One way to measure species diversity is to use the Shannon diversity index H. If a habitat consists of three species, A, B, and C, then its Shannon diversity index is H = -xlnx - ylny - zlnz where x is the percent of species A in the habitat, y is the percent of species B in the habitat, and z is the percent of species in the habitat. Use the fact that x+y+z= 1 to show that the maximum value of H occurs when x = y = z = 1⁄3 What is the maximum value of H?
Answers
The maximum value of H is ln(3).
To show that the maximum value of H occurs when x = y = z = 1/3, we need to use calculus to find the critical points of the function H(x, y, z) subject to the constraint x + y + z = 1, and then show that the point (1/3, 1/3, 1/3) is a maximum.
First, we can rewrite the expression for H(x, y, z) using the constraint x + y + z = 1 as:
H(x, y, z) = -xlnx - ylny - zlnz
= -(xlnx + ylny + zlnz)
-(-ln(x^x * y^y * z^z))
= ln(x^(-x) * y^(-y) * z^(-z))
Next, we can use Lagrange multipliers to find the critical points of H(x, y, z) subject to the constraint x + y + z = 1:
∇H(x, y, z) = λ∇(x + y + z)
= ( -ln(x) - 1 )i + ( -ln(y) - 1 )j + ( -ln(z) - 1 )k
= λi + λj + λk
Solving for x, y, z, and λ, we get:
x = y = z = 1/3 and λ = -3ln(1/3) = 3ln3
To show that (1/3, 1/3, 1/3) is a maximum, we need to use the second partial derivative test. Computing the Hessian matrix of H(x, y, z), we have:
Hessian(H) = [(-1/x^2) -1 -1 ]
[-1 (-1/y^2) -1 ]
[-1 -1 (-1/z^2)]
At (1/3, 1/3, 1/3), the Hessian matrix is diagonal with eigenvalues -2/9, -2/9, -2/9, all of which are negative. Therefore, (1/3, 1/3, 1/3) is a maximum.
The maximum value of H occurs when x = y = z = 1/3, so:
H(max) = ln((1/3)^(-1/3) * (1/3)^(-1/3) * (1/3)^(-1/3))
= ln(3^(1/3 * 3))
= ln(3)
Therefore, the maximum value of H is ln(3).
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Related Questions
Numerical problems:
a.
Convert 300 K into the celsius scale.
b
Convert 220 °C into kelvin scale.
Convert 30 °C into fahrenheit scale.
d. Convert 260 °F into °C.
Answers
8. a. 27 °C
b. 493 K
c. 86 °F
d. 12.67 °C
Answers
please help me solve this equation
Answers
Answer:
-85
Step-by-step explanation:
We are given the equation \(E^8_{n=1}\)\((-2)^{n-1}\) and are asked to evaluate it.
This can be solved using the geometric sequence, but first we have to determine our "r", "n", and \(a_{1}\).
Of course \(a_{1}\) is equal to 1. Therefore,
\(a_{1\) = 1
\(r=\)
\(n=\)
(-2) is the change per number in the sequence, therefore -2 is our "r".
\(a_{1\) = 1
\(r=-2\)
\(n=\)
The number "8" on \(E^8_{n-1}\) represents the "n". Therefore 8 is our "n".
\(a_{1}=1\)
\(r=-2\)
\(n=8\)
Use the geometric sequence sum formula and substitute :
\(a_{1} \frac{1-r^n}{1-r}\)
\(1* \frac{1-(-2)^8}{1-(-2)}\)
\(1* \frac{1-(-2)^8}{1+2}\)
\(1* \frac{1-256}{1+2}\)
\(1* \frac{-255}{3}\)
\(1 * -85\)
\(-85\)
Use the body mass indexes (BMI) for males and use the BMI measure for females listed in the accompanying table to construct boxplots. Use the boxplots to compare the data sets
Answers
By comparing the boxplots, you can observe the differences in the distribution of BMI values between males and females. Look for differences in the medians, interquartile ranges, and any outliers.
To construct boxplots for the body mass index (BMI) data sets for males and females, we first need to gather the necessary information.
To create a boxplot, you need the five-number summary of the data set: minimum value, lower quartile (Q1), median, upper quartile (Q3), and maximum value. These statistics help to visualize the distribution and compare the data sets.
