Answers
Step-by-step explanation:
4/25=0.16
6%=0.06
0.1=10%
0.875=7/8
Related Questions
What is the value of
Select one:
OA. -39
OB. -3
OC. 3
OD. 21
26² +46-9
6+4
when b= -3?
Answers
How to solve this equation
Answers
g-1(x) = ex - 1 is the inverse οf the functiοn g(x) = lοg(x + 1) as taking away 1 frοm bοth sides .
what is functiοn ?A functiοn is a mathematical fοrmula that gives each input value in a set an individual οutput value. Tο put it anοther way, a functiοn is a cοnnectiοn between twο sets where each element in the first set (referred tο as the dοmain). There are several methοds tο express a functiοn, including with a graph, a table οf values, οr a fοrmula. Mοst frequently, a fοrmula οr equatiοn is used tο represent a functiοn.
We need tο sοlve fοr x in terms οf g in οrder tο derive the inverse functiοn g-1(x) οf g(x) = lοg(x + 1). (x).
We can expοnentiate bοth sides by starting with g(x) = lοg(x + 1) and using the definitiοn οf lοgarithms:
G(x) = lοg(x + 1) = e(x)
[tex]e^{(g(x))} = x + 1[/tex]
Hence, by taking away 1 frοm bοth sides, we can isοlate x:
[tex]x = e^{(g(x))} - 1[/tex]
Inverse functiοn g-1(x) is thus:
[tex]g-1(x) = e^x - 1[/tex]
Hence, g-1(x) = ex - 1 is the inverse οf the functiοn g(x) = lοg(x + 1) as taking away 1 frοm bοth sides .
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find the diffrence. enter your answerin simplest terms usingthe slash (/) as a fraction bar
3/4 - 1/3
Answers
Answer:
5/12
Step-by-step explanation:
3/4 - 1/3
We need to use the common denominator of 12
3/4 * 3/3 = 9/12
1/3 * 4/4 = 4/12
9/12 - 4/12 = 5/12
You currently have $9,300 (Present Value) in an account that has an interest rate of 5% per year compounded annually (1 times per year). You want to withdraw all your money when it reaches $15,810 (Future Value). In how many years will you be able to withdraw all your money?
Answers
FV = PV x (1 + r)^n
where FV is the future value, PV is the present value, r is the interest rate per period, and n is the number of periods.
Plugging in the values given in the problem, we get:
$15,810 = $9,300 x (1 + 0.05)^n
Simplifying this equation, we get:
1.7 = (1.05)^n
Taking the natural logarithm of both sides, we get:
ln(1.7) = n x ln(1.05)
Solving for n, we get:
n = ln(1.7) / ln(1.05) ≈ 9.27 years
Therefore, it will take approximately 9.27 years to reach a future value of $15,810 and withdraw all the money.
FV = PV x (1 + r)^n
Where FV is the future value, PV is the present value, r is the interest rate, and n is the number of compounding periods.
Substituting the given values, we have:
$15,810 = $9,300 x (1 + 0.05)^n
Dividing both sides by $9,300, we get:
1.7 = (1.05)^n
Taking the natural logarithm of both sides, we get:
ln(1.7) = ln(1.05)^n
Using the logarithmic property that ln(a^b) = b x ln(a), we can simplify the right side:
ln(1.7) = n x ln(1.05)
Dividing both sides by ln(1.05), we get:
n = ln(1.7) / ln(1.05)
Using a calculator, we get:
n ≈ 11.97
Therefore, it will take approximately 11.97 years (or about 12 years) to reach a future value of $15,810 and be able to withdraw all the money from the account.
How do you determine which trigonometric function to use to solve a problem?
Answers
Answer: I would use tan or cot
Step-by-step explanation:
The draft financial statements of Madras, a limited liability company, for the year ended 31
December 2022 is currently under review. The following points have been raised:
a. An ex-employee has started an action against the company for wrongful dismissal. The
company’s legal team have stated that the ex-employee is not likely to succeed. The following
estimates have been given by the lawyers relating to the case:
i. Legal costs (to be incurred whether the claim is successful or not) P50,000
ii. Settlement of claim if successful P150,000
Currently no provision has been made by the company in the financial statements.
b. The company has a policy of refunding the cost of any goods returned by dissatisfied
customers, even though it is under no legal obligation to do so. This policy of making refunds
is generally known. At the year end returns totalling P48,000 have been made.
c. A claim has been made against a company for injury suffered by a pedestrian in connection
with building work by the company. Legal advisers have confirmed that the company will
probably have to pay damages of P100,000 but that a counterclaim made against the building
subcontractors for P50,000 would probably be successful.
Required:
State with reasons what adjustments, if any, should be made by the company in the financial
statements.
Answers
Based on the given information, Madras Limited Liability Company (LLC) needs to make certain adjustments in their financial statements. Firstly, the company should make a provision for the legal costs of P50,000 related to the wrongful dismissal case. Even though the legal team is confident that the ex-employee will not be successful, the provision must be made as the costs are to be incurred regardless of the outcome. This will reflect the true financial position of the company and will ensure that the financial statements present a fair view of the company's affairs.
Secondly, the company needs to make a provision for the settlement of P150,000 if the ex-employee's case is successful. This provision needs to be made because there is a likelihood of an outflow of economic resources in the future, as a result of the company losing the case. This provision will reduce the company's profit and hence its tax liability.
Thirdly, the company needs to recognize the refunds of P48,000 made to dissatisfied customers as a liability. Even though Madras LLC is under no legal obligation to make refunds, their policy of making refunds is generally known. Hence, the company has a constructive obligation to make refunds and therefore, the refunds should be recognized as a liability in the financial statements.
Lastly, the company needs to make a provision for the damages payable to the pedestrian injured on their building site. The legal advisers have confirmed that the company will have to pay damages of P100,000. Hence, the company needs to make a provision for this amount as there is a likelihood of an outflow of economic resources in the future. However, as the company has a counterclaim against the building subcontractors for P50,000, this amount can be netted off against the provision. Therefore, Madras LLC needs to make a provision of P50,000 for the net amount of damages payable to the pedestrian.
In conclusion, Madras LLC needs to make the above-mentioned adjustments to their financial statements to ensure that the financial statements present a fair view of the company's affairs. These adjustments will also ensure that the company complies with the Generally Accepted Accounting Principles (GAAP) and will help stakeholders in making informed decisions based on the financial statements.
A 17-foot pipe is cut into three sections. The longest section is two times as long as the shortest, and the middle-sized section is 5 feet longer than the shortest. Complete the diagram by
alving a and b as expressions involving x. Do not solve the problem.
Answers
The required 1-st section: 6 ft; 2-nd section: 6+9 = 15 ft; 3-rd section: 4*6 = 24 ft.
What are Sections?The 40 or 80 ft. (12-24 m) lengths of steel pipe sections, or joints, are trucked to the building site and strung out along the path in the locations where they are to be welded together.
According to question:The 45-foot conduit is divided into three pieces. The middle-sized section is 9 feet longer than the shortest segment, and the longest section is 4 times as long as the shortest. Finish the design.
x + (x+9) + 4x = 45, or
6x = 45-9
6x = 36,
x = 6.
Thus,1-st section: 6 ft; 2-nd section: 6+9 = 15 ft; 3-rd section: 4*6 = 24 ft.
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Complete Question:
A 45 foot pipe is cut into three sections. The longest section is 4 times as long as the shortest, and the middle-sized section is 9 feet longer than the shortest. Complete the diagram.
can someone help me
do the odds
please show work
Answers
Answer:
Step-by-step explanation:
To convert degrees to radians we use this formula
number of degrees x [tex]\frac{\pi}{180}[/tex]
1. [tex]-415 * \frac{\pi}{180} =[/tex][tex]\frac{-83}{36} \pi[/tex]
3. [tex]570 * \frac{\pi}{180} =[/tex][tex]\frac{19}{6} \pi[/tex]
5. [tex]473 * \frac{\pi}{180} =[/tex][tex]\frac{473}{180} \pi[/tex]
7. [tex]-799 * \frac{\pi}{180} =[/tex][tex]\frac{-799}{180} \pi[/tex]
9. [tex]44 * \frac{\pi}{180} =[/tex][tex]\frac{11}{45} \pi[/tex]
11. [tex]270 * \frac{\pi}{180} =[/tex][tex]\frac{3}{2} \pi[/tex]
13. [tex]-712 * \frac{\pi}{180} =[/tex][tex]\frac{-178}{45}[/tex]
15. [tex]-332 * \frac{\pi}{180} =[/tex][tex]\frac{-83}{45} \pi[/tex]
17. [tex]-54 * \frac{\pi}{180} =[/tex][tex]\frac{-3}{10} \pi[/tex]
19. [tex]-875 * \frac{\pi}{180} =[/tex][tex]\frac{-175}{36} \pi[/tex]
21. [tex]763 * \frac{\pi}{180} =[/tex][tex]\frac{763}{180} \pi[/tex]
23. [tex]299 * \frac{\pi}{180} =[/tex][tex]\frac{299}{180} \pi[/tex]
25. [tex]-705 * \frac{\pi}{180} =[/tex][tex]\frac{-47}{12} \pi[/tex]
27.[tex]374 * \frac{\pi}{180} =[/tex][tex]\frac{187}{90} \pi[/tex]
29. [tex]-239 * \frac{\pi}{180} =[/tex][tex]\frac{-239}{180} \pi[/tex]
Hope this helps!
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Consider the following graph of g(x). Write a formula for g(x) that describes the transformations performed on the basic function.
Answers
From the given graph the function obtained is [tex]g(x) = \sqrt{(x - 3)}+ 1[/tex].
What is a function?
In mathematics, a function is a unique arrangement of the inputs (also referred to as the domain) and their outputs (sometimes referred to as the codomain), where each input has exactly one output and the output can be linked to its input.
The graph g(x) given is a graph of function g(x) = √x.
The graph when drawn for g(x) = √x, starts from the origin (0,0).
In the graph it can be seen that the function is moved 3 units left and 1 units down.
For a function f(x) moving left with k units the formula is -
y = f(x + k)
For a function f(x) moving down with k units the formula is -
y = f(x) - k
Apply these formula to our basic function g(x) = √x.
So since the graph is moved 3 units left and 1 units down, the function will be -
[tex]g(x) = \sqrt{(x - 3)}+ 1[/tex].
Therefore, the function is obtained as [tex]g(x) = \sqrt{(x - 3)}+ 1[/tex].
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The total area of earth covered by water is about 350,000,000 km². This represents about 70% of earths total surface area. What is the approximate total surface area of earth
Answers
Answer:
The total area of earth covered by water is about 350,000,000 km². This represents about 70% of earths total surface area. What is the approximate total surface area of earth
Step-by-step explanation:
The total area of earth covered by water is about 350,000,000 km². This represents about 70% of earths total surface area. What is the approximate total surface area of earth
Some students performed an experiment in which they dropped a rubber ball from a height of 650 centimeters. They noticed that after each bounce, it reached 72% of its previous height. Which equation models the height, H, for n bounces?
a
Hn = 650(0.28)n − 1 where n = 1, 2, 3, …
b
Hn = 650(0.72)n − 1 where n = 1, 2, 3, …
c
Hn = 650(1.72)n − 1 where n = 1, 2, 3, …
d
Hn = 650(72)n − 1 where n = 1, 2, 3, …
Answers
The exponential function that models the height of the ball after n bounds is given as follows:
b) [tex]H(n) = 650(0.72)^{(n - 1)}[/tex], where n = 1, 2, 3, …
How to define an exponential function?The general format of an exponential function is given as follows:
y = ab^x.
In which the parameters are given as follows:
a is the initial value.b is the rate of change.Some students performed an experiment in which they dropped a rubber ball from a height of 650 centimeters, hence the parameter a is given as follows:
a = 650.
They noticed that after each bounce, it reached 72% of its previous height, hence the parameter b is given as follows:
b = 0.72.
Considering that the first bounce is the reference bounce, the equation is given as follows:
[tex]H(n) = 650(0.72)^{(n - 1)}[/tex]
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Find a polynomial function with real coefficients
1, 3i
f(x) =
Pls help
Answers
A polynomial function with real coefficients and roots 1, 3i is f(x) = x^3 - x^2 + 9x - 9.
What is the polynomial functionIf a polynomial function has real coefficients, then complex roots must come in conjugate pairs. That is, if 3i is a root, then -3i must also be a root. Therefore, the polynomial with roots 1, 3i, and -3i is:
(x - 1)(x - 3i)(x + 3i)
Multiplying this out using the difference of squares, we get:
(x - 1)(x^2 - (3i)^2)
Simplifying the expression using (3i)^2 = -9, we get:
(x - 1)(x^2 + 9)
Expanding the expression using FOIL, we get:
x^3 + 9x - x^2 - 9
Simplifying this expression, we get:
f(x) = x^3 - x^2 + 9x - 9
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DOM Practice Problems
1) An empty boat displaced 2525 cm^3 of water. When you placed an oar on the boat, it displaced 3878 cm^3 of water. What is the weight of the oar in grams?
2) Jane is three times older than her pet and Jane will be 13 next month. How old is Jane’s pet?
Make sure to show work and circle your answer!
*Please add a picture of the work and circle the final answer*
Answers
The weight of the fluid displaced is equivalent to the weight of the object that is submerged in the fluid.
Volume of the water displaced by the Oar = 3878 cm^3 - 2525 cm^3
= 1353 cm^3
Mass of the water = 1353 g
Weight of water displaced = Weight of the oar
Weight of oar = 1353 g
2)
Let the age of the pet be x
3x = 13
x = 13/3
x = 4 1/3 years
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Find the length of the missing side for this
right triangle.
12
9
Answers
Step-by-step explanation:
therefore 9 x 9+12
81+144
225/2
15
Rueben’s chocolate bar is 41% cocoa. If the weight of the chocolate bar is 67 grams, how many grams of cocoa does it contain? Round your answer to the nearest tenth
Answers
The weight of cocoa in Rueben’s chocolate bar is found as 27.47 grams.
Explain about the percentage of the number?Several fields employ percentage. It frequently appears in accounting and finance-related topics including taxation, sales, earnings, and interest rates. The grades of the pupils were expressed in a number of schools and institutions using percentages. Percentages are used to express probabilities, nutritional information, and download times.Percentage of cocoa in Rueben’s chocolate bar = 41%.
Weight of the chocolate bar = 67 grams.
Grams of cocoa in chocolate bar = 0.41*67
Grams of cocoa in chocolate bar = 27.47 grams
Thus, weight of cocoa in Rueben’s chocolate bar is found as 27.47 grams.
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The triangular faces of the prism shown are equilateral triangles with a perimeter of 48 cm. Each of the other faces is a square. Find the surface area of the prism
Answers
Answer:
222.4 cm
Step-by-step explanation:
16cm x 13.9 cm =222.4 cm
hope this helped! :D
Can someone please teach me how to find out if this linear, quadratic, or exponential?
Answers
First we find the difference of y values. Then Find the difference of the differences. If the difference of the differences is a constant then the data represents a quadratic function.
will the signs be reversed each time in algebraic expansion?
Answers
That note about "both signs have been reversed" only applies to the example on the right side, where –4 was distributed into the expression in parentheses.
Notice on the left example, the signs were not reversed.
When you distribute a negative number, that's when the signs will reverse. Since you're distributing a negative number, you'll be multiplying each term in parentheses by a negative number and that negative is what reverses the sign on every term.
So, yes all of the signs will be reversed each time you distribute a negative number into the parentheses, but not if you distribute a positive number.
How do you find the distance between the points shown? (C: 6, -4) and (D: 6, -8)
A. Subtract the distance between each point and the x-axis
B. Add the distance between each point and the x-axis
C. Subtract the distance between each point and the y-axis
D. Add the distance between each point and the y-axis
Answers
Therefore , the solution of the given problem of expressions comes out to be it is 4 units between locations C and D.
What does a expression actually mean?Calculations like arbitrary division, joining, multiplication variable and removal are required. If they were merged, they would produce the following: A mathematical formula, some data, and an equation. A declaration of veracity is made up of values, elements, formulae, and mathematical operations like additions, subtractions, errors, and segments. It is possible to assess and analyse words and sentences.
Here,
We can apply the distance calculation to determine the separation between two points:
=> distance= √((x2 - x1)² + y2 - y1)²)
where the coordinates for the two points are (x1, y1) and (x2, y2).
This formula allows us to determine the separation between points
C (6, -4) and D (6, -8) as follows:
=> distance = √[(6 - 6)^2 + (-8 - (-4))^2]
=> distance = √[0^2 + (-4)^2]
=> distance = √[16]
=> distance = 4
As a result, it is 4 units between locations C and D. Both of the available answer options are incorrect.
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daniel has 40 coins that are either nickels or dimes. the total value of the coins is $2.85. how many of each coin does daniel have
Answers
Answer: 17 dimes or 23 nickels
Step-by-step explanation:
Let's use two variables to represent the number of nickels and dimes that Daniel has. Let: n be the number of nickels, and d be the number of dimes.
We can write two equations based on the information given in the problem:
n + d = 40 (because Daniel has a total of 40 coins)
0.05n + 0.1d = 2.85 (because the total value of the coins is $2.85)
To solve for n and d, we can use the first equation to express one variable in terms of the other, and substitute that expression into the second equation:n + d = 40
n = 40 - d0.05n + 0.1d = 2.85
0.05(40 - d) + 0.1d = 2.85
2 - 0.05d + 0.1d = 2.85
0.05d = 0.85
d = 17
So Daniel has 17 dimes. We can substitute this value back into the first equation to solve for n:n + d = 40
n + 17 = 40
n = 23
So Daniel has 23 nickels.
Therefore, Daniel has 23 nickels and 17 dimes.
Consider the following graph of g(x). Write a formula for g(x) that describes the transformations performed on the basic function
Answers
Simplify 14×5(13×2+13×5)
Answers
let me know if this helped!
Consider a home mortgage of 200,000 at six APR of 9% for 20 years. Of the total payment over the term of the loan x% is paid toward the principal and x% is paid toward the interest.
Answers
The tοtal payment οver the term οf the lοan, 44% is paid tοward the principal and 56% is paid tοward the interest.
Tο sοlve this prοblem, we need tο use the fοrmula fοr a fixed-rate mοrtgage payment:
[tex]P = (Pv \times r) / (1 - (1 + r)^{(-n))[/tex]
Where:
Pv = present value οf the mοrtgage (in this case, $200,000)
r = mοnthly interest rate (APR divided by 12)
n = tοtal number οf payments (20 years × 12 mοnths per year = 240)
First, we need tο find the mοnthly interest rate:
r = 9% / 12 = 0.0075
Next, we can plug in the values and simplify the fοrmula:
[tex]P = (200,000\times 0.0075) / (1 - (1 + 0.0075)^{(-240))[/tex]
P ≈ $1,734.84
Therefοre, the mοnthly payment οn the mοrtgage is apprοximately $1,734.84.
Tο determine the percentage οf the payment that gοes tοward principal and interest, we need tο lοοk at the amοrtizatiοn schedule fοr the mοrtgage. This will shοw hοw much οf each payment gοes tοward principal and interest οver time.
Using an οnline amοrtizatiοn calculatοr, we can see that οver the 20-year term οf the mοrtgage, apprοximately 44% οf the tοtal payments gο tοward principal and 56% gο tοward interest.
Therefοre, οf the tοtal payment οver the term οf the lοan, 44% is paid tοward the principal and 56% is paid tοward the interest.
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HELP LIKE FR I SRSLY DK
Answers
Answer: whole = 45; part = 15%
Step-by-step explanation:
The part ( a.k.a the fraction) is the percentage.
15% can be expressed as 15/100 = 3/20.
The whole is 45.
Hope this helps! :)
the value of y varies jointly with x and the square of z, and y=90 when x=5 and z=3. find the equation represents this relationship
Answers
Answer:
Step-by-step explanation:
If y varies jointly with x and the square of z, we can express this relationship as:
y = kx z^2
where k is the constant of variation. To find the value of k, we can use the given information that y = 90 when x = 5 and z = 3:
90 = k(5)(3)^2
Simplifying, we get:
90 = 45k
Dividing both sides by 45, we get:
k = 2
Now we can substitute this value of k into our equation to get:
y = 2xz^2
Therefore, the equation that represents the relationship between y, x, and z is:
y = 2xz^2
A store sells bags of potato chips.
1/3 of the bags are barbecue-flavored chips.
3/5of the bags are cheese-flavored chips.
The rest of the bags are plain chips.
Which statement is true?
●
A More than of the bags are plain chips.
B There are no bags of plain chips.
C Exactly of the bags are plain chips.
D Less than of the bags are plain chips.
Answers
Use this to find your answer:
15 bags
There is 5 bbq
9 cheese
1 plain
The statement less than 1/2 of the bags are plain chips is true, Option D is correct.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
Let's find the fraction of bags that are plain chips.
We know that the barbecue-flavored chips make up 1/3 of the bags, and Cheese-flavored chips make up 3/5 of the bags.
The fraction of bags that are plain chips is:
1 - 1/3 - 3/5
Lets convert the fractions with same denominator.
= 15/15 - 5/15 - 9/15
=15-5-9/15
= 1/15
1/15 is less than 1/2.
Therefore, less than 1/2 of the bags are plain chips, Option D is correct.
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Estimate the intercepts
of the graph of the
function.
-4
6
4
-2
y
y=-x²-x+2
4 x
2
The x-intercepts are about
and
and the y-intercept is about
■
Answers
The estimated intercepts are:
x-intercepts: (-1, 0) and (2, 0)
y-intercept: (0, 2).
We may take an approximation reading from the graph and use it to estimate the intercepts of the graph of the function y = -x2 - x + 2:
The graph's x-intercepts are the locations where it crosses the x-axis, indicating that y = 0. We can see from the graph that the x-intercepts are about at x = -1 and x = 2.
The graph's intersection with the y-axis, known as the y-intercept, indicates that x = 0. The graph indicates that the y-intercept is roughly at y = 2.
The estimated intercepts are as follows:
x-intercepts: (-1, 0) and (2, 0)
y-intercept: (0, 2)
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A bald eagle also eats 400g of food per day.It eats fish, as well as seal carcasses that have washed up on the beach. A seal eats fish fish eating fish with 1.0 microgram per gram of DDE in their tissue. The DDE builds up in the seal's body so that the seal has 2.0 micrograms per DDE in its tissue. If the bald eagle eats 200g of seal, how much DDE does the bald eagle consume in one day.
Answers
The bald eagle cοnsumes 100 micrοgrams οf DDE in οne day.
First, we need tο calculate the amοunt οf DDE in the 200g οf seal that the bald eagle eats:
What dοes DDE stability mean?If a delay differential equatiοn (DDE) equilibrium is lοcally asymptοtically stable fοr all delays, it is said tο be tοtally stable. We οutline standards fοr DDEs with discrete time delays' absοlute stability.
Calculate the amοunt οf DDE in 1 gram οf fish:
1.0 micrοgram οf DDE per gram οf fish
Calculate the amοunt οf fish the seal eats tο accumulate 2.0 micrοgrams οf DDE in its tissue:
2.0 micrοgrams οf DDE per gram οf seal / 1.0 micrοgram οf DDE per gram οf fish = 2.0 grams οf fish per gram οf seal
Calculate the amοunt οf seal that the bald eagle eats:
200g οf seal
Calculate the amοunt οf fish the bald eagle cοnsumes frοm the 200g οf seal:
200g οf seal * (1 gram οf fish / 2.0 grams οf fish per gram οf seal) = 100g οf fish
Calculate the amοunt οf DDE the bald eagle cοnsumes frοm the fish it ate frοm the seal:
100g οf fish * 1.0 micrοgram οf DDE per gram οf fish = 100 micrοgrams οf DDE
Therefοre, the bald eagle cοnsumes 100 micrοgrams οf DDE in οne day.
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A large university accepts 70% of the students who apply.
Answers
Answer: 3,500
Step-by-step explanation: 70% of 20,000 is 14,000 25% of 14,000 is 3,500
Find cos(β/2) and cot(β/2), if sin(β)= -(√(3)/2) and pi<β<(3pi/2)
Answers
Answer:
Step-by-step explanation:
We know that the sine of an angle is negative in the third and fourth quadrants. So, since β is in the third quadrant (pi < β < (3pi/2)), we know that sin(β) is negative.
Given sin(β) = -√(3)/2, we can use the half-angle formulas for sine to find sin(β/2):
sin(β/2) = ±√[(1 - cos(β))/2]
Since β is in the third quadrant, we know that cos(β) is negative. We can use the Pythagorean identity to find cos(β):
cos²(β) + sin²(β) = 1
cos²(β) = 1 - sin²(β)
cos(β) = -√(1 - sin²(β))
cos(β) = -√(1 - (3/4))
cos(β) = -√(1/4)
cos(β) = -1/2
Now we can substitute cos(β) into the half-angle formula for sine:
sin(β/2) = ±√[(1 - cos(β))/2]
sin(β/2) = ±√[(1 - (-1/2))/2]
sin(β/2) = ±√(3/4)
sin(β/2) = ±(√3/2)
Since β is in the third quadrant, we know that cos(β/2) is negative. So we can use the half-angle formula for cosine to find cos(β/2):
cos(β/2) = ±√[(1 + cos(β))/2]
cos(β/2) = ±√[(1 - 1/2)/2]
cos(β/2) = ±√(1/4)
cos(β/2) = ±(1/2)
Since β is in the third quadrant, we know that cot(β/2) is negative. We can use the half-angle formulas for cosine and tangent to find cot(β/2):
cos(β/2) = ±√[(1 + cos(β))/2]
cos(β/2) = ±√[(1 - 1/2)/2]
cos(β/2) = ±√(1/4)
cos(β/2) = ±(1/2)
cot(β/2) = cos(β/2) / sin(β/2)
cot(β/2) = (±1/2) / (±(√3/2))
cot(β/2) = ±(1/√3)
cot(β/2) = ±(√3/3)
Therefore, cos(β/2) is either -1/2 or 1/2, and cot(β/2) is either -√3/3 or √3/3, depending on the choice of sign for the square roots.
answer: cos(β/2) = 1/2 and cot(β/2) = -sqrt(3,
Given sin()=-((3)/2, we can use the Pythagorean identity to calculate the value of cos():
cos(β) sqrt(1 - sin2()) sqrt(1 - sqrt(3)/2) = square (1 - 3/4) Equals square (1/4) = square 1/2
We know that is in the third quadrant, where cosine is negative, since pi (3pi/2). As a result, we have:
cos(β) = -1/2
Now we can calculate cos(/2) and cot(/2) using the half-angle formulas:
sqrt((1 + cos())/2) = cos(/2) = ±sqrt((1 - 1/2)/2) = ±sqrt(1/4) = ±1/2
We know that /2 is in the fourth quadrant, where tangent is negative, because is in the third quadrant. As a result, we have:
Tan(1/2) = -1/tan(1/2) = -1/sqrt((1 + cos())/1 - cos()) = -1/sqrt((1 - 1/2)/(1 + 1/2)) = -1/sqrt(1/3) = -sqrt (3)
The answers are therefore cos(/2) = 1/2 and cot(/2) = -sqrt (3). Nonetheless, we must establish the sign of cos(/2). We can exploit the fact that sine is negative in the third quadrant, where is located, to accomplish this. As a result, we have:
Sin(/2) is equal to sqrt((1 - cos())/2). = -sqrt((1 + 1/2)/2) = squared(3/4) = squared(3)/2
Given that cosine is positive and /2 is in the fourth quadrant, we have:
sqrt(1 - sin2(/2)) = cos(/2) 1/2 = sqrt(1 - 3/4) = sqrt(1/4)
Therefore, the solutions are cos(β/2) = 1/2 and cot(β/2) = -sqrt(3,