A factor of a number is called a prime factor.
The prime factor of a number is obtained by a factor tree method or by a division method.
The expression of a given number as a product of factors is called prime factorisation.
The prime number has 1 and the number itself is a factor.
The composite number will have more than one factor.
A number which is not a prime number is a composite.
1 and the number itself are not included in the prime factorisation of a composite number.
let us take the example
For the number 20, the prime factors are 2*2*5.
The above example is an example of the prime factorization of the composite number. where 1 and the number itself are not included in the prime factorisation of a composite number.
Answer:
But they can be broken down into prime numbers the process of writing out the prime numbers that make up a composite number is known as prime factorization
Step-by-step explanation:
SA bag contains 1 gold marbles, 6 silver marbles, and 21 black marbles. Someone offers to play this game: Yourandomly select one marble from the bag. If it is gold, you win $3. If it is silver, you win$2. If it is black, youlose $1.What is your expected value if you play this game?
We are given that a bag contains 1 gold marble, 6 silver marbles, and 21 black marbles. First, we need to determine the total number of marbles. The number of marbles of each color is:
[tex]\begin{gathered} N_{gold}=1 \\ N_{silver}=6 \\ N_{\text{black}}=21 \end{gathered}[/tex]The total number is then:
[tex]N_t=N_{\text{gold}}+N_{\text{silver}}+N_{\text{black}}[/tex]Substituting the values:
[tex]N_t=1+6+21=28[/tex]Therefore, there are a total of 28 marbles. Now we determine the probability of getting each of the marbles by determining the quotient of the number of marbles of a given color over the total number of marbles. For the gold marbles we have:
[tex]P_{\text{gold}}=\frac{N_{\text{gold}}}{N_t}=\frac{1}{28}[/tex]For silver we have:
[tex]P_{\text{silver}}=\frac{N_{silver}}{N_t}=\frac{6}{28}=\frac{3}{14}[/tex]For the black marbles:
[tex]P_{\text{black}}=\frac{N_{\text{black}}}{N_t}=\frac{21}{28}=\frac{3}{4}[/tex]Now, to determine the expected value we need to multiply each probability by the value that is gained for each of the colors. We need to have into account that is it is a gain we use a positive sign and if it is a lose we use a negative sign:
[tex]E_v=(3)(\frac{1}{28})+(2)(\frac{3}{14})+(-1)(\frac{3}{4})_{}[/tex]Solving the operations we get:
[tex]E_v=-0.21[/tex]Therefore, the expected value is -$0.21.
Find two solutions for the equation 4x+3y=24 , draw it's graph .
The equation of consideration is:
[tex]4x\text{ + 3y = 24}[/tex]Since two unknowns ( x and y) are given in just one equation, to get each set of the solutions, we are going to choose a value of x and get a corresponding value of y
Let x = 0, to get the value of y at point x = 0, substitute this value of x into the given equation:
[tex]4(0)\text{ + 3y = 24}[/tex][tex]3y\text{ = 24}[/tex][tex]y\text{ = }\frac{24}{3}\text{ = 8}[/tex]The first set of solutions is therefore:
[tex]x\text{ = 0, y = 8}[/tex]To get the second set of solution, let us choose x = 3 and substitute this value into the given equation:
[tex]4(3)\text{ + 3y = 24}[/tex][tex]12\text{ + 3y = 24; 3y = 24 - 12; 3y = }12;\text{ y = 12/3; y = 4}[/tex]The second set of equation is:
[tex]x\text{ = 3, y = 4}[/tex]The above is the graph showing the two sets of solutions
13 inches by 6 inches by 4 inches. what is the maximum lenght
Which of the following polynomial has roots at 3, -4, and a double root at-2?
The second polynomial [tex]f(x) = (x - 3)(x+4)(x+2)^2[/tex] has roots 3, -4 and a double root at -2.
What is root of polynomial?
The values of a variable for which the provided polynomial equals zero are referred to as a polynomial's roots. P(a) = 0 if an is the polynomial's root for x.
Since, if x is a root of the polynomial then x is a zero of the polynomial.
