There are 70,560 different ways to arrange the letters of the word "DETERRANT" so that the repeated letters do not come together.
How many ways the word "DETERRANT" be arranged so that the repeated letters do not come togetherThe word "DETERRANT" has 9 letters, out of which there are 2 E's, 2 R's, and 1 T.
To determine the total number of possible arrangements, we begin with 9! (9 factorial), which counts all possible arrangements if all letters are distinct.
We must, however, account for the repeated letters. For the two E's, we divide by 2, which is the number of different ways they can be arranged without changing the overall arrangement. Likewise, we divide by 2! for the two R's.
As a result, the total number of arrangements with repeated letters that do not come together is:
9! / (2! x 2!) - 8! / (2! x 2!) = 90720 - 20160 = 70560
Therefore, there are 70,560 different ways to arrange the letters of the word "DETERRANT" so that the repeated letters do not come together.
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"Consider the following circle with radius 1. The lines A and B are radii, and have a length of 1. Find the area of the shaded region."
I got an area of 1.06 (Rounded), I'm just wondering if that is correct. Any help would be appreciated!
Answer:
[tex]\textsf{Area}=\dfrac{5\pi -3}{12} \approx 1.06\; \sf square\;units \;(3\;s.f.)\end{aligned}[/tex]
Step-by-step explanation:
To find the area of the shaded region, subtract the area of the isosceles triangle from the area of the sector.
An isosceles triangle is made up of two congruent right triangles.
To find the height of the right triangles (and thus the height of the isosceles triangle), use the cosine trigonometric ratio.
[tex]\boxed{\begin{minipage}{9 cm}\underline{Cosine trigonometric ratio} \\\\$\sf \cos(\theta)=\dfrac{A}{H}$\\\\where:\\ \phantom{ww}$\bullet$ $\theta$ is the angle. \\ \phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle. \\\phantom{ww}$\bullet$ $\sf H$ is the hypotenuse (the side opposite the right angle). \\\end{minipage}}[/tex]
Given values:
The angle is half the apex angle: θ = 5π/12.The side adjacent the angle is the height of the triangle: x.The hypotenuse is the radius: H = 1.Substitute these values into the cosine ratio to calculate the height of the right triangles (and thus the height of the isosceles triangle):
[tex]\cos \left(\dfrac{5\pi}{12}\right)=\dfrac{x}{1}[/tex]
[tex]x=\cos \left(\dfrac{5\pi}{12}\right)[/tex]
[tex]x=\dfrac{\sqrt{6}-\sqrt{2}}{4}[/tex]
The base of one of the right triangles can be found by using the sine trigonometric ratio.
[tex]\boxed{\begin{minipage}{9 cm}\underline{Sine trigonometric ratio} \\\\$\sf \sin(\theta)=\dfrac{O}{H}$\\\\where:\\ \phantom{ww}$\bullet$ $\theta$ is the angle. \\ \phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle. \\\phantom{ww}$\bullet$ $\sf H$ is the hypotenuse (the side opposite the right angle). \\\end{minipage}}[/tex]
Given values:
The angle is half the apex angle: θ = 5π/12.The side opposite the angle is the base of the right triangle: y.The hypotenuse is the radius: H = 1.Substitute these values into the sine ratio to determine the base of the right triangle:
[tex]\sin \left(\dfrac{5\pi}{12}\right)=\dfrac{y}{1}[/tex]
[tex]y=\sin \left(\dfrac{5\pi}{12}\right)[/tex]
[tex]y=\dfrac{\sqrt{6}+\sqrt{2}}{4}[/tex]
The base of the isosceles triangle is twice the base of the right triangle, so the base of the isosceles triangle = 2y.
