Answer:
In the Pascal Expansion of [tex](x+3)^4[/tex] the third term is [tex]54x^2[/tex].
Step-by-step explanation:
Here, the given expression is given as [tex](x+3)^4[/tex]
Now, by the PASCAL'S TRIANGLE EXPANSION:
[tex](a + b)^4 = 1(a)^4 + 4(a)^3b + 6(a)^2b^2 + 4(a)b^3 + 1b^4\\= 1 + 4b + 6a^2 + 4b^3 + b^4.[/tex]
Substituting a = x and b = 3, we get:
[tex](x + 3)^4 = 1(x)^4 + 4(x)^3(3) + 6(x)^2(3)^2 + 4(x)(3)^3 + 1(3)^4\\= x^4 + 12x^3 + 54x^2 + 108x+ 81[/tex]
In the given expansion, the third term is [tex]54x^2[/tex].
Hence, in the Pascal Expansion of [tex](x+3)^4[/tex] the third term is [tex]54x^2[/tex].
Final answer:
The third term is 6 times x squared times 3 squared, which simplifies to 54x².
Explanation:
To use Pascal's Triangle to determine the third term of the expansion of (x + 3)⁴, we first need to look at the row of Pascal's Triangle that corresponds to the exponent 4. This is the fifth row if we start counting from 0 and it reads 1, 4, 6, 4, 1. We use these coefficients for each term of the expansion, which follows the pattern (a + b)ⁿ = C(n,0)*aⁿ + C(n,1)*a⁽ⁿ⁻¹⁾b + C(n,2)*a⁽ⁿ⁻²⁾b² + ... + C(n,n)*bⁿ, where C(n,k) is the binomial coefficient from Pascal's Triangle.
So, the third term in the expansion is given by the third coefficient (counting from 0) from the row, which is 6, multiplied by x raised to power 4 - 2 times 3 squared. Therefore, the third term is 6x²·3², which simplifies to 54x².
A line passes through the point (6,1) and has a slope of -3.
Write an equation in point-slope form for this line
[tex]\bf (\stackrel{x_1}{6}~,~\stackrel{y_1}{1})~\hspace{10em} \stackrel{slope}{m}\implies -3 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{1}=\stackrel{m}{-3}(x-\stackrel{x_1}{6})[/tex]
what is the percent decrease from 5000 to 4000?
Answer:
20%
Step-by-step explanation:
5000 - 4000 = 1000
[tex]\frac{1000}{5000}[/tex] × 100
[tex]\frac{1}{5}[/tex] × 100
0.2 × 100 = 20%
Check:
5000 - 20% of 5000
5000 - 2/10 × 5000
5000 - 1000
= 4000
Answer:
20% decrement
Steps:
➥5000-x% of 5000=400
➥5000-x% × 5000=4000
➥5000-4000=x% × 5000
➥1000=x% × 5000
➥x%=1000/5000
➥x%=0.2
➥x%=20/100
➥x%=20%
Points A and B lie in plane R Line AB does not lie in plane RI
always
sometimes
never
Answer:
The question has been presented in a complicated manner. I am answering the question based on my understanding. Line AB will never lie in plane R. The correct option among all the options that are given in the question is the third option.
hope I helped you answer your question:)
Answer:
never
Step-by-step explanation:
Solve.
5. During the summer, Jayden
volunteers at the library for 20 hours
each week for 7 weeks. How many
hours does Jayden volunteer in all?
How many hours did Jayden volunteer
Answer:
he volunteered 140 hours
Step-by-step explanation:
20*7=140
Paton knows she will make a selection from 36 total possible outcomes. Which statement might describe what Paton is doing? Check all that apply.
Answer: All but statement 1
Step-by-step explanation:
In order to get 36 as an outcome, the numbers need to multiply together in order to get 36. In other words, a*b*c*d*....= desired outcome number. In statement 1, 3*9=27, and therefore not 36 outcomes. The rest do multiply to 36.
Answer:
2,3,4,5
Step-by-step explanation:
Stay safe! Hope this helps!
Help finding the volume
Answer:
Using [tex]\pi =3.14[/tex] the volume of the cylinder is [tex]1471.875\ m^{3}[/tex]
Step-by-step explanation:
We are using [tex]\pi =3.14[/tex].
