The expansion of (z+1)⁶ = z⁶ + 6z⁵+ 15z⁴ + 20z³+ 15z² + 6z + 1
Pascal's Triangle:A type of number pattern is Pascal's Triangle. The triangle-shaped arrangement of integers known as Pascal's Triangle provides the coefficients for the expansion of any binomial statement.
The arrangement of the numerals creates a triangle when they are reflected. The number 1 is first placed at the top, and then the numbers are arranged in a triangle design.
The expansion of (x + y)ⁿ is is given by
(x + y)ⁿ = a₀xⁿ + a₁xⁿ⁻¹y + a₂xⁿ⁻²y² + … + a[tex]_{n -1}[/tex] xyⁿ⁻¹ + a[tex]_{n}[/tex]yⁿ
Here we have (z+1)⁶
Write out Pascal's triangle starting from 1 and up to 6 rows
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
So coefficients of expansion are 1, 6, 15, 20, 15, 6, and 1
(z+1)⁶ = (1)z⁶ + (6)z⁵(1)+ (15)z⁴(1)² + 20z³(1)³ + 15z²(1)⁴ + 6z¹(1)⁵ + (1)z⁰(1)⁶
=> (z+1)⁶ = z⁶ + 6z⁵+ 15z⁴ + 20z³+ 15z² + 6z¹ + 1
Therefore,
The expansion of (z+1)⁶ = z⁶ + 6z⁵+ 15z⁴ + 20z³+ 15z² + 6z + 1
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Scenario: Kevin is baking cupcakes for a fundraiser at his school. Yesterday, he prepared 30 cupcakes in 45 minutes. Today, he prepared 20 cupcakes in 30 minutes.
A) Does the scenario represent a direct variation?
B) If yes, explain the meaning of the constant of proportionality in the scenario.
take your time if you want :)
a) As the constants are equal, the situation represents a proportional relationship.
b) The constant of 2/3 cupcakes per minute means that in a minute, 2/3 of a cupcake is made.
What is a proportional relationship?A proportional relationship is a function in which the output variable is given by the input variable multiplied by a constant of proportionality, that is:
y = kx
In which k is the constant of proportionality.
In this problem, the constants for each situation are:
k = 30/45 = (30/15)/(45/15) = 2/3 = 20/30.
Since the constants are equal, the situation represents a proportional relationship.
The constant of 2/3 cupcakes per minute means that in a minute, 2/3 of a cupcake is made.
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Sabrina Uses a hose to fill a 20-gallon fish tank. The hose
puts out water at a rate of 3 gallons per minute. To start
with, the tank has 5 gallons of water in it.
A. How much water is in the tank after 2 minutes
of filling the tank with the hose?
B. Write an equation that represents the volume
Rewrite the function by completing the square. x2 + 4x = 0 (x + )2 =
When the completing the square is applied, the rewritten function is (x + 2)^2 = 4
How to complete the square?The equation is given as:
x^2 + 4x = 0
The above equation is a quadratic equation and the coefficient of x in the above equation is 4.
So, we have:
k = 4
Divide the variable k by 2
k/2 = 4/2
k/2 = 2
Square the above result
(k/2)^2 = 4
Next, we add 4 to both sides of x^2 + 4x = 0
x^2 + 4x + 4 = 4
Expand the equation by expressing 4x as 2x + 2x
This gives
x^2 + 2x + 2x + 4 = 4
Factorize the above equation
x(x + 2) + 2(x + 2) = 4
Factor out x + 2 from the above equation
(x + 2)(x + 2) = 4
Rewrite the equation as a perfect square of (x + 2)
(x + 2)^2 = 4
Hence, the rewritten function after completing the square is (x + 2)^2 = 4
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Help needed
65. Solve for in the logarithmic equation 3 log x = 2.
O * = 1,000,000
O * = 150
O * = 56.423
O * = 4.642
What are the three coordinates that need to be plotted?
Answer:
Step-by-step explanation:
y=2*|x-3|+2
y=|2*x-2*3|+2
y=|2x-6|+2
1)
y=2x-6
x | 2 | 3 |
y | -2| 0 |
2)
y=|2x-6|
3)
y=|2x-6|+2
y=2*|x-3|+2.
Evaluate if m= 14.
m
2 + m =
Answer:
16
Step-by-step explanation:
You replace m with 14. 2 + 14 = 16.
[tex]\huge\text{Hey there!}[/tex]
[tex]\huge\textsf{Assuming equation:}[/tex]
[tex]\mathsf{\dfrac{m}{2} + m}[/tex]
[tex]\huge\textsf{If so, let's solve for your equation:}[/tex]
[tex]\mathsf{\dfrac{m}{2} + m}[/tex]
[tex]\mathsf{= \dfrac{14}{2} + 14}[/tex]
[tex]\mathsf{= 7 + 14}[/tex]
[tex]\mathsf{= 21}[/tex]
[tex]\huge\textsf{Therefore, your answer should be:}[/tex]
[tex]\huge\boxed{\frak{21}}\huge\checkmark[/tex]
[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]
~[tex]\frak{Amphitrite1040:)}[/tex]
SOLVE FOR X AND THEN SOLVE FOR A
Answer:
Step-by-step explanation:
A + B= 180°
(5x+20)+(9x-92)=180°
14x-72=180°
14x=252
x=18
A=5(18)+20
A=110
Answer:
x = 18 , ∠ A = 110°
Step-by-step explanation:
∠ A and ∠ B are same- side interior angles and sum to 180° , that is
5x + 20 + 9x - 92 = 180
14x - 72 = 180 ( add 72 to both sides )
14x = 252 ( divide both sides by 14 )
x = 18
Then
∠ A = 5x + 20 = 5(18) + 20 = 90 + 20 = 110°
in a survey, 250 adults and children were asked whether they know how to swim. the surgery data are shown in the relative frequency tqable
Answer: D
Step-by-step explanation:
In a two-tailed hypothesis test comparing the means of independent samples, the test statistic is 3.06 and the critical values are /-2.086. True or false: the null hypothesis is not rejected.
In a two-tailed hypothesis test the null hypothesis is rejected.
According to the statement
The test statistic is 3.06 and the critical values are /-2.086.
If we find the value of t by using the formula the then it comes with Negative sign and it lies under the critical region.
That's why In a two-tailed hypothesis test the null hypothesis is rejected.
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the ecentricity of the conic section below is
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The eccentricity of the conic section below is 1, since the conic section given is a parabola.
The eccentricity of a parabola is 1.
Hope that made sense !!
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[tex]\star\tiny\pmb{_{calligraphy}}\star[/tex]
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The eccentricity of the conic section below is 1, since the conic section given is a parabola.
What is parabola?
a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. The name "parabola" is due to Apollonius, who discovered many properties of conic sections. conic section produced by the intersection of a right circular cone and a plane parallel to an element of the cone.
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Does The slopes of parallel lines have a product of -1.
Answer:
No. The slopes of parallel lines are equal.
Step-by-step explanation:
If you have two lines and their slopes are the same, then the lines are parallel.
If the two lines have slopes that are negative reciprocals (opposite signs and flipped over) then when you multiply the slopes together, you get -1... THOSE lines are perpendicular (they make right angles or make 90° angles or square angles).
Lines in the same plane that have different slopes will intersect at one point.
Using numerical integration to estimate the area of the semicircle (r=4 )and c(4,0) with n=4
The area of the semicircle = 8π
A semicircle is formed when the panel passing through the center touches both ends of the circle. Therefore, if you connect two semicircles, you get a circle
If you cut them in half or divide the circumference of the circle by 2, you will get a semi-circle. Area of semicircle The area of the semicircle is half the area of the circle. Because the area of the circle is πr2. Therefore, the area of the semicircle is 1/2 (πr2). Where r is the radius. The value of π is 3.14 or 22/7.
Semi-circular area = 1/2 (πr2)
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PLEASE HELP!!!! A bag contains 1 white (W), 3 blue (B1, B2, B3), and 2 red (R1, R2) marbles. (4 points)
a. Use a tree diagram to list all the possible outcomes for tossing a coin and then drawing a marble from the bag.
b. Are these independent or dependent events?
c. Find the probability of tossing a head and drawing a red marble. Explain your reasoning.
d. Find the probability of drawing two blue marbles if the first marble is not replaced. Explain your reasoning.
The events are independent events and probability of tossing a head and drawing a red marble is 1/7
a) The answer is present in the image attached below
b) Tossing of a coin and drawing of a marble are independent events as these two events do not interfere in the probability of each others events.
c) The probability of tossing a head and drawing a red marble is = [tex]\frac{1}{2}.\frac{2}{7}[/tex]=1/7 as we can see from the tree diagram that probability of tossing a coin and getting head is 1/2 and probability of drawing a red marble is 1/7. As they are independent events, their probability can be multiplied. Hence the probability of tossing a head and drawing a red marble is 1/7.
d) The probability of drawing two blue marbles if the first marble is not replaced and one at a time is 1/7. Probability of drawing first blue marble from the lot is 3/7 as favorable outcomes are 3 and total outcomes are 7. As the ball is not replaced the outcomes are changed. Now the number of blue balls are 2 and total number of balls are 6. So now the probability becomes 2/6. Hence the total probability becomes = 1/7
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Can someone help me on this pls? It’s urgent, so ASAP (it’s geometry)
Write formal proofs using LL Theorem.
Question 6
1) [tex]\overline{AB} \cong \overline{BD}[/tex], [tex]\overline{CD} \perp \overline{BD}[/tex], O is the midpoint of [tex]\overline{BD}[/tex], [tex]\overline{AB} \cong \overline{CD}[/tex] (given)
2) [tex]\angle ABO, \angle ODC[/tex] are right angles (perpendicular lines form right angles)
3) [tex]\triangle ABO, \triangle CDO[/tex] are right triangles (a triangle with a right angle is a right triangle)
4) [tex]\overline{BO} \cong \overline{OD}[/tex] (a midpoint splits a segment into two congruent parts)
5) [tex]\triangle ABO \cong \triangle CDO[/tex] (LL)
Question 7
1) [tex]\angle ADC, \angle BDC[/tex] are right angles), [tex]\overline{AD} \cong \overline{BD}[/tex]
2) [tex]\overline{CD} \cong \overline{CD}[/tex] (reflexive property)
3) [tex]\triangle CDA, \triangle CDB[/tex] are right triangles (a triangle with a right angle is a right triangle)
4) [tex]\triangle ADC \cong \triangle BDC[/tex] (LL)
5) [tex]\overline{AC} \cong \overline{BC}[/tex] (CPCTC)
Question 8
1) [tex]\overline{CD} \perp \overline{AB}[/tex], point D bisects [tex]\overline{AB}[/tex] (given)
2) [tex]\angle CDA, \angle CDB[/tex] are right angles (perpendicular lines form right angles)
3) [tex]\triangle CDA, \triangle CDB[/tex] are right triangles (a triangle with a right angle is a right triangle)
4) [tex]\overline{AD} \cong \overline{DB}[/tex] (definition of a bisector)
5) [tex]\overline{CD} \cong \overline{CD}[/tex] (reflexive property)
6) [tex]\triangle ADC \cong \triangle BDC[/tex] (LL)
7) [tex]\angle ACD \cong \angle BCD[/tex] (CPCTC)
Can someone help me please?
A.0.36
B.0.55
C.0.18
D.0.45
Answer:
B
Step-by-step explanation:
44/77
=0.55
in a school of 40 students, 14 liked English language only, 7 students like both English and French language and 12 also like French language only
a. how many students like English but not French?
b. many students did not like any of the two languages?
c. how many students like at least one of the language?
d. if a student is chosen at random what is the probability that he likes only French
Answer:
A. 14, because it says that only 14 liked English language ONLY
B. It’s 7.
14 liked only English, 7 liked both languages, 12 liked only French.
Add those them together.
14 + 7 + 12
21 + 12
33
33 - 40 = 7
( 33 is the number of students that liked a language and 40 is ALL the students in total )
So it would be 7.
C. 26 liked one language.
7 liked both languages and 7 didn’t like neither. You could add them together.
7 + 7 = 14
40 - 14 = 26
OR
14 + 12 = 26
( just English + just French )
D. The probability of a student chosen at random that ONLY likes French would be 12/40.
Since only 12 of the 40 students liked ONLY French, it would be 12/40 or 12 : 40
Step-by-step explanation:
I hope it helps! Have a great day!
Pan~
Answer:
a) 14
b) 7
c) 33
d) P = 12/40 = 3/10 (In decimal P=0.3)
Step-by-step explanation:
Hope this helps
Margo spends a quarter of her paycheck on a new dress, one-eighthof her paycheck on shoes, and $24 on a birthday present. if she spent $60 altogether, how much was margo’s paycheck? $96 $160 $216 $504
Answer:
$96
Step-by-step explanation:
HURRY PLS 15 MINS LEFT!!
Answer:
y is less than or equal to 2/3x + 1/5
Step-by-step explanation:
Name a postulate or theorem that can be used to prove that the two triangles are similar. Then, write a similarity statement.
Postulate: SAS
Similarity statement: [tex]\triangle CAB \sim \triangle EDF[/tex]
10. A shopkeeper bought 5 sets of sofa from a factory for $ 15800 (inclusive of a 10% value-added
tax). He then sold each set of sofa for $6794.
(a) Calculate the percentage profit made by the shopkeeper. [3]
(b) How much was the value-added tax for the 5 sets of sofa? [2]
can someone please explain to me how to solve this?
Which set of ordered pairs represents y as a function of x?
A (0, 0.5), (2, 1.5), (−5.5, 5), (−5.5, −6)
B (5, −7), (3, −7), (−9.2, −7), (9.2, −7)
C (9.3, −1), (7.3, −2), (7.3, 2), (9.3, 1)
D (8.1, 9), (9.1, 10), (10.1, 9), (9.1, 8)
Answer:
B.
Step-by-step explanation:
A C and D are not functions as they are one - to - many whereas B is many-to-one.
