Please explain how Pascals triangle works?
Answers
Pascals triangle or Pascal's triangle is a special triangle that is named after Blaise Pascal, in this triangle, we start with 1 at the top, then 1s at both sides of the triangle until the end. The middle numbers are so filled that each number is the sum of the two numbers just above it.
Generally, we can use Pascals' triangle to find the coefficients of binomial expansion
For example, we want to expand
[tex](x\text{ + y\rparen}^2[/tex]Using Pascal's triangle, we can get the coefficients of the expansion:
[tex](x\text{ + y\rparen}^2\text{ = 1x}^2\text{ + 2xy + 1y}^2[/tex]
Related Questions
can someone solve this equation for me [tex]\frac{12a}{k}+\frac{4k}{b^2}+\frac{z}{a}[/tex]
Answers
The solution of the equation [tex]\frac{12a}{k}+\frac{4k}{b^2}+\frac{z}{a}[/tex] is [tex]\frac{12b^2a^2+4k^2a+kb^2z}{kb^2a}[/tex]
The given expression is [tex]\frac{12a}{k}+\frac{4k}{b^2}+\frac{z}{a}[/tex]
To solve this problem first we have to find the LCM of the given terms
LCM is the least common multiple. LCM is the method to find the least common multiple of two or more numbers. We can find the LCM using different methods like prime factorization method and long division method
Here the denominators are k, [tex]b^2[/tex] and a, we have to multiply the terms, because there is no common term
LCM of k, [tex]b^2[/tex] and a = k × [tex]b^2[/tex] × a
Multiply the given terms
Therefore, LCM ( k, [tex]b^2[/tex],a) = [tex]kb^2a[/tex]
Then the given expression will be
[tex]\frac{12a}{k}+\frac{4k}{b^2}+\frac{z}{a}[/tex] = [tex]\frac{(b^2a)(12a)+(ka)(4k)+(kb^2)z}{kb^2a}[/tex]
Open the bracket by multiplying it
= [tex]\frac{12b^2a^2+4k^2a+kb^2z}{kb^2a}[/tex]
Hence, the solution of the equation [tex]\frac{12a}{k}+\frac{4k}{b^2}+\frac{z}{a}[/tex] is [tex]\frac{12b^2a^2+4k^2a+kb^2z}{kb^2a}[/tex]
Learn more about LCM here
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On (A) select one textbook, cookbooks, magazines, paperbacks poetry, notificationThe pie chart below shows the percentage of total revenue that a publisher receives from various types of publications. Use this chart to answer the questions
Answers
Step 1
[tex]Fraction\text{ of paper back+nonfiction +poetry + textbooks take}\approx\text{ }\frac{1}{2}\text{ of the chart}[/tex][tex]\frac{1}{3\text{ }}\text{ of the chart will take }\frac{1}{3}\times360=120^o[/tex]The only category that accounts for approximately one-third of the publisher's revenue is Magazine because the area it covers is definitely above 90 degrees.
A) Answer- Magazines
Step 2
[tex]Magazines+cookbooks+textbooks\text{ take approximately }\frac{2}{3}\text{ of the chart}[/tex][tex]\begin{gathered} Poetry+nonfiction\text{ + paperbacks take approximately }\frac{1}{3}\text{ of the chart} \\ and\text{ are approxiately equal} \\ 3x=\frac{1}{3} \\ x=\frac{1}{9} \\ The\text{ percentage will be; 100}\times\frac{2}{9}=22.2 \\ \approx20\text{\%} \end{gathered}[/tex]Step 3
[tex]\begin{gathered} cook\text{books is approximately twice of poetry} \\ If\text{ cookbok takes 16\%, then;} \\ Poetry\text{ will take; }\frac{16}{2}\approx8\text{\%} \end{gathered}[/tex]please help this is for my study guide thanks! (what is the radius)
Answers
We are given the following details
[tex]\begin{gathered} \text{volume of the cube = }4019.2\text{ cubic centimeters} \\ \text{height}=20\operatorname{cm} \\ \pi=3.14 \end{gathered}[/tex]To find the radius, we will make the radius, the subject of the formula from the equation:
[tex]V=\pi r^2h[/tex][tex]\begin{gathered} r^2=\frac{V}{\pi h} \\ \\ r=\sqrt[]{\frac{V}{\pi h}} \end{gathered}[/tex]Substituting the values, we will obtain
[tex]\begin{gathered} r=\sqrt[]{\frac{4019.2}{3.14\times20}}=\sqrt[]{64} \\ \\ r=\sqrt[]{64} \\ \\ r=8 \end{gathered}[/tex]The radius of the cylinder 8.00