We know that volume of a cylinder is [tex]\pi (r)^2h[/tex]
For this we have
Height of the cylinder is given that is [tex]18.75\ m[/tex]
And the radius of the cylinder is unknown,
But there is a hint in which we can see that the diagonal,height and diameter forms a right angled triangle (or a cone).
Where [tex]21.25\ m[/tex] is the hypotenuse.
Using Pythagoras formula.
[tex](Hypotenuse)^2=(base)^2+(perpendicular)^2[/tex]
We can find the value of diameter which is the base over here.
So,
[tex]d=\sqrt{(21.25)^2-(18.75)^2}=\sqrt{(451.56-351.56)}=\sqrt{(100)}=10\ m[/tex]
Then radius [tex]\frac{diameter}{2} =\frac{10}{2}=5\ m[/tex]
Now using the above values and plugging it on to the formula.
Volume [tex]=\pi (r)^2h=(3.14)(5)^2\times 18.75=1471.875\ m^3[/tex]
So the volume of the cylinder is [tex]1471.875\ m^3[/tex]
A gardener at a nursery is filling pots with soil to prepare to
transplant seedlings into these larger pots. Each new pot needs about 27 cubic inches
of soil. The amount of soil s (in cubic inches) it takes to fill p pots is given by the
function s = 27p
a.
The gardener is filling the pots from a bag of soil that contains 3456 cubic
inches of soil. How many pots can be filled from the bag?
b.
Suppose the gardener needs to fill at most 109 pots. How much soil (in cubic
inches) will the gardener need?
Answer:
a. 127 pots
b. 2,943 cubic inches
Step-by-step explanation:
To find the amount of pots of soil can be filled, you can use the equation given. s = 3456 in part a
a.
s = 27p
3456 = 27p
divide 3456 by 27 to isolate p
p = 127 pots
Instead of substituting for s like in part a, you can now substitute for p, since you know that you need to fill 109 pots.
b.
s = 27 * 109
s = 2943 cubic inches
The gardener can fill 128 pots with the 3456 cubic inches of soil available. If the gardener needs to fill at most 109 pots, they will require 2943 cubic inches of soil.
To solve both parts of the question, we will use the function s = 27p, where s represents the amount of soil (in cubic inches) and p represents the number of pots.
Part a: Calculating the Number of Pots
The gardener has a bag containing 3456 cubic inches of soil. To find out how many pots can be filled from the bag, we need to divide the total amount of soil by the amount needed per pot:
p = 3456 cubic inches / 27 cubic inches per pot = 128 pots
Part b: Soil Needed for a Maximum Number of Pots
If the gardener needs to fill at most 109 pots, the total amount of soil required can be calculated using the provided function:
s = 27 cubic inches per pot * 109 pots = 2943 cubic inches of soil
A piece of cloth costs 200$.if the piece was 5m longer, and the cost of each metre of clth was 2$ less, the cost of the piece would have remained unchanged. How long is the piece and what is its original number of persons.
Answer:
20 m long,
each piece of cloth for $10
Step-by-step explanation:
Let x m be the length of the piece of cloth.
Original cloth:
Length = x m
Cost = $200
Cost per meter [tex]\dfrac{\$200}{x}[/tex]
Changed cloth:
Length = x + 5 m (5 m longer)
Cost = $200
Cost per meter [tex]\dfrac{\$200}{x}-\$2[/tex] (each metre of cloth was $2 less)
Hence,
[tex]\left(\dfrac{200}{x}-2\right)\cdot (x+5)=200[/tex]
Multiply the equation by x:
[tex](200-2x)(x+5)=200x\\ \\200x+1,000-2x^2-10x=200x\\ \\-2x^2-10x+1,000=0\\ \\x^2+5x-500=0\\ \\D=5^2-4\cdot(-500)=25+2,000=2,025\\ \\x_{1,2}=\dfrac{-5\pm\sqrt{2,025}}{2}=\dfrac{-5\pm 45}{2}=-25,\ 20[/tex]
The length cannot be negative, so x = 20 m, each metre for $10.
