Therefore , the solution of the given problem of probability comes out to be -y^(β-1)/[(β-1)Β(1, β)] .
What is probability exactly?The primary goal of the structures in the style known as parameters is the determination of the likelihood that a claim is accurate or that a specific event will occur. Any number between 0 and 1, at which 1 typically denotes certainty but also 0 typically denotes potential, can be used to symbolise chance. A probability diagram shows the chance that a specific event will occur.
Here,
We are aware that X's PDF is provided by:
If x is between 0 and 1 and 0 otherwise, f(x) Equals 1/B
where represents the size factor.
The formula for Yn's cumulative distribution function (CDF) is
=> F(y) = P(Yn ≤ y) = [P(Xi > y)]ⁿ = [1 - P(Xi ≤ y)]ⁿ
Using the CDF of the beta distribution, we have:
=> F(y) = [1 - Β(1, β; y)]ⁿ
We can approximate Yn by a normal distribution with mean and variance given by the following formulas using the continuity adjustment and central limit theorem:
=> E(Yn) = Β(2, β; 1)/Β(1, β) and Var(Yn) = E(Yn) - [Β(1, β; 1)/Β(1, β)]²
Therefore, we have:
=> (Yn - E(Yn))/sqrt(Var(Yn)) → N(0,1)
With the formulas for E(Yn) and Var(Yn) substituted, we obtain:
=> (Yn - Β(2, β; 1)/Β(1, β))/sqrt[Β(2, β; 1)/Β(1, β) - [Β(1, β; 1)/Β(1, β)]²] → N(0,1)
We need to work out the solution to determine b:
where is the CDF's usual standard value.
After applying the limit and taking the logarithm of both sides, we obtain:
=> F(y) = lim n log[n] = lim n n log
=> [1 - B(1, β; y)] = lim n -n (y, n)/(1, n)
L'Hopital's law gives us:
=> lim n -d/dy [n -(1, y)/n -(1, y)] = lim nn y(-1), ((1, )) ^2Β(2, β; y)
=> -y^(β-1)/[(β-1)Β(1, β)]
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Carmen's swimming pool has the shape of a rectangular prism
with a length of 20 feet, a width of 10 feet, and a depth of 5
feet. The pool will be filled with water until the water level is 1 1/2 feet from the top of the pool. Which volume of water will be filled into the pool?
A. 1000 cubic feet
B. 700 cubic feet
C. 500 cubic feet
D. 300 cubic feet
[tex]1\dfrac{1}{2} = \dfrac{3}{2} = 1.5[/tex]
[tex]20 (10) (5-1.5) = 200(3.5) = 700 \ cubic \ feet[/tex]
Answer:
(B) 700 cubic feet
Step-by-step explanation:
The height of water in the pool is 5 - 1 1/2 = 3 1/2 feet.
The volume of water needed to fill the pool up to 3 1/2 feet is:
20 feet x 10 feet x 3 1/2 feet = 700 cubic feet
Therefore, the answer is (B) 700 cubic feet.
Work out the value of 5 cubed - 10 squared.
Give your answer as a power of 5
Answer:
5 cubed is 5 x 5 x 5 = 125.
10 squared is 10 x 10 = 100.
So, 5 cubed - 10 squared = 125 - 100 = 25.
We can also express 25 as a power of 5 by noting that 25 = 5 squared. Therefore:
5 cubed - 10 squared = 5 squared x 5 - 10 squared = 5^(2+1) - 10^(2) = 5^3 - 10^(2) = 125 - 100 = 25.
So, the answer is 5 squared or 5^2.
Step-by-step explanation:
Answer:
25
Step-by-step explanation:
5 cubed - 10 squared
[tex]5^{3} - 10^{2} \\ = 125 - 100\\= 25[/tex]
Hope this helps!
Brainliest and a like is much appreciated!
5 Exam-style Samia invests £3000 in an account for one year.
At the end of the year interest is added to her account.
Samia pays tax on the interest at a rate of 20%.
She pays £7.80 tax.
Work out the percentage interest rate for the account.
The percentage interest rate added to the account at the end of the year is 1.3%
What percentage interest rate for the account?Tax is the compulsory levy paid by citizens to the government for buying goods and services.
Amount Samia invest = £3,000
Amount of interest = x
Rate of tax = 20%
Amount of tax = £7.80
So,
20% of x = £7.80
0.2x = £7.80
x = £7.80/0.2
x = £39
Percentage of interest rate:
x% of £3000 = £39
x% × 3000 = 39
x% = 39/3000
x% = 0.013
x% = 1.3%
Hence, 1.3% is the percentage of Samia investment added to the account.
