Answers
Answer:
a) See attachment.
b) Domain: (2, ∞)
Range: (-∞, ∞)
c) As x → 2⁺, f(x) → -∞
As x → ∞, f(x) → ∞
d) f(3) = -5
e) x = 4
Step-by-step explanation:
Given logarithmic function:
[tex]f(x)=\log_2(x-2)-5[/tex]
Part aAs the argument of a log function can only take positive arguments, the domain is restricted to x > 2 and there is a vertical asymptote at x = 2.
To find the x-intercept, set the function to zero and solve for x:
[tex]\implies \log_2(x-2)-5=0[/tex]
[tex]\implies \log_2(x-2)=5[/tex]
[tex]\textsf{Apply log law}: \quad \log_ab=c \iff a^c=b[/tex]
[tex]\implies x-2=2^5[/tex]
[tex]\implies x-2=32[/tex]
[tex]\implies x=34[/tex]
Therefore, the x-intercept is (34, 0).
Substitute x = 10 into the function to find a second point on the curve:
[tex]\implies f(10)=\log_2(10-2)-5[/tex]
[tex]\implies f(10)=\log_2(8)-5[/tex]
[tex]\implies f(10)=\log_2(2)^3-5[/tex]
[tex]\implies f(10)=3\log_2(2)-5[/tex]
[tex]\implies f(10)=3(1)-5[/tex]
[tex]\implies f(10)=3-5[/tex]
[tex]\implies f(10)=-2[/tex]
Therefore, a second point on the curve is (10, -2).
Part bAs the argument of a log function can only take positive arguments, the domain is restricted to x > 2:
Interval notation: (2, ∞)The range is unrestricted:
Interval notation: (-∞, ∞)Part cAs x approaches x = 2 from the positive side, the function approaches negative infinity:
As x → 2⁺, f(x) → -∞As x approaches positive infinity, the function approaches positive infinity.
As x → ∞, f(x) → ∞Part dFrom inspection of the graph:
f(3) = -5Check by evaluating f(3) algebraically:
[tex]\implies f(3)=\log_2(3-2)-5[/tex]
[tex]\implies f(3)=\log_2(1)-5[/tex]
[tex]\implies f(3)=0-5[/tex]
[tex]\implies f(3)=-5[/tex]
Part e
From inspection of the graph:
f(x) = -4 ⇒ x = 4Check by evaluating f(x) = -4 algebraically:
[tex]\implies \log_2(x-2)-5=-4[/tex]
[tex]\implies \log_2(x-2)=1[/tex]
[tex]\textsf{Apply log law}: \quad \log_ab=c \iff a^c=b[/tex]
[tex]\implies x-2=2^1[/tex]
[tex]\implies x-2=2[/tex]
[tex]\implies x=4[/tex]
Related Questions
I need help giving out how to find the IQR of a data set
Answers
Order the data from least to greatest.
Find the median.
Calculate the median of both the lower and upper half of the data.
The IQR is the difference between the upper and lower medians.
Use the algebraic tests to check for symmetry with respect to both axes and the origin.
y = x^4 - x^2 + 3
Answers
Axis of Symmetry: This function is not symmetric with respect to the x-axis, y-axis, or the origin.
To check for symmetry with respect to both the x-axis and the y-axis, we need to check whether the function is even or odd. To do this, we can substitute -x for x in the equation and check if the equation remains the same.
y = x^4 - x^2 + 3
y = (-x)^4 - (-x)^2 + 3
y = x^4 - x^2 + 3
Since the equation remains the same, the function is neither even nor odd and therefore is not symmetric with respect to either the x-axis or the y-axis.
To check for symmetry with respect to the origin, we need to check if the equation is symmetric when x and y are both substituted with -x or -y.
y = x^4 - x^2 + 3
y = (-x)^4 - (-x)^2 + 3
y = x^4 - x^2 - 3
Since the equation is not the same, the function is not symmetric with respect to the origin.
