Answers
Recall the binomial theorem.
[tex](a+b)^n = \displaystyle \sum_{k=0}^n \binom nk a^{n-k} b^k[/tex]
1. The binomial expansion of [tex]\left(1+\frac x3\right)^7[/tex] is
[tex]\left(1 + \dfrac x3\right)^7 = \displaystyle\sum_{k=0}^7 \binom 7k 1^{7-k} \left(\frac x3\right)^k = \sum_{k=0}^7 \binom 7k \frac{x^k}{3^k}[/tex]
Observe that
[tex]k = 1 \implies \dbinom 71 \left(\dfrac x3\right)^1 = \dfrac73 x[/tex]
[tex]k = 2 \implies \dbinom 72 \left(\dfrac x3\right)^2 = \dfrac73 x^2[/tex]
When we multiply these by [tex]8-9x[/tex],
• [tex]8[/tex] and [tex]\frac73 x^2[/tex] combine to make [tex]\frac{56}3 x^2[/tex]
• [tex]-9x[/tex] and [tex]\frac73 x[/tex] combine to make [tex]-\frac{63}3 x^2 = -21x^2[/tex]
and the sum of these terms is
[tex]\dfrac{56}3 x^2 - 21x^2 = \boxed{-\dfrac73 x^2}[/tex]
2. The binomial expansion is
[tex]\left(2a - \dfrac b2\right)^8 = \displaystyle \sum_{k=0}^8 \binom 8k (2a)^{8-k} \left(-\frac b2\right)^k = \sum_{k=0}^8 \binom 8k 2^{8-2k} a^{8-k} b^k[/tex]
We get the [tex]a^6b^2[/tex] term when [tex]k=2[/tex] :
[tex]k=2 \implies \dbinom 82 2^{8-2\cdot2} a^{8-2} b^2 = 28 \cdot2^4 a^6 b^2 = \boxed{448} \, a^6b^2[/tex]
Related Questions
factor the gratest common factor: -5k^2+20k-30
Answers
Answer:
±5
Step-by-step explanation:
-5k²+20k-30
if u look at the equation ±5 are the greatest common factors so we ±5(±k²±4k±6)
pls solve this. this is of set class 9.
Answers
Here is the answer :
people who like both the songs is 35 and
people who like only folk songs is 100.
what is the formula to calculate area of square
Answers
Answer:
Step-by-step explanation:
A square is made up of 4 sides that are equal in length. They can all be called "x" or "s" or whatever variable you want. The formula for the area of a square (and also the area of a rectangle) is Area = side × side.
If you called your sides "x", then the formula is A = x × x or A = x². Another way to note the area of either a square or a rectangle is to call the "bottom" the base and one of the sides the height and then the area formula looks like this:
A = b × h
or
A = length × width
They are all the same thing.
A steel hex nut has two regular hexagonal bases and a cylindrical hole with a diameter of 1.6 centimeters through the middle. the apothem of the hexagon is 2 centimeters. a cylinder is cut out of the middle of a hexagonal prism. the hexagon has an apothem with a length of 2 centimeters and base side lengths of 2.3 centimeters. the prism has a height of 2 centimeters. the cylinder has a diameter of 1.6 centimeters. the equation for the area of a regular hexagon = one-half (apothem) (perimeter). what is the volume of metal in the hex nut, to the nearest tenth? use 3.14 for π.