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What is the factor of the equation
Answer:
(2x - 3) (x + 3)
Step-by-step explanation:
Let's check
(2x - 3) (x + 3)
2x² + 6x - 3x - 9
2x² + 3x - 9
So, (2x - 3) (x + 3) is the correct answer.
A tractor dealer puts a markup of 21% on cost on a part for which it paid $420. Find (a) the selling price as a percent of cost, (b) the selling price, and (c) the markup
the selling price as a percent of cost is 120.52%, b- the selling price is $508.20 and the markup is 20.95% of cost.
(a) To find the selling price as a percent of cost, we need to first find the selling price and then divide it by the cost and multiply by 100%.
The markup on cost is 21%, which means the dealer sells the part for:
420 + 0.21*420 = $508.20
So the selling price is $508.20 and the cost is $420.
The selling price as a percent of cost is:
508.20/420 * 100% = 120.52%
So the selling price is 120.52% of the cost.
(b) The selling price is $508.20.
(c) The markup is the difference between the selling price and the cost, expressed as a percentage of the cost:
Markup = (Selling price - Cost)/Cost * 100%
= (508.20 - 420)/420 * 100%
= 20.95%
So the markup is 20.95% of the cost.
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which division equation describes the situation
The division equation that describes the situation here is as follows:
4 ÷ 2/3 = 6.
Define division?Together with addition, subtraction, and multiplication, division is one of the four basic mathematical operations. The act of separating anything into smaller groups so that each one has the same number of items is known as division.
This operation is used by mathematicians to organise and evenly distribute resources.
As we can see in the question, that there are 6 groups.
Now 6 groups of 2/3 parts.
Now the whole part has been given as = 4.
This can also be stated as:
The whole part (4) has been divided into 2/3 small parts to get 6 equal parts.
Therefore, the expression as per division can be:
4 ÷ 2/3 = 6.
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The sides of a triangle are 43,96 , and 89. Use the Pythagorean Theorem to determine if the triangle is right, acute, or obtuse.
Using Pythagorean theorem, the triangle is an acute triangle.
What is Pythagorean theorem?
The Pythagorean theorem is a fundamental result in Euclidean geometry that describes the relationship between the three sides of a right-angled triangle.
In equation form, this can be written as:
[tex]c^2 = a^2 + b^2[/tex]
where c is the length of the hypotenuse, and a and b are the lengths of the other two sides (also known as the legs) of the right-angled triangle.
If the triangle is not a right triangle, then the inequality [tex]c^2 < a^2 + b^2[/tex]holds for an acute triangle and [tex]c^2 > a^2 + b^2[/tex] holds for an obtuse triangle.
Using this formula, we can check if the given triangle is right, acute, or obtuse:
a = 43, b = 96, and c = 89
[tex]c^2 = 89^2 = 7921[/tex]
[tex]a^2 + b^2 = 43^2 + 96^2 = 1849 + 9216 = 11065[/tex]
Since [tex]c^2 < a^2 + b^2[/tex], we can see that:
[tex]c^2 < a^2 + b^2[/tex]
So the triangle is an acute triangle.
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To find the area of the blue parallelogram, you can move the red triangle to the green triangle to make a rectangle. After doing this, what can you conclude about the formula for the area of a parallelogram?
The area of a parallelogram can be obtained from the formula for a rectangle by using the same base and height, leading to the formula Area of parallelogram = base x height.
By moving the red triangle to the green triangle, we create a rectangle with the same base and height as the original parallelogram. Therefore, the area of the original parallelogram is equal to the area of the rectangle, which is given by the formula:
Area of rectangle = base x height
We can conclude that the formula for the area of a parallelogram is also given by:
Area of parallelogram = base x height
This is because a parallelogram can be divided into two congruent triangles, and the area of each triangle is half the area of the parallelogram. Therefore, the area of a parallelogram is equal to the base times the height, which is the same as the formula for the area of a rectangle.