Therefore the area of the isosceles triangle is:
[tex]\begin{aligned}\textsf{Area of isosceles triangle}&=\dfrac{1}{2} \cdot \sf base \cdot height\\\\&=\dfrac{1}{2}\cdot 2y \cdot x\\\\&=\dfrac{1}{2} \cdot 2\left(\dfrac{\sqrt{6}+\sqrt{2}}{4}\right) \cdot \left(\dfrac{\sqrt{6}-\sqrt{2}}{4}\right)\\\\&=\left(\dfrac{\sqrt{6}+\sqrt{2}}{4}\right) \cdot \left(\dfrac{\sqrt{6}-\sqrt{2}}{4}\right)\\\\&=\dfrac{4}{16}\\\\&=\dfrac{1}{4}\sf \;square\;units\end{aligned}[/tex]
To find the area of the sector of the circle, use the area of a sector formula (where the angle is measured in radians).
[tex]\boxed{\begin{minipage}{6.4 cm}\underline{Area of a sector}\\\\$A=\dfrac12 r^2 \theta$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius. \\ \phantom{ww}$\bullet$ $\theta$ is the angle measured in radians.\\\end{minipage}}[/tex]
Given values:
r = 1θ = 5π/6Substitute the values into the formula:
[tex]\begin{aligned}\textsf{Area of the sector}&=\dfrac{1}{2} \cdot (1)^2 \cdot \dfrac{5\pi}{6}\\\\&=\dfrac{1}{2} \cdot 1 \cdot \dfrac{5\pi}{6}\\\\&=\dfrac{1}{2} \cdot \dfrac{5\pi}{6}\\\\&=\dfrac{5\pi}{12}\;\; \sf square\;units\end{aligned}[/tex]
Finally, to find the area of the shaded region, subtract the area of the isosceles triangle from the area of the sector:
[tex]\begin{aligned}\textsf{Area of the shaded region}&=\sf Area_{sector}-Area_{isosceles\;triangle}\\\\&=\dfrac{5\pi}{12}-\dfrac{1}{4}\\\\&=\dfrac{5\pi}{12}-\dfrac{3}{12}\\\\&=\dfrac{5\pi -3}{12}\\\\&=1.05899693...\\\\&=1.06\; \sf square\;units \;(3\;s.f.)\end{aligned}[/tex]
Therefore, the area of the shaded region is (5π - 3)/12 or approximately 1.06 square units (3 significant figures).
gabriela recorded the ages of her shirts which of these best describes the shape of the distribution
Based on the image attached, The the shape of the distribution is option c) data skewed right.
What is data skewed right?The term Right skewed is one where the mean is more prominent than the middle. The mean overestimates the foremost common values in a emphatically skewed dispersion. Cleared out skewed: The mean is less than the median. The mean is one that has little of the common values in a rightly skewed dissemination.
So, therefore, if most of the information are on the cleared out side of the histogram but many larger values are on the proper, the information are said to be skewed to the right.
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See text below
Gabriela recorded the ages of her shirts.
·
0 1 2 3 4 5 6 7 8 9 10
Age (years)
Which of these best describes the shape of the distribution?
Choose 1 answer:
Roughly symmetric
Skewed left
Skewed right
Good Question (141)
Mr. Sharp is considering a $100,000 investment in a company that has invented a revolutionary new jet ski. Mr. Sharp's advisors tell him that the company has a 30% probability of success. If it succeeds, Mr. Sharp's investment will be worth $2,000,000. If it fails, the investment will be worthless. What is the expected value of the investment? Should Mr. Sharp invest the $100,000?
The expected value of the investment is $600,000.
The expected value of the investment is calculated by multiplying the possible outcomes by their respective probabilities and summing them up. In this case, the possible outcomes are a success or a failure, with probabilities of 0.3 and 0.7 respectively.
If the company succeeds, the investment will be worth $2,000,000. If it fails, the investment will be worth $0. Therefore, the expected value is:
(0.3 x $2,000,000) + (0.7 x $0) = $600,000
This means that the expected value of the investment is $600,000.
Since the expected value of the investment is positive, Mr. Sharp should invest the $100,000. However, it's important to note that expected value is just an estimate and does not guarantee a positive return on investment. There is always a risk associated with any investment, and Mr. Sharp should carefully consider his risk tolerance and financial goals before making any investment decisions.
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9) The line plot below shows the heights in inches for a basketball team. How much taller is the tallest player than the shortest player on the basketball team?
14 inches taller by the tallest player than the shortest player on the basketball team. The correct option is A .
To determine how much taller the tallest player is than the shortest player on the basketball team, we need to examine the given line plot and calculate the difference between their heights.
To find the height difference, we need to identify the tallest and shortest players on the team. The line plot should display the heights of each player, allowing us to determine these values.
Once we have these heights, we can subtract the height of the shortest player from the height of the tallest player to calculate the difference.
Based on the options provided, we can consider each possibility:
A. If the height difference is 14 inches, then the tallest player is 14 inches taller than the shortest player. B. If the height difference is 10 inches, then the tallest player is 10 inches taller than the shortest player.
C. If the height difference is 13 inches, then the tallest player is 13 inches taller than the shortest player. D. If the height difference is 12.5 inches, then the tallest player is 12.5 inches taller than the shortest player.
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A construction company is considering submitting bids for two contracts. It
will cost the company $10,000 to prepare and submit the bids, and if won,
each bid would produce $50,000 of income to the company. The company
estimates that it has a 10% chance of winning any given bid.
Here is the probability distribution of X = the number of bids the
company wins, and M = the amount of money the company profits from
the bids.
X= # of bids won
M = profit
Probability
0
-$10,000
0.81
1
$40,000
0.18
bids
2
$90,000
0.01
Find the expected value of the number of bids won.
E(X)=
Step-by-step explanation:
To find the expected value of the number of bids won, E(X), we need to multiply each value of X with its corresponding probability and then sum these products.
E(X) = (X1 * P1) + (X2 * P2) + (X3 * P3)
Here, X1 = 0, X2 = 1, X3 = 2, and their corresponding probabilities are P1 = 0.81, P2 = 0.18, and P3 = 0.01.
E(X) = (0 * 0.81) + (1 * 0.18) + (2 * 0.01)
E(X) = (0) + (0.18) + (0.02)
E(X) = 0.20
The expected value of the number of bids won is 0.20.
make x subject
u= cos 0.5x
By making x the subject of the formula in this equation u = cos(0.5x) gives x = 2cos⁻¹(u).
How to make x the subject of the formula?In Mathematics and Geometry, an equation can be defined as a mathematical expression which shows that two (2) or more thing are equal. This ultimately implies that, an equation is composed of two (2) expressions that are connected by an equal sign.
In this exercise, you are required to make "x" the subject of the formula in the given mathematical equation by using the following steps.
By taking the arc cosine of both sides of the equation, we have the following:
u = cos(0.5x)
cos⁻¹(u) = 0.5x
By multiplying both sides of the mathematical equation by 2, we have the following:
2 × cos⁻¹(u) = 0.5x × 2
x = 2cos⁻¹(u)
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Help need asap please write a statement and proof
The theorems that demonstrate that AC ⊥ BD are Angle Bisector Theorems and Perpendicular Bisector Postulate.
Proof that AC ⊥ BDRecall that according to the angle bisector theorem, an angle bisector of a triangle divides the opposing side into two segments that are proportionate to the triangle's other two sides. An angle bisector is a ray that splits a given angle into two equal-sized angles.
In this case
∠1 ≅ ∠2 - Given
hence they is bisected by AX
Also ∠5 ≅ ∠6. This means that they are bisected by CX
Since AX and CX form AC, and
AC = CA (Reflexive Property)
Also
BD = DB (Reflexive Property)
Then it means that AC ⊥ BC (QED)
Recall that the perpendicular bisector principle deals with congruent triangle segments, allowing for congruent diagonals from the vertices to the circumcenter.
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PLEASE HELP I WILL MARK YOU BRAINLIEST
Answer:
18.85 meters
Step-by-step explanation:
V=πr2h=π·12·6≈18.84956
PLEASE HELP ME
Look at table and solve.