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HELP ASAP
14. Chung wants to determine the favorite hobbies.
among the teachers at his school. How could he
generate a representative sample? Why would it
be helpful to generate multiple samples?
Answers
Chung is going to be able to get the representative sample by carrying out a random selection.
It would be helpful to get the multiple samples because it is a way w=of getting the true representation of sample.
Why would it be helpful to generate multiple samples?Chung could generate a representative sample by randomly selecting a group of teachers from the school population and surveying them about their hobbies.
It would be helpful to generate multiple samples because, with only one sample, there's always a chance that the sample is not truly representative of the population. By generating multiple samples, Chung can compare the results and see if there is a consistent pattern among the samples, which would increase the likelihood that the results are representative of the population.
Additionally, by comparing multiple samples, Chung can also estimate the degree of variability in the population and the sample, this allow him to have more robust conclusions.
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This is very very confusing
Answers
Linear pair: The measure of two adjacent (together) angles in a line add up to 180°
And that the sum of the interior angles of a triangle always add up to 180°
—-
For the angle 1:
180-50-40 = angle 1
Angle 1 = 90
Angle 2:
There it says that angle ABC = 120, so if angle 1 = 90
Angle 2 = 30. (120-90)
Angle CTD:
Linear pair of 50
Angle CTD = 130
Angle CDA:
The shape is a parallelogram, so opposite sides are congruent (same measure)
Angle CDA: 120
I hope that this is not confusing for you anymore
Have a good day ^^
Which of the following is the equation that represents the graph?
[tex]a)y=-\frac{2}{3}* -6\\b)y=-\frac{3}{2}*-6\\c)y=-\frac{2}{3} *-4\\d) y=-\frac{3}{2}*-4[/tex]
Answers
The linear function that models the graph is given as follows:
a) y = -2x/3 - 4.
How to define the linear function?The slope-intercept definition of a linear function is given as follows:
y = mx + b.
In which:
m is the slope, representing the rate of change.b is the intercept, representing the value of y when x = 0.The graph crosses the y-axis at y = -4, hence the intercept b is obtained as follows:
b = -4.
When x increases by 6, y decays by 4, hence the slope m is obtained as follows:
m = -4/6
m = -2/3.
Hence the function is defined as follows:
y = -2x/3 - 4.
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susan is conducting a survey about the electric bills of households in her city. from the electric company, she found that the population mean is $98.75 with a standard deviation of $10.45. susan has a sample size of 60. complete the equation that susan can use to find the interval in which she can be 68% sure that the sample mean will lie. 98.75 88.3 109.2 60 10.45 7.75 reset next
Answers
As per the given standard deviation, the interval in which she can be 68% sure that the sample mean will lie is 98.75 ± 10.45/ √60
In math the term called standard deviation is defined as a statistic that measures the dispersion of a dataset relative to its mean and is calculated as the square root of the variance.
Here we have given that the population mean is $98.75 with a standard deviation of $10.45 and the a sample size of 60.
Now, here we need to find the equation that Susan can use to find the interval in which she can be 68% sure that the sample mean will lie.
Here the distribution of the sample mean is approximately normal distribution with mean $98.75 and standard deviation approximately written as 10.45/√60
Here we have Symbolically written as,
=> X~ N(98.75, 10.45/√60) 2
Here by using empirical rule, then we find the limits for sample mean for which we're 68% sure that the sample mean will lie in that interval as one standard deviation away from its mean, then it can be written as
=> P (98.75 -10.45/√60 < x <98.75 +10.45/√60) ≈ 68%
Hence the equation is written as
=>98.75 ± 10.45/ √60
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A salesperson engaged in activities that constitute violations of the NRS 645, including blockbusting and discrimination on the basis of disability. The salesperson also cashed a $25,000 earnest money check from a prospective buyer and used the proceeds to buy a new car. The salesperson’s employing broker was unaware of all of these activities. What is the impact on the salesperson’s broker when the salesperson’s violations are brought to the attention of the real estate commission?
Answers
The impact on the salesperson’s broker when the salesperson’s violations are brought to the attention of the real estate commission is that the employing broker may be disciplined by the real estate commission for failure to supervise.
What is real estate commission?Real estate commission is defined as the agency that is responsible for the stating the rules and regulations regarding the selling and acquisition of land and other related properties.
The rules and regulations enacted by them helps to prevent different forms of violations which includes blockbusting and discrimination on the basis of disability.
Therefore, salesperson’s broker would be disciplined by the real estate commission for failure to supervise the salesperson.
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true/false. in the greek geocentric model, the retrograde motion of a planet occurs when: in the greek geocentric model, the retrograde motion of a planet occurs when: earth is about to pass the planet in its orbit around the sun. the planet actually goes backward in its orbit around earth. the planet is aligned with the moon in our sky.
Answers
According to the Greek geocentric paradigm, a planet moves retrogradely when it actually goes backward in its orbit around Earth. So the option b is correct.
The geocentric paradigm, also known as geocentrism, which is frequently and particularly shown by the Ptolemaic system, is an outdated explanation of the cosmos that places Earth at its center. The Sun, Moon, stars, and planets all orbit Earth according to the majority of geocentric models.
Since humans have been gazing up at the stars for thousands of years, our perception of what the Universe looks like has undergone a significant transformation.
In time, humanity also realized that the Earth was a sphere and developed logical justifications for how other heavenly bodies behaved. By classical antiquity, scientists had developed theories about how the planets moved and how all the celestial bodies interacted. This led to the development of the now-defunct Geocentric model of the cosmos, which described how the Sun, Moon, and firmament revolved around our earth.
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The complete question is:
In the Greek geocentric model, the retrograde motion of a planet occurs when (a) Earth is about to pass the planet in its orbit around the Sun. (b) the planet actually goes backward in its orbit around Earth. (c) the planet is aligned with the Moon in our sky.
PLS HELPP
Which line is parallel to the liney=4x+5?
A)y=-4x+2
B) y=4x+3
C)4x+5
D)y=-1/4x+9
Answers
Answer:
c
Step-by-step explanation:
it is 4×+5 which is easy as u tines it by fixing the sum into parts and its eisier
Please help thanks!
Answers
Carl is preparing to clean out the gutters on his
house. The gutters are 24 feet from the ground.
The bottom of the ladder and the ground make a
60 degree angle. How far away from the house
should the ladder be placed in order to reach the
gutters?
Answers
Answer:
The gutters are 24 ft above the ground. For safety, the distance a ladder reaches up a wall should be four times the distance from the bottom of the ladder to the base of the side of the house. Therefore, the ladder must be 6 ft from the base of the house.
Which inequality is represented by this graph?
Answers
Answer:
x > 0
Step-by-step explanation:
An open circle means less than or greater than the number it is on.
A closed circle means less than or equal to, or greater than or equal to the number it is on.
This graph has an open circle on zero with the arrow pointing to the right.
so x>0
let $n$ be a positive integer such that every digit in $n$ is a $1$ or a $2,$ and $n$ is divisible by both $4$ and $9.$ what is the smallest possible value of $n?$ https://web2.0calc/questions/
Answers
The smallest possible number with digits 1 and 2, divisible by 4 and 9 is 22212.
A number that is divisible by both 4 and 9 must be divisible by their least common multiple (LCM), which is 36. To find the smallest possible value of n such that every digit is a 1 or a 2 and the number is divisible by 36, we can start with the smallest possible number that is divisible by 36 and only contains 1s and 2s, which is 11112.
As the number is divisible by 4 and contains only 1 and 2, the last two digits is 12. Now it is divisible by 9 so sum of digits is 9 or multiple of 9.
1 + 2 = 3
So we need digits sum equal 9-3 = 6 to be divisible by 9, so number is 22212.
