Answer:
There are Even, Odd and None of them and this does not depend on the degree but on the relation. An Even function: [tex]f(-x)=f(x)[/tex] And Odd one: [tex]-f(x)=f(-x)[/tex]
Step-by-step explanation:
1) Firstly let's remember the definition of Even and Odd function.
An Even function satisfies this relation:
[tex]f(-x)=f(x)[/tex]
An Odd function satisfies that:
[tex]-f(x)=f(-x)[/tex]
2) Since no function has been given. let's choose some nonlinear functions and test with respect to their degree:
[tex]f(x)=x^{2}-4, g(x)= x^{5}+x^{3}[/tex]
[tex]f(x)=x^2 -4\Rightarrow f(-x)=(-x)^{2}-4\Rightarrow f(-x)=x^{2}-4\therefore f(x)=f(-x)[/tex]
[tex]g(-x)=-(x^{5}+x^{3})\Rightarrow g(-x)=-x^{5}-x^{3}\Rightarrow g(-x)=-g(x)[/tex]
3) Then these functions are respectively even and odd, because they passed on the test for even and odd functions namely, [tex]f(-x)=f(x)[/tex] and [tex]-f(x)=f(x)[/tex] for odd functions.
Since we need to have symmetry to y axis to Even functions, and Symmetry to Odd functions, and moreover, there are cases of not even or odd functions we must test each one case by case.
Final answer:
The degree of a polynomial indicates whether it is even or odd based on the highest power of x. Even functions exhibit symmetry across the y-axis, while odd functions show symmetry with respect to the origin. The derivative of an even function is odd due to the horizontal flip property during differentiation, and the integral of an odd function over a symmetric interval is zero.
Explanation:
To determine whether the degree of a function is even or odd, and whether the function itself is even or odd, one must understand the definitions and properties of even and odd functions. An even function satisfies the property f(-x) = f(x), implying symmetry across the y-axis. A classic example is the cosine function, cos x, or any power function x^n where n is an even number. Conversely, an odd function is defined by the property f(-x) = -f(x), indicating symmetry with respect to the origin. The sine function, sin(x), and x^n where n is odd, are examples of odd functions.
Regarding derivatives, it is interesting to note that the derivative of an even function results in an odd function due to the horizontal flip property of the derivative. This is because differentiation involves a limit process that inherently flips the sign of any even function's symmetric components, resulting in an odd function.
Moreover, both even and odd functions display specific behaviors when integrated over symmetric intervals: the integral of an odd function is zero due to its antisymmetric nature while even functions do not necessarily share this property. For instance, when considering a function expressed as a product, such as f(x) = (x^3 - 3x)e^{-x^2}, where one function is odd and the other even, the resulting function will be odd, based on the product of their respective eigenvalues with respect to the inversion operator.
Company X sells leather to company Y for $60,000. Company Y uses the leather to make shoes, selling them to consumers for $180,000. The total contribution to gross domestic product (GDP) is
Answer: $180,000
Step-by-step explanation:
Gross Domestic Product (GDP) is the total monetary value of all finished goods and services made within a country during a specific period. It can be used to estimate the size and growth rate of the country's economy.
In the case above Company X sell leather which is not a finished good to Company Y, so it will not contribute to the gross domestic product (GDP). Company Y sells leather shoes which is a finished good to the consumers, which will contribute to the GDP.
Therefore the total contribution to GDP is $180,000
PLEASE ANSWER; MAY NOT BE HARD
Find the sum of all positive 3-digit numbers whose last digit is 2
Answer:
Step-by-step explanation:
102+202+302+402+502+602+702+802+902(4518)
+112+212+312+...+ 812+912(4608)
+122+222+322+...+822+922(4698)
+132+232+332+...+932(4788)
..........................................
+192+292+392+...+992(5328)
4518+4608+4698+...+5328
n=10
[tex]s=\frac{10}{2}(4518+5328)\\=5(9846)\\=49230[/tex]
Final answer:
To find the sum of all positive 3-digit numbers ending in 2, we calculate the total for each digit's place and sum them up, resulting in a total sum of 8280.
Explanation:
The problem requires finding the sum of all positive 3-digit numbers with a last digit of 2. To calculate this, we can identify that the first such number is 102 and the last is 992. There are 90 such numbers because they correspond to the tens digit going from 0 to 9 for each of the nine possible hundreds digits (1-9).
Since each number ends in 2, we can think of them as (100x + 10y + 2), where x is the hundreds digit (1 through 9) and y is the tens digit (0 through 9). To find the sum, we calculate the sum of the hundreds digits times their frequency, the sum of the tens digits times their frequency, and add 2 times the number of terms (90). The formula would be:
Sum = (Sum of hundreds values) * 10 * 9 + (Sum of tens values) * 1 * 90 + 2 * 90
The hundreds values are 1 through 9, whose sum is 45, and the tens values are 0 through 9, whose sum is 45 as well. Plugging these values into the formula, we get:
Sum = 45 * 10 * 9 + 45 * 1 * 90 + 2 * 90 = 4050 + 4050 + 180 = 8280.
One card is selected from a deck of cards. Find the probability of selecting a black card or a jack.
The probability of selecting a black card or a jack is 15/26.
Given that, one card is selected from a deck of cards.
What is the probability?Probability can be defined as the ratio of the number of favourable outcomes to the total number of outcomes of an event.
We know that, probability of an event = Number of favourable outcomes/Total number of outcomes
Total number of outcomes =52
The number of black cards in a deck =26
The number of jack cards in a deck =4
Probability of an event = 26/52 +4/52
= 30/52
= 15/26
Therefore, the probability of selecting a black card or a jack is 15/26.
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Final answer:
The probability of selecting a black card or a jack from a standard deck of 52 cards is 7/13.
Explanation:
To find the probability of selecting a black card or a jack from a standard deck of 52 cards, we need to consider the number of favorable outcomes and the total number of possible outcomes.
In a deck, there are 26 black cards (13 clubs and 13 spades) and a total of 4 jacks.
However, since two of the jacks are black, we must avoid counting them twice.
The probability becomes:
P(Black or Jack) = P(Black) + P(Jack) - P(Black and Jack)
P(Black) = 26/52, P(Jack) = 4/52, and P(Black and Jack) = 2/52
Thus, the probability is:
P(Black or Jack) = (26/52) + (4/52) - (2/52) = 28/52 = 7/13
Therefore, the probability of selecting a black card or a jack from a standard deck is 7/13.
The third term in a geometric sequence is -81. The common ratio is 1/3
What is the second term of the sequence?
If you answer, can you explain it?
Answer:
Step-by-step explanation:
The formula for the nth term of a geometric sequence is expressed as follows
Tn = ar^(n - 1)
Where
Tn represents the value of the nth term of the sequence
a represents the first term of the sequence.
n represents the number of terms.
From the information given,
r = 1/3
T3 = - 81
n = 3
Therefore,
- 81 = a× 1/3^(3 - 1)
-81 = a × (1/3)^2
-81 = a/9
a = -81 × 9 = - 729
The exponential equation for this sequence is written as
Tn = - 729 * (1/3)^(n-1)
Therefore, to find the second term,T2, n = 2. It becomes
T2 = - 729 * (1/3)^(2-1)
T2 = - 729 * (1/3)^1
T2 = - 729 * (1/3)
T2 = - 243
At the price of $3 a pound of pork, Jason buys 8 pounds of pork and Noelle buys 10 pounds of pork. When the price rises to $5 a pound, Jason buys 5 pounds of pork and Noelle buys 7 pounds of pork. What is the market demand at $5?
Answer:
Market demand at $5 is 12 pork.
Step-by-step explanation:
In a market, the sum of individual demand for a product from buyers is known as market demand.
It is give that the at the price of $3 a pound of pork, Jason buys 8 pounds of pork and Noelle buys 10 pounds of pork.
So, market demand at $3 is
8 + 10 = 18
When the price rises to $5 a pound, Jason buys 5 pounds of pork and Noelle buys 7 pounds of pork.
So, market demand at $5 is
5 + 7 = 12
Therefore, the market demand at $5 is 12 pork.
Lena ordered 12 copies of the same book for his book club members. The book cost $19 each and the other has a 15 shipping charge what is the total cost of Lena's order$
Answer:
Step-by-step explanation:
Total copies of books ordered by Lena for his book club members is 12. The cost of the book is $19 each. Since the books are the same, the total cost of the books will be
19 × 12 = $228
the order has a $15 shipping charge. It means that the total amount that Lena would pay for the 12 books is total cost of the books + shipping fee. it becomes
228 + 15 =
=$243
A circle has its center at (0,0) and passes through the point (0,9). What is the standard equation of the circle?
x² + y² = 0
x² + y² = 9
x² + y² = 9²
Answer:
Step-by-step explanation:
I believe the answer is x² + y² = 9²
if I'm in a plane flying at 512 miles per hour and a plane flies below me in the opposite direction, will it appear to fly slow or fast
Answer:
It appears to fly faster than its actual speed.