Once you have the five-number summary for both males and females, draw two number lines and place the respective summaries on each line. Then, construct a box around the quartiles and draw lines (whiskers) extending to the minimum and maximum values. This will create two boxplots side by side.
These differences can indicate variations in BMI between the genders.
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The first, second and fourth terms of a proportion are 16, 24 and 54 respectively. Find the third
term.
(a) 8
(b) 36
(c) 24
(d) 7.1
Answers
Answer:
Third term is 36
Step-by-step explanation:
According to the definition of proportion, if two sets of given numbers are in the same ratio, then the ratios are said to be directly proportional to each other
We use the symbol :: to denote proportion
So if
a:b :: c:d
Then a/b = c/d
We are given that first number = 16, second is 24 and the last is 54
Let the third number be x
We have that
16:24 :: x : 54
In other words
16/24 = x/ 54
Simplifying left side
2/3 = x/54
x = 2/3 x 54 = 36
Peter says 3/4 of a pizza is always the same as 6/8 of a pizza. Nadia says while 3/4 and 6/8 are equivalent fractions, 3/4 and 6/8 of a pizza could represent different amounts.
PLEASE HELP ASAP I HAVE 3 MINS LEFT!!!
PLEASE ANSWER THE PICTURE I SENT YOU BECAUSE IT HAS TO DO WITH THE QUESTION.
JUST DO NUMBER 2
Answers
Peter is right and Nadia is wrong.
What are equivalent fractions?Equivalent fractions are fractions that represent the same value, even though they look different.
Given is that Peter says 3/4 of a pizza is always the same as 6/8 of a pizza. Nadia says while 3/4 and 6/8 are equivalent fractions, 3/4 and 6/8 of a pizza could represent different amounts.
{ 1 } -
3/4 = (3 x 2)/(4 x 2) = 6/8
Peter is right.
Now -
3/4 = 6/8
So it could not represent two different quantities. So, Nadia is wrong.
{ 2 } -
3/4 = 6/8 = 9/12 = 12/15
These all represent the same quantity. Division of all the expressions will result in same value.
Therefore, Peter is right and Nadia is wrong.
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A pair of jeans is on sale for $56 at 35% off. What is the original price of the jeans? Show your work to receive full credit
Answers
Which Choice is equivalent to the product below when X > 0
Answers
Answer: this is equal to √(3x²/12) = √(x²/4) = x/2
Step-by-step explanation:
Answer:
C.) 1/3
Step-by-step explanation:
What is 161 divided by 23
Answers
Answer:
\(161 \div 23 = 26.78\)
How Do I Do Goldbach’s Conjecture.
Answers
suppose that m is a proper ideal in a commutative ring r. we proved in class that if r/m is simple, then m is a maximal ideal in r. give a careful direct proof of the converse direction. that is, prove that if m is maximal, the
Answers
Here is a careful direct proof of the converse direction of the ideal in a commutative ring.
Suppose that M is a proper ideal in a commutative ring R. We proved in class that if R/M is simple, then M is a maximal ideal in R. Now let's prove the converse, that if M is maximal, then R/M is simple.
To prove this, we must consider the following two facts:
* If M is a maximal ideal in R, then for any ideal N of R, either M is contained in N or N is contained in M.
* If R/M is simple, then for any nonzero ideal N of R/M, N = R/M.
Now, let N be any ideal of R. If M is contained in N, then the image of N in R/M is equal to N. Since R/M is simple, this means that N = R/M.
If M is not contained in N, then N is not contained in M. Since M is maximal, this means that N must be equal to R. The image of N in R/M is therefore also equal to R. Since R/M is simple, this means that N = R/M.
In either case, we have shown that for any ideal N of R, either M is contained in N or N is contained in M. This means that M is a maximal ideal in R.
Therefore, if M is maximal, then R/M is simple.
Thus, it's proved that "If M is a maximal ideal, then R/M is a simple ideal".
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if you roll two fair six-sided dice, what is the probability that the sum is 9 99 or higher?
Answers
The probability that the sum of two fair six-sided dice is 9 or 99 or higher is 5/36.