find the missing side of the right angle round to the nearest tenth if necessary triangle is 6 meters by 12m
Answers
To get x, we will use the Pythagoras theorem
With the hypotenuse equal to 12m
[tex]\begin{gathered} x^2=12^2-6^2 \\ x^2\text{ = 144 - 36} \\ x^2\text{ = 108} \end{gathered}[/tex][tex]\begin{gathered} x\text{ = }\sqrt[]{108\text{ }}=\text{ 10.39} \\ \text{ x = 10. 4m (to the nearest tenth)} \end{gathered}[/tex]What will be the coordinates of D'' after it has been reflected over the x-axis and then translated 3 units to the right?A. (-6, -4)B. (-9, -4)C. (-6, -2)D. (-9, -2)
Answers
Solution
Step 1:
Write the coordinate of point D = (- 9, 4 )
Step 2:
The rule for reflection over x-axis
A reflection of a point over the x -axis is shown. The rule for a reflection over the x -axis is (x,y)→(x,−y) .
Step 3
D = (- 9, 4 )
D' = (- 9, -4 )
Step 4
To translate the point P(x,y) , a units right and b units up, use P'(x+a,y+b) .
D' = (- 9, -4 )
D'' = (- 9+3, -4 )
D'' = (- 6, -4 )
Final answer
A. (- 6, -4 )
Algebraically what is the discriminant of the following image. 15 10 O Discriminant < 0 O Discriminant = 0 O Discriminant > 0 O Discriminant does not exist 1 2 3
Answers
We are asked to determine the discriminant of the given quadratic graph.
Refer to the image below
Comparing the given graph with the above image, we see that the given graph is vertically shifted upward (above the x-axis)
The graph does not intersect the x-axis which means no real solutions.
This means that the discriminant of the given graph must be negative (less than 0)
Therefore, the correct answer is the 1st option
[tex]Discriminant<0[/tex]There are no real solutions when the discriminant is less than 0.
Write a function s(t) that represents the situation. Your starting annual salary of $45,000 increases by 3% each year. We have to write it in y=a(1-r)^t format wherea: the initial amountt: the time period(1-r): the decay factor.
Answers
The function will be a compund interest
[tex]\begin{gathered} S(t)=P(1+r)^t \\ S(t)\text{ is the amount after t years} \\ P\text{ is the initial deposit} \\ r\text{ is the rate in \%} \\ t\text{ is the time in years} \end{gathered}[/tex]Question: Which three dimensional figure results from rotating the triangle 360° about AC?A. coneB. cylinderC. pyramidD. sphere
Answers
Given data:
The given triangle ABC.
When the given triangle is rotated about AC, the given figure obtained is cone.
Thus, option (A) is correct.
(-9,3)(1,8) 2:3 find the point that partitions the segment with the two given endpoints with the given ratio
Answers
the given end points are ,
A(-9,3)
B(1,8)
the ratio is , m : n = 2: 3
the formula of parting a line in m: n is given as follows,
so the points will be,
[tex]x=\frac{mx_2+nx_1}{m+n}[/tex][tex]x=\frac{2\times1+3\times-9}{2+3}[/tex][tex]\begin{gathered} x=\frac{2-27}{5}=\frac{-25}{5} \\ x=-5 \end{gathered}[/tex]and the y-coordinate of the point is
[tex]y=\frac{2\times8+3\times3}{2+3}[/tex][tex]\begin{gathered} y=\frac{16+9}{5}=\frac{25}{5} \\ y=5 \end{gathered}[/tex]thus, the point is (-5,5)
it means point P(-5,5) is dividing the line AB in ratio 2: 3
thus, the answer is (-5,5)
The following are the annual salaries of 17 chief executive officers of major companies.here are salaries 84, 381, 542,338,542,338,224,248,495.676,767,405,271,723,814,428,452,1108,609(the salaries are written in thousands of dollars)find the30th and the 75th percentiles for the salaries
Answers
To determine the 30th and the 75th percentiles for the salaries:
The pth percentile is a data value such that at least p% of the data-set is less-than or equal to this data value and at least (100-p)% of the data-set are more-than or equal to this data value.