That means f(x) = 0
Consider, the first polynomial, [tex]f(x) = (x+3)(x-4)(x-2)^2[/tex]
plug x = 3, [tex]f(3) = (9)(-1)(1)^2 = -9[/tex] ≠ 0
Plug x = -4, [tex]f(-4) = (-1)(-8)(36) = 368[/tex] ≠ 0
Plug x = -2, [tex]f(-2) = (1)(-6)(16) = -96[/tex] ≠ 0
Therefore, 3, -4 and -2 are not the roots of the first polynomial.
Now consider the second polynomial, [tex]f(x) = (x-3)(x+4)(x+2)^2[/tex]
Plug x = 3, [tex]f(3) = 0[/tex]
Plug x = -4, [tex]f(-4) = 0[/tex]
Plug x = -2, [tex]f(-2) = 0[/tex]
Therefore, x = 3, -4, -2 are the roots of the second polynomial.
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a card is from a standard deck of cards is chosen at random, then a coin is tossed. what is the probability of getting ace and tails!?
A standard deck of cards consists of 52 cards with 4 aces.
The probability of getting an ace in a pack of the standard deck is;
[tex]P_1=\frac{4}{52}=\frac{1}{13}[/tex]Find the inverse of the function. Is the inverse a function? Simplify your answer.F(x)=2x-1f^-1(x)=
The definition of the inverse function is
[tex]\begin{gathered} f(f^{-1}(x))=x \\ \text{and} \\ f^{-1}(f(y))=y \end{gathered}[/tex]In our case,
[tex]f(x)=2x-1[/tex]Then,
[tex]\begin{gathered} f^{-1}(f(x))=x \\ \Rightarrow f^{-1}(2x-1)=x \\ \Rightarrow f^{-1}(x)=\frac{x+1}{2} \end{gathered}[/tex]We need to verify this result using the other equality as shown below
[tex]\begin{gathered} f^{-1}(x)=\frac{x+1}{2} \\ \Rightarrow f(f^{-1}(x))=f(\frac{x+1}{2})=2(\frac{x+1}{2})-1=x+1-1=x \\ \Rightarrow f(f^{-1}(x))=x \end{gathered}[/tex]Therefore,
[tex]\Rightarrow f^{-1}(x)=\frac{x+1}{2}[/tex]The inverse function is f^-1(x)=(x+1)/2.
We say that a relation is a function if, for x in the domain of f, there is only one value of f(x).
In our case, notice that for any value of x, there is only one value of (x+1)/2=x/2+1/2.
The function is indeed a function, it is a straight line on the plane that is not parallel to the y-axis.
The inverse f^-1(x) is indeed a function
graph at least one full cycle of the following trig function, lable the amplitude midline and maximum and the intervals f(x)=2sin(x-pi/2)-1
In one city, the probability that a person will pass his or her driving test on the first attempt is 0.63. 11 people are selected at random from among those taking their driving test for the first time. What is the probability that among these 11 people, the number passing the test is between 2 and 4 inclusive?
The probability that among these 11 people, the number passing the test is between 2 and 4 inclusive is 0.0665
What is Probability?
Probability gives us the information about how likely an event is going to occur
Probability is calculated by Number of favourable outcomes divided by the total number of outcomes.
Probability of any event is greater than or equal to zero and less than or equal to 1.
Probability of sure event is 1 and probability of unsure event is 0.
Now
Binomial distribution of probability will be used
Here, n = 11, p = 0.63,
P(X = x) = [tex]{n\choose x} p^x(1-p)^{n-x}[/tex]
P(X= 2) = [tex]{11\choose 2} 0.63^2(1-0.63)^{11-2}[/tex]
= 0.0028
P(X = 3) = [tex]{11\choose 3} 0.63^3(1-0.63)^{11 - 3}\\[/tex]
= 0.0144
P(X = 4) = [tex]{11\choose 4} 0.63^4(1-0.63)^{11 - 4}\\\\[/tex]
= 0.0493
The probability that among these 11 people, the number passing the test is between 2 and 4 inclusive = 0.0028 + 0.0144 + 0.0493
= 0.0665
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Find measure angle ABD and measure angle CBD #C 2x A B
As we see in the figure, BD bisects the right angle ABC and thus, we find out that ∠ABD = 60° and ∠CBD = 30°. Thus, option 1 is correct.