To find the probability of whether the sum of two fair six-sided dice is 9 or 99 or higher, we need to determine the number of favorable outcomes and the total number of possible outcomes.
The possible outcomes when rolling two six-sided dice range from 2 (minimum sum) to 12 (maximum sum). We want to calculate the probability of getting a sum of 9 or 99 or higher.
To determine the favorable outcomes, we need to identify all the combinations of dice rolls that meet our criteria. Here are the favorable outcomes:
(3, 6) = sum of 9
(4, 5) = sum of 9
(5, 4) = sum of 9
(6, 3) = sum of 9
(6, 6) = sum of 12
Therefore, we have 5 favorable outcomes.
Now, let's calculate the total number of possible outcomes. Since each die has 6 possible outcomes (numbers 1 to 6), the total number of possible outcomes for rolling two dice is 6 * 6 = 36.
The probability of getting a sum of 9 or 99 or higher is given by:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 5 / 36
Therefore, the probability that the sum of two fair six-sided dice is 9 or 99 or higher is 5/36.
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Determine whether each of these statements is true or false.a) 0 ∈ ∅ b) ∅∈{0}c) {0}⊂∅ d) ∅⊂{0}e) {0}∈{0} f ) {0}⊂{0}g) {∅} ⊆ {∅}
Answers
a) True, b) False, c) False, d) True, e) False, f) False, g) True
a) The statement is true. The symbol "∅" represents the empty set, which does not contain any elements. Since 0 is not an element of the empty set, the statement "0 ∈ ∅" is false.
b) The statement is false. The symbol "{0}" represents a set containing the element 0. The empty set, represented by "∅", does not contain any elements. Therefore, the empty set is not an element of the set {0}.
c) The statement is false. The symbol "{0}" represents a set containing the element 0. However, the empty set, represented by "∅", does not contain any elements. Therefore, the set {0} is not a subset of the empty set.
d) The statement is true. The empty set, represented by "∅", does not contain any elements. Every set is a subset of the empty set, including the set {0}.
e) The statement is false. The symbol "{0}" represents a set containing the element 0. In this case, {0} is not an element of itself. Therefore, the statement "{0}∈{0}" is false.
f) The statement is false. The symbol "{0}" represents a set containing the element 0. In this case, {0} is not a proper subset of itself. Therefore, the statement "{0}⊂{0}" is false.
g) The statement is true. The symbol "{∅}" represents a set containing the empty set. In this case, {∅} is a subset of itself because it contains the element ∅. Therefore, the statement "{∅} ⊆ {∅}" is true.
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Last year you bought a pair of boots for $84 last year. Your friend bought the same pair of boots this year for $120. What is the percent of change in the cost of the boots.
Answers
Answer:
30%
Step-by-step explanation:
70% of 120 is 84. So take the remainder of 70 (30, as %'s go up to 100.) and that's the percent left. We can "fact check this" the other way around as well.
30% of 120 = 36.
84 + 36 = 120.
The price of the boots have changed/raised 30%.
Hope this helps and have a nice day!
Circle the correct answer. Carl wants to plan a garden that is 1 1 2yards long with an area of 1 3 2square yards. The width of the garden should be
Answers
Carl wants to plan a garden that is 1 1/2 yards long with an area of 132 square yards, the width of the garden should be 88 yards.
To find the width of the garden, we can use the formula for the area of a rectangle: Area = length × width. In this case, the given length of the garden is 1 1/2 yards and the area is 132 square yards.
To calculate the width, we need to divide the area by the length. Let's convert the length to an improper fraction for easier computation.
1 1/2 yards can be written as 3/2 yards. Now we can calculate the width:
Width = Area / Length
Width = 132 square yards / 3/2 yards
To divide by a fraction, we can multiply by its reciprocal:
Width = 132 square yards * 2/3 yards
Multiplying the numerators and denominators gives us:
Width = (132 * 2) / (3 * 1) yards
Width = 264 / 3 yards
Width = 88 yards
Therefore, the width of the garden should be 88 yards.
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Write the correct answer. Carl wants to plan a garden that is 1 1 2yards long with an area of 1 3 2square yards. The width of the garden should be ?
find the volume common to two spheres, each with radius r, if the distance between their centers is r/2.