Answers
Subtracting the volume of the cylinder from the volume of the prism, the volume of metal in the hex nut to the nearest tenth exists [tex]$$23.6 cm^3[/tex]
How to estimate the volume of metal in the hex nut?Diameter of the cylinder be d = 1.6 cm
Apothem of the hexagon be a = 2 cm
Thickness of the steel hex nut be t = 2 cm
Volume of the prism be [tex]V_p[/tex]
Volume of the cylinder be [tex]V_c[/tex]
Volume of metal in the hex nut,
[tex]$$V = V_p - V_c[/tex]
To estimate the volume of a prism,
[tex]$$V_p = A_b h[/tex]
Ab = n L a / 2
Number of the sides, n = 6
The side of the hexagon be L
Height of the prism, h = t = 2 cm
Central angle in the hexagon, A = 360°/n
A = 360°/6 = 60°
[tex]$tan (\frac{A}{2} )=(\frac{L/2}{a})[/tex]
simplifying the value of L, we get
[tex]$tan (\frac{60}{2} )=(\frac{L/2}{2})[/tex]
[tex]$tan 30}=(\frac{L/2}{2})[/tex]
[tex]$tan (\frac{\sqrt{3}}{3} )=(\frac{L/2}{2})[/tex]
Solving for L/2:
[tex]$\frac{2 \sqrt{3}}{3} =\frac{L}{2}[/tex]
Solving the value of L, we get
[tex]$2\frac{2 \sqrt{3}}{3} =L[/tex]
[tex]$\frac{4 \sqrt{3}}{3} =L[/tex]
[tex]$L=4 \sqrt{3}/3 cm[/tex]
Ab = n L a / 2
Substitute the values in the above equation, we get
[tex]$A_b=\frac{6 (4 \sqrt{3}/3)(2)}{2}[/tex]
[tex]$$A_b=24 \sqrt{3}/3 $$cm^2[/tex]
[tex]$A_b=8 \sqrt{3} cm^2[/tex]
[tex]V_p = A_b h[/tex]
substitute the values in the above equation, we get
[tex]$V_p=(8 \sqrt{3})(2)[/tex]
[tex]$V_p=16 \sqrt{3} cm^3[/tex]
[tex]$$V_p=16 (1.732) cm^3[/tex]
[tex]$$V_p=27.712 cm^3[/tex]
To estimate the volume of cylinder,
[tex]$V_c[/tex] = (π[tex]d^2[/tex]/4) h
Here, π = 3.14 and d = 1.6 cm
Height of the cylinder, h = t = 2 cm
substitute the values in the above equation, we get
[tex]$V_c=[3.14 (1.6) / 4] (2)[/tex]
[tex]$V_c=[3.14 (2.56) / 4] (2)[/tex]
[tex]$V_c=(2.0096) (2)[/tex]
[tex]$$V_c=4.019 cm^3[/tex]
Substitute the values in the equation, we get
[tex]$$V=V_p-V_c[/tex]
[tex]$$V=27.712 - 4.019[/tex]
[tex]$$V=23.693 cm^3[/tex]
[tex]$$V=23.6 cm^3[/tex]
Therefore, the volume of metal in the hex nut, to the nearest tenth exists [tex]$$23.6 cm^3[/tex].
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The mean if four numbers is 25 if three numbers are 36, 12 and 24 what is the fiurth number
Answers
correct answer is 28.
If the mean of four numbers is 25 and three numbers are 36, 12 and 24 then fourth number is 28.
Find fourth number?given terms are:
first number =36
second number=12
third number=24
fourth number=?
let us assume the fourth number be x.
what is mean?we know the formula to find mean is:
Mean=addition of given terms / total number of terms
According to given statement, we put the values in above equation and we get,
25=(36+12+24+x) / 4
By rearranging, we get
36+12+24+x=100
72+x=100
x=100-72
x=28
so, the fourth number is 28.
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what is the volume of a triangular pyramid that is 5 feet tall and has a base area of 9 square feet
Answers
The volume of the triangular prism as described is; 15 cubic foot.
What is the volume of the triangular pyramid?A triangular pyramid simply refers to any geometric solid with a triangular base, and all three lateral faces are also triangles with a common vertex
It follows from the formula for calculating the volume of a triangular prism that;
Volume = (1/3) × base area × height.
Consequently,
volume of the prism is; = (1/3) ×9 × 5
Volume = 15 cubic foot
Therefore, volume of the triangular prism as described is; 15 cubic foot.
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What is the Value of x
Answers
Answer:
x = 11
Step-by-step explanation:
A segment parallel to a side of a triangle cuts off proportional segments on the sides.
48/6 = (3x + 7)/5
8 = (3x + 7)/5
3x + 7 = 40
3x = 33
x = 11
Can anyone help me with this? I'm not very good at math... =.=
Answers
yes if horizal is mirrored
Solve question in image (yr 8 math, 40 pt)
Answers
Answer:
y = 3
Step-by-step explanation:
(5y - 3)/4 + 6 = 3y
5y - 3 + 24 = 12y
7y = 21
y = 3
Check.