Step-by-step explanation:
In general, if talking about velocities, the direction of the movement should also be taken into account. For example, if two objects move in opposite directions, a person inside one object observes that the other one moving in the opposite goes faster than its actual speed (because the velocities are summed up). If they are moving in the same direction the opposite phenomenon is true (the velocities are subtracted).
Finally, in this example the planes move in opposite direction, therefore, a plane flying in the opposite direction will appear to fly faster.
The total cost of producing a type of car is given by C(x)=12000−40x+0.04x2, where x is the number of cars produced. How many cars should be produced to incur minimum cost?
Answer:
Step-by-step explanation:
C'(x)=-40+0.08 x
C'(x)=0 gives
-40+0.08 x=0
x=40/0.08=500
C"(x)=0.08>0 at x=500
so C(x) is minimum if x=500
so 500 cars need to be produced for minimum cost.
or we can solve by completing the squares.
c(x)=12000+0.04(x²-1000 x+250000-250000)
=12000+0.04(x-500)²-0.04×250000
=0.04 (x-500)²+12000-10000
=0.04(x-500)²+2000
c(x) is minimum if x=500
To minimize the cost based on the provided quadratic cost function, 500 cars should be produced.
Explanation:This a problem of optimization in the arena of Calculus. The cost function C(x) = 12000-40x+0.04x2 is a quadratic function, and the minimum cost occurs at the vertex of the parabola described by this function.
For any quadratic function f(x)=ax2 +bx + c, minimum or maximum value occurs at x = -b/2a.
In this case, a = 0.04 and b = -40.
So minimum cost occurs when x = -(-40) / 2*0.04 = 500.
So, to incur minimum cost, 500 cars should be produced.
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What does the fundamental theorem of algebra illustrate?
Answer:
The fundamental theorem of algebra guarantees that a polynomial equation has the same number of complex roots as its degree.
Step-by-step explanation:
The fundamental theorem of algebra guarantees that a polynomial equation has the same number of complex roots as its degree.
We have to find the roots of this given equation.
If a quadratic equation is of the form [tex]ax^{2}+bx +c=0[/tex]
Its roots are [tex]\frac{-b+\sqrt{b^{2}-4ac } }{2a}[/tex] and [tex]\frac{-b-\sqrt{b^{2}-4ac } }{2a}[/tex]
Here the given equation is [tex]2x^{2}-4x-1[/tex] = 0
a = 2
b = -4
c = -1
If the roots are [tex]x_{1} and x_{2}[/tex], then
[tex]x_{1}[/tex] = [tex]\frac{-2+\sqrt{(-4)^{2}-4\times 2\times (-1)}}{2\times 2}[/tex]
= [tex]\frac{4 +\sqrt{24}}{4}[/tex]
= [tex]\frac{2+\sqrt{6} }{2}[/tex]
[tex]x_{2}[/tex] = [tex]\frac{-2-\sqrt{(-4)^{2}-4\times 2\times (-1)}}{2\times 2}[/tex]
= [tex]\frac{4 +\sqrt{8}}{4}[/tex]
= [tex]\frac{2-\sqrt{6} }{2}[/tex]
These are the two roots of the equation.
Complete the statement to describe the expression (a+b)(d+e)(a+b)(d+e)left parenthesis, a, plus, b, right parenthesis, left parenthesis, d, plus, e, right parenthesis. The expression consists of factors, and each factor contains 4 terms.
Final answer:
To complete the statement describing the expression (a+b)(d+e)(a+b)(d+e), we expand it by multiplying each term. Simplifying the expression, we get a²*d² + 2a²de + 2abde + b²e².
Explanation:
To complete the statement describing the expression (a+b)(d+e)(a+b)(d+e), we need to expand it. This can be done by multiplying each term in the first factor by each term in the second factor and then multiplying the result by the third factor. So the expression becomes:
(a+b)(d+e)(a+b)(d+e) = (a*d + a*e + b*d + b*e)(a*d + a*e + b*d + b*e)
We can simplify this further by combining like terms:
(a*d + a*e + b*d + b*e)(a*d + a*e + b*d + b*e) = a²*d² + 2a²de + 2abde + b²e²
a set of cards includes 15 yellow cards, 10 green cards and 10 blue cards. find the probability of each event when a card is chosen at random not yeallow or green
Answer:
P(not yellow or green)=\frac{2}{7}[/tex]
Step-by-step explanation:
a set of cards includes 15 yellow cards, 10 green cards and 10 blue cards
Total cards= 15 yellow + 10 green + 10 blue = 35 cards
Probability of an event = number of outcomes divide by total outcomes
number of outcomes that are not yellow or green are 10 blue cards
So number of outcomes = 10
P(not yellow or green)= [tex]\frac{10}{35} =\frac{2}{7}[/tex]
The probability of choosing a card that is neither yellow nor green from the set is 2/7, as there are 10 blue cards and a total of 35 cards.
Explanation:The question asks for the probability of choosing a card that is neither yellow nor green from a set containing 15 yellow cards, 10 green cards, and 10 blue cards. To find this probability, we must consider only the blue cards, as they are not yellow or green. The total number of blue cards is 10, and the total number of cards is 35 (since 15 + 10 + 10 = 35).
To calculate the probability, we use the formula:
P(Blue card) = Number of blue cards / Total number of cards = 10 / 35 = 2/7
Thus, the probability of randomly choosing a card that is not yellow or green (i.e., a blue card) is 2/7.
Alice sleeps an average of 9 hours per night. A cat can sleep up to 20 hours per day. About how many more hours does a cat sleep in 1 month that Alice?
Answer:
Cat sleeps 420 hours than Alice in a month
Step-by-step explanation:
Given:
Number of hours Alice sleeps per night = 9 hours
Number of hours Cat sleeps per night = 20 hours
To Find:
How many more hours does a cat sleep in 1 month that Alice
Solution:
Let
The total number of hours for which Alice Sleeps in one month be x
The total number of hours for which Cat sleeps in one month be y
Step 1: Number of hours for which Alice Sleeps in one month
X = number of days in a month X Number of hours Alice sleeps per night
X = 30 X 9
X = 180 Hours
Step 2: Number of hours for which Cat Sleeps in one month
y= number of days in a month X Number of hours cat sleeps per night
y = 30 X 20
y = 600 Hours
Now ,
=> y – x
=>600 – 180
=>420 hours
Mr. Johnson currently has a square garden. It is in his garden and into a range of 5 feet shorter than three times shorter than times it width. He decides that the perimeter should be 70 feet. Determine the dimensions, in feet, of his new garden
Answer:
The Dimension of new garden is [tex]25 \ feet\ \times 10\ feet.[/tex]
Step-by-step explanation:
Given:
Perimeter of new garden = 70 feet.
Let the length of the new garden be 'l'.
Also Let the width of the new garden be 'w'.
We need to find the dimension of new garden.
Now Given:
Length is 5 feet shorter than three times it width.
framing the equation we get;
[tex]l =3w-5 \ \ \ \ equation\ 1[/tex]
Now we know that;
Perimeter of rectangle is equal to twice the sum of length and width.
framing in equation form we get;
[tex]2(l+w)=70[/tex]
Now Diving both side by 2 using Division property of equality we get;
[tex]\frac{2(l+w)}2=\frac{70}{2}\\\\l+w =35[/tex]
Now Substituting equation 1 in above equation we get;
[tex]3w-5+w=35\\\\4w-5=35[/tex]
Adding both side by 5 Using Addition Property of equality we get'
[tex]4w-5+5=35+5\\\\4w=40[/tex]
Now Diving both side by 4 using Division property of equality we get;
[tex]\frac{4w}{4}=\frac{40}{4}\\\\w=10\ ft[/tex]
Now Substituting the value of 'w' in equation 1 we get;
[tex]l =3w-5\\\\l =3\times10-5\\\\l = 30-5\\\\l= 25\ ft[/tex]
Hence The Dimension of new garden is [tex]25 \ feet\ \times 10\ feet.[/tex]
Carol puts some green cubes and red cubes in a box. The ratio is 2:1. She adds 12 more cubes to the red cubes in the box and the ratio becomes 4:5. How many green cubes were in the box?
Answer:
16 green cubes are in the box.