Answer:
The piece is 20 m long and costs $10 per meter
Step-by-step explanation:
The total cost of the cloth in $ can be expressed as
C = xu
Where x is the length of the cloth in m, and u is the unit cost in $/m
The problem states that if the length was 5 meters more and the unit cost was $2 less, the cost would not change, that is
C = (x+5)(u-2)
Operating:
C = xu-2x+5u-10
Since C = xu, then
-2x+5u=10 [1]
We also know that
xu=200 [2]
Isolating u in [1] we get
[tex]u=\frac{10+2x}{5}[/tex]
Replacing into [2]
[tex]x\frac{10+2x}{5}=200[/tex]
Simplifying
[tex]10x+2x^2=1000 \\=> x^2+5x-500=0[/tex]
Factoring
(x-20)(x+25)=0
Which gives x=20, x=-25. We can only use the positive value of x, so
x=20 m and therefore
u=200/20=10$
So the piece is 20 m long and costs $10 per meter
There are 472 students in different
grades. Each grade has about the same
number of students. Select all the
statements that are reasonable
estimates for the number of students in
each grade.
n 50 students, because 472 = 6 is
about 450 = 5.
80 students, because 472 = 6 is
about 480 - 6.
100 students, because 472 4 6 is
about 500 = 5.
150 students, because 472 = 6 is
about 450 = 3.
200 students, because 472 = 6 is
about 1,200 + 6 = 200.
Final answer:
Reasonable estimates for the number of students per grade when there are 472 students total are 50, 80, or 100 students per grade.
Explanation:
The student's question asks for reasonable estimates for the number of students in each grade if there are 472 students in different grades with about the same number of students in each grade.
Let's examine the options given:
50 students, because 472 ÷ 9 is about 450 ÷ 9, which yields 50 students per grade.80 students, because 472 ÷ 6 is indeed close to 480 ÷ 6, which would result in about 80 students per grade.100 students, because 472 ÷ 5 is close to 500 ÷ 5, translating to 100 students per grade.150 students is not a reasonable estimate because 472 ÷ 3 is about 157, which is significantly off from 150.200 students is not reasonable either since 472 ÷ 2 is 236 and 1,200 ÷ 6 is not closely related to this problem.Therefore, reasonable estimates for the number of students per grade are 50 students, 80 students, and 100 students.
A sequence has a common ratio of Three-halves and f(5) = 81. Which explicit formula represents the sequence? f(x) = 24(Three-halves) Superscript x minus 1 f(x) = 16(Three-halves) Superscript x minus 1 f(x) = 24(Three-halves) Superscript x f(x) = 16(Three-halves) Superscript x
Answer:
[tex]f(x)=16\, (\frac{3}{2} )^{x-1}[/tex]
Step-by-step explanation:
Since we are given the common ratio (3/2), all we need to find to define the geometric sequence, is its multiplicative factor ([tex]a[/tex]) that corresponds to the first term of the sequence - remember that all consecutive terms will be generated by multiplying this first value repeatedly by the common ratio (3/2) as shown below:
[tex]f(1)=a\\f(2)=a\,*\,(\frac{3}{2} )\\f(3)=a\,*\,(\frac{3}{2} )^2\\f(4)=a\,*\,(\frac{3}{2} )^3\\f(5)=a\,*\,(\frac{3}{2} )^4[/tex]
Since we are given the information that [tex]f(5)=81[/tex] we can use this to find the value of the first term:
[tex]f(5)=\,a\,*\,(\frac{3}{2} )^4\\81=\,a\,*(\frac{81}{16} )\\a\,*\,81=\,81\.*\,16\\a\,=\,16[/tex]
Notice as well that the first term doesn't contain the common ratio, the second term contains the common ration (3/2) to the power one, the third one contains the common ratio to the power two, the fourth one contains it to the power three, and so forth. So the exponent at which the common ratio appears is always one unit less than the order (x) of the term in question. This concept helps us finalize the expression for the sequence's formula:
[tex]f(x)=16\,*\,(\frac{3}{2} )^{x-1}[/tex]
Answer:
B
The other answer was too long.