(5*3 - 3)/4 + 6 = 3*3
(15 - 3)/4 + 6 = 9
12/4 + 6 = 9
3 + 6 = 9
9 = 9
Answer: y = 3
Step-by-step explanation:
Given equation
[tex]\frac{5y~-~3}{4} ~+~6~=~3y[/tex]
Multiply 4 on both sides (to eliminate fraction)
[tex]\frac{5y~-~3}{4} *4~+~6*4~=~3y*4[/tex]
[tex](5y~-~3)~+~24~=~12y[/tex]
Expand parenthesis and combine like terms
[tex]5y~-~3~+~24~=~12y[/tex]
[tex]5y~+~21~=~12y[/tex]
Subtract 5y on both sides
[tex]5y~+~21~-~5y~=~12y~-~5y[/tex]
[tex]21~=~7y[/tex]
Divide 7 on both sides
[tex]21~/~7~=~7y~/~7[/tex]
[tex]\Large\boxed{y=3}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
The growth of algae can be modelled by the function f(t) = 2¹.
Find the value of t such that f(t) = 128.
Answers
Answer:
t = 7
Step-by-step explanation:
You can use your knowledge of powers of 2, or you can use logarithms to find the value of t.
Powers of 22^1 = 2
2^2 = 4
2^4 = 16
128 = 16×4×2 = (2^4)(2^2)(2^1) = 2^(4+2+1) = 2^7
Now, the equation is ...
f(t) = 2^t = 2^7
Equating exponents, we have ...
t = 7
LogarithmsTaking the log of both sides of the equation ...
2^t = 128
we have ...
t×log(2) = log(128)
t = log(128)/log(2) = 7 . . . . . divide by the coefficient of t
The value of t is 7.
__
Additional comment
The relevant rule of exponents is ...
(a^b)(a^c) = a^(b+c)
The relevant rule of logarithms is ...
log(a^b) = b×log(a)
The functions f(x) = −(x − 1)2 5 and g(x) = (x 2)2 − 3 have been rewritten using the completing-the-square method. apply your knowledge of functions in vertex form to determine if the vertex for each function is a minimum or a maximum and explain your reasoning.
Answers
The vertex of the function f(x) exists (1, 5), the vertex of the function g(x) exists (-2, -3), and the vertex of the function f(x) exists maximum and the vertex of the function g(x) exists minimum.
How to determine the vertex for each function is a minimum or a maximum?Given:
[tex]$\mathrm{f}(\mathrm{x})=-(\mathrm{x}-1)^{2}+5$[/tex] and
[tex]$\mathrm{g}(\mathrm{x})=(\mathrm{x}-2)^{2}-3$[/tex]
The generalized equation of a parabola in the vertex form exists
[tex]$y=a(x-h)^{2}+k[/tex]
Vertex of the function f(x) exists (1, 5).
Vertex of the function g(x) exists (-2, -3).
Now, if (a > 0) then the vertex of the function exists minimum, and if (a < 0) then the vertex of the function exists maximum.
The vertex of the function f(x) exists at a maximum and the vertex of the function g(x) exists at a minimum.
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Answer:
- For the function f(x) = -(x - 1)^2/5, the vertex is a maximum point.
- For the function g(x) = x^4 - 3, the vertex is a minimum point.
Step-by-step explanation:
To determine if the vertex for each function is a minimum or a maximum, we need to analyze the leading coefficient of each function.
Let's start with the function f(x) = -(x - 1)^2/5. This function is already in vertex form, where the vertex is (1, 0).
Since the leading coefficient is negative (-1/5), the graph of f(x) opens downward. In this case, the vertex is a maximum point. You can visualize it as a peak on a graph.
Now, let's move on to the function g(x) = (x^2)^2 - 3. This function can be rewritten in vertex form as g(x) = x^4 - 3.
The leading coefficient is positive (1), so the graph of g(x) opens upward. In this case, the vertex is a minimum point. You can visualize it as a valley on a graph.
In summary:
- For the function f(x) = -(x - 1)^2/5, the vertex is a maximum point.
- For the function g(x) = x^4 - 3, the vertex is a minimum point.
Remember, the sign of the leading coefficient determines whether the vertex is a minimum or maximum point.
