a. By the binomial theorem,
[tex]\displaystyle\left(x^2+\frac1x\right)^n=\sum_{k=0}^n\binom nk(x^2)^{n-k}\left(\frac1x\right)^k=\sum_{k=0}^n\binom nkx^{2n-3k}[/tex]
which produces a constant term when [tex]2n-3k=0[/tex], or [tex]2n=3k[/tex]. [tex]2n[/tex] is even for any choice of [tex]n[/tex], so [tex]n[/tex] must be any positive integer for which [tex]2n[/tex] is an even multiple of 3 i.e. a multiple of 6. Then [tex]n[/tex] can be any of the integers in the sequence {6, 12, 18, ...}.
b. Let [tex]n=6m[/tex] for some positive integer [tex]m[/tex]. Then
[tex]\displaystyle\left(x^2+\frac1x\right)^{6m}=\sum_{k=0}^{6m}\binom {6m}k(x^2)^{6m-k}\left(\frac1x\right)^k=\sum_{k=0}^{6m}\binom{6m}kx^{12m-3k}[/tex]
and the constant term occurs when [tex]12m-3k=0[/tex], or [tex]4m-k=0[/tex], or [tex]k=4m[/tex]. At this value of [tex]k[/tex], we get the term
[tex]\dbinom{6m}{4m}x^{12m-3(4m)}=\dfrac{(6m)!}{(4m)!(6m-4m)!}=\dfrac{(6m)!}{(4m)!(2m)!}[/tex]
Which represents the solution(s) of the system of equations, y = x2 – x + 1 and y = x? Determine the solution set by graphing.
Answer:
(1 , 1)
Step-by-step explanation:
Given in the question two equations
Equation 1
y = x² – x + 1
Equation 2
y = x
Equate both equations
x² – x + 1 = x
0 = x² – x + 1 - x
x² – 2x + 1 = 0
x² - x - x + 1 = 0
x(x - 1) -1(x - 1) = 0
(x - 1)(x - 1) = 0
(x-1)² = 0
x - 1 = 0
x = 1
Plug x = 1 in first equation
y = 1² – 1 + 1
y = 1
Answer:
The answer is option A, or (1,1)
Step-by-step explanation:
I just took the test
A Ferris wheel has a radius of 35 m. Its center is 36 m above the ground. It rotates once every 60 s. Suppose you get on the bottom at t = 0 .
Write an equation that expresses your height as a function of elapsed time.
h = 36 cos 2π (t - 30) / 60 + 35
h = 60 cos 2π (t - 35) / 36 +60
h = 35 cos π (t - 30) / 60 + 36
h = 35 cos 2π (t - 30) / 60 +36
The equation expressing the height as a function of elapsed time for the provided Ferris wheel scenario is: h = 35 cos 2π (t - 30) / 60 + 36. This assumes the highest point is reached halfway through the rotation and uses a cosine function to map the cycle.
Explanation:The subject of this question is mathematics, specifically it's about trigonometric functions and their applications to real world scenarios. The question is seeking for an equation that can express the height of a point on a Ferris wheel as a function of elapsed time.
Here's the solution: The highest height will be the radius of the Ferris wheel (35m) plus the height of the wheel's center above the ground (36m), which is a total of 71m. This height will be reached halfway through the rotation period of the wheel (30s). So, as the function of time, height can be represented by a cosine function shifted to the right by 30s with a period of 60s. Therefore, the equation to represent this scenario correctly is:
h = 35 cos 2π (t - 30) / 60 + 36
In this equation, 'h' stands for height, 't' represents time, 'cos' is the cosine function, 'π' is pi, and '35', '60', '30', and '36' are specific constants representing the radius of the Ferris wheel, the period of rotation, the time shift, and the wheel's center height respectively.
Learn more about Trigonometric Functions here:https://brainly.com/question/31540769
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Kevin caught some fish. Of them 4/9 were herring, and 2/9 salmon. What was the total amount of fish, amount of herring and amount of salmon if there were 12 flounders?