From the given figure, we have
∠ABD = 4x° ---- (1)
∠CBD = 2x° ---- (2)
∠ABC = 90° ---- (3)
We have to find out the values of the ∠ABD and ∠CBD.
As given in the figure, we can see that BD bisects ∠ABC into ∠ABD and ∠CBD. So, we can say that -
∠ABD + ∠CBD = ∠ABC
=> 4x° + 2x° = 90° [From equation (1), (2), (3)]
=> 6x° = 90°
=> x° = 15° ---- (4)
Substituting equation (4) in equations (1) and (2), we get
∠ABD = 4x° and ∠CBD = 2x°
=> ∠ABD = 4*15° and ∠CBD = 2*15°
=> ∠ABD = 60° and ∠CBD = 30°
Since BD bisects the right angle ABC, we find out that ∠ABD = 60° and ∠CBD = 30°. Thus, option 1 is correct.
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Rewrite in simplest terms: 10(7p + 6) – 5(5p + 4)
Answer:
Step-by-step explanation:
10(7p + 6) – 5(5p + 4)=70p+60-25p-20=45p+40=5(9p+8)
Two matrices can always be multiplied if the have the same dimensions. True False
SOLUTION:
Case: Matrices multiplication
Given:
Two matrices can always be multiplied if they have the same dimensions.
Method:
From the image above, if and only if the number of items of columns matches the number of items of the columns, then it is possible to multiply.
Final answer:
True,
Two matrices can always be multiplied if they have the same dimensions
Write in point slope and convert to slope intercept form: a line with a slope -5 that goes through the point (1.-7)
Weare asked to use the "point-slope" form of a line that has slope -5 and goes though the point (1, -7) on the plane.
Therefore we use the form:
y - yp = m (x - xp)
where "m" is the slope, and xp and yp are the coordinates of the point on the plane the line goes through. So in our case we have:
y - (-7) = -5 (x - 1)
now we proceed to remove parenthesis using distributive property:
y + 7 = -5 x + 5
and finally express the equation in slope-intercept form by isolating "y" on the left:
Subtract 7 from both sides and combine:
y = -5 x + 5 - 7
y = -5 x - 2
The wholesale price for a pillow is $4.50 . A certain department store marks up the wholesale price by 90% . Find the price of the pillow in the department store. Round your answer to the nearest cent, as necessary.
The wholesale price for a pillow is $4.50 .
The store marks up the wholesale price by 90%.
That is, 90% of $4.50 has been aadded with the wholesale price as the store price (SP).
Therefore we have, '
[tex]Sp=4.50+\frac{90}{100}\times4.50=8.55[/tex]Thus, the store price is $8.55
A) Angle CDE measures 80 degrees.B)Angle CDE measures 100 degrees C) The sum of the measures of the arcs from E to C, one passing through D and passing through b is 360D)The arcs from E to C passing through D measures 100 degreesE) Angle BCD measures 50 degrees F) The arc from B to D passing through C measures 100
Given the figure of a cyclic quadrilateral
We will check whether the given statements are true or false.
A) Angle CDE measures 80 degrees.
True
Because the sum of the opposite angles has a sum of 180
B) Angle CDE measures 100 degrees
False
C) The sum of the measures of the arcs from E to C, one passing through D and passing through b is 360
True
Because the sum of the central angles of the circle = 360
The two arcs are forming the complete circle.
D)The arcs from E to C passing through D measure 100 degrees
False
Because the measure of the arc = 2 times the angle CBE = 200
E) Angle BCD measures 50 degrees
False
Because the measure of the angle BCD = 180 - 50 = 130
The sum of the opposite angles = 180
F) The arc from B to D passing through C measures 100
True
Because the inscribed angle opposite the arc = 50
So, the measure of the arc = 2 times the opposite inscribed angle
Solve the equation for a: z = ma – b
From the given question
There are given that the equation:
[tex]z=ma-b[/tex]Now,
For finding the value of a, first, add b in both sides of the equation
So,
[tex]\begin{gathered} z=ma-b \\ z+b=ma-b+b \\ z+b=ma \end{gathered}[/tex]Then,
Divide by m on both sides the above equation
[tex]\begin{gathered} z+b=ma \\ \frac{z+b}{m}=\frac{ma}{m} \\ a=z+b \end{gathered}[/tex]Hence, the value of a is z + b.