3/5 with an exponent of 4
Step-by-step explanation:
[tex]\text{If is}\ \dfrac{3}{5}^4,\ \text{then}\ \dfrac{3}{5}^4=\dfrac{3\cdot3\cdot3\cdot3}{5}=\dfrac{81}{5}=16\dfrac{1}{5}\\\\\text{If is}\ \left(\dfrac{3}{5}\right)^4=\left(\dfrac{3}{5}\right)\left(\dfrac{3}{5}\right)\left(\dfrac{3}{5}\right)\left(\dfrac{3}{5}\right)=\dfrac{3\cdot3\cdot3\cdot3}{5\cdot5\cdot5\cdot5}=\dfrac{81}{625}[/tex]
sin x= 1.5
value of x
Answer:
x = π - sin^(-1)(3/2) + 2 π n_1 for n_1 element Z
or x = 2 π n_2 + sin^(-1)(3/2) for n_2 element Z
Step-by-step explanation:
Solve for x:
sin(x) = 1.5
1.5 = 3/2:
sin(x) = 3/2
Take the inverse sine of both sides:
Answer: x = π - sin^(-1)(3/2) + 2 π n_1 for n_1 element Z
or x = 2 π n_2 + sin^(-1)(3/2) for n_2 element Z
add the polynomials
3x^5-2x^3+4x
+ 4x^6-5x^5+4x^3-x^2
Answer: B
4x^{6} - 2x^{5} + 4x^{3} -x^{2} +4x
Step-by-step explanation:
3x⁵ - 2x³ + 4x + 4x⁶ - 5x⁵ + 4x³ - x² =
= 4x⁶ + 3x⁵ - 5x⁵ - 2x³ + 4x³ - x² + 4x
= 4x⁶ - 2x⁵ + 2x³ - x² + 4x
A hockey puck slides across an ice pond at a constant speed of 18 meters per second. Complete the table to determine the distance the hockey puck slides at different times.
Answer:
1 18
2 36
3 54
4 72
Step-by-step explanation:
18 m/s every second means you multiply the time by rate (to get the distance traveled in 2 seconds, multiply 18m/s by 2 to get 36)
A function has y-intercept of -3 and a slope of 1/5. Which equation below describes the function?
A.
y = -3x + 1/5
B.
y = 1/5x2 - 3
C.
y = 1/5 + x - 3
D.
y = 1/5x - 3
y= mx+b slope intercept form
slope is also m
b is the y-intercept
y-intercept= -3
slope=1/5
answer:
D. y = 1/5x - 3
Whats the slope (1,2) (3,-4)
Answer:
the slope is -3
m=-3
i tried my best :(
can you please help me i don't understand this
Answer:
10) Supplementary; 11) Supplementary; 12) Complementary
Step-by-step explanation:
Sum of supplementary angles equals 180° while sum of complementary angles equals 90°.
For question 10
x + 70° = 180° (sum of straight lines)
x = 180° - 70° = 110°
Hence, 70° and 110° are supplementary angles.
For question 11,
x + 85° = 180° (sum of straight lines)
x = 180° - 85° = 95°
Hence, 85° and 95° are supplementary angles.
For question 12,
x + 32° = 90° (sum of right angles)
x = 90° - 32° = 58°
Hence, 32° and 58° are complementary angles.
(3.6 x 10^-5) + (3.5 x 10^-5)
Write your answer in scientific notation.
Answer:
(3.6 × 10^-5) + (3.5 × 10^-5) = 7.1 × 10^-5
57 first-year teachers and 19 experienced teachers attended a school staff meeting meeting. What percentage of the teachers in the meeting were first-year teachers?
Write your answer using a percent sign (%).
Answer:
75%
Step-by-step explanation:
57 / (57+19) = 0.75
Final answer:
About 75% of the teachers in the meeting were first-year teachers, as calculated by dividing the number of first-year teachers by the total number of teachers and then multiplying by 100.
Explanation:
To determine what percentage of the teachers in the meeting were first-year teachers, we use the formula 'part over total, times one hundred, equals percent'. In this scenario, the 'part' is the number of first-year teachers which is 57, and the 'total' is the sum of first-year and experienced teachers, which is 57 + 19 = 76. The calculation then becomes 57/76 times 100, to find that percentage.
Carrying out the calculation:
(57 / 76) times 100 = 0.75 times 100 = 75%
Therefore, approximately 75% of the teachers at the meeting were first-year teachers.
what is 776.732 in word form ?