Answers
The volume common to two spheres, each with radius r, if the distance between their centres is r/2 is V = (11/12)×π×r³.
The attached diagram shows 2 circumferences with radius r and separated centres by r/2.
Let´s call circumferences 1 and 2; by symmetry, rotating area A will produce a volume V₁ identical to a V₂, Obtained by rotating area B ( both around the x-axis), then the whole volume V will be:
V = 2× V₁
V₁ = ∫π×y²×dx (1)
Now
( x - r/2)² + y² = r² the equation of circumference 1
y² = r² - ( x - r/2)²
Plugging this value in equation (1)
V₁ = ∫π×[ r² - ( x - r/2)²]×dx with integrations limits 0 ≤ x ≤ r/2
V₁ = π×∫ ( r² - x² + (r/2)² - r×x )×dx
V₁ = π× [ r²×x - x³/3 + (r/2)²×x - (1/2) × r × x²] evaluate between 0 and r/2
V₁ = π× [(5/4)×r²×x - x³/3 - (1/2) × r × x²]
V₁ = π× [(5/4)×r² × ( r/2 - 0 ) - (1/3)×(r/2)³ - (1/2) × r × (r/2)²]
V₁ = π× [ (5/8)×r³ - r³/24 - r³/8]
V₁ = π× (11/24)×r³
Then
V = 2× V₁
V = 2×π×11/24)×r³
V = (11/12)×π×r³
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What is (14x-2)-(11x-21) as a simplified expression
Answers
Answer: (14) x (-2)- (11) x (-21)
Step-by-step explanation: Hope I was I right!
If a polynomial function, f(x), with rational coefficients has roots 3 and startroot 7 endroot, what must also be a root of f(x)?
Answers
The conjugate which is 3 - √7 is also a root of f(x).
What is the root of a polynomial function?The root of a polynomial function f(x) is the value of x for which f(x) = 0.
Now if a polynomial function has a root x = a + √b then the conjugate of x which is x' = a - √b is also a root of the function, f(x).
What must also be a root of f(x)?
Given that the polynomial function, f(x), with rational coefficients has roots
3 + √7, then by the above, the conjugate which is 3 - √7 is also a root of f(x).
So, 3 - √7 is also a root of f(x).
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give an arithmetic sequence of 20,16,12find it’s common difference and the 25th term
Answers
Answer: Common difference = -4, and 25th term = -56
Step-by-step explanation:
The common difference, d, is 20 - 16 or 16 - 12, which is -4.
Using the formula \(a_{n}=a_{1}+(n-1)d\), \(a_{25}=20+(20-1)*-4\)
= 20 + 19 * -4,
= 20 - 76
= -56
consider parallelogram vwxy below use the information given in the figure to findx m
Answers
In the given parallelogram vwxy: x = 3; m∠Y = 65° and ∠YVX = 61° by (alternate interior angle).
Explain about the parallelogram?A quadrilateral with the opposing sides parallel is called a parallelogram (and thus opposite angles equal).
A parallelogram with all right angles is known as a rectangle, and a quadrilateral having equal sides is known as a rhombus. Rectangles and squares are both particular varieties of parallelograms since a square is just a degenerate instance of a rectangle.In the given parallelogram vwxy:
Parallel sides are equal.
So, 2x = 6
x = 6/2 = 3
Opposite pair of angles are also equal.
So,
∠W = ∠Y = 65°
Now,
∠YVX = 61° (alternate interior angle).
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Will give brainliest, like, and comment to best answer
What is 645 divided by 50?
Answers
Answer:
645 ÷ 50 = 12.9
I hope this helps and that you have a wonderful day ^‿^
Answer:
the answer is 129
Step-by-step explanation:
The sparrow tennis team outscored its opponents 5:2 last season. If their opponents scored 40 points, how many points did The sparrows score?
Answers
Answer:
100?