Use the distributive property to remove the parenthesis (X+7)12
Answer
Use the distributive property to remove the parenthesis
[tex]\begin{gathered} a(b+c) \\ ab+ac \end{gathered}[/tex]Now , Given
[tex]\begin{gathered} (x+7)12 \\ x\times12\text{ +7}\times12 \\ 12x+84 \end{gathered}[/tex]The final answer
[tex]12x+84[/tex](4+3i)-(2-i)Simplify, leave in a+bi form
The given expression is
(4 + 3i) - (2 - i)
We would simplify the terms inside the parentheses. The negative sign(- 1) outside the second parentheses would be used to multiply each term inside. We have
4 + 3i + - 1 * 2 + - 1 * - i
4 + 3i - 2 + i
By collecting like terms, we have
4 - 2 + 3i + i
2 + 4i
Thus, the answer in a + bi form is
2 + 4i
Waterworks is a company that manufactures and sells paddle boards. It's profit P, in hundreds of dollars earned, is a function of the number of paddle boards sold x, measured in thousands. Profit is modeled by the function P(x)=-2x^3+34x^2-120x. What do the zeros of the function tell you about the number of paddle boards that waterworks should produce?
areAs given by the question
There are given that the profit function
[tex]P(x)=-2x^3+34x^2-120x[/tex]Now,
The zeros are the x values where the graph intersects the x axis.
Then,
To find the zeroes, replace P(x) with 0 and solve for x.
Then,
The zeroes of the given function is:
[tex]\begin{gathered} P(x)=-2x^3+34x^2-120x \\ 0=-2x(x^2-17x^{}+60) \\ x^2-17x^{}+60=0 \\ (x-12)(x-5)=0 \\ x=0,\text{ 12, 5} \end{gathered}[/tex]Hence, the zeroes of the function is 0, 12, 5.
main and Range HAMAD SALIM 12 Range 12 The graph below represents the function y = f(x). State the domain and range of this function. 1 I US
Remember that the domain is the data set of all possibles values of x and the range is the data set of all possibles values of y
so
Looking at the graph
Domain is the interval for x
[-5,8}
All real numbers greater than or equal to -5 and less than or equal to 8
Range is the interval for y
[-3,2]
All real numbers greater than or equal to -3 and less than or equal to 2
12. Write the equation of the line that is perpendicular to the line x - 4y = 20 and passes through the point (2,-5).
Two perpendicular lines have reciprocal and opposite slopes.
First we have to write the given line in the slope-intercept form:
[tex]y=mx+b[/tex]Where m is the slope and b is the y-intercept.
We have this equation:
[tex]x-4y=20[/tex]To write it in the slope-intercept form we have to clear y:
[tex]\begin{gathered} x-20=4y \\ \downarrow \\ y=\frac{1}{4}x-5 \end{gathered}[/tex]The slope is 1/4 and the y-intercept is -5.
The slope of the perpendicular line will be the opposite and reciprocal of 1/4, that's -4.
For now we have the perpendicular line's equation:
[tex]y_p=-4x+b[/tex]There are a lot of lines that are perpendicular to the given line, but only one that passes through (2, -5). We use this point to find the y-intercept by replacing x = 2 and y = -5 into the expression above and solving for b:
[tex]\begin{gathered} -5=-4\cdot2+b \\ -5=-8+b \\ -5+8=b \\ b=3 \end{gathered}[/tex]The y-intercept of the perpendicular line is 3.
The equation of a line perpendicular to the given line that passes through the point (2,-5) is
[tex]y_p=-4x+3[/tex]Which of the following is the equation c^(4d+1)=7a-b written in logarithmic form?
We have the expression:
[tex]c^{(4d+1)}=7a-b[/tex]We can apply logarithm to both sides. We would use it in order to get "4d+1". Then, we would apply logarithm with base c. This is beacuse of the definition of logarithm:
[tex]\log _c(x)=y\Leftrightarrow c^y=x[/tex]If we apply this to our expression, we get:
[tex]\begin{gathered} c^{(4d+1)}=7a-b \\ \log _c(c^{(4d+1)})=\log _c(7a-b) \\ 4d+1=\log _c(7a-b) \end{gathered}[/tex]If we rearrange both sides, we get the expression in Option B (we have to switch the sides):
[tex]\begin{gathered} 4d+1=\log _c(7a-b) \\ \log _c(7a-b)=4d+1 \end{gathered}[/tex]Answer: Option B
2.Evaluate the following mixed numbers, then simplify.7 1/2 divide 1 1/8
Given the numbers : 7 1/2 and 1 1/8
We will divide them
so,
[tex]\begin{gathered} 7\frac{1}{2}\div1\frac{1}{8} \\ \\ =\frac{15}{2}\div\frac{9}{8} \\ \\ =\frac{15}{2}\times\frac{8}{9} \\ \\ =\frac{8}{2}\times\frac{15}{9}=4\times\frac{5}{3}=\frac{20}{3}=6\frac{2}{3} \end{gathered}[/tex]Pls help with this math problem pl
Using the slope intercept equation, the equation of the line in fully simplified slope intercepted form is y=4x−4.
In the given question we have to write the equation of the line in fully simplified slope intercepted form.
As we know that slope intercept form of equation of line is given by
y=mx+c
where m=slope
c=intercept of the line (i.e point where line cut y-axis )
From graph we can easily find two point of the line that is (1,0)(0,−4).
From the point x(1)=1, y(1)=0, x(2)=0, y(2)=−4
Slope (m)=(y(2)−y(1))/(x(2)−x(1))
m=(−4−0)/(0−1)
m=-4/−1
m=4
As we know that c is a point where line cut y axis so c=−4
Hence, slope-intercept form of equation is y=4x−4.
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8.9.Find the slopes of the lines that are (a) parallel and (b) perpendicular to the line through the pairof points.(3, 3) and (-5, -5)OA-3335B 0; 0C 1; -1OD -1; 1Determine whether the lines are parallel, perpendicular, skew, or neither.
We know that the equation of the line that pass through the pair of points (3, 3) and (-5, -5) is x = y, so the slope of a paralell line is 1 and a perpendicular line is -1.
So the answe is C. 1, -1.
Solve by substitution 4x + 2y =-14 x -2y =4
In order to solve by subdtitution, first, solve the second equation for x:
x - 2y = 4 add 2y both sides
x = 4 + 2y
next, replace the previous expression for x into the first equation and solve for y:
4x + 2y = -14 replace x=4+2y
4(4 + 2y) + 2y = -14 apply distribution property
16 + 8y + 2y = -14 subtract 16 both sides
8y + 2y = -14 - 16 simplify like terms both sides
10y = -30 divide by 10 both sides
y = -30/10
y = -3
next, replace y=-3 into x = 4 + 2y
x = 4 + 2y = 4 + 2(-3) = 4 -6 = -2
x = -2
Hence, the solution to the given system of equations is:
x = -2
y = -3
find the two dimensional diagonal. Write your answer as a radical.
Using the pythagoras theorem,
[tex]\begin{gathered} c^2=b^2+a^2 \\ 6^2=3^2+a^2 \\ a^2=36-9 \\ a^2=27 \\ a=\sqrt[]{27} \\ a=5.19 \end{gathered}[/tex]Background Layout - Theme Transition 910 78 45 111 112 113 11 USE THE GIVEN INFORMATION TO ANSWER EACH QUESTION BELOW. 5(4) From the choices at the right, drag the expression that could be used to find the area of each piece 132 Andre needs to paint three square pieces of wood in the sizes shown. He has them arranged so that they meet to form a right triangle A: B: C: 13 Type to record the number of square centimeters Andre will need to paint on each piece 12(4) INTRO TO PYTHAGOREAN THEOREM A: B: C: 122 C 13 cm 123 A 5 cm 3 Add the area of piece A and the area of piece B together. What does this prove about the side lengths in a right triangle? 12 cm 52 B DRAG THESE Mong the Middle LLC, 2019
The area of a square is the squared side, it means
[tex]A=l^2[/tex]It means, for A, which has a side of 5, the area is
[tex]5^2[/tex]For B, which side is 12, its area is
[tex]12^2[/tex]For C, the area is
[tex]13^2[/tex]Andre has to paint (solve each power):
[tex]\begin{gathered} A=25 \\ B=144 \\ C=169 \end{gathered}[/tex]Once we add the areas of A and B we realize that the sum is equal to the area of C, it proves the pythagorean theorem that says that the sum of the squared length of the legs equals the squared length of the hypotenuse
I need help with this question please Identify the binomial that is a factor of the polynomial
(x-2)
1) Let's use the Rational Roots Theorem so that we can factor this Polynomial and find the factors that make up this Polynomial.
2) Taking all the factors of the constant and the leading coefficients we have:
[tex]P(x)=3x^3-11x^2-2x+24[/tex]
Let's enlist these factors:
[tex]\begin{gathered} 24\colon\pm1,\pm2,\pm4,\pm3,\pm6,\pm8,\pm12,\pm24 \\ 3\colon\pm1,\pm3 \end{gathered}[/tex]2.2) Let's pick any number on the numerator and divide it by any number of the denominator, to get possible roots:
[tex]\begin{gathered} \frac{\pm1,\pm2,\pm4,\pm3,\pm6,\pm8,\pm12,\pm24}{\pm1,\pm3}=\pm1,\pm2,\pm\frac{4}{3}, \\ \end{gathered}[/tex]Proceeding with that let's do a Synthetic Division, testing 2
[tex]\begin{gathered} \frac{3x^3-11x^2-2x+24}{(x-2)}= \\ (x-2)(3x^2-5x-12) \\ (x-2)(3x+4)(x-3) \end{gathered}[/tex]Note that we have three factors. After factoring out
3) Hence, the answer is (x-2)
The Harris school district gives awards to its schools based on overall student attendance. The data for attendance are shown in the table below, where Min represents the fewest days attended and Max represents the most days attended for a single student:SchoolMinMaxRangeMeanMedianIQRσHigh School A1071808216915048.533.6High School B921807214113944.527.0High School C1041807516115154.532.4Part A: If the school district wants to award the school that has the most consistent attendance among its students, which high school should it choose and why? Justify your answer mathematically. (5 points)Part B: If the school district wants to award the school with the highest average attendance, which school should it choose and why? Justify your answer mathematically. (5 points) (10 points)
Answer to part A :
The school that gets the award for the highly consistent attendance among its students is school B . Because its range is the smallest that is there exists a smaller variation in the highest and the lowest value and also because the interquartile range is the smallest means that the data is spread out the least.
Answer to part B :
The school that gets the award for the highest average attendance among its students is High school A . This is because its mean and median are the highest which means that there are more kids attended school on average.
Rachel is conducting a study in her cognitive psychology lab about people's ability to remember rhythms. She played a short Rhythm to 425 randomly chosen people. One minute later, she asked him to repeat it by clapping. If 121 people were able to successfully reproduce the rhythm, estimate the proportion of the population (including the margin of error) that would be able to successfully reproduce the rhythm. Use a 95% confidence interval.
Given:
Sample Size (n) = 425
No. of Success = 121
Find: estimate the proportion of the population
Solution:
Let's calculate first the success proportion in the sample by dividing no. of success over the total number of people then multiply it by 100.
[tex]\frac{121}{425}\times100\%=28.47\%[/tex]Our sample proportion p = 28.47%.
Then, for the margin of error, the formula is:
[tex]MOE=z\sqrt{\frac{p(1-p)}{n}}[/tex]where z = critical value, p = sample proportion, and n = sample size.
For our z-value, since we are using a 95% confidence interval, the value of z = 1.645.
[tex]MOE=1.645\sqrt{\frac{.2847(1-.2847)}{425}}[/tex]Then, solve.
[tex]MOE=1.645\sqrt{\frac{0.203648}{425}}[/tex][tex]MOE=1.645(0.02189)[/tex][tex]MOE=0.036[/tex]Let's multiply the MOE by 100%.
[tex]0.036\times100\%=3.6\%[/tex]Therefore, about 28.47% ± 3.6% or between 24.87% to 32.07% of the population would be able to successfully reproduce the rhythm.