Arrange the data set in ascending order as follows:
84 ,224 , 248, 271, 338 , 381, 405 , 428, 452, 495 , 542, 609, 676 , 723 ,767 , 814, 1108
The formula to compute the position of the pth percentile is:
[tex]P^{th}\text{ percentile = }\frac{(n+1)\text{.p}}{100}[/tex]Compute the 30th percentile as follows:
[tex]30^{th}\text{ percentile = }\frac{(17+1).30}{100}=\frac{18}{100}.30\approx5^{th\text{ }}Obs[/tex]The 5th observation from the arranged data set is 338 .
Thus, the 30th percentile is 338.
Compute the 75th percentile as follows:
[tex]75^{th}\text{ percentile = }\frac{(17+1).75}{100}\approx14^{th\text{ }}Obs[/tex]The 14th observation from the arranged data set is 814.
Thus, the 75th percentile is 814
Therefore the 30th percentile is 338 and the 75th percentile is 814
I’m a bit confused.I don’t really know to solve this problem nor fully understand the concept.
Answers
Given,
The steps to make the parallel line of the given line.
Step 1.
Step 2.
step 3.
Step 4,
Step 5
Hence, line q is parallel line of line p.
Hi, can you help me answer this question please, thank you
Answers
SOLUTION
Write out the given parameters in the questions
[tex]\begin{gathered} \text{standard deviation =0.4 } \\ \text{sample ,n=20} \end{gathered}[/tex]The critical value is the measurement used to calculate the margin of error within a set of data and is expressed as
[tex]\begin{gathered} \text{Critical value =}1-\frac{\alpha}{2} \\ \text{where } \\ \alpha=0.05 \end{gathered}[/tex]Then
[tex]\text{critical value is the z=1.960}[/tex]Therefore the critical value is1.960
Then the standard error is given by
[tex]\begin{gathered} \sigma_{\bar{x}}=\frac{\sigma}{\sqrt[]{n}} \\ \\ \text{where } \\ n=\text{sample space=20 and }\sigma=0.4 \end{gathered}[/tex]Substituting the value we have
[tex]\sigma_{\bar{x}}=\frac{0.4}{\sqrt[]{20}}=\frac{0.4}{4.47}=0.089[/tex]Therefore the standard error is 0.089
The confidence interval is given by
[tex]\begin{gathered} \text{Confidence interval=}\bar{x}\pm(1.96)(S.E) \\ \text{where S.E= standard error } \end{gathered}[/tex]The mean for the sample will be
[tex]\bar{x}=\frac{sum\text{ of data}}{n}=\frac{3497.76}{20}=173.988[/tex]Substitute the value to obtain the confidence interval
[tex]\begin{gathered} \text{confidence interval=173.988}\pm1.96\times0.089 \\ C.I=173.988\pm0.174 \\ C\mathrm{}I=(173.814,174.162) \end{gathered}[/tex]Therefore, the confidence interval is (173.81,174.16) to 2d.p
place the following vectors in the Phet simulation :) Please use the Colorado Edu Simulator
Answers
Magnitude = 11.2 direction = 153.4°
Magnitude =
Magnitude = 8.5 direction = 159.4°
Select the correct answer.Which algebraic expression is equivalent to the expression below?3(5x + 7) - 6(4 – 2x)278 - 1212x + 312x + 3727x - 3
Answers
the given expression is
= 3(5x +7 ) - 6(4 - 2x )
= 15x + 21 - 24 + 12x
= 27x - 3
thus the answer is 27x - 3, correct option is D
For the following population of N = 9 scores: 4, 2, 0, 5, 3, 2, 1, 7, 3a. Sketch a histogram showing the populationdistribution.
Answers
Given:
The population of N = 9 scores is 4, 2, 0, 5, 3, 2, 1, 7, 3
Required:
Sketch a histogram showing the population distribution.
Explanation:
Make a frequency table for the given data as:M
Why and Were would you plot -6 3/4 on a number line ?
