Answers
The photo is not visable, can you type the question iin the comments I WILL solve it :)
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Find a Cartesian equation for the curve and identify it.
r = 4csc(θ)
circle line parabola ellipse
Answers
The Cartesian equation for the curve r = 4csc(θ) is (x^2 + y^2) * y = 4, and it represents a semi-cubical parabola.
To find a Cartesian equation for the curve r = 4csc(θ) and identify it, we can follow these steps:
Step 1: Rewrite the equation using the definition of cosecant
Cosecant (csc) is the reciprocal of sine, so we can rewrite the equation as:
r = 4/sin(θ)
Step 2: Express r and θ in Cartesian coordinates
We know that r^2 = x^2 + y^2 and y = r * sin(θ). We can use these relationships to replace r and θ with x and y in our equation. First, we solve for sin(θ):
sin(θ) = y / r
Step 3: Substitute sin(θ) and r in the equation
Now, replace sin(θ) and r in the original equation:
r = 4/(y/r)
Step 4: Simplify and solve for y
To get a Cartesian equation, multiply both sides by y:
r * y = 4
Since y = r * sin(θ), we can replace y with r * sin(θ):
r * (r * sin(θ)) = 4
Step 5: Replace r with x^2 + y^2
Since r^2 = x^2 + y^2, we can replace r in the equation:
(x^2 + y^2) * (y) = 4
Step 6: Identify the curve
This equation represents a curve in the Cartesian plane. By analyzing the equation, we can see that it's a semi-cubical parabola.
In conclusion, the Cartesian equation for the curve r = 4csc(θ) is (x^2 + y^2) * y = 4, and it represents a semi-cubical parabola.
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The figure shows two regular pentagons. What is the perimeter of the larger pentagon?
A. 5n
B. 10n
C. 15n
D. 60n
Answers
Answer:
As shown in picture,
perimeter of inside pentagon: p1 = n + n + n + n + n = 5n
ratio between inside pentagon and outside pentagon: r = 6/(6 + 6) = 1/2
=> perimeter of outside pentagon: p2 = 2 x p1 = 2 x 5n = 10n
=> Option B is correct
Hope this helps!
:)
A rectangular solid with sides a and b, and with a height of 20 cm has a volume of 120 cm³. Find the formula which defines b in terms of a. Why is this formula an indirect proportion? What is the domain of this function? A rectangular solid with sides a and b, and with a height of 20 cm has a volume of 120 cm³. Find the formula which defines b in terms of a. Why is this formula an indirect proportion? What is the domain of this function?
Answers
120 = ab(20)
Simplifying this equation, we get:
b = 6/a
Therefore, the formula that defines b in terms of a is b = 6/a.
This formula is an indirect proportion because as the value of a increases, the value of b decreases, and vice versa. In other words, the product of a and b is constant (in this case, 120), so if one of the variables increases, the other must decrease to maintain the same product.
The domain of this function is all positive real numbers except for a = 0, since division by zero is undefined. Therefore, the domain of the function is (0, ∞).
The table displays data collected, in meters, from a track meet.
one third 2 4 1
7 two thirds four fifths five halves
What is the median of the data collected?
Answers
Answer:
The answer to your question is, 2 meters.
Step-by-step explanation:
We know that a median is the middle number in a sorted, ascending or descending list of numbers.
How to caluclate it:
Collected date: 1, 4, 6, 2, 4, 3, 2/3, 1/3, 1/2
Rearreange the date in accending order:
1/3, 1/2, 2/3, 1, 2, 3, 4, 4, 6
Next we should always pick the middle of the number in the arranged data..:
Media would = 2 meters
Thus, your answer is, 2 meters
Prove or disprove that for all sets A, B, and C, we have a) Ax (BUC) = (A × B) U (A x× C). b) Ax (Bn C) = (A x B) n (A x C).