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Help me right one gets brainiest
Answer:
7x = 7
with solution of x = 1
This is the first choice
Step-by-step explanation:
Choice 1
7x = 7 => x = 7/7 = 1
Choice 2
7x = 7x provides no additional info and therefore there are an infinite number of solutions; any value of will satisfy
Choice 3
x + 1 = x + 1; infinite solutions for reasons cited under choice 2
Choice 4
x + 1 = x + 2
Eliminating x from both sides gives 1 = 2
Such an equation will have no solution; no value of x can satisfy the equation
Correct answer: Choice 1: 7x = 7
What is the volume of the cylinder? Round to the nearest hundredth and approximate using π = 3. 14. Cylinder with a segment from one point on the circular base to another point on the base through the center labeled 3. 2 feet and a height labeled 3. 8 feet
the volume of the cylinder is approximately 30.19 cubic feet.
To find the volume of a cylinder, we use the formula:
V = πr²h
where V is the volume, r is the radius of the circular base, h is the height, and π is a constant that approximates the ratio of the circumference of a circle to its diameter, which is approximately 3.14.
In this case, we're given that the cylinder has a height of 3.8 feet, and a segment from one point on the circular base to another point on the base through the center labeled 3.2 feet. This segment is the diameter of the circular base, which means that the radius of the base is half of that, or 3.2/2 = 1.6 feet.
Plugging these values into the formula, we get:
V = πr²h
V = 3.14 x (1.6 feet)² x (3.8 feet)
V = 3.14 x 2.56 square feet x 3.8 feet
V = 30.1864 cubic feet
Rounding to the nearest hundredth, we get:
V ≈ 30.19 cubic feet
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A house painter charges a fee for supplies and an hourly fee for the time spent painting. A paint job
that takes 3 h costs $140. A paint job that takes 5 h costs $200.
Answer:
y=30x+50
Step-by-step explanation:
3x30+50=140
5x30+50=200
Find the interest on 20,000 at 6% interest for 3 years
answer the question picture is there
Answer:
Step-by-step explanation:
1. 500
2. 10
3. 25000
Angle Relationships & PT Quiz Review Are the angles are complementary, supplementary, or neither?
1. m<1=91°, m <2 = 89°
2: m< 3 = 17°, m<4 = 73°
3. m< 5 = 124° m <6 = 66°
4. m <7 = 33° m< 8 = 148°
5. m< 9 = 52° m <10 = 38°
pleasee help fast
Pls help! This is due today!
Answer:
3(a−1)
———
a+1
Step-by-step explanation:
A population of 250 wild turkeys decreases by 2. 2% per year. At the end of 8 years, there will be approximately 209 turkeys in the population
Each year, the wild turkey population, which was once 250, drops by 2.2%. There will be around 209 turkeys left after 8 years.
Wild turkey populations are declining at a 2.2% annual pace. The initial population of 250 turkeys is anticipated to drop to around 209 turkeys after 8 years. Using the exponential decay formula, we can determine the rate of decline annually: P is the population at a certain moment, P0 is the starting population, r is the rate of reduction expressed as a decimal, and t is the time in years. P = P0 (1 - r)t. The answer to the r equation is r = (1 - (P/P0)(1/t)) = 0.022. As a result, the population is declining by 2.2% annually.
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Part A
SAVINGS A company has a bonus incentive for its employees. The company pays employees an initial signing bonus of $1000 and invests that amount for the employees. Suppose the investment earns 8% interest compounded quarterly.
a. If an employee receiving this incentive withdraws the balance of the account after 5 years, how much will be in the account? Round to the nearest cent.
Part B
b. If an employee receiving this incentive withdraws the balance of the account after 35 years, how much will be in the account? Round to the nearest cent.
After answering the given query, we can state that So, after 35 years, the amount account would have a value of $10,062.07.
What is amount ?aggregate attempting to determine the amount, total number, or duration needed. The amount in front of you or being thought about is very active. the final outcome, its significance, or its meaning. three accountings: principal, interest, and the third. Word versions include amounts, amounting, and amounted. supple word How much something is, how much you have, how much you need, or how much you get is its amount. He needs that much cash to get by.
a. We must first determine the quarterly interest rate in order to determine the account amount after five years:
r = 8% / 4 ≈ 2% every three months.
The amount after five years can then be determined using the compound interest formula:
[tex]A = P(1 + r/n)^{(nt)[/tex]
where P = $1,000 as the original investment
2% is the quarterly interest rate (r).
N = 4 times per year that interest is added (quarterly)
If t = amount of years, then 5
[tex]A = \$1000(1 + 0.02/4)^{(4*5)} = $1,221.50[/tex]
So, after five years, the account would have a total of $1,221.50.
b. We employ the same method as before to determine the account balance after 35 years:
[tex]A = \$1000(1 + 0.02/4)^{(4*35)} = $10,062.07[/tex]
So, after 35 years, the account would have a value of $10,062.07.
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John goes to the casino and plays a slot machine. The probability that he wins on the first spin is 2/5. For all subsequent spins, the probability of John winning will be 5/6 if John wins in the preceding round, and the probability of John winning will be 1/5 if John did not win in the preceding round.(a) John plays 3 rounds. Find the probability that he wins on the third round, given that he only won two rounds of the three. (4 marks)(b) Let X denote the number of rounds John need to play before he finally wins for the first time. Comment on the suitability of modelling X after the geometric distribution. Compute P(X = 5). (3 marks)(c) John visits the casino on 20 separate days, he played exactly 10 rounds on each day. Let Y denote the number of days (out of 20) that John does not win anything. State any necessary assumptions required in order to suitably model Y after the binomial distribution. State clearly the parameters of this binomial distribution as well. (4 marks)(d) Assume that your assumptions in Question 2(c) hold. Compute E(Y) and Var(Y ). (4 marks)
For (a), the probability that John wins on the third round given that he only won two rounds of the three is 5/6.
For (b), the suitability of modelling X after the geometric distribution is appropriate because the geometric distribution is the probability distribution of the number of Bernoulli trials needed to get one success. The probability of John winning on the fifth round is $(\frac{2}{5})^4(\frac{5}{6})=\frac{50}{648}$
For (c), any necessary assumptions required in order to suitably model Y after the binomial distribution is that the trials (i.e. rounds of play) are independent, and that the probability of success is the same for each trial. The parameters of this binomial distribution are: n = 10, p = 1/5.
For (d), assuming the assumptions in Question 2(c) hold, the expected value of Y (E(Y)) is 8 and the variance of Y (Var(Y)) is 2.4.
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suppose that a new employee starts working at $7.22 per hour and receives a 4% raise each year. After time t, in years, his hourly wage is given by the equation y=$7.03(1.03)^t. find the amount of time after which he will be earning $10.00 per hour.
After approximately 9.95 years, the employee will be earning $10.00 per hour.
What is property of logarithms?
The properties of logarithms are a set of rules that can be used to manipulate logarithmic expressions, including the product, quotient, power, and change of base properties.
We are given that the employee's hourly wage after t years is given by the equation [tex]y=7.03(1.03)^t[/tex]. We need to find the amount of time t after which he will be earning $10.00 per hour. So we set y = $10.00 and solve for t as follows:
$10.00 = $[tex]7.03(1.03)^t[/tex]
Divide both sides by $7.03:
1.4246 = [tex]1.03^t[/tex]
Take the natural logarithm of both sides:
ln(1.4246) = ln([tex]1.03^t[/tex])
Using the property of logarithms that ln([tex]a^b[/tex]) = b * ln(a), we can simplify the right-hand side:
ln(1.4246) = t * ln(1.03)
Solve for t:
t ≈ 9.95 years
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Given the polynomial f(x) = x^3 + 3x^2 − x − 3, which of the following is true?
(x + 3) is a factor since f(3) = 0.
(x + 3) is a factor since f(−3) = 0.
(x − 3) is a factor since f(3) = 0.
(x − 3) is a factor since f(−3) = 0.
The true statement is that (x + 3) is a factor of f(x) = x³ + 3x² - x - 3 since f(-3) = 0.
How to find the factor of a polynomial ?A polynomial is an expression that consists of variables, terms, exponents and constants.
The factors are the polynomials which are multiplied to produce the original polynomial.
The factorisation of a polynomial is breaking the polynomial as a products.
Therefore,
f(x) = x³ + 3x² - x - 3
Hence, let's use the factor (x + 3)
Therefore,
f(-3) = (-3)³ + 3(-3)² - (-3) - 3
f(-3) = -27 + 27 + 3 - 3
f(-3) = 0
Therefore, the factor is (x + 3) since f(-3) = 0
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Suppose that 20% of all copies of grade 7 Science text book fail a certain binding strength test. Let X denote the number among 15 randomly selected copies that fail the test.
b. What is the probability that at least 8 fail the test?
binomial distribution
Question: Suppose that 20% of all copies of grade 7 Science textbook fail a certain binding strength test. Let X denote the number among 15 randomly selected copies that fail the test.b. What is the probability that at least 8 fail the test?To find the probability that at least 8 fail the test, we will use the binomial probability formula.The binomial distribution is a statistical distribution that describes the probability of success or failure in an experiment or survey with two possible outcomes.Suppose a trial consists of n independent experiments, each of which has two possible outcomes: either success with probability p or failure with probability q = 1 – p. If X is the number of successes in n trials, then X is said to follow a binomial distribution with parameters n and p. The probability that X = x is given by;P(X = x) = nCxpx(1-p)n-xwhere nCx = n! / x! (n-x)! is the binomial coefficient which counts the number of ways that x successes can be chosen from n trials.Using the formula, we have;P(X >= 8) = P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)Here n = 15, p = 0.2 and q = 0.8P(X >= 8) = P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)= 15C8 (0.2)^8 (0.8)^7 + 15C9 (0.2)^9 (0.8)^6 + 15C10 (0.2)^10 (0.8)^5 + 15C11 (0.2)^11 (0.8)^4 + 15C12 (0.2)^12 (0.8)^3 + 15C13 (0.2)^13 (0.8)^2 + 15C14 (0.2)^14 (0.8)^1 + 15C15 (0.2)^15 (0.8)^0= 0.0094 + 0.0267 + 0.0524 + 0.0864 + 0.1174 + 0.1312 + 0.1184 + 0.0352= 0.5771Therefore the probability that at least 8 fail the test is 0.5771 (rounded to four decimal places).Answer: 0.5771
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Dumisani earn 42 480 per month. He splits his earnings in the ratio 7:5. And then saves the lesser amount. How much does he save
If Dumisani earn $42480 per month and he splits his earnings in the ratio 7:5, then he saves $17,700 per month.
A ratio is a way of comparing two quantities or values. It expresses the relationship between two numbers or values in terms of their relative sizes or amounts.
To split Dumisani's earnings in the ratio 7:5, we need to divide his earnings into 7 + 5 = 12 equal parts.
Each part is therefore:
42480 ÷ 12 = 3,540
Dumisani saves the lesser amount, which is in the ratio of 5 parts to 7 parts.
So he saves:
5 × 3,540 = $ 17,700
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5. Explain the properties of logarithms that you
would need to use to solve the following:
log₂2x + log₂4 = 5
Answer:
x = 4
Step-by-step explanation:
log a + log b = log ab
log(base b) x = y, then b^y = x
log₂2x + log₂4 = 5
log₂ (2x × 4) = 5
log₂ 8x = 5
8x = 2^5
8x = 32
x = 4
1. Explain what Marc did in steps 4 and 5.
2. Why did he do this?
3. Create your own radical equation and explain how to solve it.
4. Is there an extraneous solution to your equation?
Which inequality describes the graph?
Answer:
Step-by-step explanation:
I think y ≤ 3 - 3x
(Please answer quickly!!!!) Given 7.05(−18.2), find the product.
−128.31
−12.83
77.55
578.10
Answer:
-128.31
First choice
Step-by-step explanation:
Using a calculator:
7.05(−18.2) = 7.05 x -18.2 = -128.31
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The figures converted from fraction to decimal to percent (Questions 9-14) are given as follows:
Percent Decimal Fraction
9) 316.2%. 3.162 3 4861/30000
10) 6.94%. 0.0694 347/5000.
11) 15.27%. 0.1527 1527/10000
12) 217 2761/3367% 2.178 1089/500
13) 723 12/21% 7.236 1809/250
14) 87% 0.87 87/100
9) Given the compound fraction: 3 4861/30000
To convert 3 4861/30000 to a decimal, we first need to convert the mixed number to an improper fraction:
3 4861/30000 = (30000*3 + 4861)/30000
= 94861/30000
To convert this fraction to a decimal, we divide the numerator by the denominator:
94861/30000 ≈ 3.162
To convert 3 4861/30000 to a percentage, we multiply the decimal by 100:
3.162 x 100 ≈ 316.2%
So, 3 4861/30000 as a percentage is approximately 316.2%
10) Given the decimal 0.0694
To convert 0.0694 to a fraction, we can write it as the numerator over a power of 10, where the power of 10 has the same number of digits as the number of decimal places in the original number:
0.0694 = 694/10000
We can simplify this fraction by dividing both the numerator and the denominator by their greatest common factor, which is 2:
694/10000 = (3472)/(50002) = 347/5000
So, 0.0694 as a fraction is 347/5000.
To convert 0.0694 to a percentage, we multiply the decimal by 100:
0.0694 x 100 = 6.94%
So, 0.0694 as a percentage is 6.94%.