A saleslady is paid a commission of 3% on goods worth 100,000.she is also paid a salary of 11000.in a certain month she sold 360 shoes at460 each.the following month she had a salary increase of 20%.her total earnings were 22,200 calculate total amount of money received from sales of shoes that month
The total amount of money received from sales of shoes that month = $4968
Calculating sales commissionSalary = $11000
Number of shoes sold = 360
Cost of 1 shoe = $460
Total cost of shoes sold = 360 x 460
Total cost of shoes sold = $165600
Since $165600 is greater than $100000, she will earn a commission of 3%
Commission = (3/100) x $165600
Commission = $4968
The commission is the amount of money received from sales of shoes. Other earnings are salary or increment
Therefore, the total amount of money received from sales of shoes that month = $4968
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1. Write the next 3 terms in
patterns.
these
121,11,1,1/11
Answer:
Your simply dividing by 11
Step-by-step explanation:
0.008...
0.0007...
0.00006...
All you have to do is turn these numbers into fractions!
Answer:
1/121, 1/1331, 14641
Step-by-step explanation:
121, 11 ,1 , 1/11
We need to find the next three terms in the pattern
We are taking each term and dividing by 11
121/11 =11
11/11 =1
1/11 = 1/11
1/11 divided by 11
1/(11*11) = 1/121
1/121 * 1/11 = 1/1331
1/1331 * 1/11 = 14641
The graph represents the cost of catering a dinner as a function of the number of guests. On a coordinate plane, a graph titled Dinner Catering Cost shows Guests Served on the x-axis and Total cost in dollars on the y-axis. A piecewise function has 3 lines. The first line has an open circle at (0, 100) and goes up to a closed circle at (40, 600). The second line has an open circle at (40, 700) and goes up to a closed circle at (80, 1300). The third line has an open circle at (80, 1600) and goes up to (100, 2000). What is the catering cost for a dinner with 40 guests? $400 $600 $700 $800
Considering the piece-wise function described, the catering cost for a dinner with 40 guests is of $600.
What is a piece-wise function?A piece-wise function is a function that has different definitions, depending on the input.
In this problem, considering between 0 and 40 guests, the definition is the open circle at (0, 100) and goes up to a closed circle at (40, 600), hence the catering cost for a dinner with 40 guests is of $600.
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Answer:
600
Step-by-step explanation:
edge
What is the solution of startfraction 1 over c minus 3 endfraction minus startfraction 1 over c endfraction = startfraction 3 over c (c minus 3) endfraction?
The solution of the given equation can be all real numbers, except c = 0 and c = 3.
Lets simplify the given problem,
1/c-3 - 1/c = 3/ (c-3)
As the denominators cannot be equal to zero (0) because a division by zero (0) is undefined.
Next, we would multiply both sides of the given by the lowest common multiple (LCM).
The lowest common multiple (LCM) is c(c - 3).
c-[c-3] = 3
c-c+3 = 3
0 = 3-3
0=0
Therefore, the solution of the given equation can be all real numbers, except c = 0 and c = 3.
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Can someone help me with these problems please
Calculate the (modeled) probability P(E) using the given information, assuming that all outcomes are equally likely.
S = {1, 3, 5, 7, 9}, E = {1, 5, 7}
The probability P(E) is 13/15.
According to the statement
we have given that the S = {1, 3, 5, 7, 9}, E = {1, 5, 7}
And we have to find the probability P(E).
So, For this purpose
Recall the formula for the probability of an event E in case when all outcomes are equally likely:
P(E)= n(E) /n(S)
in which S is the sample space.
But we have the S = {1, 3, 5, 7, 9},
so, n(S) = Sum of all outcomes / number of outcomes
n(S) = 1+3+5+7+9 /5
n(S) = 25 /5
n(S) = 5 and
For n(E) = Sum of all outcomes / number of outcomes
n(E) = 1+5+7 /3
n(E) = 13 /3
n(E) = 13 /3
Substitute these values in the above written formula then
P(E)= n(E) /n(S)
P(E)= (13/3)/ 5
P(E)= 13 /15
So, The probability P(E) is 13/15.
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HELPP A PERSON OUT PLSS! i dont really understand how to do ittt!!
giving you all my points so help pleaseeee
bestbroi
If light travels at 10,000 km in 3.0 x 10² seconds,
how long will it take light to travel one meter?
(1 km = 1 x 10³m)
Answer:
0.03 seconds
Step-by-step explanation:
10,000km = 300secs
1km = x seconds
Cross multiply the equation
10,000x = 300
Divide both sides by 10,000
10,000x = 300
10,000 10,000
x = 0.03 seconds
There are 20 teachers and 705 students in cory's school. what is the ratio of teachers and students
We know that ratios can be shown with both fractions and colons.
20:705
or
[tex]\frac{20}{705}[/tex]
It also looks like we can simplify this by dividing by 5.
20/5 = 4
705/5 = 141.
This means that for every 4 teachers, there are 141 students.
The ratio for teachers to students is:
4 : 141