Answers
Answer:
a. Proven: Ax (BUC) = (A × B) U (A x C)
b. Disproven: Ax (Bn C) ≠ (A x B) n (A x C)
Step-by-step explanation:
(a) The equality holds. This can be proven by showing that an element (x, y) belongs to both sides of the equation. An element belongs to Ax (BUC) if and only if it belongs to (A × B) U (A x C), satisfying the condition for equality.
(b) The equality does not hold. Counterexamples can be provided to demonstrate that there exist elements that belong to one side of the equation but not the other. Thus, the statement is disproven, indicating that the sets on both sides are not necessarily equal.
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If you help me I will give you 100 brainly points !!
Val is a runner in a 5-kilometer outdoor race that consists of a 0.25-kilometer sprint, a stream crossing part, and two obstacle parts. Each obstacle part of the race is 1.3 kilometers. How long is the stream part of the race?
Answers
Answer:
Stream is 2.15 kilometers of the race.
Step-by-step explanation:
If the two obstacles are 1.3 kilometers each, and the sprint is 0.25 kilometers of the race, we simply have to add these together and subtract from the total distance of 5 kilometers. 1.3+1.3+0.25 is 2.85 kilometers. 5-2.85=2.15.
Answer:
2.15 km
Step-by-step explanation:
The obstacle parts of the race are 2*1.3 = 2.6 kilometers long.
The total length of the race is 5 kilometers.
The stream part of the race is 5-0.25 - 2.6 = 2.15 kilometers long.
So the answer is 2.15 km
Solve this system of linear equations. Separate
the x- and y-values with a comma.
17x = -60 - 3y
5x = -6 + 3y
Answers
Answer:
(-3, -3)
Step-by-step explanation:
1.) Rewrite the second equation so 3y is on one side of the equation:
3y=6+5x
2.) Substitute the given value of 3y (replacing 3y with 6+5x, since we know they equal each other) into the equation 17x=-60-3y
Should end up with this:
17x=-60-(6+5x)
3.) Solve 17x=-60-(6+5x)
Calculate Difference: 17x=-66-5x
Combine Like Terms: 22x = -66
Divided both sides by 22 to isolate and solve for x: -3
So We know x=-3, now we got to find the y value. We can use either the first or second equation to find y value, so lets use the second.
3y=6+5x
1.) We know that x=-3, so we can simply substitute x in the equation
3y=6+5x with -3
3y=6+5(-3)
2.) Solve 3y=6+5(-3)
Combine Like Term: 3y=6+-15
Combine Like Term Even More: 3y= -9
Divide by 3 on both sides to isolate and solve for y: y=-3
So now we know y=-3 and once again we know x=-3, so if we format that
(-3,-3)
^ ^
x y
nine points are on a circle. after drawing all chords by connecting each and every pair of the nine points, four of the chords are randomly selected. the probability that the four chords form a quadrilateral can be expressed as , where p and q are relatively prime integers. what is p q?
Answers
The probability of randomly selecting four chords that form a quadrilateral is 14/126, which simplifies to 1/9.
To calculate the probability that the four randomly selected chords from the nine points on a circle form a quadrilateral, we need to determine the total number of ways to select four chords and the number of ways to form a quadrilateral.
The total number of ways to choose four chords can be calculated using combinations. Since there are nine points and we want to choose four chords, the number of ways to select four chords is given by C(9, 4) = 126.
To form a quadrilateral, we need to choose four chords that do not intersect at a single point. The number of ways to do this can be determined by counting the number of non-crossing chords in the circle. The formula to calculate the number of non-crossing chords is given by the Catalan number C(n/2), where n is the number of points. In this case, n = 9, so C(9/2) = C(4) = 14.
Therefore, the probability of randomly selecting four chords that form a quadrilateral is 14/126, which simplifies to 1/9.
The numerator and denominator, p and q, are 1 and 9, respectively. So, p/q = 1/9.
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The heights of people in a certain population are normally distributed with a mean of 64 inches and a standard deviation of 3.1 inches. Determine the sampling distribution of the mean for samples of size 39.
Answers
The sampling distribution of the mean for samples of size 39 has a mean of 64 inches and a standard deviation of approximately 0.496 inches.
We are required to determine the sampling distribution of the mean for samples of size 39, given that the heights of people in a certain population are normally distributed with a mean of 64 inches and a standard deviation of 3.1 inches.
The sampling distribution of the mean is also normally distributed. To find the mean and standard deviation of the sampling distribution, you'll use the following formulas:
1. Mean of the sampling distribution (μx) = Mean of the population (μ)
2. Standard deviation of the sampling distribution (σx) = Standard deviation of the population (σ) divided by the square root of the sample size (n)
Applying these formulas:
1. μx = μ = 64 inches
2. σx = σ / √n = 3.1 inches / √39 ≈ 0.496
So, the mean is 64 inches and a standard deviation is approximately 0.496 inches.
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A bag of 12 ounces of popcorn costs $4.80. What is the unit rate? (in others words what is the cost for one ounce) *
Answers
Answer:
0.4
Step-by-step explanation:
$4.80 divided by 12 = 0.4
Answer:
2.50 per ounce
Step-by-step explanation:
divide 12 by 4.80
Which statements are true regarding the diagram? Check all that apply.
1. ΔXYZ ≅ ΔLMN
2. ∠Y ≅ ∠M
3. ∠X ≅ ∠L
4. ∠Z ≅ ∠L
5. YZ ≅ ML
6. XZ ≅ LN
Answers
Answer:
1, 2, 3, 6
Step-by-step explanation:
I don't really know how to explain this but if the only change between the two triangles is that it was reflected, then there was no change of any of the angles or side lengths, you just have to make sure you match up the rights angles and the right sides.
Answer:
Hi how are you doing today Jasmine
An airship propelled by some mechanical device travels five miles in ten minutes in the direction of the wind, but requires one hour to go back again to the starting point against the same wind. How long would it have taken to go the whole ten miles in a calm day without any wind? Hint: You don't have to use equations to solve this problem. You can solve it perfectly using math reasoning.
Answers
An airship propelled by some mechanical device travels five miles in ten minutes in the direction of the wind, but requires one hour to go back again to the starting point against the same wind. How long would it have taken to go the whole ten miles in a calm day without any wind? Hint: You don't have to use equations to solve this problem.
we have that
5 miles -------> 10 minutes (direction of the wind)
5 miles ------> 1 hour (against the same wind)
therefore
10 miles ------> ? (without any wind)
so
Let
x -----> speed of the wind
y -----> speed of the airship
direction of the wind
speed=d/t ------> 5/10=0.5 miles per min
against the same wind
speed=5/60=1/12 miles per min
so
x+y=0.5 ------> x=0.5-y -----> equation 1
y-x=1/12 -----> equation 2
solve the system
substitute equation 1 in equation 2
y-(0.5-y)=1/12
2y=(1/12)+1/2
2y=7/12
y=7/24=0.2917 miles per min
Find the value of x
x=(1/2)-7/24
x=5/24=0.2083 miles per min
therefore
10 miles without any wind
speed=d/t ------> t=d/speed
t=10/(7/24)
t=34.29 minno files please!!!!!!!!!!!
Answers
Answer:
(-1/2, 3)
Explaination:
Multiply the second equation by 2, then add the equations together.
(4x-y=-5)
2(-2x+3y=10)
Becomes:
4x-y=-5
-4x+6y=20
Then add the equations to eliminate x.
5y=15
then solve 5y-15 for y. (Divide both sides by 5)
5y/5 = 15/5
y= 3
Okay, now we have found y where going to plug it in an equation to solve for x
Going to chosse equation : 4x-y=-5
Subsitue 3 for y in 4x - y = - 5
4x - 3 = -5
( Add 3 to both sides)
4x - 3 + 3 = - 5 + 3
4x = -2
(Divide both sides by 4 )
4x/4 = -2/4
x= -1/2
Divide the following. Put
your answer in scientific
notation.
(3.4 X 10^-1) / (4)
Answers
Answer:
8.5 × 10-2
Step-by-step explanation:
I believe this is it
Mia heard that her favorite band was coming to the school. She immediately told her 3 best friends. Each person that found out was also excited and went and told 3 more people. This continued for the whole lunch period. 
Answers
First iteration: Mia and her 3 friends tell 3 people each, so now 1 + 3 x 3 = 10 people know.
Second iteration: Those 9 new people tell 3 people each, so now 10 + 9 x 3 = 37 people know.
Third iteration: Those 27 new people tell 3 people each, so now 37 + 27 x 3 = 118 people know.
Fourth iteration: Those 81 new people tell 3 people each, so now 118 + 81 x 3 = 361 people know.
And so on. As you can see, the number of people who know about the band grows very quickly, thanks to the exponential growth.
Suppose 4,700 was invested with an 5. 3 interest rate how much will be in account after 8 years
Answers
After 8 years, the amount in the account will be approximately $7,228.62.
To calculate the future value of an investment, we can use the formula for compound interest:
Future Value = Principal * (1 + Interest Rate)^Time
In this case, the principal (initial investment) is $4,700, the interest rate is 5.3% (or 0.053 as a decimal), and the time is 8 years. Plugging these values into the formula:
Future Value = 4700 * (1 + 0.053)^8
Calculating the future value:
Future Value = 4700 * (1.053)^8 ≈ $7,228.62
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To find an estimate for π I could...
Answers
Answer:
You could use 3.14 or 22/7
Step-by-step explanation:
It should be what you're asking, if not, tell me and I'll give you another answer.
:)
Answer:
use 22/7 or 355/133, but 355/133 is not very commonly used and is unwieldy.
Step-by-step explanation:
help this is worth a lot of points
Answers
Answer:
ok
Step-by-step explanation:
theresa bought a new desktop computer. one side of the desktop screen is 14 inches and the other side is 18 inches. what is the length of the diagonal of the desktop screen? answer choices are rounded to the nearest inch.
Answers
Rounded to the nearest inch, the length of the diagonal of the desktop screen is approximately 23 inches.
To find the length of the diagonal of the desktop screen, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Given:
One side of the desktop screen = 14 inches
The other side of the desktop screen = 18 inches
Let's denote the length of the diagonal as d.
Using the Pythagorean theorem, we have:
d² = 14² + 18²
d² = 196 + 324
d² = 520
Taking the square root of both sides to solve for d:
d ≈ √520
d ≈ 22.803
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Find the area of each regular polygon. Leave answer in simplest form.
Answers
The areas of the regular polygons are listed below:
Case 8: A = 166.277
Case 10: A = 166.277
Case 12: A = 779.423
Case 14: A = 905.285
Case 16: A = 678.964
Case 18: A = 332.554
Case 20: A = 1122.369
Case 22: A = 166.277
How to determine the area of a regular polygon
In this problem we must determine the areas of eight regular polygons, whose formula is now shown below:
A = 0.5 · (n · l · a)
a = 0.5 · l / tan (180 / n)
Where:
a - Apothemal - Side lengthn - Number sidesNow we proceed to determine the area of each polygon:
Case 8:
l = 2 · a · tan (180 / n)
l = 2 · 4√3 · tan 30°
l = 8√3 · (√3 / 3)
l = 8
A = 0.5 · (n · l · a)
A = 0.5 · 6 · 8 · 4√3
A = 166.277
Case 10:
a = 0.5 · l / tan (180 / n)
a = 0.5 · 8 / tan 30°
a = 4 / (√3 / 3)
a = 4√3
A = 0.5 · (n · l · a)
A = 0.5 · 6 · 8 · 4√3
A = 166.277
Case 12:
a = 0.5 · l / tan (180 / n)
a = 0.5 · 10√3 / tan 30°
a = 5√3 / (√3 / 3)
a = 15
A = 0.5 · (n · l · a)
A = 0.5 · 6 · 10√3 · 15
A = 779.423
Case 14:
l = 2 · a · tan (180 / n)
l = 2 · (28√3 / 3) · tan 30°
l = (56√3 / 3) · (√3 / 3)
l = (56 · 3 / 9)
l = 56 / 3
A = 0.5 · (n · l · a)
A = 0.5 · [6 · (56 / 3) · (28√3 / 3)]
A = 905.285
Case 16:
l = 2 · a · tan (180 / n)
l = 2 · 14 · tan 30°
l = 28 · √3 / 3
l = 28√3 / 3
A = 0.5 · (n · l · a)
A = 0.5 · 6 · (28√3 / 3) · 14
A = 678.964
Case 18:
l = 2 · a · tan (180 / n)
l = 2 · 8 · tan 60°
l = 16√3
A = 0.5 · (n · l · a)
A = 0.5 · 3 · 16√3 · 8
A = 332.554
Case 20:
a = 0.5 · l / tan (180 / n)
a = 0.5 · 12√3 / tan 30°
a = 6√3 / (√3 / 3)
a = 18
A = 0.5 · (n · l · a)
A = 0.5 · 6 · 12√3 · 18
A = 1122.369
Case 22:
a = 0.5 · l / tan (180 / n)
a = 0.5 · 8 / tan 30°
a = 4 / (√3 / 3)
a = 4√3
A = 0.5 · (n · l · a)
A = 0.5 · 6 · 8 · 4√3
A = 166.277
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Select the equivalent expression.
Umm so yeah... happy pride month royals!
Answers
Answer:
Step-by-step explanation:
The first answer is correct because you are really not changing anything about the question
The function f is defined by f(x)=x²-7. What is the value of f(3) ? A -16 B -2 C 2 D 16
Answers
Answer:
Option C is the answer
Are polynomials closed under addition and subtraction?
Answers
Polynomials form a system like to that of integers hence they are closed under the operations of addition and subtraction.
Exponents of polynomials are whole numbers.
Hence the resultant exponents will be whole numbers , addition is closed for whole numbers. As a result, polynomials are closed under addition.
If an operation results in the production of another polynomial, the resulting polynomials will be closed.
The outcome of subtracting two polynomials is a polynomial. They are also closed under subtraction as a result.
The word polynomial is a Greek word. We can refer to a polynomial as having many terms because poly means many and nominal means terms. This article will teach us about polynomial expressions, polynomial types, polynomial degrees, and polynomial properties.
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!!Urgent!! Pls help!! Parallel lines p and q are cut by two non-parallel lines, m and n, as shown in the figure.
The value of x is -----
degrees, and the value of y is ----
degrees.
Answers
x = 180 -102
x = 78
y = 115
A sales person is given a choice of two salary plans. plan 1 is a weekly salary of 500 plus 3% commission of sales. plan 2 is a straight commission of 5% of sales. how much in sales must she make in a week for both plans to result in the same salary
Answers
Let's write and equation for both cases.
In plan 1, it is 500 plus 3% comission. 3% is equivalent to multiply the amount it sale in the week, "x", by 0.03.
So, the equation for plan 1 is:
\(p_1=0.03x+500\)For plan 2, there is no fixed value, only comission of 5%, which is equivalent of multiplying the weekly sales, "x", by 0.05, so the equation is:
\(p_2=0.05x\)For both plans to result in the same salary, we have:
\(\begin{gathered} p_1=p_2 \\ 0.03x+500=0.05x \\ 0.05x-0.03x=500 \\ 0.02x=500 \\ x=\frac{500}{0.02} \\ x=25,000 \end{gathered}\)So, for both plans to result in the same salary, she have to sale 25,000 in a week.
a student has a class that is supposed to end at 9:00am and another that is supposed to begin at 9:15am. suppose the actual ending time of the 9am class is normally distributed random variable (x1) with a mean of 9:02 and a standard deviation of 2.5 minutes and that the starting time of the next class is also a normally distributed random variable (x2) with a mean of 9:15 and a standard deviation of 3 minutes. suppose also that the time necessary to get from one class to another is also a normally distributed random variable (x3) with a mean of 10 minutes and a standard deviation of 2.5 minutes. what is the probability that the student makes it to the second class before the second lecture starts? (hint: assume x1, x2 and x3 are independent also think linear combinations)
Answers
The probability that the student makes it to the second class before it starts is very close to 0.
To find the probability that the student makes it to the second class before it starts, we can use the concept of linear combinations of random variables and the properties of normal distributions.
Let's define the random variable X as the total time it takes for the student to transition from the end of the first class to the start of the second class. Since X is a linear combination of independent normally distributed random variables (X1, X2, X3), we can use their means and variances to calculate the mean and variance of X.
The mean of X is the sum of the means of X1, X2, and X3:
μX = μ1 + μ2 + μ3 = 9:02 + 9:15 + 10 = 28:17 minutes.
The variance of X is the sum of the variances of X1, X2, and X3:
σX^2 = σ1^2 + σ2^2 + σ3^2 = (2.5)^2 + (3)^2 + (2.5)^2 = 15.25 minutes^2.
Now, we need to calculate the probability that X is less than or equal to 0, meaning the student arrives before the second lecture starts. Since X follows a normal distribution, we can standardize the variable and calculate the probability using the standard normal distribution table.
Z = (0 - μX) / σX = (0 - 28:17) / √15.25 ≈ -9.43.
Using the standard normal distribution table or a calculator, we can find the probability corresponding to Z = -9.43. The probability is essentially 0, as the value is significantly far in the left tail of the standard normal distribution.
Therefore, the probability that the student makes it to the second class before it starts is very close to 0.
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Write a word problem that would have a proof drawing with no new tens and no new hundreds. Explain how you chose your numbers. Solve using a numerical method and proof drawing
Answers
Word problems: Since they encourage students to apply their knowledge of many topics to "real-life" situations, word problems are a crucial component of the core modules.
Additionally, they help pupils get familiar with terms like "more," "fewer," "subtract," "difference," "totally," "equal," "sharing," "multiplied," and "reduced."
A word problem is a description of a situation in real life when a computation is necessary to solve a problem.
It is important for students to learn how to answer word problems because it allows them to apply mathematical ideas to a variety of real-life situations.
Students must therefore be conversant with the vocabulary associated with the mathematical symbols they are used to in order to comprehend the word problem. In this tutorial, let's learn more about word problems in depth.
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Write the first trigonometric function in terms of the second for\thetain the given quadrant. sec\theta, sin\theta;\thetain Quadrant III sec\theta=
Write the first trigonometric function in terms of the second for\thetain the given quadrant. sec\thetasin\theta;\thetain Quadrant I sec\theta=
Answers
The first trigonometric function in terms of the second for θ in Quadrant III is secθ = -1/cosθ.
In Quadrant III, the values of secθ and sinθ are both negative. To find the first trigonometric function in terms of the second, we need to express secθ in terms of sinθ.
Let's start by recalling the definitions of secθ and sinθ. The secant function (secθ) is defined as the reciprocal of the cosine function (cosθ), while the sine function (sinθ) represents the ratio of the length of the side opposite to the angle θ to the length of the hypotenuse in a right triangle.
Since we are dealing with Quadrant III, the x-coordinate is negative, and the y-coordinate is positive. Therefore, the cosine of θ (cosθ) is negative, while the sine of θ (sinθ) is positive.
To find secθ in terms of sinθ, we can use the reciprocal property of secant. The reciprocal of cosθ is 1/cosθ. However, in Quadrant III, we have a negative cosine, so we can express secθ as -1/cosθ. This negative sign indicates the change in sign due to the quadrant.
Step 3:
In Quadrant III, where both secθ and sinθ are negative, we can express secθ in terms of sinθ using the formula secθ = -1/cosθ. This means that the secant function of an angle in Quadrant III is equal to the negative reciprocal of the cosine function.
To understand why this is the case, we need to consider the properties of trigonometric functions in Quadrant III. In this quadrant, the x-coordinate is negative, and the y-coordinate is positive. As a result, the cosine function (cosθ) is negative, indicating that the angle θ lies on the left side of the unit circle.
By definition, secθ is the reciprocal of cosθ. However, in Quadrant III, where cosθ is negative, taking the reciprocal alone wouldn't suffice. To account for the change in sign, we introduce a negative sign in front, resulting in secθ = -1/cosθ.
This negative sign ensures that the secant function aligns with the quadrant and reflects the change in the sign of the x-coordinate. Thus, we can express secθ in terms of sinθ as -1/cosθ in Quadrant III.
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Elle placed a bucket under a leaky roof. The bucket was 3/10 full after 2/5 of an hour. How much water is leaking per hour?
Answers
Answer:
The amount of water leaking per hour is 3/4 of the bucket volume
Step-by-step explanation:
The given parameters are;
The volume of water in the bucket = 3/10 of the bucket volume
The time duration of the leak = 2/5 hour
Therefore;
The leak rate = Volume/Time
The amount of water leaking per hour = (3/10 of the bucket volume)/(2/5 hour)
The leak rate = The amount of water leaking per hour = 3/4 of the bucket volume/hour.
The amount of water leaking per hour is \(\frac{3}{4}\) of the bucket volume.
Given ;
The volume of water in the bucket = \(\frac{3}{10}\) of the bucket volume
The time duration of the leak = \(\frac{2}{5}\) hour
Therefore;
The leak rate = \(\frac{volume }{time}\)
The amount of water leaking per hour = ( \(\frac{3}{10}\) 0f the bucket volume)/(\(\frac{2}{5}\) hour)
= ( \(\frac{3}{10} . \frac{5}{2}\) ) volume per hour
The leak rate = The amount of water leaking per hour = 3/4 of the bucket volume per hour.
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find an open cover of the interval [0, 10) with no finite subcover.
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The question asks to find an open cover of the interval [0, 10] with no finite subcover. Let us consider the following open cover of the interval [0, 10]: { (−1,1),(0,2),(1,3),(2,4),(3,5),…,(8,10),(9,11),(10,12) }. This open cover of the interval [0, 10] has no finite subcover.
In other words, it is not possible to choose finitely many sets from this open cover so that their union still covers the interval [0, 10).The reason behind this is that each of the sets in the above open cover contains an interval of length 1, except for the set (1,1). It follows that if one chooses finitely many sets from the above open cover, the union of those sets will have a length less than 10 and hence will not cover the entire interval [0, 10]. To explain the above solution in more detail, an open cover of a set is a collection of open sets whose union contains the set. In this case, we are looking for an open cover of the interval [0, 10] that has no finite subcover. A finite subcover of an open cover is a collection of finitely many sets from the open cover whose union still contains the set. The fact that the open cover we have given has no finite subcover means that it is not possible to choose finitely many sets from this open cover so that their union still covers the interval [0, 10).The open cover we have given consists of sets of the form (n,n+2) for n = 1,0,1,...,9,10. Notice that each of these sets contains an interval of length 1, except for the set (1,1), which has length 2. It is easy to see that any finite subcover of this open cover can only cover an interval of length less than 10, since there are only finitely many sets in the subcover. Hence, the only way to cover the entire interval [0, 10] is to take all the sets from the open cover, which is not a finite subcover. Therefore, the open cover we have given has no finite subcover.
Thus, we have found an open cover of the interval [0, 10] with no finite subcover. The open cover we have given consists of sets of the form (n,n+2) for n = 1,0,1,...,9,10. We have shown that any finite subcover of this open cover can only cover an interval of length less than 10, and hence the open cover has no finite subcover.
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