OA=
Please help asap!! Thanks so much :))
Answers
In the given diagram, the value of the dashed side of rhombus OABC is 5
Distance between two pointsFrom the question, we are to determine the length of the dashed line (OA), in rhombus OABC
In the diagram, we can observe that the length of OA is the distance between point A and the origin (O).
Using the formula for calculating distance between two points,
d =√[(x₂-x₁)² + (y₂-y₁)²]
In the diagram,
The coordinate of the origin is (0, 0)
The coordinate of point A is (3, 4)
Thus,
x₁ = 0
x₂ = 3
y₁ = 0
y₂ = 4
Putting the parameters into the formula, we get
OA =√[(3-0)² + (4-0)²]
OA =√(3² + 4²)
OA =√(9+16)
∴ OA =√25
OA = 5
Hence, in the given diagram, the value of the dashed side of rhombus OABC is 5
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[tex]\frac{12}{20}[/tex] = ______% = ______ hundredths
Answers
The percentage form of given fraction is 60% and the hundredths form is 0.60
According to the statement
we have given that the a fraction and we have to find the percentage of that fraction and write in the form hundredths.
So, For this purpose,
The given fraction is 12/20.
Then the definition of the percentage is that
The Percentage, a relative value indicating hundredth parts of any quantity.
so, the percentage of given fraction is :
Percentage fraction = 12/20 * 100
After solving it, The percentage fraction will become:
Percentage fraction = 60%
and Now convert into the hundredths form then
In the hundredths form it will become
from 60% to 0.60.
So, The percentage form of given fraction is 60% and the hundredths form is 0.60
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Identify the standard form of the equation by completing the square.
4x2 − 9y2 − 8x + 36y − 68 = 0
Answers
Answer:
[tex]\dfrac{(x-1)^2}{9}-\dfrac{(y-2)^2}{4}=1[/tex]
Step-by-step explanation:
Given equation:
[tex]4x^2-9y^2-8x+36y-68=0[/tex]
This is an equation for a horizontal hyperbola.
To complete the square for a hyperbola
Arrange the equation so all the terms with variables are on the left side and the constant is on the right side.
[tex]\implies 4x^2-8x-9y^2+36y=68[/tex]
Factor out the coefficient of the x² term and the y² term.
[tex]\implies 4(x^2-2x)-9(y^2-4y)=68[/tex]
Add the square of half the coefficient of x and y inside the parentheses of the left side, and add the distributed values to the right side:
[tex]\implies 4\left(x^2-2x+\left(\dfrac{-2}{2}\right)^2\right)-9\left(y^2-4y+\left(\dfrac{-4}{2}\right)^2\right)=68+4\left(\dfrac{-2}{2}\right)^2-9\left(\dfrac{-4}{2}\right)^2[/tex]
[tex]\implies 4\left(x^2-2x+1\right)-9\left(y^2-4y+4\right)=36[/tex]
Factor the two perfect trinomials on the left side:
[tex]\implies 4(x-1)^2-9(y-2)^2=36[/tex]
Divide both sides by the number of the right side so the right side equals 1:
[tex]\implies \dfrac{4(x-1)^2}{36}-\dfrac{9(y-2)^2}{36}=\dfrac{36}{36}[/tex]
Simplify:
[tex]\implies \dfrac{(x-1)^2}{9}-\dfrac{(y-2)^2}{4}=1[/tex]
Therefore, this is the standard equation for a horizontal hyperbola with:
center = (1, 2)vertices = (-2, 2) and (4, 2)co-vertices = (1, 0) and (1, 4)[tex]\textsf{Asymptotes}: \quad y = -\dfrac{2}{3}x+\dfrac{8}{3} \textsf{ and }y=\dfrac{2}{3}x+\dfrac{4}{3}[/tex][tex]\textsf{Foci}: \quad (1-\sqrt{13}, 2) \textsf{ and }(1+\sqrt{13}, 2)[/tex]solve the equation (39+5=5a=9)
Answers
Answer:
EVALUATE
false
Step-by-step explanation:
Complex number
(39+5=5a=9)
5107465
State the value of digit in figures ?
Answers
Place value of 5 in 5107465 is equal to 5,000,000.
According to the statement
we have given that the a value and we have find the place value of a digit 5 in this numbers.
So, For this purpose,
The given digit is 5107465
And we have find the value of digit 5 in it so,
we know that,
A Place value is the position or place of a digit in a given number .
As we can see that, in given number 5107465:-
5 is at one million place = 5 * 1,000,000 = 5,000,000
1 is at hundred thousand place = 1 * 100000 = 100000
0 is at ten thousand place = 0 * 10000 = 0
7 is at thousand place = 7 * 1000 = 7000
4 is at hundred place = 4 * 100 = 400
6 is at tens place = 6 * 10 = 60
5 is at unit place = 5 * 1 = 5
Therefore,
Place value of 5 in 5107465 is equal to 5,000,000.
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18 1 / 2 miles of highway 1-35 Is undergoing repairs. The construction workers repairs The construction workers repair 2 1⁄4 Miles of hideaway each week Approximately How many weeks will it take for the repairs to be completed
Answers
Answer:
The exact amount of time would be 8 2/9 weeks, but if we are only counting full weeks, it would be finished in 9 weeks.
Step-by-step explanation:
y = xm
18 1/2 = 2 1/4x Change both numbers into improper fractions
18 1/2 would be 37/2 The denominator (bottom number) stays 2. To get the top number you would (2 x 18) + 1 or 37. That makes 37/2 equivalent to 18 1/2.
Now do the same thing to 2 1/4. The bottom number stays the same. We get the top number (4 x 2) + 1 or 9. That makes 9/4 equivalent to 2 1/4.
We know have
37/2 = 9/4 x We can solve for x by multiplying both sides of the equation by 4/9
37/2(4/9) = (9/4)(4/9)x
148/18 = x If we divide both the top and the bottom by 2 we would get 74/9 = x If we divide 74 by 9 we get 8 2/9
PLEASE HELP IM STUCK
Answers
Answer:
6x + 4y = 16
Step-by-step explanation:
2(3x + 2y = 8)
2(3x + 2y) = 2(8)
6x + 4y = 16
Answer:
6x+4y=16
Step-by-step explanation:
2 (3x+2y=8)
2×3x+2×2y=2×8
=6x+4y=16 ans.
I don’t understand what I did wrong
Answers
Step-by-step explanation:
You got your fraction messed up
It should be -3/2 or -1.5
Which statement is true about the value of Dante’s assets and liabilities?
He has $50,000 more in assets than in liabilities.
He has $50,000 more in liabilities than in assets.
He has $110,000 more in liabilities than in assets.
He has $160,000 more in assets than in liabilities.
Answers
Answer: te correct answer is A
Step-by-step explanation:
He has $50,000 more in assets than in liabilities.
Answer:
1. He has $50,000 more in assets than in liabilities.
Step-by-step explanation:
Use the recursive formula to find the first five terms in the arithmetic sequence.
Answers
Answer:
3, 1, - 1, - 3, - 5
Step-by-step explanation:
using the recursive formula and f(1) = 3 , then
f(2) = f(1) - 2 = 3 - 2 = 1
f(3) = f(2) - 2 = 1 - 2 = - 1
f(4) = f(3) - 2 = - 1 - 2 = - 3
f(5) = f(4) - 2 = - 3 - 2 = - 5
first 5 terms are 3, 1, - 1, - 3, - 5
7. [Decimal +,-]
16
9.74
+2 6.80
Answers
Answer:
169.74+26.80
= 196.54
the correct decimal answer is 196.54
Find the surface area of the prism.
Answers
The find the area of the triangles: 8x6= 48
Then add to get an area of 240 yd^2
Assume the four intersecting lines are parallel. In this figure, A = 36, B = 24, and C = 60 . If D = 33,
what are the measurements of E and F?
Answers
Using the proportionality theorem, the measurements are: c. E = 22 and F = 55.
What is the Proportionality Theorem?If two transversals intersect three or more parallel lines, they divide the lines in such a way that the smaller segments are proportional to each other or have ratios that are equal to each other.
Given the following:
A = 36,
B = 24,
C = 60
D = 33.
Since all lines are parallel, then:
A/D = B/E = C/F
Find the measure of E using the ratio, A/D = B/E:
36/33 = 24/E
Cross multiply
E = (24 × 33)/36
E = 22
Find the measure of F using the ratio, A/D = C/F:
36/33 = 60/F
Cross multiply
F = (60 × 33)/36
F = 55
The answer is: c. E = 22 and F = 55.