There were 3 different fish caught, Salmon, Herring and Flounder.
4/9 were Herring, 2/9 were Salmon, 4/9 +2/9 = 6/9
This means 3/9 were Flounders ( 3/9 + 6/9 = 9/9 = 1)
3/9 can be reduced to 1/3.
1/3 of the fish were Flounders.
Divide the amount of flounders by the portion caught:
12 / 1/3 = 12 * 3/1 = 36 total fish.
Now you have total number of fish, multiply the total by each portion for each type of fish.
Total fish = 36
Herring = 4/9 x 36 = 16
Salmon = 2/9 x 36 = 8
The period of this function is
2π
π/8
π/4
π/2
I’m not quite sure what the function is but if it’s sine the period is pi/8
ANSWER
The period is
[tex] \frac{\pi}{8} [/tex]
EXPLANATION
The given function completes four cycles on the interval
[tex]
[ - \frac{\pi}{4} , \frac{\pi}{4} ][/tex]
Therefore the period is
[tex] = \frac{ \frac{\pi}{4} - \frac{\pi}{4} }{4} [/tex]
Simplify the numerator:
[tex] = \frac{ \frac{\pi}{2}}{4} [/tex]
Simplify further to get:
[tex]= \frac{\pi}{8} [/tex]
Determine the area of a sector with a central angle of 90° and a radius of 10 meters.
A. 25 meters2
B. 90π meters2
C. 100π meters2
D. 25π meters2
Area of circle
= 90/360 x π x (10)^2
= 1/4 π x 100
= 25 π meter^2
Final answer:
The area of a sector with a central angle of 90° and a radius of 10 meters is calculated using the formula A = θ/360 × πr². Substituting the given values, the area is found to be 25π m². The correct answer is D. 25π meters².
Explanation:
Finding the Area of a Sector
To find the area of a sector with a central angle of 90° in a circle with a radius of 10 meters, we use the formula for the area of a sector, which is A = θ/360 × πr², where θ is the central angle in degrees, π is the mathematical constant approximately equal to 3.14159, and r is the radius. Since the central angle θ is 90°, and the radius r is 10 meters, we have:
A = 90°/360° × π × (10 m)²
= 0.25 × π × 100 m²
= 25π m².
Therefore, the correct answer is D. 25π meters².
Need help
Choices
4
3
2
1
Answer:
1
Step-by-step explanation:
The amplitude is the maximum distance away from the middle of the wave. Here we can see that the middle of the wave is the x axis and the farthest point (largest difference between the y coordinate of the x-axis, or the line y = 0, and the wave) is 1 unit away.
Answer:
1
Step-by-step explanation:
Please help...........
Answer:
x=18
Step-by-step explanation:
The two triangles are similar. The ratio of the sides are the same.
20/20=40/(40+x)
Cross multiply and solve. Hint: the easiest way to solve is to simplify, then solve.
Solve: 110°20'
PLS TRY to translate degrees and minutes to degrees! BRAINLIEST for 1st answer :)
Answer:
110 1/3 or 100.33 degrees (to the nearest hundredth).
Step-by-step explanation:
There are 60 ' (minutes) in a degree.
So 20' = 20/60 = 1/3 of a degree.
Answer:
110 1/3°
Step-by-step explanation:
Recall that 1° = 60'. Then 20' is 1/3 of 1°, or (1/3)°.
Then 110° 20' = 110° + (1/3)° = 110 1/3°.
What is the maximum number of turns in the graph of this function?
[tex]f(x) = x^4-x^3+3x+1[/tex]
Answer:
The maximum number of turns is 3
Step-by-step explanation:
The given function is
[tex]f(x)=x^4-x^3+3x+1[/tex]
The degree of this polynomial is 4.
If the degree of a given polynomial is n, then the polynomial has at least n-1 turns.
Therefore the number of turns of this 4th degree polynomial is at least 3.
Use the drop-down menus to identify the key values of the box plot. The median is . The minimum is . The maximum is . The lower quartile (Q1) is . The upper quartile (Q3) is .
ANSWERS:
C
A
E
B
D
Answer: The lower quartile is Q1) is your minimum & your maximum is quartile Q3) bc is the larger one
Step-by-step explanation:
c
a
e
b
d
is none of your answer.
Answer:
person above is correct
Step-by-step explanation:
show them some love
Decide which trigonometric ratio to use. Solve for x in the triangle below. Round your answer to the hundredths place.
Options
4.59
0.07
6.55
5.60
Answer: 4.59
sin(35)=x/8
8*sin(35)=x
x=4.59
The value of x in the given triangle round to the nearest hundredths is:
4.59
Step-by-step explanation:From the given triangle we observe that the side x is the opposite side corresponding to angle 35°.
i.e. we will use sine trignometric ratio in the triangle to get the value of x.
We know that:
[tex]\sin 35=\dfrac{x}{8}ddd[/tex]
Since the sine of an angle is the ratio of opposite side to the hypotenuse of the right angled triangle.
i.e.
[tex]x=8\times \sin 35\\\\i.e.\\\\x=8\times 0.5736\\\\i.e.\\\\x=4.588[/tex]
which is approximately equal to 4.59 units.
The center is a(?,?)
The radius is _____ units.
Answer:
The center is at [tex](2,4)[/tex]
3 units
Step-by-step explanation:
You can observe in the graph a circle.
The x-coordinate and the y-coordinate of the center of the circle shown in the figure are:
[tex]x=2[/tex] and [tex]y=4[/tex]
Therefore, the point of the center of the circle is:
[tex](2,4)[/tex]
You can observe that the diameter goes from -1 to 5, then the diameter is:
[tex]diameter=1units+5units\\\\diameter=6units[/tex]
The radius is half the diameter. Therefore, you can say that the radius of this circle is:
[tex]radius=\frac{diameter}{2}\\\\radius=\frac{6units}{2}\\\\radius=3units[/tex]
ANSWER
The center is (2,4)
The radius is 2√2 units
EXPLANATION
Each of the ticks is 1 unit
Counting two ticks from the origin to the right and 4 units up will land us at the center of the circle.
Hence the center is (2,4)
The circle passes through (0,2).
The radius is
[tex]r = \sqrt{ {(2 - 0)}^{2} + {(4 - 2)}^{2} } [/tex]
[tex]r = \sqrt{8} = 2 \sqrt{2} [/tex]
How do I algebraically solve this? (solve for x)
Check the picture below.
It took Otis 6 hours to travel to the Grand Canyon. Along the way he took 18 minutes to get gasoline and 53 minutes to eat. How much time did Otis spend driving? With work
The answer is 4 hours and 49 minutes HERE IS MY WORK
4 hours and 49 minutes
Which of the following graphs shows the preimage P(x)=|x| and the image I(x)=12⋅P(x)?
Answer:
The picture where the red image is the skinniest
Step-by-step explanation:
The graph P(x) is the parent graph for all absolute functions. It has a vertex of (0,0) and has the following points:
x f(x)
-2 2
-1 1
0 0
1 1
2 2
The image of l(x) = 12P(x) changes the points of the function to
x f(x)
-2 24
-1 12
0 0
1 12
2 24
This makes the graph much skinnier. The graph with the skinniest red graph is the graph.
Answer:
The second graph is the right answer.Step-by-step explanation:
The parent function is
[tex]P(x)=|x|[/tex]
(Remember, a parent function refers to the simplest function of its type)
The image function is
[tex]I(x)=12P(x)[/tex]
Which is [tex]I(x)=12|x|[/tex].
Observe in the image attached, the image function is vertically stretched by a factor of 12.
Therefore, the right answer is the second graph.
PLEASE HELP ASAP!!! CORRECT ANSWER ONLY PLEASE!!!
The results of a poll show that the true proportion of students who prefer the new schedule is likely in the interval (0.195,0.245).
What is the point estimate of the proportion of students who prefer the new schedule?
Enter your answer, as a decimal, in the box.
Answer:
0.22
Step-by-step explanation:
The point estimate is the average of the upper and lower bounds of the interval.
6 / 2 (1 + 2) = 9
PLEASE EXPLAIN HOW IT IS 9!!! I followed PEMDAS, and I keep getting 1
Answer:
9
Step-by-step explanation:
6 / 2 (1 + 2) = 9
PEMDAS
Parentheses first
1+2 =3
Substitute this in
6 / 2 (3) = 9
Multiplication and Division from left to right
Division first
6/2 =3
Substitute this in for 6/2
3 (3) =9
3*3 =9
What is the recursive rule for the sequence?
Answer:
C
Step-by-step explanation:
Since we are dealing with recursive equations, we can use a simple format.
[tex]a_{1}=x \\ \\ a_{n}=a_{n-1}+y[/tex]
x is the starting value and y is the change.
We can see that the change is -5.6 since 5.6 is being subtracted from each term each time.
So the equation would be
[tex]a_n=a_{n-1}-5.6[/tex]
We can check this by plugging in the numbers.
Let's see if -8.3 works, which is the 2nd term in the equation.
[tex]a_2=a_{2-1}-5.6 \\ \\ -8.3=a_{1}-5.6 \\ \\ -8.3=-2.7-5.6 \\ \\ -8.3=-8.3[/tex]
Therefore, the recursive rule is the third one.
use the equation to answer the following question y=(x-3(x+2)/(x+4)(x-4)(x+2)
a. Find all points of discontinuity
b. Determine whether each point is removable(hole) or non-removable (vertical asymptote)
c.find the equation of the horizontal and vertical asymptotes for the rational function if any
Answer:
See below
Step-by-step explanation:
The given rational function is;
[tex]y=\frac{(x-3)(x+2)}{(x+4)(x-4)(x+2)}[/tex]
The given function is not continuous where the denominator is equal to zero.
[tex](x+4)(x-4)(x+2)=0[/tex]
The function is discontinuous at [tex]x=-4,x=4,x=-2[/tex]
b) The point at x=-2 is a removable discontinuity(hole) because (x+2) is common to both the numerator and the denominator.
The point at x=-4 and x=4 are non-removable discontinuities(vertical asymptotes)
c) The equation of the vertical asymptotes are x=-4 and x=4
To find the equation of the horizontal asymptote, we take limit to infinity.
[tex]\lim_{x\to \infty}\frac{(x-3)(x+2)}{(x+4)(x-4)(x+2)}=0[/tex]
The horizontal asymptote is y=0
A theater can hold 160 giants or 240 elves. If 109 giants are inside how many elves can also be admitted.
Answer:
it depends
Step-by-step explanation:
If the tradeoff between giants and elves is linear, then about 76 elves can be admitted. If it is non-linear, we need more information about the tradeoff.
For example, elves may fit in the spaces between giants in such a way that they take up proportionately less room. Or, elves may repel giants in such a way that they take up proportionately more room.
See the attached graph for some possibilities.
_____
The linear equation can be written as ...
g/160 + e/240 = 1
Then for g=109, we can solve for "e" as ...
109/160 + e/240 = 1
240(1 -109/160) = e = (3/2)(51) = 76.5
If the tradeoff is linear, 76 elves can be admitted.
Though it is true I enjoy your writing style, I don't agree with your main point of view about this one. I do delight in your website nevertheless. Gedefkkfdefd
Answer:
34.
Step-by-step explanation:
Prove or disprove the identity. if you find the identity is true, state the first line of the proof. if you find the identity is false, write the correct equation by replacing the right side. sec2 x(1 – sin2 x) = csc2 x cos x
Answer:
False
Step-by-step explanation:
Using the trigonometric identities
• sin²x + cos²x = 1
• secx = [tex]\frac{1}{cosx}[/tex]
Consider the left side
sec²x(1 - sin²x)
= [tex]\frac{1}{cos^2x}[/tex] × cos²x = 1
The correct identity is
sec²x(1 - sin²x) = 1
Please help me with this
Answer:
8.4
Step-by-step explanation:
hope it helped
Can someone help with these 8 problems and show your work please.
Answer:
Part 1) option a. [tex]y=(x+1)^{2}[/tex]
Part 2) option c. [tex]y(x)=10x+1[/tex]
Part 3) option a. Yes , d=-2
Part 4) option b. [tex]y=2x+4[/tex]
Part 5) option b. [tex]m=-2[/tex]
Part 6) option c. [tex]y=4x+14[/tex]
Part 7) option c. [tex]y=4x+5[/tex]
Part 8) option a. y=2x-1 and y=x+1
Step-by-step explanation:
Part 1)
we know that
If a ordered pair satisfy a function, then the function pass through the ordered pair
Verify each function with the points (1,4), (2,9) and (3,16)
case a) we have
[tex]y=(x+1)^{2}[/tex]
For x=1, y=4
[tex]4=(1+1)^{2}[/tex]
[tex]4=4[/tex] ----> is true
For x=2, y=9
[tex]9=(2+1)^{2}[/tex]
[tex]9=9[/tex] ----> is true
For x=3, y=16
[tex]16=(3+1)^{2}[/tex]
[tex]16=16[/tex] ----> is true
therefore
The function pass through the three points
case b) we have
[tex]y=(x+3)^{2}[/tex]
For x=1, y=4
[tex]4=(1+3)^{2}[/tex]
[tex]4=16[/tex] ----> is not true
therefore
The function not pass through the three points
case c) we have
[tex]y=7x-5[/tex]
For x=1, y=4
[tex]4=7(1)-5[/tex]
[tex]4=2[/tex] ----> is not true
therefore
The function not pass through the three points
Part 2)
Let
y------> the number of laps
x-----> the number of hours
we know that
The linear equation that represent this situation is
[tex]y(x)=10x+1[/tex]
Part 3) we have
{4,2,0,-2,-4,-6,...}
Let
a1=-4
a2=2
a3=0
a4=-2
a5=-4
a6=-6
we know that
a2-a1=2-4=-2 -----> a2=a1-2
a3-a2=0-2=-2 ----> a3=a2-2
a4-a3=-2-0=-2 -----> a4=a3-2
a5-a4=-4-(-2)=-2----> a5=a4-2
a6-a5=-6-(-4)=-2----> a6=a5-2
therefore
Is an arithmetic sequence, the common difference is -2
Part 4) we know that
The y-intercept of the graph is (0,4)
The x-intercept of the graph is (-2,0)
therefore
the function is [tex]y=2x+4[/tex]
because
For x=0 -----> y=2(0)+4 -----> y=4
For y=0 ----> 0=2x+4 --------> x=-2
Part 5) we know that
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
we have
[tex]A(3,5)\ B(2,7)[/tex]
substitute the values
[tex]m=\frac{7-5}{2-3}[/tex]
[tex]m=-2[/tex]
Part 6) we know that
The equation of the line into slope point form is equal to
[tex]y-y1=m(x-x1)[/tex]
we have
[tex]m=4[/tex]
[tex]point(-3,2)[/tex]
substitute the values
[tex]y-2=4(x+3)[/tex]
Convert to slope intercept form
[tex]y=4x+12+2[/tex]
[tex]y=4x+14[/tex]
Part 7) we know that
If two lines are parallel, then their slopes are the same
The equation of the given line is [tex]y=4x-2[/tex]
so
The slope of the given line is [tex]m=4[/tex]
therefore
The line [tex]y=4x+5[/tex] is parallel to the given line
Because the slope is equal to [tex]m=4[/tex]
Part 8) we know that
If a ordered pair is a solution of a system of equations, then the ordered pair must satisfy both equations of the system
Verify each case for (2,3)
case a)
y=2x-1 -----> equation 1
y=x+1 -----> equation 2
Substitute the value of x and the value of y in each equation and then compare the results
Verify equation 1
3=2(2)-1
3=3 -----> is true
Verify equation 2
3=2+1
3=3 -----> is true
therefore
The point (2,3) is a solution of the system of equations case a
case b)
y=2x+1 -----> equation 1
y=x-1 -----> equation 2
Substitute the value of x and the value of y in each equation and then compare the results
Verify equation 1
3=2(2)+1
3=5 -----> is not true
therefore
The point (2,3) is not a solution of the system of equations case b
case c)
y=4x-5 -----> equation 1
y=2x -----> equation 2
Substitute the value of x and the value of y in each equation and then compare the results
Verify equation 1
3=4(2)-5
3=3 -----> is true
Verify equation 2
3=2(2)
3=4 -----> is not true
therefore
The point (2,3) is not a solution of the system of equations case c
Which of the following ordered pairs represents an output of 5 and an input of -8?
(-8,-8)
(-8,5)
(5, 5)
(5, -8)
Answer:
(-8,5)
Step-by-step explanation:
The output corresponds to the y-value of an ordered pair while the input corresponds to the x-value of an order pair.
An ordered pair is written like this (x,y)
HELP PLEASE!!! I WILL FIVE STAR!!
Answer:
Step-by-step explanation:
You can't answer c. There is not enough information. Other than there are 14 boys that went, you know nothing about how many went.
Girls going = (3/5) * 20 = 12
Girls not going = 20 - 12 = 8
Please help me out please
Answer:
h = 10.92 m.
Step-by-step explanation:
The small triangle on the left is similar to the whole triangle, so:
12/20 = h/18.2
h = 12*18.2 / 20
h = 10.92 m.
Please help with #5 only
Answer:
1680Step-by-step explanation:
[tex]\sum\limits_{n=1}^{42}2n-3\\\\a_n=2n-3\\\\a_{n+1}=2(n+1)-3=2n+2-3=2n-1\\\\a_{n+1}-a_n=(2n-1)-(2n-3)=2n-1-2n+3=2=constant\\\\\text{Therefore it's an arithmetic sequence with first term}\\\\a_1=2(1)-3=2-3=-1\\\\\text{and common difference}\\\\d=2[/tex]
[tex]\text{The formula of a sum of terms of an aithmetic sequence:}\\\\S_n=\dfrac{2a_1+(n-1)d}{2}\cdot n\\\\\text{Substitute:}\\\\n=42,\ a_1=-1,\ d=2:\\\\S_{42}=\dfrac{2(-1)+(42-1)(2)}{2}\cdot42=[-2+(41)(2)](21)=(-2+82)(21)\\\\=(80)(21)=1680[/tex]
1. x/5 = 10 *
2. x + 2 = 15 *
3. 5x = 40 *
4. 2y = 44 *
5. r - 15 = 30 *
Answer:
Step-by-step explanation:
1) x/5 = 10
multiply by 5 : x = 50
2) x + 2 = 15
add -2 : x = 13
3) 5x = 40
divid by 5 : x =8
4) 2y = 44
divid by 2 : y = 22
5. r - 15 = 30
add 15 ; 5r =45
divid by 5 : r = 9
Point B is the image of point A when point A is rotated about the origin. What is known about point A and B?
A:Point B is the result of a 180° rotation.
B:Point A and B have the same x-coordinate.
C:Point B is the result of a 90° counterclockwise rotation.
D:Point A and point B are the same distance from the origin.
Answer:
D:Point A and point B are the same distance from the origin.
Step-by-step explanation:
hope this helps :)
Answer:
D
Step-by-step explanation:
Hit the THANKS button pls >.<
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