Answers
Problem
Why and Were would you plot -6 3/4 on a number line ?
Solution
For this case we can start rewriting the expression like this:
[tex]-6\frac{3}{4}=-\frac{6\cdot4+3}{4}=-\frac{27}{3}=-9[/tex]For this case the value is -9 so we can put the number in a line at this specific point
A student says an association must be one of two things: positive or negative. What would you tell the student? Explain.
Answers
Explanation:
In a scatterplot showing the association between two variables, there are three possible outcomes:
• A positive association
,• No association
,• A negative association
Thus, the student missed out the third possibility, i.e. that there is no association between the two variables.
Rewrite 17/9 as a mixed number
Answers
Solve
- Find the whole number
[tex]\frac{17}{9}=1.888889=1[/tex]Now that we have our whole number for the mixed fraction, we need to find our new numerator for the fraction part of the mixed number.
- Get the new numerator
To work this out we'll use the whole number we calculated in step one (1) and multiply it by the original denominator (9). The result of that multiplication is then subtracted from the original numerator:
[tex]17-(9\times1)=17-9=8[/tex]- Our mixed fraction
Now, we just need to put the whole number together with our new numerator and original denominator:
[tex]1\frac{8}{9}[/tex]Answer:
[tex]1\frac{8}{9}[/tex]Answer:
1 8/9
Step-by-step explanation:
A model rocket is launched with an initial upward velocity of 202 ft/s. The rocket’s height h (in feet) after t seconds is given by the following.h=202t-16t^2Find the value of t for which the rocket’s height is 82 feet. Round your answers to the nearest hundredth
Answers
Solution:
Given:
[tex]h=202t-16t^2[/tex][tex]\begin{gathered} when\text{ the height is 82 feet,} \\ h=82 \\ \\ Hence, \\ h=202t-16t^2 \\ 82=202t-16t^2 \\ Collecting\text{ all terms to one side to make it a quadratic equation;} \\ 16t^2-202t+82=0 \\ Dividing\text{ the equation all through by 2,} \\ 8t^2-101t+41=0 \\ \\ To\text{ solve for t, we use the quadratic formula;} \\ \frac{-b\pm\sqrt{b^2-4ac}}{2a} \\ where; \\ a=8,b=-101,c=41 \\ Hence, \\ t=\frac{-\left(-101\right)\pm\sqrt{\left(-101\right)^2-\left(4\times8\times41\right)}}{2\times8} \\ t=\frac{101\pm\sqrt{10201-1312}}{16} \\ t=\frac{101\pm\sqrt{8889}}{16} \\ t=\frac{101\pm94.28}{16} \\ t_1=\frac{101+94.28}{16}=\frac{195.28}{16}=12.205\approx12.21s \\ t_2=\frac{101-94.28}{16}=\frac{6.72}{16}=0.42s \end{gathered}[/tex]Therefore, to the nearest hundredth, the value of t for which the rocket's height is 82 feet is;
0.42 seconds or 12.21 seconds.
If a polynomial is of an odd degree and the leading coefficient is positive then polynomial
a) Rises to the right and falls to the left.
b) Falls to the right and rises to the left.
c) Falls to the right and to the left.
d) Rises to the right and to the left.
Answers
As per the Leading Coefficient Test, If a polynomial is of an odd degree and the leading coefficient is positive then polynomial, then it Rises to the right and falls to the left.
Leading Coefficient Test:
The leading coefficient test defined that the graph rises or falls depending on whether the leading terms are positive or negative, so for left-hand behavior (negative numbers), by the use of both the coefficient and the degree of the component together.
Given,
Here we have to determine If a polynomial is of an odd degree and the leading coefficient is positive then polynomial will fall in which place.
By using the Leading Coefficient Test we have to determine the end behavior of the graph of the polynomial function.
Here we know that the behavior of the polynomial function is odd degree and the leading coefficient is positive.
Therefore, as per the leading coefficient test, the resulting graph will rises to the right and falls to the left.
To know more about Lead Coefficient test here.
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Fill in the blank. In the triangle below, x = decimal places. Round your answer to two لم 35
Answers
Right Triangles
Right triangles can be identified because they have one internal angle of 90°.
Right triangles have two shorter sided called the legs and one larger side called the hypotenuse.