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Please help, ignore the 7
Answers
Answer: 17
Step-by-step explanation:it goes -1 then x2 so after ten is 9 then 18 then 17
-1, x2
So:
3, 2, 4, 3, 6, 5, 10, 9, 18, 17
Answer: 17
5 A college student is paying for tuition through private loans. Two lenders have approved the student for a $25,000 loan. Offer 1: 5.99% annual simple interest, with a total account balance of $32,487.50 after a 60-month term Offer 2: 3.75% annual interest compounded monthly for a 66-month term Assuming no payments are made, what is the difference in the account balances at the end of the loan terms? Round your answer to the nearest penny. $1,245.00 $1,770.87 $2,964.36 $3,319.94
Answers
The difference in the account balances at the end of the loan terms is, $1770.87. So option B is correct
What is simple interest?Simple interest is an amount which is calculated for a certain period of time and for a certain rate, in simple interest calculation there are three factors which we consider: principal amount(P), rate(R) and time(T).
Every time we calculate simple interest the principal values will be the same unlike compound interest.
Formula; Simple Interest(I) = P × R × T/100
Given that,
Offer 1- 5.99% annual simple interest, with a total account balance of $32,487.50 after a 60-month term Offer
Offer 2- 3.75% annual interest compounded monthly for a 66-month term
In offer 1,
Total amount A = $32487.50
In offer 2,
Total amount =
[tex]A = P[ 1 + r/n]^{nt}[/tex]
[tex]A = 25000[ 1 + 0.0375/12]^{12*5.5}[/tex]
A = $30,716.63
Difference = 32487.50 - 30716.63
= $1770.87
Hence, the difference in account balance is $1770.87
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Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.
(a) f(x) = log2(x) (b) g(x) = ∜x (c) h(x) = 2x^3/(1 - x^2) (d) u(t) = 1 - 1.1t + 2.54t^2 (e) v(t) = 5^t (f) w(θ) = sin θ cos^2 θ
Answers
(a) log₂(x) is a logarithmic function, (b) ∜x is a root function, (c) 2x^3/(1 - x^2) is a rational function, (d) 1 - 1.1t + 2.54t^2 is a polynomial of degree 2, (e) 5^t is an exponential function and (f) sin θ cos^2 θ is a trigonometric function.
A power function has the form f(x) = x^n, where n is a constant.
A root function has the form f(x) = √x or f(x) = x^(1/n), where n is a constant.
A polynomial is an algebraic function that can be written in the form f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0, where a_i are constants and n is the degree of the polynomial.
A rational function is a function that can be written as the ratio of two polynomials, i.e., f(x) = (p(x))/(q(x)), where p(x) and q(x) are polynomials and q(x) ≠ 0.
An algebraic function is a function that can be expressed as the solution to a polynomial equation, i.e., it can be written as a polynomial, a root, or a combination of polynomials and roots.
A trigonometric function is a function that involves the ratios of sides in a right triangle, such as sine, cosine, tangent, etc.
An exponential function has the form f(x) = a^x, where a is a positive constant greater than 1.
A logarithmic function has the form f(x) = logₐx, where a is a positive constant greater than 1.
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State if the three numbers can be the measures of
the sides of a triangle.
4,7,8
Select the correct response:
O True
O False
Answers
True. The three numbers can be the measures of the sides of a triangle.
What is the triangle inequality theorem?
The triangle inequality theorem is a fundamental principle in Euclidean geometry that states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This means that in any triangle, no side can be longer than the sum of the other two sides.
The three numbers 4, 7, and 8 can be the measures of the sides of a triangle. This is because of the triangle inequality theorem which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In other words, for any triangle, the sum of two sides must always be greater than the third side.
For example, in this case,
4 + 7 = 11 > 8,
7 + 8 = 15 > 4,
4 + 8 = 12 > 7
Hence, 4, 7, and 8 can be the measures of the sides of a triangle, so the correct response would be "True".
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De una botella que contenía 1litro de agua, Juan bebió 1/4 de litro.Expresa con forma de número decimal la cantidad de agua que queda en la botella
Answers
The fraction of water left from the bottle which was 1 liter is 3/4 liter.