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workout the value of (3.5x10⁶)÷(5x10-³) and show your workout.
Answers
The value of the given expression is 7.0 × 10⁸
Simplifying an expressionFrom the question, we are to determine the value of the given expression
The given expression is
(3.5x10⁶)÷(5x10-³)
The expression can be simplified as follows
(3.5x10⁶)÷(5x10⁻³)
= (3.5 ÷ 5) × (10⁶ ÷ 10⁻³)
= (0.7) × (10⁶⁻⁽⁻³⁾)
= 0.7 × (10⁶⁺³)
= 0.7 × (10⁹)
= 7.0 × 10⁸
Hence, the value of the given expression is 7.0 × 10⁸
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Jamie has 2 dimes, 4 nickels and 8 pennies. in how many different ways can she make 26 cents?
Answers
Answer:
(2 dimes + 6 pennies)
(4 nickels + 6 pennies)
(1 dime + 2 nickels + 6 pennies)
( 2 dimes + 1 nickels + 1 penny)
(1 dime + 3 nickels + 1 penny)
5 different ways that Jamie can make 26 cents.
What is the solution of square root 1-3x = x+3?
O x = -8 or x = -1
O x=-8
O x=-1
O no solution
Answers
Answer: x = -1
Step-by-step explanation:
To remove the radical on the left side of the equation, square both sides of the equation.
√1−3x2=(x+3)2
Simplify each side of the equation.
1−3x=x2+6x+9
Solve for x
x=−1−8
Exclude the solutions that do not make true
√1−3x=x+3
x=−1
In the circle below, if arc AB is congruent to arc CD, chord AB = 16x - 2 and chord CD = 14x + 8, find x.
Answers
Answer:
d. x = 5
Step-by-step explanation:
Chords that subtend congruent arcs are congruent.
SetupChord lengths are the same:
AB = CD
16x -2 = 14x +8
Solution2x = 10 . . . . . . add 2-14x
x = 5 . . . . . . . . divide by 2
If A c B and A N B= • then which of the following can be concluded about the sets A and B!
Answers
Step-by-step explanation:
Set A is a subset of set B if all the elements of A are also the elements of Set B.. (common element)
The above set have 0 common elements so.. if there is no common element and are subsets then it means that there arent any elements in both the sets
so, the answer for the question would be "both the Sets A and B are the empty sets"
At which points on the curve y = 1 60x3 − 2x5 does the tangent line have the largest slope?
Answers
The tangent line has the largest slope at x = 3√2 and x = -3√2 on the curve y = 1 + 60x³ − 2x⁵.
First, let's find the derivative of the given function y = 1 + 60x³ − 2x⁵ using the power rule for differentiation:
dy/dx = 0 + 3(60)x² - 5(2)x⁴
= 180x² - 10x⁴
To find the critical points, we set the derivative equal to zero and solve for x:
180x² - 10x⁴ = 0
Factoring out common terms, we get:
10x²(18 - x²) = 0
Setting each factor equal to zero, we have:
10x² = 0 or 18 - x² = 0
From the first equation, we find x = 0.
From the second equation, we have:
18 - x² = 0
x² = 18
Taking the square root, we get:
x = ±√18
= ±3√2
So the critical points are x = 0, x = 3√2, and x = -3√2.
Now we need to evaluate the slope at these critical points. We can do this by plugging each x-value into the derivative:
When x = 0:
dy/dx = 180(0)² - 10(0)⁴ = 0
When x = 3√2:
dy/dx = 180(3√2)² - 10(3√2)⁴ = 180(18) - 10(216) = 3240 - 2160 = 1080
When x = -3√2:
dy/dx = 180(-3√2)² - 10(-3√2)⁴ = 180(18) - 10(216) = 3240 - 2160 = 1080
The slope is 0 when x = 0 and 1080 when x = 3√2 or x = -3√2.
Therefore, the tangent line has the largest slope at x = 3√2 and x = -3√2 on the curve y = 1 + 60x³ − 2x⁵.
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The complete question is as follows:
At which points on the curve y = 1 + 60x³ − 2x5⁵ does the tangent line have the largest slope?