Step-by-step explanation:
Answer:
100
Step-by-step explanation:
5/7/2/7*40
5/7*7/2*40
=100
Laurie was scuba diving. each time she dove 5 feet deeper, she would stop and clear her ears. each time that this totaled 20 feet deeper, she would stop and check her instruments to make sure they were working properly. after laurie had checked her instruments 3 times, she was at her maximum depth. which expression best represents laurie’s maximum depth? (4) (negative 5) (3) (negative 4) (negative 5) (negative 3) (negative 5) (20) (3) (negative 5) (negative 20) (negative 3)
Answers
x is 4, which gives us the expression (4)(-5)(3) using algebraic equation.
What is algebraic equation?An algebraic equation or polynomial equation is an equation of the form P=0 where P is a polynomial with coefficients in some field, often the field of the rational numbers.
Given Data
In each step she goes 5 feet deeper, hence (-5) must be a part of the expression.
She checked her instruments 3 times, so +3 must be a part of the expression.
Her total depth was 60 feet so
-60= x(-5)(-3)
X= 4
Therefore x is 4, which gives us the expression (4)(-5)(3)
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two adjacent sides of a rhombus form a $60$-degree angle. if each side of the rhombus measures $2$ cm, what is the area of the rhombus, in square centimeters? express your answer in simplest radical form.
Answers
The area of the rhombus with each side 2cm and angle between two adjacent sides is 60° is equals to the 2√3 cm².
We have, rhombus ABCD, with length of rhombus = 2 cm
The angle between two adjacent sides of a rhombus = 60°
We have to calculate the area of the rhombus ABCD. A rhombus is a type of quadrilateral with both pairs of opposite sides parallel and all sides the same length, i.e., an equilateral parallelogram. The area of a rhombus with and height is, A = base× height. Draw an altitude/prependicular from D to E (refer in the above figure). This form a 30-60-90 right triangle. Now, angle DAE = 60°, angle ADE =30°. Since AB = AD = 2cm , then the side opposite the 30° angle is half of the hypotenuse, or half of AD = 2. The side opposite angle ADE is AE, so,
AE = 1 cm. Using the Pythagoras threoem,
AD² = AE² + DE²
=> DE² = AD² - AE²
=> DE² = 2² - 1 = 3
=> DE = √3
Thus, remaining side, DE is equal to √3 cm. The height of the figure is equal to √3 and the base equals to 2cm. So,
Area of rhombus, A = base × height
=> A = 2× √3
Hence, the total area is 2√3 cm².
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how many feet of granite was tunneled through to make tunnel no. 6 through the sierra nevada mountains?
Answers
Nearly 1,659 feet of granite was tunnelled through to make tunnel no. 6 through the sierra Nevada mountains.
Early snowfall prevented the Central Pacific from starting construction on Tunnel No. 6, or the Summit Tunnel, in August 1865. It was built using a variety of engineering and construction methods and was located more than seven thousand feet above sea level.
When the workmen finally broke through, they discovered that they were only two inches off from the calculations that were used to locate its end points and central shaft. The length of the tunnel that was built through the Sierra Nevada mountains is therefore given as nearly 1,659 feet of granite was tunnelled through to make tunnel no. 6 through the Sierra Nevada mountains.
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Use the marginal tax rate chart to answer the question. Tax Bracket Marginal Tax Rate $0–$10,275 10% $10,276–$41,175 12% $41,176–$89,075 22% $89,076–$170,050 24% $170,051–$215,950 32% $215,951–$539,900 35% > $539,901 37% Determine the effective tax rate for a taxable income of $180,900. Round the final answer to the nearest hundredth.
19.50%
20.00%
21.11%
32.00%
Answers
For a taxable income of $1,80,900, the effective tax rate is 21.11%.
what is meant by marginal tax?Marginal tax is the tax rate applied to an individual’s last dollar of income. It is the additional tax that must be paid on any additional income earned. In other words, it is the tax rate that applies to an individual’s income after all deductions. For example, if an individual’s income is $50,000 and the marginal tax rate is 25%, then the individual must pay an additional $12,500 in taxes on top of the $25,000 in taxes already paid on their $50,000 income.
Marginal tax rates are progressive, meaning that as an individual’s income increases, the marginal tax rate will increase as well. The marginal tax rate is important to understand because it helps individuals determine how much additional income they will need to earn in order to make up for the taxes they will have to pay on that income. It is also important to understand how marginal tax rates can affect tax planning strategies.