Step-by-step explanation:
No. of red cubes at first (x):
4(x + 12) = 5(2x)
4x + 48 = 10x
6x = 48
x = 8
No. of green cubes:
= 8 * 2
= 16
Answer: 16 green cubes are in the box.
20 red cubes are in the box in the end.
Proof (the ratio in the end is 4:5):
= 16 is to 20
= 16/4 is to 20/4
= 4 is to 5
HELP!!!!!!!!!
Your goal is to save at least $350.00 over the next 6 weeks. How much money must you save each week in order to meet that goal? Write and solve an inequality.
A) 6+x[tex]\geq[/tex]360;x[tex]\geq[/tex]354
B) 60x[tex]\leq[/tex]360;x[tex]\leq[/tex]10
C) x/6[tex]\leq[/tex]360;x[tex]\leq[/tex]2160
D) 6x[tex]\geq[/tex]360;x[tex]\geq[/tex]60
D) 6x≥360; x≥60
Step-by-step explanation:
The goal is to save at least $350 over the next 6 weeks.
Let the amount to save per week be x
x *6 should be equal or more than the goal.This is
6x ≥ 360
However, dividing the goal amount by number of weeks to get the amount to save per week gives;
360/6 =60
so x≥ 60
The inequality is thus : 6x ≥360;x≥60
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13. Write an equation for the given function given the amplitude, period, phase shift, and vertical shift.
amplitude: 4, period 4 phase shift = vertical shift = -2
Answer:
[tex]y=4sin(\frac{2\pi(t+\frac{4}{3}\pi ) }{4\pi } )-2[/tex]
Step-by-step explanation:
Let's start with the original function.
[tex]y=a sin\frac{2\pi t}{T}[/tex]
We can immediately fill in the amplitude 'a' and period 'T' , as the question defines these for us, and provides values for 'a' and 'T', 4 and 4[tex]\pi[/tex] respectively.
[tex]y=4sin(\frac{2\pi t}{4\pi } )[/tex]
Now we only have phase shift and vertical shift to do. Vertical shift is very easy, you can just add it to the end of the right side of the expression. A positive value will shift the graph up, while a negative value will move shift the graph down. We have '-2' as our value for vertical shift, so we can add that on as so:
[tex]y=4sin(\frac{2\pit }{4\pi } )-2[/tex]
Now phase shift the most complicated of the transformations. Basically, it is just movement left or right. A negative phase shift moves the graph right, a positive phase shift moves the graph left (I know, confusing!). Phase shift applies directly to the x variable, or in this case the t variable. To achieve a -4/3 pi phase shift, we need to input +4/3 pi into the function, because of the aforementioned negative positive rule. Here is what the function looks like with the correct phase shift:
[tex]y=4sin(\frac{2\pi(t+\frac{4}{3}\pi ) }{4\pi } )-2[/tex]
This function has vertical shift -2, phase shift -4/3 [tex]\pi[/tex], amplitude 4, and period 4[tex]\pi[/tex].
Desmos.com/calculator is a great tool for learning about how various parts of an equation affect the graph of the function, If you want you can input each step of this problem into desmos and watch the graph change to match the criteria.
HELP ASAP!!! BRAINLIEST!
its a b and c they all equal 45
Answer:
1 * 45
5 * 9
3 * 15
Hope this helps!
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Consider the rational expression (IMAGE ATTACHED)
3x^2−3/
3x^2+2x−1
Which statements are true?
Answer:
3x² is a term in the numeratorx + 1 is a common factorThe denominator has 3 termsStep-by-step explanation:
You can identify terms and count them before you start factoring. Doing so will identify 3x² as a term in the numerator, and will show you there are 3 terms in the denominator.
When you factor the expression, you get ...
[tex]\dfrac{3x^2-3}{3x^2+2x-1}=\dfrac{3(x^2-1)}{(3x-1)(x+1)}=\dfrac{3(x-1)(x+1)}{(3x-1)(x+1)}[/tex]
This reveals a common factor of x+1.
So, the above three observations are true of this rational expression.
The cost of a peanut butter bar is $0.07 more than the cost of a chocolate bar. If you buy 5 peanut butter bars and 6 chocolate bars, the total cost is $6.40. How much does the chocolate bar cost?
$0.61
$0.55
$0.54
$0.62
Hello!