Answer: seven hundred seventy six and seven thousand thirty two thousandths
Step-by-step explanation:
Answer:
seven hundred and seventy six and seven hundred thirty two thousandths
Which describes the cross section of the square prism that passes through the vertices A, B, C, and D shown below?
a rectangle
a square
a kite
a trapezoid
Answer:
Trapezoid
Step-by-step explanation:
PLEASE HELPPPP ! 16^x=64^x+4
Answer:
x = -12
Step-by-step explanation:
Set logs up on both sides-
log 16^x = log 64^x+4
The variable goes behind because of the exponent log property (exponents are permitted to go behind the log to multiply)-
x * log 16 = x + 4 * log 64
Divide both logs out-
log 64/log 16 = 1.5
Simplify-
x = x+4(1.5)
1.5x + 6 = x
Remove 1.5-
1 - 1.5 = -.5x
Simplify-
-.5x = 6
Divide both sides-
6/-.5x = -12
Hence, the answer would be -12
What is 2/8+1/6 in the simplest form.
Answer:
20/48
Step-by-step explanation:
Answer:
I think it would be 1/4+1/6
Step-by-step explanation:
2/8 can be simplified by 2 because 2 goes into 2 once so 1 and 2 goes into 8, 4 times so that equals 1/4 and I believe the 1/6 is already simplified.
Molly has walked 80% of the way to school.She has walked 16 blocks.How many blocks is her whole trip to school?
Answer:
20 blocks
Step-by-step explanation:
80% = 0.8
16 / 0.8 = 20
Rebecca started with 30 pieces of candy and gave away x
pieces. Ana started with 45 pieces of candy and gave away
twice as many pieces as Rebecca did. How many pieces of
candy did Rebecca give away if they had the same number of
pieces of candy left?
Answer:
The correct answer would be 15.
Step-by-step explanation:
30-15=15
45-2(15)=15
Which only lists multiples of 16?
O
1, 2, 4, 8, 16
16, 24, 32, 40
16, 32, 48, 64
1, 2, 4, 8, 12, 16
16, 32, 48, 64
16 = 16x1
32 = 16x2
48 = 16x3
64 = 16x4
Answer:
C
Step-by-step explanation:
EAM
on of the system of equations,
2x + 2y = 16
10x - 2y = 24
Answer:
x=10/3, y=14/3. (10/3, 14/3).
Step-by-step explanation:
2x+2y=16
10x-2y=24
---------------
x+y=8
10x-2y=24
-----------------
x=8-y
10(8-y)-2y=24
80-10y-2y=24
80-12y=24
12y=80-24
12y=56
y=56/12=14/3
x=8-14/3=24/3-14/3=10/3
x=10/3, y=14/3.
-3(y+3)=2y+3
help plzs!!!!!!......
Answer:
For the given expression -3( y + 3 ) = 2 y + 3 the value of y = -2.4.
Step-by-step explanation:
Here, the given expression to be simplified : -3( y + 3 ) = 2 y + 3
Now, by DISTRIBUTIVE PROPERTY:
A (B +C) = AB + AC
Simplifying the given expression:
-3( y + 3 ) = 2 y + 3 ⇒ -3 ( y) - 3 ( 3) = 2 y + 3
or, -3y - 9 = 2y + 3
⇒ 3 y + 2 y = -3 - 9
or, 5 y = -12 ⇒ y = -12/5 = -2.4
⇒ y = - 2.4
Hence,for the given expression -3( y + 3 ) = 2 y + 3 the simplified value of y = -2.4
What is a quarter of 200
Answer:
50
Step-by-step explanation:
200/2=100
100/2=50
A quarter of something is just another word for a fourth of something (1/4th).
We can solve this using two different ways.
Solution #1:
1/4 = 0.25
200 * 0.25 = 50 (quarter of 200)
Solution #2:
200 / 4 = 50 (quarter of 200)
______
Best Regards,
Wolfyy :)
Which statement shows the associative property of addition?
(d + 4) + 17 = 17 + (d + 4)
13 + 5a = 5a + 13
7⋅(4⋅x)=(7⋅4)⋅x
(18 + 6) + y = 18 + (6 + y)
Answer:
(18 + 6) + y = 18 + (6 + y)
Step-by-step explanation:
Answer:
the answer is (18 + 6) + y = 18 + (6 + y )
Step-by-step explanation:
i did the quiz and got it right!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
hope this helps!!!!!!!!!!!!!!!!!!!11