To determine the effective tax rate for a taxable income of $180,900, calculate the sum of the marginal tax rates for each tax bracket that the $180,900 falls into. The taxable income of $180,900 is greater than $89,075 and less than $170,050. The marginal tax rates for these two tax brackets are 22% and 24%, respectively. The sum of the marginal tax rates is 46%. Divide the sum by the taxable income of $180,900 to get the effective tax rate of 21.11%. Round the final answer to the nearest hundredth to get 21.11%.
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Which fraction is not equivalent to 5/6 ?
Answers
Answer:
4/12
Step-by-step explanation:
Let y be a binomial random variable with n = 10 and p = 0.4. (a) Use Table 1 to obtain P(3
Answers
To solve this problem, we can use Table 1 (which is a table of values for the cumulative distribution function of the binomial distribution).
(a) To find P(3 < y < 7), we need to calculate the probability of getting between 4 and 6 successes (inclusive) in 10 trials, where the probability of success in each trial is 0.4.
Using Table 1, we can find these probabilities by looking up the values for the cumulative distribution function (CDF) of the binomial distribution. Specifically, we need to find P(y ≤ 6) and P(y ≤ 3), and then subtract the latter from the former to get the probability of getting between 4 and 6 successes.
To find P(y ≤ 6), we look up the row for n = 10 and p = 0.4, and then find the value in the column for y = 6. This value is 0.9688.
To find P(y ≤ 3), we look up the row for n = 10 and p = 0.4, and then find the value in the column for y = 3. This value is 0.3823.
Subtracting P(y ≤ 3) from P(y ≤ 6), we get:
P(4 ≤ y ≤ 6) = P(y ≤ 6) - P(y ≤ 3)
= 0.9688 - 0.3823
= 0.5865
Therefore, the probability of getting between 4 and 6 successes (inclusive) in 10 trials, where the probability of success in each trial is 0.4, is 0.5865.
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Write an
equation for the linear function f satisfying the given conditions.
1) f(-1) = -5 and f(5) = 7
Answers
Answer: f(x) = 2x - 3
===================================================
We're given f(-1) = -5 and f(5) = 7
This means we know the two points (-1,-5) and (5,7) are on the line.
Find the slope
m = (y2-y1)/(x2-x1)
m = (7-(-5))/(5-(-1))
m = (7+5)/(5+1)
m = 12/6
m = 2
The slope is 2.
Use this along with (x1,y1) to find the equation. We'll use point slope form
y - y1 = m(x - x1)
y - (-5) = 2(x - (-1))
y + 5 = 2(x + 1)
y = 2x + 2 - 5
y = 2x - 3
f(x) = 2x - 3
If S ⊆ R is nonempty, show that S is bounded if and only if there exists a closed bounded interval I such that S ⊆ I.
Answers
S is bounded since we have found real numbers a and b such that for all s in S, a ≤ s ≤ b. Therefore, we have proven that S is bounded if and only if there exists a closed bounded interval I such that S ⊆ I.
First, let's define what it means for a set to be bounded. A set S is bounded if there exist real numbers M and m such that for all s in S, m ≤ s ≤ M.
Now, let's prove the statement "S is bounded if and only if there exists a closed bounded interval I such that S ⊆ I."
Forward direction: Assume that S is bounded. Then, by definition, there exist real numbers M and m such that for all s in S, m ≤ s ≤ M. Let I = [m, M]. I is a closed bounded interval since it contains its endpoints and is closed. Furthermore, since for all s in S, m ≤ s ≤ M, it follows that S is a subset of I. Therefore, S is bounded and there exists a closed bounded interval I such that S ⊆ I.
Backward direction: Assume that there exists a closed bounded interval I such that S ⊆ I. Let a and b be the endpoints of I. Then, for all s in S, a ≤ s ≤ b. Therefore, S is bounded since we have found real numbers a and b such that for all s in S, a ≤ s ≤ b.
Therefore, we have proven that S is bounded if and only if there exists a closed bounded interval I such that S ⊆ I.
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The outside temperature decreases by 7°F in 3 hours. What is the change in temperature per each hour? (Express as an integer) PLEASE HELP!!!
Answers
Answer:It drops 7 degrees per 6 hours
Step-by-step explanation: