Therefore, the quadratic polynomial with sum of zeroes 2/3 and product of zeroes 1/3 is:
3x² - 3x + 1
What is quadratic equation?
Quadratic equations are the polynomial equations of degree 2 in one variable of type f(x) = ax² + bx + c = 0 where a, b, c, ∈ R and a ≠ 0. It is the general form of a quadratic equation where 'a' is called the leading coefficient and 'c' is called the absolute term of f (x).
Let the quadratic polynomial be of the form ax² + bx + c.
We know that the sum of the zeroes is given by -b/a, and the product of the zeroes is given by c/a.
We are given that the sum of the zeroes is 2/3 and the product of the zeroes is 1/3.
So, we have the following system of equations:
-2b/3a = 2/3 (since the sum of zeroes is 2/3)
c/a = 1/3 (since the product of zeroes is 1/3)
Simplifying the first equation, we get:
b/a = -1
Substituting this in the second equation, we get:
c/a = 1/3
Multiplying both sides by a, we get:
c = a/3
So, we can choose any value for a, and then compute b and c accordingly.
Let's choose a = 3 for simplicity. Then, we have:
c = a/3 = 1
b/a = -1, so b = -3
Therefore, the quadratic polynomial with sum of zeroes 2/3 and product of zeroes 1/3 is:
3x² - 3x + 1
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There are two types of trees to plant in the yard type A trees are 36 inches tall and grows 8 inches per year type B are 18 inches tall but grow 10 inches per year when will the trees be the same height
As a result, both varieties of trees will be the same height after 9 years. We can change both equations to a = 9 to verify this: Height of the type A tree is 36 + 8(9) or 36 + 72 inches, whereas the height of the type B tree is 18 + 10(9) or 18 + 90 inches.
What function do height and distance serve in everyday life?Trigonometry includes heights and distances, and it has numerous uses in practical daily life. It utilised to compute height of towers, structures, mountains, etc., and distance between any two objects such celestial bodies others. ,sys,s tos.as to .... and.
Let's use "a" to denote the number of years after planting the trees.
A type A tree will reach the following height after "a" years:
Height of type A tree = 36 + 8a
After "a" years, the height of a type B tree will be:
Height of type B tree = 18 + 10a
We must set the two types of trees' heights equal to one another and solve for "a" to determine when they will reach the same height:
36 + 8a = 18 + 10a
Subtracting 8a from both sides, we get:
36 = 18 + 2a
Subtracting 18 from both sides, we get:
18 = 2a
Dividing both sides by 2, we get:
a = 9
Therefore, after 9 years, both types of trees will be the same height.
To check this, we can substitute a = 9 into both equations:
Height of type A tree = 36 + 8(9) = 36 + 72 = 108 inches
Height of type B tree = 18 + 10(9) = 18 + 90 = 108 inches
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Question:
You have two types of trees to plant in your yard: type A trees are 36 inches tall and grow 8 inches per year, while type B trees are 18 inches tall and grow 10 inches per year. At what point in time will the trees be the same height? How tall will the trees be at that time?
for positive integers n. which elements of this sequence are divisible by 5? what about 13? are any elements of this sequence divisible by 65
No element in this sequence can be divided by 5, 13, or 65.
This sequence's elements are not all divisible by 5, 13, or 65.
For positive integers n, we define the sequence a1 = 2n - 3.
We must determine if 2n - 3 is divisible by 5 for various values of n in order to determine whether members of this sequence are divisible by 5.
2ⁿ mod 5 equals 2ⁿ mod 1 = 2ⁿ mod 2 = 4ⁿ mod 3 = 3ⁿ mod 4 = 1, etc.
None of the items in this sequence can be divided by 5, as we can see from the fact that 2ⁿ mod 5 is not necessarily 0.
When divided by 13, 2ⁿ mod 13 equals 2ⁿ mod 1 mod 13 = 2, 2n mod 2 mod 4 mod 8 mod 3 mod 13 = 3, etc.
Since 2ⁿ mod 13 is not necessarily 0, none of the sequence's elements are divisible by 13 as a result.
When 65 is divided by 5*13, 2n mod 65 equals 2n mod 65 times 2, 2ⁿ
mod 65 times 4, 2ⁿ mod 65 times 8, 2ⁿ mod 65 times 3, etc.
None of the items in this sequence are divisible by 65 since 2ⁿ mod 65 is not necessarily 0.
Hence, No element in this sequence can be divided by 5, 13, or 65.
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The complete question is:
Consider the sequence a₁ = = 2¹-3=-1₁ -2²-3=1, 0₂= 03 =2³-3=5, 04-2¹-3=13, ⠀ a₁ = 2" - 3, defined for positive integers n. Which elements of this sequence are divisible by 5?
What about 13? Are any elements of this sequence divisible by 65= 5. 13? Why or why not?
Hence, determine the circumstances of the base base of a coffee tin
Answer:
We can write the diameter and circumferance of base as -
D = 2√(750ρ/πh)
C = 2π√(750ρ/πh)
Step-by-step explanation:
What is function?
A function is a relation between a dependent and independent variable.
Mathematically, we can write → y = f(x) = ax + b.
Given is to find the diameter and height of the tin can.
Assume the density of coffee as {ρ}. We can write the volume of the tin can as -
Volume = mass x density
Volume = 750ρ
We can write -
πr²h = 750ρ
r = √(750ρ/πh)
D = 2r
D = 2√(750ρ/πh)
Now, we can write the circumferance as -
C = 2πr
C = 2π√(750ρ/πh)
Therefore, we can write the diameter and circumferance of base as -
D = 2√(750ρ/πh)
C = 2π√(750ρ/πh)
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Weekly CPU time used by an accounting firm has probability density
function (measured in hours) given by
f(x) = { 3/64 * x^2
(4 − x) 0 ≤ x ≤ 4
0 Otherwise }
(a) Find the F(x) for weekly CPU time.
(b) Find the probability that the of weekly CPU time will exceed two hours
for a selected week.
(c) Find the expected value and variance of weekly CPU time.
(d) Find the probability that the of weekly CPU time will be within half an
hour of the expected weekly CPU time.
(e) The CPU time costs the firm $200 per hour. Find the expected value
and variance of the weekly cost for CPU time.
Using probability, we can find that:
E(Y)= 2.4, Var (Y) = 0.64
E(Y) = 480, Var(Y) = 25,600
Define probability?The probability of an event is the proportion of favourable outcomes to all other potential outcomes. To determine how likely an event is, use the following formula:
Probability (Event) = Positive Results/Total Results = x/n
Given,
The weekly CPU time is as follows:
f(y) where, 0≤y≤4
Here, probability density function is 4-y is correct, or else we get negative expected values.
We have to find E(Y) and var(Y)
E(Y) = 2.4
var (Y) = E(Y²)-(E(Y)) ²
= 6.4 - (2.4) ²
= 0.64
The CPU time is costing the firm $200 per hour.
Now, we find E(Y) and var(Y) of the weekly cost for the CPU time.
Y = 200E
E(Y) = 200 × 2.4
= 480
var(Y) = 200V(Y)
= 200 × 0.64
= 25600
We can observe that the weekly cost is not exceeding $600 as weekly cost for CPU time = 480.
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What is 0.1 in exponent form
Answer:
pretty-pretty sure its 1/10
Step-by-step explanation:
Convert to a mixed number by placing the numbers to the right of the decimal over
10
. Reduce the fraction.
1
10
Sarah is a psychologist at an practise. she earns a basic salary of R3000 per month as well as 20% commission on income up to R 5000. She receives an additional 10% bonus on top of the normal commission rate on earning above R 5000. If Sarah did work to the value of R 12000 ,how much did she earn in total???
Answer:
Sarah earns R6 100 in total for her work to the value of R12 000.
Step-by-step explanation:
To calculate Sarah's earnings, we need to break down her income into two parts: the commission she earns on income up to R5 000, and the bonus commission she earns on income above R5 000.
Commission on income up to R5 000:
Sarah's basic salary is R3 000 per month, and she earns 20% commission on income up to R5 000. So for the first R5 000 of income, Sarah's commission is:
[tex]\text{Commission on income up to} \ R5, 000 = 20\% \ \text{of} \ R5, 000 = R1 ,000[/tex]
Bonus commission on income above R5 000:
Sarah also receives a 10% bonus on top of the normal commission rate on earning above R5 000. So for the amount earned above R5 000, her commission is:
[tex]\text{Commission on income above} \ R5, 000 = (20\% + 10\%) of (R12, 000 - R5, 000) = 30\% \ \text{of} \ R7, 000 = R2 ,100[/tex]
Total earnings:
Sarah's total earnings are the sum of her basic salary and the commission she earns:
Total earnings = Basic salary + Commission on income up to R5 000 + Commission on income above R5 000
[tex]\text{Total earnings} = R3, 000 + R1 ,000 + R2 ,100[/tex]
[tex]\text{Total earnings} = 6,100[/tex]
Therefore, Sarah earns R6 100 in total for her work to the value of R12 000.
Select the correct answer.
Consider the function f(x) = 3° and the function g, which is shown below.
g(x) = f(x) - 2 = 3° _ 2
How will the graph of g differ from the graph of f?
O A.
The graph of g is the graph of fshifted 2 units down.
О B.
The graph of g is the graph of f shifted 2 units up.
O c.
The graph of g is the graph of f shifted 2 units to the left.
O D.
The graph of g is the graph of f shifted 2 units to the right.
Therefore , the solution of the given problem of function comes out to be option A is right response the graph of g is a 2 unit downshifted version of the graph of f.
Explain function.The midterm exam will include questions in variable design, mathematics, each topic, and both actual and hypothetical locations. a catalog of the connections between various components that work together to produce the same outcome. A service is made up of many unique parts that work together to produce unique outcomes for each input. Additionally, each mailbox has a unique address, which could be an enclave.
Here,
The graph of f is an increasing curve that passes through the point because the exponential growth function f(x) = 3x depicts growth where the base 3 is higher than 1. (0,1).
The curve of f is shifted down by 2 units to yield the function g(x) = f(x) - 2 = 3x - 2. This indicates that to acquire the corresponding y-coordinates of the graph of g, all the y-coordinates of the graph of f must be decreased by 2.
As a result, choice A—the graph of g is the graph of f moved down by two units—is correct.
So, A is the right response. The graph of g is a 2 unit downshifted version of the graph of f.
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According to Money magazine, Maryland had the highest median annual household income of any state in 2018 at $75,847.† Assume that annual household income in Maryland follows a normal distribution with a median of $75,847 and standard deviation of $33,800.
(a) What is the probability that a household in Maryland has an annual income of $90,000 or more? (Round your answer to four decimal places.)
(b) What is the probability that a household in Maryland has an annual income of $50,000 or less? (Round your answer to four decimal places.)
The required probability that a household in Maryland with annual income of ,
$90,000 or more is equal to 0.3377.
$50,000 or less is equal to 0.2218.
Annual household income in Maryland follows a normal distribution ,
Median = $75,847
Standard deviation = $33,800
Probability of household in Maryland has an annual income of $90,000 or more.
Let X be the random variable representing the annual household income in Maryland.
Then,
find P(X ≥ $90,000).
Standardize the variable X using the formula,
Z = (X - μ) / σ
where μ is the mean (or median, in this case)
And σ is the standard deviation.
Substituting the given values, we get,
Z = (90,000 - 75,847) / 33,800
⇒ Z = 0.4187
Using a standard normal distribution table
greater than 0.4187 as 0.3377.
P(X ≥ $90,000)
= P(Z ≥ 0.4187)
= 0.3377
Probability that a household in Maryland has an annual income of $90,000 or more is 0.3377(rounded to four decimal places).
Probability that a household in Maryland has an annual income of $50,000 or less.
P(X ≤ $50,000).
Standardizing X, we get,
Z = (50,000 - 75,847) / 33,800
⇒ Z = -0.7674
Using a standard normal distribution table
Probability that a standard normal variable is less than -0.7674 as 0.2218. This implies,
P(X ≤ $50,000)
= P(Z ≤ -0.7674)
= 0.2218
Probability that a household in Maryland has an annual income of $50,000 or less is 0.2218.
Therefore, the probability with annual income of $90,000 or more and $50,000 or less is equal to 0.3377 and 0.2218 respectively.
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Please order the following fractions from least to greatest: 5/6, 2/3, 5/9, 5/12, 6/5
Answer:
5/12, 5/9, 2/3, 5/6, 6/5
Step-by-step explanation:
5/12= 0.41666667, 5/9= 0.55555556, 2/3= 0.66666667, 5/6= 0.83333333, 6/5= 1.2
Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested. Find the probability of obtaining a reading less than -2.74°C. Round your answer to 4 decimal places
Answer:
Step-by-step explanation:
We are given that the temperature readings are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. Let X be the temperature reading of a single thermometer selected at random. Then, X ~ N(0, 1).
We need to find the probability of obtaining a reading less than -2.74°C, which can be expressed mathematically as P(X < -2.74).
Using standard normal distribution tables or a calculator, we can find that the z-score corresponding to -2.74°C is:
z = (x - μ) / σ = (-2.74 - 0) / 1 = -2.74
The probability can be calculated as:
P(X < -2.74) = P(Z < -2.74) ≈ 0.0030 (rounded to 4 decimal places)
Therefore, the probability of obtaining a reading less than -2.74°C is approximately 0.0030.
Answer:
We need to find the probability of obtaining a reading less than -2.74°C from a normal distribution with a mean of 0°C and a standard deviation of 1.00°C.
Using the standard normal distribution, we have:
z = (x - μ) / σ
where:
x = -2.74°C (the reading we want)
μ = 0°C (the mean)
σ = 1.00°C (the standard deviation)
Substituting the values, we get:
z = (-2.74 - 0) / 1.00 = -2.74
Using a standard normal distribution table or calculator, we find that the probability of obtaining a z-score less than -2.74 is approximately 0.0030.
Therefore, the probability of obtaining a reading less than -2.74°C from the batch of thermometers is approximately 0.0030.
PLEASE HELP, 30 POINTS!
Answer:
1097.28 centimeters
Step-by-step explanation:
You have to multiply length given by 91.44
Answer:
1097.28 cm
Step-by-step explanation:
1 yd = 36 in
=> 12 yrd = 12 x 36 = 432 in
1 in = 2.54 cm
=> 432 in = 432 x 2.54 = 1097.28 cm
Which equation calculates the total amount of milk needed to make 7 milkshakes if each milkshake requires 3 4 cup of milk? A. 7 × 3 4 = 21 28 cups B. 7 + 3 4 = 7 3 4 cups C. 7 × 1 4 = 7 4 , or 1 3 4 cups D. 7 × 3 4 = 21 4 , or 5 1 4 cups
Answer:
D
Step-by-step explanation:
7/1 times 3/4 equals 21/4 cups of milk
Oliver's normal rate of pay is $10.40 an hour.
How much is he paid for working 5 hours overtime one Saturday at time-and-a-half?
Decide if the events A and B are mutually exclusive or not mutually exclusive. A card is drawn from a standard deck of 52 playing cards.
Event A: The Result is a club
Event B: The result is a king
Are they mutually exclusive or not mutually exclusive?
7. Convert 37/6 to a mixed number.
A. 6¹6
B. 623
C. 65
D.556
[tex] \boxed{ \red{ \mathfrak{A.\: 6 \frac{1}{6} }}}[/tex]
Find the 6th term of the geometric sequence described below.
m₁ = -3(-5)-1
Show your work here
Hint: To add an exponent (z"), type "exponent" or press "A"
Answer:
m₆ = 9375
Step-by-step explanation:
Given sequence is
[tex]m,_i = -3(-5)^{i - 1}[/tex]
To find the 6th term, all you have to do is substitute i = 6 and compute
For i = 6 we get
[tex]m_6 = -3(-5)^{6 - 1}\\= -3(-5)^5\\\\= -3(-3125) \\\\= 9375[/tex]
A negative number raised to an odd number is negative that is why (-5)⁵ is negative
solve using systems answer in a ordered pair
y = –x + 3
y = 4x – 2
Answer:
(1,2)
Step-by-step explanation:
Pre-SolvingWe are given the following system of equations:
y = -x + 3
y = 4x - 2
And we want to solve it, with the answer in an ordered pair.
SolvingBecause both systems are equal to y, we can set both of the equations equal to each other, and solve for x in that way.
This is possible due to transitivity, which states that if a=b, and b=c, then a=c.
Hence,
-x + 3 = 4x - 2 (same as y=y)
We can add x to both sides.
3 = 5x - 2
Add 2 to both sides.
5 = 5x
Divide both sides by 5.
1 = x
Now, we can use this value to find y.
Substitute 1 as x in either y = -x + 3 or y = 4x - 2
Taking y = -x + 3 for instance:
y = -1 + 3 = 2
So, we now that x=1, y=2.
As an ordered pair, that is (1,2).
which of the following numeric measures would be most likely to produce invalid statistical analysis?
The most likely to produce invalid statistical analysis of numeric measures is Pain rating as: none = 0; slight = 1; much = 2, as it is an ordinal scale of measurement, which does not have equal intervals between the categories. So, the correct answer is B).
It means that the differences between the categories are not necessarily equivalent, and therefore, any statistical analysis based on this scale may not accurately reflect the true relationship between variables.
The other options (blood pressure in mmHg, oxygen saturation in percentage, neonatal birth weight in kilograms) are measured on interval or ratio scales, which have equal intervals between values and can be used for meaningful statistical analysis. so, the correct option is B).
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____The given question is incomplete, the complete question is given below:
Which of the following numeric measures would be most likely to produce invalid statistical analysis?A)Analysis of patients' blood pressures in mmHgB)Pain rating as: none = 0; slight = 1; much = 2C)Assessment of oxygen saturation in percentageD)Analysis of neonatal birthweight in kilogr
How many functions are there from the set {1, 2, . . . , n}, where n is a positive integer, to the set {0, 1} a) that are one-to-one? b) that assign 0 to both 1 and n? c) that assign 1 to exactly one of the positive integers less than n?
The number of functions ,
(a) that are one-to-one are 0.
(b) that assign 0 to both 1 and n are 2ⁿ⁻²,
(c) that assign "1" to exactly one of positive-integers less than n are 2.(n-1).
Part(a) : We have to find total number of "one-to-one" functions from the set {1,2,......,n} to {0,1}.
⇒ If n=1 then there are 2 possible functions depending whether 1 is mapped to "0" or "1" ,So there are 2 such functions.
⇒ If n=2 then domain is {1,2} then there are 2 choices for first element in domain.
Then, since one choice is taken there is one choice for second element in the domain. So, if n=2 we have 2×1 = 2 functions.
⇒ If value of n is greater than 2 then domain will be {1,2,....n} then only two value of this domain will be mapped to codomain {0,1} to provide a one-to-one function and
So, domain will not be used fully so there does not exist any one-to-one function.
Part(b) : Every element in the domain {1, 2, . . . , n} has two options in codomain {0, 1},
So, there are total of "2n" functions from domain to co-domain.
Since, the function assigns 0 to both 1 and n.
There are "n-2" elements left in domain which can be assigned 0 or 1.
So, for "n-2" elements in domain and there are "2ⁿ⁻²" functions from domain to codomain.
Part(c) : The domain set has "n" elements and codomain set has "2" elements.
So, each of "n" elements from domain has 2 choices in function and thus we get "2n" total functions.
There are "n-1" elements less than "n" in domain.
Now, by the condition that exactly "1" positive-integer less than "n" maps to "1".
So, all other remaining less than n (i.e. n-2) must be map to 0.
We find this number in ⁿ⁻²C₁ ways = n-2;
So, total number of ways in which elements less than "n" can be mapped is = n-2(mapped to 0) +1(mapped to 1) = n-1
Also, "n" can be mapped to either "0" or "1" which means., nth element have two-choices.
So, there are 2.(n-1) possible functions.
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Walmart was having a sale on video games. They offered a 15% discount on a game that was originally priced at $30. After the sale, the discounted price of the game was increased by 10%. What is the new price of the game after this increase?
Answer:
28.05$
Step-by-step explanation:
The game got a discount of 15%.
New price is (30$) ( 0.85 ) = 25.5$ (since the discount is 15% you only pay for the 85% of the original price)
Then an increase of 10%. this is:
New price: 25.5$ (1.10) = 28.05
New price is 28$ with 5 cents
for populations that are not known to be normally distributed which of the following is true within the central limit theorem
The sampling distribution of the sample mean is approximately normal for large sample sizes, regardless of the distribution of the population is the best definition of the Central Limit Theorem. So the option e is correct.
The Central Limit Theorem, which states that the distribution of sample means approaches a normal distribution as the sample size increases. This means that the sample mean will be normally distributed, even if the population from which the sample is drawn is not normally distributed. This is useful because it can be used to make inferences and predictions about the population based on the sample data. So the option e is correct.
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The complete question is:
Which one of the following statements is the best definition of the Central Limit Theorem?
(a) In large populations, the distribution of the population mean is approximately normal.
(b) For non-normally distributed populations, the sampling distribution of the sample mean will be approximately normal, regardless of the sample size.
(c) If the distribution of the population is non-normal, it can be normalized by taking a large sample size.
(d) For large sample sizes, the sampling distribution of the population mean is approximately normal, regardless of the distribution of the population.
(e) The sampling distribution of the sample mean is approximately normal for large sample sizes, regardless of the distribution of the population.
Suppose we want to choose 5 letters, without replacement, from 15 distinct letters
[tex]\text{order does not matter}[/tex]
[tex]\text{sample space}= \text{15 letters}[/tex]
[tex]\text{no repetition}[/tex]
[tex]\text{P(A)}= \text{15C5}= \text{3003 ways}[/tex]
Find the derivative of f(x) = -2x^3 by the limit process…
Answer:
f'(x) = -6x^2
f'(-5) = -150
f'(0) = 0
f'(√17) = -102
PLEASEE HELP!
Draw an angle that is 90 degrees. Make sure to draw the symbol on the angle.
Step-by-step explanation:
This is just the corner of a rectangle or a square :
ection A-Classwork Let's make mathematical sentences of each of the following stateme Statements Mathematical sentences a) The sum of x and 4 is 7. b) The difference of y and 5 is 4. c) Two times x is 10. d) Two times y added to 3 is 9. e) f) x is more than 2 by 1. 3 is less than y by 2.
a) The sum of x and 4 equals 7: x + 4 = 7
b) The difference of y and 5 equals 4: y - 5 = 4
c) Two times x equals 10: 2x = 10
d) Two times y added to 3 equals 9: 2y + 3 = 9
f) If y > 3 + 2 or if 3 = y - 2 then y is smaller than y by 2
Define equationAn equation is a statement in mathematics that two expressions are equivalent. It has two sides that are divided by the equals sign (=). One or more terms, such as integers, variables, constants, and mathematical operations like addition, subtraction, multiplication, division, and exponentiation, may be included on each side of the equation.
a) The sum of x and 4 equals 7:
x + 4 = 7
b) The difference of y and 5 equals 4:
y - 5 = 4
c) Two times x equals 10:
2x = 10
d) Two times y added to 3 equals 9:
2y + 3 = 9
e) If x exceeds 2 by 1, then either x = 3 or x > 2 + 1.
f) If 3 is smaller than y by 2 then either y > 3 + 2 or 3 = y - 2
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which statemnt is ture when the dimensions of a two-dimensional figures are dilated by a scale factor of 2
When a shape is dilated, the size of the shape changes. The true statement is (d) The scale factor is 2.5.
Dilation:
Dilation is the process of changing the size of an object or shape by reducing or increasing its size by a specific scale factor. For example, a circle with a radius of 10 units shrinks to a circle with a radius of 5 units. Applications of this method are in photography, arts and crafts, sign making and more.
According to the Question:
How to determine the scale factor
In figure A, we have:
Length = 0.6
In figure B, we have:
Length =1.5
The scale factor is then calculated as:
K = 1.5/0.6
Dividing the equation:
k = 2.5
Hence, the true statement is (d) The scale factor is 2.5.
Complete Question:
The first figure is dilated to form the second figure. Which statement is true?
The scale factor is 0.4.
The scale factor is 0.9.
The scale factor is 2.1.
The scale factor is 2.5.
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4.5. Using Linear Scale
Solve the following scenarios.
8. You have a map that is missing a scale. The distance from Point A to Point B is
five inches on the map, and after driving it, you know it is 250 miles in reality.
The scale of the map is 1 inch to 50 miles, or 1:50.
What is distance?
Distance is defined as the space between two points in space.
To find the scale of the map in inches per mile, we can use the ratio of the distance on the map to the actual distance:
5 inches on the map / 250 miles in reality
Simplifying this ratio gives:
1 inch on the map / 50 miles in reality
Therefore, the scale of the map is 1 inch to 50 miles, or 1:50.
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Let X be a random variable whose probability density function is given by else (a) Write down the moment generating function for X (b) Compute the first and second moments.
a) The moment generating function of X is
M(t) =
{ (1/(2-t)) e^(-2t) + (1/(2-t)) (1/(1+t)) if t<1
{ infinity if t>=1
b) The first moment (mean) of X is 3/4.
The second moment (expected value of X^2) of X is 7/8
(a) The moment generating function (MGF) of a random variable X with probability density function f(x) is defined as M(t) = E(e^(tX)), where E(.) denotes the expected value operator. Therefore, the MGF of X is
M(t) = E(e^(tX)) = ∫[0,∞) e^(tx) f(x) dx
Substituting the given probability density function f(x), we get
M(t) = ∫[0,∞) e^(tx) (e^(-2x) + (e^-x)/2) dx
Simplifying and integrating by parts, we get
M(t) = [(1/(2-t)) e^(-2t) + (1/(2-t)) (1/(1+t))] for t<1, and
M(t) = infinity for t>=1
Therefore, the MGF of X is
M(t) =
{ (1/(2-t)) e^(-2t) + (1/(2-t)) (1/(1+t)) if t<1
{ infinity if t>=1
(b) To compute the first moment (i.e., the mean or expected value) of X, we take the first derivative of the MGF at t=0
E(X) = M'(0) = d(M(t))/dt | t=0
Differentiating the MGF and simplifying, we get
E(X) = 3/4
To compute the second moment (i.e., the expected value of X^2), we take the second derivative of the MGF at t=0
E(X^2) = M''(0) = d^2(M(t))/dt^2 | t=0
Differentiating the MGF again and simplifying, we get
E(X^2) = 7/8
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The given question is incomplete, the complete question is:
Let x be a random variable whose probability density function is given by f(x) = e^(-2x) + (e^-x)/2 when x>0 f(x) = 0 when else, a) write down the moment generating function X (b) Compute the first and second moments
what is the proof that complex numbers with absolute value 1 constitute a group under multiplication?
To prove that complex numbers with absolute value 1 constitute a group under multiplication, they must satisfy the four axioms of a group which includes
ClosureAssociativityIdentity elementInverseWhat are complex numbers?
In mathematics, a complex number is described as an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and can be expressed in the form a + bi, where a and b are real numbers.
The four axioms of a group includes:
Closure: The product of any two complex numbers with absolute value 1 is another complex number with absolute value 1.
Associativity: The product of any three complex numbers with absolute value 1 is the same irrespective of the order in which the multiplication is performed.
Identity: There exists an element in the group, denoted by 1, such that 1 times any element in the group is equal to that element.
Inverse: there exists another element in the group, denoted by the reciprocal or inverse, such that their product is equal to the identity element for each element in the group.
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Where i = sqrt(- 1) which of the following complex numbers is equal to (6 - 5i) - (4 - 3i) + (2 - 7i) ? A (4 - 9i)/25 B 4 - i C 9i - 4 D 4 - 9i E 4 + 9i
Answer: A) 4 - 9i/25
Step-by-step explanation:
We can simplify the expression (6 - 5i) - (4 - 3i) + (2 - 7i) by combining the real and imaginary parts separately:
Real part: (6 - 5i) - (4 - 3i) + (2 - 7i) = 6 - 4 + 2 - (-5i + 3i + 7i) = 4 - 5i
Imaginary part: 0
Therefore, the complex number equal to (6 - 5i) - (4 - 3i) + (2 - 7i) is 4 - 5i.
None of the answer choices matches this result exactly, but we can simplify 4 - 5i further:
(4 - 5i)/1 = (4 - 5i)/sqrt(1*1) [multiply the numerator and denominator by 1]
= (4/sqrt(1)) - (5/sqrt(1))i [divide the real and imaginary parts by 1]
= 4 - 5i
Therefore, the answer is A) (4 - 9i)/25. We can verify this by multiplying the numerator and denominator of this fraction by 25:
(4 - 9i)/25 = (4/25) - (9/25)i
Now, we can see that this is equivalent to 4 - 5i, which is the simplified form of the original expression.