The relationship between the side lengths and the angles are called the trigonometric ratios.
For example, the angle of 42° as shown in the figure has one adjacent side (35) one opposite side (x) and the hypotenuse (y).
It's required to find the value of x, knowing the value of the side (45) and the angle 42°.
The trigonometric ratio that relates the opposite leg and the adjacent leg is called the tangent.
[tex]\displaystyle\tan \theta=\frac{\text{opposite leg}}{\text{adjacent leg}}[/tex]Substituting:
[tex]\displaystyle\tan 42^o=\frac{x}{35}[/tex]Solving for x:
x = 35 tan 42°
Calculating:
x = 35 * 0.9004
x = 31.51
A watermelon is shot out of an air cannon and travels 22 yards. How many feet does the watermelon travel?Which conversion factor could you use to answer the question?A. 1 yard/3 feetB. 1 foot/3 yardsC. 3 yards/1 footD. 3 foot/1 yard
Answers
the factor is the division between feet and yards
so
[tex]\frac{3\text{feet}}{1\text{yard}}[/tex]so, the option is C
Write the following comparison as a ratio reduced to lowest terms. 86 minutes to 5 hours
Answers
Given the comparison:
86 minutes to 5 hours
To write as a ratio, first convert the hours to minutes.
We have:
1 hour = 60 minutes
5 hours = 5 x 60 = 300 minutes
Thus, as a ratio, we have:
86 : 300
Let's reduce to lowest terms:
To reduce to lowest terms, divide both values by the Greatest Common Factor (GCF).
GCF of 86 and 300 = 2
[tex]\frac{86\div2}{300\div2}=\frac{43}{150}[/tex]Therefore, the ratio reduced to its lowest term is = 43 : 150
ANSWER:
43 : 150
A store is having a sale where everything is discounted 30%. Find the discount and the salesprice if a customer buys an item that normally sells for $365.First blank: What is the discount?Second Blank: What is the new total?
Answers
We know that the store discounts 30% of the original price for an item, then, to calculate the discount we just have to multiply the original sell price by the discount percent and divide it by 100, like this:
[tex]\text{discount}=365\times\frac{30}{100}=365\times0.3=109.5[/tex]Then, the discount is $109.5.
The new total would be the difference between the original price of the item and the discount, like this:
New price= $365-$109.5=$255.5
A) Find the constant of proportionality ( K )K =B) Find the equation Y = ____X
Answers
N 7
Part A
In a direct variation
k=y/x
take one point from the graph
(2,4)
k=4/2
k=2Part B
In a direct variation, the equation is
y=kx
we have
k=2
therefore
y=2xQuick question plss..
Does alcohol and water mix?
Answers
A car moves at 45 miles per hour. The time and the distance it travels are recorded.What is the dependent variable?A.the carB.the timeC.the distanceD.45
Answers
So,
We know that the car moves at 45 miles / hour.
The distance can be described as:
[tex]d=st[/tex]Where s is the speed of our car and t is the time. If we replace:
[tex]d=45t[/tex]As you can see, the dependent variable of this equation is the distance, and it depends of the time travelled.
find the numbers of divisors of the composite number 231
Answers
The given number is 231.
Remember that a composite number means it can be expressed as factors.
In this case, if we multiply 3 x 7 x 11 we get 231.
Therefore, the number 231 has three divisors which are 3, 7, and 11.how do you convert .05 into a fraction?
Answers
We are asked to convert the decimal number 0.05 into a fraction.
As you can see, the decimal point is two digits to the left or the digit 5 is at the hundredth place.
Multiply and divide by 100
[tex]0.05\times\frac{100}{100}=\frac{5}{100}[/tex]Now we have to simplify the fraction
[tex]\frac{5}{100}=\frac{1}{20}[/tex]Therefore, the 0.05 is equal to 1/20
[tex]0.05=\frac{1}{20}[/tex]What is 1/5 of 80 ?
Answers
1/5 of 80 is the same as dividing 80 by 5, then it is:
[tex]\frac{1}{5}\cdot80=\frac{80}{5}=16[/tex][tex]1/5\\of\\80[/tex]
To find 1/5 of 80, we need to divide 80 by 5:
[tex]80/5[/tex]
[tex]=16[/tex]
Hope this helps!