Given information
Tip = 9
Bottom = 21, ?
Determine the sum
9 × 6 = 54
Find the other bottom part
54 - 21 = 33
[tex]\Large\boxed{B.~33}[/tex]
Question 3
Given information
Tip = ?
Bottom = 60, 60
Determine the sum
60 + 60 = 120
Find the tip
120 ÷ 6 = 20
[tex]\Large\boxed{E.~20}[/tex]
Question 4
Given information
Tip = ?
Bottom = 18, 30
Determine the sum
18 + 30 = 48
Find the tip
48 ÷ 6 = 8
[tex]\Large\boxed{C.~8}[/tex]
Question 5
Given information
Tip = 12
Bottom = ?, 38
Determine the sum
12 × 6 = 72
Find the other bottom part
72 - 38 = 34
[tex]\Large\boxed{B.~34}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
cos 90 - 2sin45 + 2tan180
Answer:
- [tex]\sqrt{2}[/tex]
Step-by-step explanation:
cos90° - 2sin45° + 2tan180°
= 0 - ( 2 × [tex]\frac{\sqrt{2} }{2}[/tex] ) + 2(0)
= 0 - [tex]\sqrt{2}[/tex] + 0
= - [tex]\sqrt{2}[/tex]
can someone give the exact answer for all the blanks
Considering the given function, we have that:
f(a) = 4 - 2a + 6a²f(a + h) = 6a² + 12ah + 6h² - 2a - 2h + 4.[f(a + h) - f(a)]/h = 12a + 6h - 2.What is the function for this problem?The function is:
f(x) = 4 - 2x + 6x².
When x = a, we have that:
f(a) = 4 - 2a + 6a².
When x = a + h, we have that:
f(a + h) = 4 - 2(a + h) + 6(a + h)² = 6a² + 12ah + 6h² - 2a - 2h + 4.
For the fraction, we have that:
[f(a + h) - f(a)]/h = [6a² + 12ah + 6h² - 2a - 2h + 4 - 4 + 2a - 6a²]/h = [12ah + 6h² - 2h]/h = h(12a + 6h - 2)/h
Hence:
[f(a + h) - f(a)]/h = 12a + 6h - 2.
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Prove the following relations.
Cos²A + cos²A.cot²A=cot²A
(1-sin²A)(1+cot²A)=cot²A
Step-by-step explanation:
Let's solve for the first one,
[tex]R.H.S. = Cos²A + cos²A.cot²A[/tex]
Step 1- Take Cos²A common,
[tex] = Cos²A( 1+ cot²A)[/tex]
[tex] \small \sf \: Now \: we \: know \: that \\ \small (1 + Cot^{2} \theta = Cosec^{2} \theta)[/tex]
Step 2 - Substituting above value in step 1,
[tex] = Cos^{2} A(Cosec^2A)[/tex]
[tex] \small \sf \: We \: know \: that \: Cosec \theta= \frac{1}{Sin \theta} [/tex]
Step 3 - Substitute Cosec²A with 1/Sin²A
[tex] = \frac{Cos^2A}{Sin^2A} [/tex]
[tex] \small \sf \: Now \: the \: basic \: trigonometric \: function \: is \\ \small \frac{Cos\theta}{Sin \theta} = Cot\theta[/tex]
Step 4 - Replacing the product of step 3 with above function,
[tex] = {Cot}^{2} A[/tex]
Hence proved,
[tex]R.H.S = L.H.S[/tex]
﹌﹌﹌﹌﹌﹌﹌﹌﹌﹌﹌﹌﹌﹌﹌﹌﹌
Now let's prove the second relation,
[tex](1-sin²A)(1+cot²A)=cot²A[/tex]
Let's solve for R.H.S.,
[tex]R.H.S.= (1-sin^{2} A)(1+cot^{2} A)[/tex]
As I told above,
[tex]\small(1+cot^{2} \theta) = cosec^{2} \theta[/tex]
Another important relation is,
[tex] \small sin^2\theta + cos^2\theta= 1 \: or \: \\ \small cos^2\theta = 1 - sin^2\theta[/tex]
Replacing both the above brackets with this relation we get,
[tex] = Cos^2A \cdot Cosec^2A [/tex]
Now follow the step 3 and step 4 of first question and you will get,
[tex]R.H.S. = L.H.S.[/tex]
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Some important trigonometric relations that you must learn,
[tex]Sin^2\theta + Cos^2\theta = 1 \\ 1 + Cot^2\theta = Cosec^2\theta \\ 1+Tan^2\theta = Sec^2\theta[/tex]
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Find a z-score for a data value of 19 if the mean of a set of data is 37 and the standard deviation is 8.6.
The z-score for a data value of 19 if the mean of a set of data is 37 and the standard deviation is 8.6 is -2.09
How to determine the z-score of the data value?From the question, the given parameters about the distribution are
Mean value of the set of data = 37
Standard deviation value of the set of data = 8.6
The actual data value = 19
The z-score of the data value is calculated using the following formula
z = (x - mean value)/standard deviation
Substitute the given parameters in the above equation
z = (19 - 37)/8.6
Evaluate the difference of 19 and 37
z = -18/8.6
Evaluate the quotient of -18 and 8.6
z = -2.09
Hence, the z-score for a data value of 19 if the mean of a set of data is 37 and the standard deviation is 8.6 is -2.09
Read more about z-score at:
https://brainly.com/question/2283546
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The legs of a 45-45-90 triangle has lengths of 21. What is the length of the hypotenuse?
Answer:
[tex]21 \sqrt{2} [/tex]
Step-by-step explanation:
See attached image.