11) Given the decimal - 0.1527
To convert 0.1527 to a percentage, we multiply the decimal by 100:
0.1527 x 100 = 15.27%
So, 0.1527 as a percentage is 15.27%.
To convert 0.1527 to a fraction, we can write it as the numerator over a power of 10, where the power of 10 has the same number of digits as the number of decimal places in the original number:
0.1527 = 1527/10000
12) Given the compound percentage - 217 2761/3367%
To convert 217 2761/3367% to a decimal, we first need to convert the mixed number to an improper fraction:
217 2761/3367% = (3367*217 + 2761)/3367% = 733400/3367%
743032/3367% = 217.82001782%
217.82001782% = 2.1782001782
220.680724681% [tex]\approx[/tex] 2.178
Thus, in fraction:
2.178 = 2.178/1
Multiply to remove 3 decimal places. Here, you multiply top and bottom by 10³ = 1000
= 2.178/1 x 1000/1000
= 2178/1000
= 1089/500
13) Given the compound fraction - 723 12/21%
To convert 723 12/21% to a decimal, we first need to convert the mixed number to an improper fraction:
723 12/21% = (21*723 + 12)/21% = (15195/21)%
(15315/21)% = 7.23571428571
15315/21% [tex]\approx[/tex] 7.236
Converting 7.236 to fraction:
7.236 = 7.236/1
Multiply to remove 3 decimal places. Here, you multiply top and bottom by 10³ = 1000
= 7.236/1 x 1000/1000
= 7236/1000: Divide numerator and denominator by 4
= (7236÷ 4)/(1000÷4)
= 1809/250
14) Given 0.87:
To convert 0.87 to a percentage, we multiply the decimal by 100:
0.87 x 100 = 87%
So, 0.87 as a percentage is 87%.
To convert 0.87 to a fraction, we can write it as the numerator over a power of 10, where the power of 10 has the same number of digits as the number of decimal places in the original number:
0.87 = 87/100
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The Gross Domestic Product (GPD) of a country in the first quarter of 2014 was $1.1395x10^7. Rewrite the GDP in standard notation.
113,950,000,000 ([tex]1.1395x10^7[/tex] is scientific notation, so you need to move the decimal to the left 7 places and add the relevant commas to get 113,950,000,000 in standard notation).
What is notation?Notation is a method of representing information, usually in a concise and organized way. It can be used to represent mathematical equations, algorithms, music, dance, and other concepts. Notation is often used as a shorthand for communicating ideas to others, as it allows for a more precise and efficient way of conveying information. Notation also allows for the precise and consistent representation of a concept, allowing for greater accuracy and understanding.
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Please please help me with this!
Answer:
288π in³
Step-by-step explanation:
V-sphere = 4/3πr³
D = 12in r = D/2 = 12/2 = 6 in
V = (4/3)(6)³π = (4/3)(216)π = 288π in³
Consider the following theorem. Theorem: For any integer , is odd if and only if + 1 is even.
Construct a proof for the theorem by selecting sentences from the following scrambled list and putting them in the correct order. Note that each statement will be used at most once (or not at all).
Since 2 | 2( + 1), 2 | ( + 1).
Suppose is odd.
Therefore, + 1 is even.
Suppose + 1 is even.
Therefore, + 1 is odd.
Thus, for some integer , + 1 = 2 + 1.
Thus, for some integer , = 2.
Thus, for some integer , = 2 + 1.
Therefore, is odd.
Thus, for some integer , + 1 = 2.
Now + 1 = (2 + 1) + 1 = 2 + 2 = 2( + 1).
Now = (2 - 1) = 2( - 1) + 1 .
Proof:
(⇒)
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(⇐)
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The answer of the given question based on the theorem For any integer , is odd if and only if + 1 is even the answer is for any integer , is odd if and only if + 1 is even.
What is Theorem?A theorem is statement in mathematics that has been proven to be the true based on logical reasoning and the mathematical principles. In other words, theorem is a proposition that can be demonstrated or proved to be true through logical argument.
Proof:
(⇒) Suppose is odd.
Thus, for some integer , = 2 + 1.
Now + 1 = (2 + 1) + 1 = 2 + 2 = 2( + 1).
Therefore, + 1 is even.
(⇐) Suppose + 1 is even.
Then, for some integer , + 1 = 2.
Thus, for some integer , = 2 - 1 = 2( - 1) + 1 .
Since 2 | 2( + 1), 2 | ( + 1).
Therefore, is odd.
Thus, we have shown that for any integer , is odd if and only if + 1 is even.
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