How to calculate the fraction?Fraction simply means a piece of a whole. In this situation, the number is represented as a quotient such that the numerator and denominator are split. In this situation, in a simple fraction, the numerator as well as the denominator are both integers.
In this case, from the bottle which was 1 liter, Juan drank 1/4. The amount that is left will be:
= 1 - 1/4
= 3/4
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A bowl at a delicatessen contains a mix of 15 apples, 12
oranges, and 9 pears. What is the probability of a customer
who gets one piece of fruit from the bowl randomly choosing
an orange or a pear?
Answers
The probability of a customer who gets one piece of fruit from the bowl randomly choosing an orange or a pear is 21/36
What is Probability ?
Probability can be defined as the ratio of number of favourable outcomes and total number of outcomes
Given ,
A bowl at a delicatessen contains a mix of 15 apples, 12 oranges, and 9 pears
so , we have to find the probability of a customer who gets one piece of fruit from the bowl randomly choosing an orange or a pear.
Choosing orange or a pear ,
= 12c1 + 9c1
= 12+9
= 21
Total = 36.
Probability = 12 + 9 / 36
= 21 / 36 .
Therefore , The probability of a customer who gets one piece of fruit from the bowl randomly choosing an orange or a pear is 21/36
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2. An Olympic sized pool holds 6.6 x 105 gallons. How many liters is that? (1 gallon = 3.79 Liters)
a. 1.7 x 105 L
b. 2.5x 106 L
c. 1.7 x 104 L
d. 2.5 x 107 L
Answers
It will take 51 days to fill the Olympic sized pool.
What is relation between time and work?Time and work have a relationship that is
Work Completed is equal to Time x Rate of Work.
Alternately, Rate of Work = Time Taken / 1.
Alternatively, Time Taken = 1 / Rate of Work
Considering that,
Pool volume is 6.6043 x 105 gallons, or 6,600,430 gallons.
9 litres of water are delivered through a hose every minute.
Therefore, the amount of water delivered by hose every hour is equal to (9 * 60) gallons.
Water provided by hosen each day is equal to (9 * 60 * 24) gallons, or 12,960 gallons.
Thus, the amount of time needed to fill the pool is equal to 6,600,430/12960.
There are 50.9 days needed to fill the pool.
51 days were needed to fill the swimming pool.
The Olympic-sized pool will therefore fill in 51 days.
The Complete Question is Olympic sized pool.
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P(a)=.40 p(b)=.50 P(A
and b )=.10. Whats p(b/a)
Answers
Answer:
The probability of (B|A) is 0.25 which is correct option(B).
What is probability?
The probability is defined as the possibility of an event is equal to the ratio of the number of outcomes and the total number of outcomes.
P(A)=0.40
P(B)= 0.50
P(A ∩ B) = 0.10
In order to find the probability of (B|A),
Condi
Step-by-step explanation:
which of the following sentences are conditional statements? [more than one of the statements may be conditional.] group of answer choices i will order a pizza tonight only if we eat dinner at home. rain delayed the game. are you the only one home? discrete mathematics is required for a computer science transfer degree from foothill college.
Answers
I will order a pizza tonight only if we eat dinner at home is a conditional statement.
Sentences are groups of words that convey complete and logical meaning. They have subjects and predicates. It can be in form of exclamation and questions consisting of main and subordinating clauses.
Conditional statements are sentences where the subordinating clause influences whether the action that will be performed or not in the main clause. In these statements, the subordinating clause is also called the antecedent, and the main clause is called the consequent.
There are two types of Conditional statements, they are Implicative and predictive statements. Implicative sentences are based on facts whereas predictive statements depend on a situation dependent on a hypothetical case.
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a statistical study inbolves a data set of the different types of shoe designs by the designer calbvivn klein collected consistentyly over the years from 1987 to 2014 quizzlet
Answers
Answer:
c
Step-by-step explanation: got it right on test
(a) Describe a in words a sequence of transformations that maps AABC to AA'B'C'
(b) Write an ordered-pair rule for each transformation in the sequence.
Answers
Transformation includes changing the position of a shape.
The sequence of metamorphosis is 90 degrees counterclockwise gyration, followed by a vertical restatement to the right by 2 units.
What's meant by sequence?
A grouping of two or further particulars in a logical sequence. The following arrangement of two or further effects chronological arrangement. an arrangement of analogous particulars or generalities in chronological sequence. The length of the sequence is determined by the number of particulars. In discrepancy to a set, the same particulars might appear further than formerly in a sequence at colorful points, and unlike a set, the order is important. The formal description of a series is a function from natural figures to the rudiments in each position. An listed family, which is a function from any indicator set, can be allowed of as a conception of the idea of a sequence.
The ordered brace- rule for each are :
(x,y)⇒ (-y,x) and (x,y)⇒(x+2,y)
The sequence of metamorphosis.
From the figure, we observed that ABC was rotated and also restated
So, the sequence of metamorphosis on ABC are
90 degrees counter clockwise gyration.
Vertical restatement to the right by 2 units.
The ordered brace rule :
When a point is rotated 90 degrees counter clockwise, the rule of metamorphosis is :
x, y)->(- y, x)
When a point is restated 2 units right, the rule of metamorphosis is :
x, y)->( x 2, y)
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Transformation includes changing the position of a shape.
The sequence of metamorphosis is 90 degrees counter clockwise gyration, followed by a vertical restatement to the right by 2 units.
What's meant by sequence?A grouping of two or further particulars in a logical sequence.The following arrangement of two or further effectschronological arrangement. an arrangement of analogous particulars or generalities in chronological sequence.The length of the sequence is determined by the number of particulars.In discrepancy to a set, the same particulars might appear further than formerly in a sequence at colorful points, and unlike a set, the order is important.The formal description of a series is a function from natural figures to the rudiments in each position.An listed family, which is a function from any indicator set, can be allowed of as a conception of the idea of a sequence.The ordered brace- rule for each are :
(x,y)⇒ (-y,x) and (x,y)⇒(x+2,y)
The sequence of metamorphosis.
From the figure, we observed that ABC was rotated and also restated
So, the sequence of metamorphosis on ABC are
90 degrees counter clockwise gyration.
Vertical restatement to the right by 2 units.
The ordered brace rule :
When a point is rotated 90 degrees counter clockwise, the rule of metamorphosis is :
x, y)->(- y, x)
When a point is restated 2 units right, the rule of metamorphosis is :
x, y)->( x 2, y)
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3w-4z=8 2w+3z=-6 solve.
Answers
Answer:
w = 0, z = − 2
Step-by-step explanation:
The solution for the given system of equations is: w = 0 and z = -2.
What is a system of equations?
A system of equations is a collection of two or more equations with the same set of unknowns. For a system to have a unique solution, the number of equations must equal the number of unknowns.
The equations are as follows:
[tex]3w-4z=8[/tex]-------- (1)
[tex]2w+3z=-6[/tex]-------(2)
The solution for the above equation,
Multiply the equation 1 by 2,
⇒([tex]3w-4z=8[/tex] )*2 --------- (3)
Multiply the equation 2 by 3,
⇒([tex]2w+3z=-6[/tex] )*3 --------(4)
Now solve equations 3 and 4 to get,
⇒ [tex]-17z=34[/tex]
⇒ [tex]z= -2[/tex]
Now substitute the value of z in equation 1,
⇒ [tex]3w-4(-2)=8[/tex]
⇒ [tex]\\3w=0[/tex]
⇒ [tex]w=0\\[/tex]
Hence we can conclude that the solution for the given system of equations is w = 0 and z = -2.
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Select the figure with an angle of 300°
on a circle in standard position.
Answers
The figure of the coterminal angle of 300° on a circle in standard position is attached in the solution.
What are Coterminal angles?Coterminal angles are angles in standard position (angles with the initial side on the positive x-axis) that have a common terminal side.
Coterminal angles are two angles in the standard position where one angle is a multiple of 360 degrees larger or smaller than the other. That is, if angle A has a measure of M degrees, then angle B is co-terminal if it measures M +/- 360n, where n=0,1,2,3, ...
Starting from that position, 300° is drawn by moving around the circle to the left "counter-clockwise" and ending at the 300° measure.
The negative coterminal angle is the angle that would end in the same place if we went to the right instead of right on the circle. It is equal to the original 300° minus 360°
Coterminal angles Ac to angle A may be obtained by adding or subtracting k⋅360° or k × (2π)
Hence, Ac = A + k⋅360°, if A is given in degrees.
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24-hour digital clock is gaining three minutes every hour,if you set it to the correct time at 6 p.m. on Saturday,when will it show the correct time again?
Answers
after 20 hours the clock will show 3*20 = 60 min = 1 hr more
since the clock is analogue after 12 hours it will show the same time hat is 6 am and 6 pm are shown the same way.
after 20*12 = 240 hr = 10 days the clock will show 12 hr more but since 6 am and 6 pm are shown the same after 240 hr = 10 days the clock will show the correct time again.
the correct time will be shown on Monday 6 pm.
5. The store is having a sale
on fruit. You can buy 6 kiwi for
$1.50. You can also get 5
mango for $3.00. What is the
total cost of buying 2 kiwis
and 3 mangoes?
Answers
Answer:
Step-by-step explanation:
1.50/6 = 0.25
3/5 = 0.6
0.25*2 = 0.5
0.6*3 = 1.8
1.80+0.50 = 2.30
$2.30
What are the coordinates of the point on the directed line segment from ( − 10 , 9 ) (−10,9) to ( − 4 , 3 ) (−4,3) that partitions the segment into a ratio of 1 to 5 help plss
Answers
We can find the coordinates of the point that partitions a line segment into a ratio of 1 to 5 by using the formula:
(x, y) = (x1 + r * (x2 - x1), y1 + r * (y2 - y1))
Where (x1, y1) and (x2, y2) are the coordinates of the endpoints of the line segment, and r is the ratio of the distance from the starting point to the point of interest, to the total distance of the line segment. In this case, r = 1/(1 + 5) = 1/6.
We know the coordinates of the two endpoints of the line segment:
(x1, y1) = (-10, 9)
(x2, y2) = (-4, 3)
So, we can substitute these values into the formula and find the coordinates of the point that partitions the line segment into a ratio of 1 to 5:
(x, y) = (-10 + (1/6) * ((-4) - (-10)), 9 + (1/6) * (3 - 9))
= (-10 + (1/6) * (-6), 9 + (1/6) * (-6))
= (-10 -1/6, 9 -1)
= (-10/6 -1/6, 3/2 -1)
= (-16/6, 3/2 -1)
= (-8/3,-1/2)
So, the coordinates of the point on the directed line segment from (-10, 9) to (-4, 3) that partitions the segment into a ratio of 1 to 5 is (-8/3,-1/2)
It is important to notice that this point is a vector and not a point in the Cartesian plane
express a cosine and sine function in terms of an imaginary exponential. conversely, express the imaginary exponential in terms of cosine and sine
Answers
A cosine and sine function can be expressed in terms of an imaginary exponential using Euler's formula: e^(iθ) = cos(θ) + i*sin(θ).
This formula can be derived by using the Taylor series for e^x, and replacing x with iθ.
A cosine and sine function can be expressed in terms of an imaginary exponential using Euler's formula: e^(iθ) = cos(θ) + i*sin(θ).
Conversely, an imaginary exponential can be expressed in terms of cosine and sine by solving for θ in the above equation. This can be done by taking the inverse cosine and inverse sine of both sides to get θ = cos^(-1)(e^(iθ)) = sin^(-1)(i*sin(θ)). Combining these equations gives θ = sin^(-1)(i*(e^(iθ) - cos(θ))).
Learn more about Euler's formula here:
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