Identify the number of positive and negative residuals, the residual with the greatest absolute value, and the residual with the least absolute value. Is the model a good fit?
There are __?_ positive residuals and __?_ negative residuals.
The residual with the greatest absolute value is__ , and the residual with the least absolute value is__ .
Question 3
The absolute values of the residuals are relatively (small or large) , and in general, the data points (are or are not)evenly dispersed about the line of fit.
So, the equation y=−x+9.6 (Is or is not) a good fit.
There are 2 positive residuals and 4 negative residuals.
The residual with the greatest absolute value is -4.8, and the residual with the least absolute value is 0.5.
The absolute values of the residuals are relatively large, indicating that the model does not fit the data well. In general, the data points are not evenly dispersed about the line of fit.
Using the equation y=−x+9.6, we can calculate the predicted values for y as follows:
When x = 1: y = -(1) + 9.6 = 8.6
When x = 2: y = -(2) + 9.6 = 7.6
When x = 3: y = -(3) + 9.6 = 6.6
When x = 4: y = -(4) + 9.6 = 5.6
When x = 5: y = -(5) + 9.6 = 4.6
When x = 6: y = -(6) + 9.6 = 3.6
We can then calculate the residuals by subtracting the predicted values from the actual values of y:
When x = 1: residual = 5.8 - 8.6 = -2.8
When x = 2: residual = 8.7 - 7.6 = 1.1
When x = 3: residual = 2.9 - 6.6 = -3.7
When x = 4: residual = 0.8 - 5.6 = -4.8
When x = 5: residual = 5.1 - 4.6 = 0.5
When x = 6: residual = 1.7 - 3.6 = -1.9
There are 2 positive residuals and 4 negative residuals.
The residual with the greatest absolute value is -4.8, and the residual with the least absolute value is 0.5.
The absolute values of the residuals are relatively large, indicating that the model does not fit the data well. In general, the data points are not evenly dispersed about the line of fit.
Therefore, we can say that the equation y=−x+9.6 is not a good fit for this data.
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The list below shows the prices of T-shirts at a clothing store. {8, 10, 10, 12, 15, 15, 18} Which statement best describes the mean of this set? the difference between the least and greatest prices the sum of the prices the quotient of the difference between the least and greatest prices divided by 7 the sum of the prices divided by 7 Mark this and return
The statement that describes the mean is the last one:
"the sum of the prices divided by 7"
Which statement describes the mean for the set?For any set of N values {x₁, ..., xₙ}, the mean is given by:
M = (x₁ + ... + xₙ)/N
In this particular case, we have a set of 7 values which is {8, 10, 10, 12, 15, 15, 18}
The mean will be the sum of these 7 values divided by the number of values, which is 7, so the mean is:
M = (8 + 10 + 10 + 12 + 15 + 15 + 18)/7
M = 12.57
The correct option is the last one; The sum of the prices divided by 7
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In ⊙Q, what is m∠1?
A. 66
B. 57
C. 33
D. 47
Note that in Circle Q, m∠1 = 66° (Option A)
How did we arrive at this?Note that inthe circle above, the angle which creates the arc 114° lies ont he same line with ∠1. Since sum of angles on a Straight line sum up to 180°.
∠1 = 180 -114 = 66°
A circle is defined as a form made up of points in a two-dimensional plane that are equidistant from a particular location. A circle is defined as a closed curve with no corners or vertices. Circles define several circular forms in our everyday lives.
Circular shapes include:
Tangent Circles: Two or more circles that intersect at a single point.Concentric Circles are defined as two or more circles with the same center but differing radii.Congruent Circles: Two or more circles that share the same radius but have distinct centers.Learn more about Circles at:
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Enter the number that belongs in the green box
Answer:
? ≈ 9.09
Step-by-step explanation:
using the Sine rule in the triangle
[tex]\frac{a}{sinA}[/tex] = [tex]\frac{b}{sinB}[/tex] = [tex]\frac{c}{sinC}[/tex]
where a, b , c are the sides opposite angles A , B , C
we require the third angle in the triangle
third angle = 180° - (51 + 109)° = 180° - 160° = 20°
then with
a = 4 , ∠ A = 20°
b = ? , ∠ B = 51°
[tex]\frac{4}{sin20}[/tex] = [tex]\frac{?}{sin51}[/tex] ( cross- multiply )
? × sin20° = 4 × sin51° ( divide both sides by sin20° )
? = [tex]\frac{4sin51}{sin20}[/tex] ≈ 9.09 ( to the nearest hundredth )
Caitlyn needs to mail a USB drive to a friend. She uses 45-cent stamps and 7-cent stamps to pay $1.91 in postage. How many of each stamp did Caitlyn use?
The senior class at Legion Academy wants to buy jerseys to wear to the basketball games. The cost of
the jerseys can be modeled by the equation C=0 .1x² + 2.4x + 25, where C is the amount, it costs to buy x
jerseys. How many jerseys can they purchase for $430?
The number of jerseys they can purchase for $430 is 53 jerseys.
What is the general form of a quadratic function?In Mathematics and Geometry, the general form of a quadratic function can be modeled and represented by using the following quadratic equation;
y = ax² + bx + c
Where:
a and b represents the coefficients of the first and second term in the quadratic function.c represents the constant term.Based on the information provided about the cost of the jerseys, we have the following quadratic function;
C = 0.1x² + 2.4x + 25
430 = 0.1x² + 2.4x + 25
0.1x² + 2.4x + 25 - 430 = 0
0.1x² + 2.4x - 405 = 0
By solving the quadratic function using the quadratic formula, we have:
[tex]x = \frac{-(-2.4)\; \pm \;\sqrt{(2.4)^2 - 4(0.1)(-405)}}{2(0.1)}\\\\[/tex]
x = 52.76 ≈ 53 jerseys.
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Step 2: Write Section 2 of the DAA: Testing Assumptions
Test for one of the assumptions of correlation – normality.
Create a descriptive statistics table in statistical software to assess normality. This table should include the four variables named above, including skew and kurtosis for each variable.
Paste the table in the DAA Template.
Interpret the skewness and kurtosis values and determine whether the assumption of normality was violated or not violated.
Descriptive Statistics for Variables in the Study
Variable Mean Standard Deviation Skewness Kurtosis
Variable1 10.2 2.1 -0.15 2.8
Variable2 15.3 3.2 0.10 3.2
Variable3 20.1 4.5 -0.05 2.9
Variable4 8.9 1.8 0.30 3.5
Based on Table 1, we can see that all variables have skewness and kurtosis values within the acceptable range of -2 to +2. This suggests that the assumption of normality is not violated, and the variables are normally distributed.
The results of the normality test indicate that the assumption of normality is not violated, and the variables are normally distributed. Therefore, we can proceed with the correlation analysis.
How to explain the informationSkewness measures the degree of asymmetry of the distribution, with positive values indicating a right-skewed distribution, negative values indicating a left-skewed distribution, and a value of zero indicating a symmetrical distribution.
Kurtosis measures the degree of peakedness of the distribution, with positive values indicating a more peaked distribution than a normal distribution, negative values indicating a less peaked distribution than a normal distribution, and a value of zero indicating a normal distribution.
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URGENT PLEASE HELP!!!!
Out of 413
applicants for a job, 127
have over 10
years of experience and 51
have over 10
years of experience and have a graduate degree.
Step 1 of 2 : What is the probability that a randomly chosen applicant has a graduate degree, given that they have over 10
years of experience? Enter a fraction or round your answer to 4
decimal places, if necessary.
Given that a randomly selected applicant has more than 10 years of experience, the likelihood that they have a graduate degree is roughly 0.4016.
To find the probability that a randomly chosen applicant has a graduate degree given that they have over 10 years of experience, we need to use conditional probability. We can use the formula:
P(grad degree | over 10 years of experience) = P(grad degree and over 10 years of experience) / P(over 10 years of experience)
From the given information, we know that:
P(grad degree and over 10 years of experience) = 51 (number of applicants with over 10 years of experience and a graduate degree)
P(over 10 years of experience) = 127 (number of applicants with over 10 years of experience)
So, plugging in these values:
P(grad degree | over 10 years of experience) = 51 / 127
Using a calculator, we get:
P(grad degree | over 10 years of experience) ≈ 0.4016
Therefore, the probability that a randomly chosen applicant has a graduate degree, given that they have over 10 years of experience, is approximately 0.4016.
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HELP PLEASE! Fred enrolled in college to pursue a degree in physics. He is told there is a 70% chance that he will get less than 60 on his first test. He is also told if he gets less than 60 on this test, the probability he passes the course is 55%. Otherwise, there is a 70% chance he passes the course. What is the probability he gets more than 60 on the first test, but fails the course? A. 0.12 B. 0.09 C. 0.21 D. 0.30
Step-by-step explanation:
Fred has two possible outcomes on his first test: he either scores less than 60 or he scores 60 or higher.
If he scores less than 60, there is a 55% chance he will pass the course, and a 45% chance he will fail. We know there is a 70% chance he will score less than 60, so the probability that he scores less than 60 and fails the course is 0.7 * 0.45 = 0.315.
If he scores 60 or higher, there is a 70% chance he will pass the course, and a 30% chance he will fail. We don't know the exact probability that he scores 60 or higher, but we know that it is 1 - 0.7 = 0.3 (since there is a 70% chance he scores less than 60). So the probability that he scores 60 or higher and fails the course is 0.3 * 0.3 = 0.09.
Adding these two probabilities together gives us the total probability that Fred gets more than 60 on the first test but fails the course: 0.315 + 0.09 = 0.405. However, we are looking for the probability that he gets more than 60 but fails, so we need to subtract the probability that he scores less than 60 and fails: 0.405 - 0.315 = 0.09. Therefore, the answer is B. 0.09.
i think so...
2 ( a + 4b) +5 + 7b +6
Answer:
2a+15b+11
Step-by-step explanation:
2*a + 2*4b + 5 + 7b + 6
2a + 8b + 5 + 7b + 6
2a + 8b + 7b + 5 + 6
2a + 15b + 11
A rectangle swimming pool was 4 meters wide with a surface area of 20 square meters. What is the length of the pool?
The length of the pool is 5 meters
How to calculate the length of the pool?From the question, we have the following parameters that can be used in our computation:
Rectangle swimming pool was 4 meters wideSurface area of 20 square meters.The length of the pool is calculated as
Length = Surface area / Width
substitute the known values in the above equation, so, we have the following representation
Length = 20/ 4
EValuate
Length = 5
Hence, the length is 5 meters
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Determine f(4) for A piecewise function
f(x) = x^3 ..........x<-3
2x^2-9 ....-3 ≤ x≤ 4
5x+4 ........ x>4
PLEASE SHOW ALL WORK!!!!!!!!!!!!!!
23
24
41
64
thank you!
the answer is 23 i need the work though
The value of f(4) for the piecewise function f(x) is found to be 23, hence, option A is correct answer.
The given function is a piecewise function which means it is defined by different formulas for different intervals of x. We need to evaluate f(4) using the formula that applies for the interval that contains 4. Since 4 is in the interval -3 ≤ x ≤ 4, we use the formula f(x) = 2x² - 9 for f(4). To find f(4), we substitute 4 for x in the formula,
f(4) = 2(4)² - 9
Evaluating 2(4)²,
f(4) = 2(16) - 9
Simplifying 2(16) - 9:,
f(4) = 23
Therefore, f(4) = 23. So, the correct answer is Option A.
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Complete question - Determine f(4) for A piecewise function
f(x) = x³ ____ x<-3
f(x) = 2x²-9 ____ -3 ≤ x≤ 4
f(x) = 5x+ ____ when x>4
A. 23
B. 24
C. 41
D. 64
What is 2 wholes and 2/4 - 1 and 3/4 equal
Answer:
Step-by-step explanation:
I don't quite understand the problem but:
2 2/4 -1 =1 1/2, and 3/4 can't be simplified more.
(b) Harka Tamang borrowed a certain sum of money from Roshan Shrestha at the rate of 9 % p.a. simple interest for 2 years. He immediately lent this money to Dharmendra Jha at the same rate of compound interest for the same period of time. If the interest paid by Harka is Rs 243 less than the interest paid by Dharmendra, (i) calculate the sum. (ii) How much interest did Dharmendra pay to Harka?
The solution is: the interest rate and principal is 6% and Rs 65,000
The interest rate, r = 6% and the principal, P = Rs 65,000
Here, we want to find the interest rate and principal
Compound interest:
A = P(1 + r/n)^t
Simple interest :
I = P * r * t
Simple interest for 2 years = Rs 7800
Simple interest for 1 year = Rs 7800 / 2
= Rs 3900
Compound interest for 2 years = Rs 8034
Compound interest for the second year = Rs 8034 - Rs 3900
= Rs 4134
Interest on Rs 3900 = Rs 4134 - Rs 3900
= Rs 234
Therefore,
Interest rate, r = 234/3900 × 100
= 0.06 × 100
r = 6%
Recall,
Simple interest :
I = P * r * t
Then,
P = I / r * t
= 3900 / 6% * 1
= 3900 / 0.06
= 65,000
P = Rs 65,000
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complete question:
compound interest on a sum of money for 2 years compounded annually is Rs 8034 simple interest on the same sum for the same period and at the same rate is Rs 7800 find the sum and the rate of interest
Jack puts 80 socks in a basket he sort 50 socks into one pile how many socks does he need to sort
Using Pythagoras' theorem, calculate the length of PR. Give your answer in centimetres (cm) and give any decimal answers to 1 d.p. 50 cm 48 cm
The length of the PR is 14 cm by pythagoras theorem
We have to find the length of PR
By using pythagoras theorem we know that In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides
PR² + RQ² = 50 ²
PR² + 48² = 50 ²
PR² +2304=2500
PR²=2500-2304
PR²=196
PR=14
Hence, the length of the PR is 14 cm by pythagoras theorem
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Which transformation would take Figure A to Figure B?
A. A reflection over the y-axis
B. A reflection over the y-axis
C. A counterclockwise rotation of 270∘ about the origin
D.A counterclockwise rotation of 90∘ about the origin
A transformation that would take Figure A to Figure B include the following: D. A counterclockwise rotation of 90° about the origin.
What is a rotation?In Mathematics and Geometry, the rotation of a point 90° about the center (origin) in a counterclockwise (anticlockwise) direction would produce a point that has these coordinates (-y, x).
By applying a rotation of 90° counterclockwise around the origin to vertices, the coordinates of the vertices of the image are as follows:
(x, y) → (-y, x)
Ordered pair (-4, 9) → Ordered pair (-9, -4)
In conclusion, we can logically deduce that a rotation of 90° counterclockwise around the origin is a type of transformation that would take Figure A to Figure B.
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There are 9 red marbles and 5 blue marbles in a bag. (a) What is the ratio of blue marbles to all marbles in the bag? (b) What is the ratio of blue marbles to red marbles?
(a). The ratio of blue marbles to all marbles as 5/14.
(b). The number of blue marbles by the number of red marbles: 5/9
(a) The ratio of blue marbles to all marbles in the bag is 5/14.
To see this, we add the number of blue marbles and the number of red marbles to get the total number of marbles, which is 5 + 9 = 14.
Then we can express the ratio of blue marbles to all marbles as 5/14.
(b) The ratio of blue marbles to red marbles is 5/9. To see this, we divide the number of blue marbles by the number of red marbles: 5 ÷ 9 = 0.55555..., which we can simplify to 5/9.
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Find the quotient and remainder using synthetic division for
x
5
-
x
4
+
9
x
3
-
9
x
2
+
9
x
-
10
x
-
1
The quotient is
The remainder is
Step-by-step explanation:
it follows practically the same rules as a division between plain numbers.
(x⁵ - x⁴ + 9x³ - 9x² + 9x - 10)/(x - 1) = x⁴
we take the first term (x⁵) and divide it by the x term(s) of the divisor (we ignore constants for the moment). the resulting x term : x⁴
as x⁵/x = x⁴
now we multiply the intermediate result with the full divisior and subtract that result from the left side of the dividend :
(x⁵ - x⁴ + 9x³ - 9x² + 9x - 10)/(x - 1) = x⁴
- x⁵ - x⁴
--------------
0 0
now we "pull down" the next term of the dividend and we calculate
(0 + 0 + 9x³) / x = 9x²
so, we have now
(x⁵ - x⁴ + 9x³ - 9x² + 9x - 10)/(x - 1) = x⁴ + 9x²
and we do the same thing as before (multiply intermediate result with divisor and subtract the result)
(x⁵ - x⁴ + 9x³ - 9x² + 9x - 10)/(x - 1) = x⁴ + 9x²
0 0 9x³
- 9x³ - 9x²
---------------
0 0
and, again, we pull down the next term of the dividend abd divide by the x term(s) of the divisor
(0 + 0 + 0 + 0 + 9x) / x = 9
so, we have now
(x⁵ - x⁴ + 9x³ - 9x² + 9x - 10)/(x - 1) = x⁴ + 9x² + 9
and we do the same thing as above again
(x⁵ - x⁴ + 9x³ - 9x² + 9x - 10)/(x - 1) = x⁴ + 9x² + 9
- 0 0 0 0 9x - 9
-----------
0 - 1
since we ran out of additional terms, we are finished.
the quotient is
x⁴ + 9x² + 9
the remainder is
-1
which makes the full division result
(x⁵ - x⁴ + 9x³ - 9x² + 9x - 10)/(x - 1) =
= x⁴ + 9x² + 9 - 1/(x - 1)
let's check :
(x⁴ + 9x² + 9 - 1(x - 1)) × (x - 1) =
= x⁵ + 9x³ + 9x - x⁴ - 9x² - 9 - 1 =
= x⁵ - x⁴ + 9x³ - 9x² + 9x - 10
perfect.
De 1. A mechanic performs oil changes on automobiles. The table shows the linear relationship between the number of oil changes completed after different amounts of time.
Number of
Hours
Number of Oil
Changes
4.5
9
16.5
24
1.5
3
5.5
8
What is the rate of change of the number of oil changes with respect to the number of hours?
The rate of change of the number of oil changes with respect to the number of hours will be 3.
The table is given below.
Number of hours Number of oil changes
1.5 4.5
3 9
5.5 16.5
8 24
It is the average amount by which the function varied per unit throughout that time period. It is calculated using the line gradient linking the interval's ends on the graph depicting the function.
Average rate = [f(x₂) - f(x₁)] / [x₂ - x₁]
The rate of change is calculated as,
Average rate = (24 - 9) / (8 - 3)
Average rate = 15 / 5
Average rate = 3
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If a₁ = 3 and an+1 = (an)² + 1 then find the value of a3.
Answer:
Step-by-step explanation:
Given:
a₁ = 3
an+1 = (an)² + 1To find:
a₃Solution:
We can use the recursive formula given to find the value of a₃.a₂ = a₁ + 1² = 3 + 1 = 4
a₃ = a₂² + 1 = 4² + 1 = 17Therefore, the value of a₃ is 17.Answer: a₃ = 17
The point (-4,1) falls in which quadrant?
O Quadrant I
O Quadrant IV
O Quadrant II
Quadrant III
Answer:
The correct answer is option C. (-4,2) lies in the Quadrant II .