To be quick and simple, your answer would be $0.55
The cost of the chocolate bar in the given scenario is $0.55. This was determined by solving a two-variable system of linear equations from the information provided.
Explanation:This problem is a classic example of a system of linear equations, specifically two-variable linear equations. Here, we need to find the cost of one chocolate bar and one peanut butter bar, and we have two pieces of information that can be translated into equations. The first information is that a peanut butter bar costs $0.07 more than a chocolate bar. The second is that 5 peanut butter bars and 6 chocolate bars total $6.40. We'll use these equations to solve for the variables.
Let's denote the cost of the chocolate bar as x and the cost of the peanut butter bar as y. Then, from the information given, we can form two equations:
y = x + $0.075y + 6x = $6.40Substitute the first equation into the second to solve for x:
5(x + $0.07) + 6x = $6.405x + $0.35 + 6x = $6.4011x + $0.35 = $6.4011x = $6.05x = $0.55So the cost of the chocolate bar is $0.55.
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The price of the dinner for the both of them was $30. They tipped their server 20% of that amount. How much did each person pay, if they shared the price of dinner and the tip equally?
Each person will pay 19.5 dollars.
Step-by-step explanation:
Given
Total bill for dinner = b=$30
First of all we will calculate the 30% of dinner bill to find the amount of tip
So,
[tex]Tip = t = 30\%\ of\ 30\\= 0.30*30\\=9[/tex]
the tip is $9
The total bill including tip will be:
[tex]= 30+9 = \$39[/tex]
Two persons have to divide the tip and dinner equally so,
Each person's share = [tex]\frac{39}{2} = 19.5[/tex]
Hence,
Each person will pay 19.5 dollars.
Keywords: Fractions, division
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Really need help with this .
Answer:
Step-by-step explanation:
The attached photo shows the diagram of quadrilateral QRST with more illustrations.
Line RT divides the quadrilateral into 2 congruent triangles QRT and SRT. The sum of the angles in each triangle is 180 degrees(98 + 50 + 32)
The area of the quadrilateral = 2 × area of triangle QRT = 2 × area of triangle SRT
Using sine rule,
q/SinQ = t/SinT = r/SinR
24/sin98 = QT/sin50
QT = r = sin50 × 24.24 = 18.57
Also
24/sin98 = QR/sin32
QR = t = sin32 × 24.24 = 12.84
Let us find area of triangle QRT
Area of a triangle
= 1/2 abSinC = 1/2 rtSinQ
Area of triangle QRT
= 1/2 × 18.57 × 12.84Sin98
= 118.06
Therefore, area of quadrilateral QRST = 2 × 118.06 = 236.12
Answer:
216 square units
Step-by-step explanation:
Apparently, we're supposed to ignore the fact that the given geometry cannot exist. The short diagonal is too short to reach between the angles marked 98°. If Q and S are 98°, then R needs to be 110.13° or more for the diagonals to connect as described.
__
The equal opposite angles of 98° suggests that the figure is symmetrical about the diagonal RT. That being the case, diagonal RT will meet diagonal QS at right angles. Then the area is half the product of the lengths of the diagonals:
(1/2)×18×24 = 216 . . . . square units
_____
In a quadrilateral, the area can be computed as half the product of the diagonals and the sine of the angle between them. Here, we have assumed the angle to be 90°, so the area is simply half the product of diagonal measures.
If the function b(t) gives the number of boats it takes t people to cross a river, what is the appropriate domain?
Answer:
whole numbers
Step-by-step explanation:
The domain is the number of people. The smallest number of people you could have would be 0 people so the appropriate domain is whole numbers.
Make me the brainliest
Answer:
whole numbers
Step-by-step explanation:
Find the imaginary part of\[(\cos12^\circ+i\sin12^\circ+\cos48^\circ+i\sin48^\circ)^6.\]
Answer:
The imaginary part is 0
Step-by-step explanation:
The number given is:
[tex]x=(\cos(12)+i\sin(12)+ \cos(48)+ i\sin(48))^6[/tex]
First, we can expand this power using the binomial theorem:
[tex](a+b)^k=\sum_{j=0}^{k}\binom{k}{j}a^{k-j}b^{j}[/tex]
After that, we can apply De Moivre's theorem to expand each summand:[tex](\cos(a)+i\sin(a))^k=\cos(ka)+i\sin(ka)[/tex]
The final step is to find the common factor of i in the last expansion. Now:
[tex]x^6=((\cos(12)+i\sin(12))+(\cos(48)+ i\sin(48)))^6[/tex]
[tex]=\binom{6}{0}(\cos(12)+i\sin(12))^6(\cos(48)+ i\sin(48))^0+\binom{6}{1}(\cos(12)+i\sin(12))^5(\cos(48)+ i\sin(48))^1+\binom{6}{2}(\cos(12)+i\sin(12))^4(\cos(48)+ i\sin(48))^2+\binom{6}{3}(\cos(12)+i\sin(12))^3(\cos(48)+ i\sin(48))^3+\binom{6}{4}(\cos(12)+i\sin(12))^2(\cos(48)+ i\sin(48))^4+\binom{6}{5}(\cos(12)+i\sin(12))^1(\cos(48)+ i\sin(48))^5+\binom{6}{6}(\cos(12)+i\sin(12))^0(\cos(48)+ i\sin(48))^6[/tex]
[tex]=(\cos(72)+i\sin(72))+6(\cos(60)+i\sin(60))(\cos(48)+ i\sin(48))+15(\cos(48)+i\sin(48))(\cos(96)+ i\sin(96))+20(\cos(36)+i\sin(36))(\cos(144)+ i\sin(144))+15(\cos(24)+i\sin(24))(\cos(192)+ i\sin(192))+6(\cos(12)+i\sin(12))(\cos(240)+ i\sin(240))+(\cos(288)+ i\sin(288))[/tex]
The last part is to multiply these factors and extract the imaginary part. This computation gives:
[tex]Re x^6=\cos 72+6cos 60\cos 48-6\sin 60\sin 48+15\cos 96\cos 48-15\sin 96\sin 48+20\cos 36\cos 144-20\sin 36\sin 144+15\cos 24\cos 192-15\sin 24\sin 192+6\cos 12\cos 240-6\sin 12\sin 240+\cos 288[/tex]
[tex]Im x^6=\sin 72+6cos 60\sin 48+6\sin 60\cos 48+15\cos 96\sin 48+15\sin 96\cos 48+20\cos 36\sin 144+20\sin 36\cos 144+15\cos 24\sin 192+15\sin 24\cos 192+6\cos 12\sin 240+6\sin 12\cos 240+\sin 288[/tex]
(It is not necessary to do a lengthy computation: the summands of the imaginary part are the products sin(a)cos(b) and cos(a)sin(b) as they involve exactly one i factor)
A calculator simplifies the imaginary part Im(x⁶) to 0
Find the rate of change for x³. You need to work out the change in f(x)=x³ when x is increased by a small number h to x+h. So you will work out f(x+h)-f(x). Then do some algebra to simplify this. Then divide this by h to get the average rate of change of f(x) between x and x+h. The average rate of change of f(x) from x to x+h is:
Answer:
3x² +3xh +h²
Step-by-step explanation:
[tex]\dfrac{f(x+h)-f(x)}{h}=\dfrac{(x+h)^3-x^3}{h}=\dfrac{(x^3+3x^2h+3xh^2+h^3)-x^3}{h}\\\\=\dfrac{3x^2h+3xh^2+h^3}{h}=3x^2+3xh+h^2[/tex]
Diana is painting statues. She has \dfrac{7}{8} 8 7 start fraction, 7, divided by, 8, end fraction of a liter of paint remaining. Each statue requires \dfrac{1}{20} 20 1 start fraction, 1, divided by, 20, end fraction of a liter of paint. How many statues can she paint?
Answer:
Number of statues that can be painted are 17
Step-by-step explanation:
Initially Diana has [tex]\frac{7}{8}[/tex] liters of paint remaining.
Every statue requires [tex]\frac{1}{20}[/tex] liters of paint for painting.
We have to find how many statues we will be able to paint with this remaining paint.
To get the number of statues,
Number of statues = [tex]\frac{Paint remaining}{Paint required for 1 statue}[/tex]
number of statues = [tex]\frac{\frac{7}{8} }{\frac{1}{20} }[/tex]
= [tex]\frac{35}{2}[/tex] = 17.5
Since the number of statues is not an integer the maximum number of statues that can be painted are 17.
Answer: 35/2
Step-by-step explanation:
Assume that a surveyor stands at the top of a mountain that is "h" feet tall. If the distance (in feet) that he can see is defined by d = 3200.2 SQRT(h), then answer the following. (a) How far can the surveyor see from the top of a 2000-foot mountain? (b) How tall is the mountain, if the surveyor can see 15 miles? (Note: 1 mile equals 5280 feet.)
Answer:
a) d = 143,117 ft
b) h = 612.45 ft
Step-by-step explanation:
If height of the mountain = h
And distance till the surveyor can see = d = 3200.2 SQRT (h)
Refer to attached file for graphical representation
Then;
A) If h=2000 ft
Then d =3200.2 √ (2000)
d = 3200.2 (44.72)
d = 143,117 ft
B) If d = 15 miles
1mile = 5280 ft
15 mile = 15*5280
15 mile = 79,200 ft
Therefore;
d = 79,200 ft
Since,
d =3200.2 √ (h)
79,200 = 3200.2 √ (h)
79200/3200.2 =√ (h)
√ (h) = 24.75
{√ (h)} ² = (24.75) ²
h = 612.45 ft
demochares has ived a fourth of his life as a boy, a fifth as a youth, a third as a man, and has spend 13 years in his dotage. how old is he?
Answer: 60 years
Step-by-step explanation:
Let x denotes the age of Demochares .
Time he spent as a boy = [tex]\dfrac{x}{4}[/tex]
Time he spent as a youth = [tex]\dfrac{x}{5}[/tex]
Time he spent as a man= [tex]\dfrac{x}{3}[/tex]
Time he spent in dotage= 13 years
As per given , we have the following equation:
[tex]x=\dfrac{x}{4}+\dfrac{x}{5}+\dfrac{x}{3}+13[/tex]
[tex]x=\dfrac{15x+12x+20x}{60}+13[/tex] [Take LCM]
[tex]x=\dfrac{47x}{60}+13[/tex]
[tex]x-\dfrac{47x}{60}=13[/tex]
[tex]\dfrac{60x-47x}{60}=13[/tex]
[tex]\dfrac{13x}{60}=13[/tex]
[tex]x=13\times\dfrac{60}{13}=60[/tex]
Hence, he is 60 years old.
Howdy! Id love to have these questions answered asap! Thank you for the help!
1) Which angle is not coterminal to 120 degrees?
A. 840
B. -180
C. 480
2) Use the unit circle and the reference angle to determine which of the following trigonometric values is correct when theta = -90
A. Cos theta = undefined
B. Sin theta = -1
C. Tan = 0
Answer:
1. B.
2. B.
Step-by-step explanation:
Trigonometry
1) Coterminal angles can be found by adding or subtracting 360° (or 2\pi radians) to a given angle. If we have 120°, adding 360° gives 480°, adding again 360° gives 840°. There is no way to get -180°, so this option is not a coterminal angle to 120°
2)
A. [tex]Cos (-90^o)=0[/tex], and not undefined
B. [tex]Sin (-90^o)=-1[/tex]. This is correct
C. [tex]Tan (-90^o)[/tex] is undefined, not zero
Thus the only correct option is B.
On a coordinate plane, a dashed straight line has a negative slope and goes through (0, 2) and (4, 0). Everything below and to the left of the line is shaded. Which point is a solution to the linear inequality y < Negative one-halfx + 2? (2, 3) (2, 1) (3, –2) (–1, 3)
Answer:
Option C.
Step-by-step explanation:
A dashed straight line has a negative slope and goes through (0, 2) and (4, 0).
The given inequality is
[tex]y<-\dfrac{1}{2}x+2[/tex]
We need find the point which is a solution to the given linear inequality.
Check the given inequality for point (2, 3).
[tex]3<-\dfrac{1}{2}(2)+2[/tex]
[tex]3<1[/tex]
This statement is false. Option 1 is incorrect.
Check the given inequality for point (2, 1).
[tex]1<-\dfrac{1}{2}(2)+2[/tex]
[tex]1<1[/tex]
This statement is false. Option 2 is incorrect.
Check the given inequality for point (3, -2).
[tex]-2<-\dfrac{1}{2}(3)+2[/tex]
[tex]-2<0.5[/tex]
This statement is false. Option 3 is correct.
Check the given inequality for point (-1,3).
[tex]3<-\dfrac{1}{2}(1)+2[/tex]
[tex]3<1.5[/tex]
This statement is false. Option 4 is incorrect.
Therefore, the correct option is C.
Answer:
C
Step-by-step explanation:
(2,1)