We have proved that p² is greater than or equal to 4πa for any closed curve in the plane that does not self-intersect and has total length p and area a.
To prove that p² ≥ 4πa for a closed curve that does not self-intersect and has total length p and area a, we can use the isoperimetric inequality.
The isoperimetric inequality states that for any simple closed curve in the plane, the ratio of its perimeter to its enclosed area is at least as great as the ratio for a circle of the same area.
That is:
p / a ≥ 2π
Multiplying both sides by a, we get:
p² / a ≥ 2πa
Since a is positive and the left-hand side is non-negative, we can multiply both sides by 4π to obtain:
4πa(p² / a) ≥ 8π²a²
Simplifying, we get:
p² ≥ 4πa.
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Based on the isoperimetric inequality, we have successfully proven that for a closed, non-self-intersecting curve with perimeter p and area a, the inequality p² ≥ 4πa holds true.
To prove that p² ≥ 4πa for a closed, non-self-intersecting curve with perimeter p and area a, we will use the isoperimetric inequality.
Step 1: Understand the isoperimetric inequality
The isoperimetric inequality states that for any closed curve with a given perimeter, the maximum possible area it can enclose is achieved by a circle. Mathematically, it is given as A ≤ (P² / 4π), where A is the area enclosed by the curve and P is the perimeter.
Step 2: Apply the inequality to the given curve
For our closed curve with perimeter p and area a, we have a ≤ (p² / 4π) according to the isoperimetric inequality.
Step 3: Rearrange the inequality
To prove that p² ≥ 4πa, we simply need to rearrange the inequality from step 2. Multiply both sides by 4π to obtain 4πa ≤ p².
Step 4: Conclusion
Based on the isoperimetric inequality, we have successfully proven that for a closed, non-self-intersecting curve with perimeter p and area a, the inequality p² ≥ 4πa holds true.
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give the volume of the triangular prism shown
A-440in^3
B-528in^3
C-880in^3
D-264in^3
Answer:
D. 264 in³
Step-by-step explanation:
The missing edge dimension can be found using the Pythagorean theorem. Then the volume can be found using the appropriate formula.
Edge dimensionThe missing dimension and the two given for the top triangle will satisfy the Pythagorean theorem. If the missing dimension is represented by 'a', then we have ...
a² +8² = 10²
a = √(100 -64) = 6 . . . . inches
VolumeThe area of the top triangle is ...
A = 1/2bh . . . . where b and h are the base and height dimensions
A = 1/2(6 in)(8 in) = 24 in²
The volume of the triangular prism is ...
A = Bh . . . . . where B is the area of the triangular base, and h is the prism height
A = (24 in²)(11 in) = 264 in³
The volume of the triangular prism shown is 264 in³.
What does the following expression mean? x <= y
As per inequality, it means that the value of x can be less than or equal to y.
Inequality in Mathematics
The two expressions that make up an equation or an inequality are connected in a mathematical statement. The equal sign (=) in an equation denotes the equality of the two expressions. The symbols >, <, ≤, or ≥ are used to denote an inequality, when the two expressions aren't always equal.
Type of Inequality
The given expression, x <= y denotes the type of inequality where the possible values of x are either less than y or equal to less. The value of x cannot be greater than y in any case.
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You have one type of chocolate that sells for $1.50/lb and another type of chocolate that sells for $6.50/lb. You would like to have mix the $1.50/lb type with 10 lbs of the $6.50/lb type to make a chocolate mixture that sells for $2.50/lb. How much of the $1.50/lb type will you need to obtain the desired mixture
14 of the $1.50/lb type will you need to obtain the desired mixture.
According to the statement
Chocolate A that sells for $1.50/lb
Chocolate B that sells for $6.50/lb
Then in equation form the quantity of mixture will
A + B = X
then
$1.50A + $6.50(10) = X
1.50A + 65 = X -(1)
after solving the equation the value of a and b will become
A = 14.
So, 14 of the $1.50/lb type will you need to obtain the desired mixture.
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TIME REMAINING
43:25
Triangles X Y Z and X prime Y prime Z prime are shown.
ΔXYZ was reflected over a vertical line, then dilated by a scale factor of One-half, resulting in ΔX'Y'Z'. Which must be true of the two triangles? Select three options.
△XYZ ~ △X'Y'Z'
AngleXZY ≅ AngleY'Z'X'
YX ≅ Y'X'
XZ = 2X'Z'
mAngleYXZ = 2mAngleY'X'Z'
Answer:
△XYZ ~ △X'Y'Z'AngleXZY ≅ AngleY'Z'X'XZ = 2X'Z'Step-by-step explanation:
Reflection over any line is a rigid transformation that does not change angles or lengths. Dilation is a transformation that changes lengths by the same scale factor, but has no effect on angles. Dilated figures are similar to the original.
ApplicationWhen ΔXYZ is reflected over a vertical line, no lengths or angles are changed. When the reflected triangle is dilated by half, we get ...
Y'X' = (1/2)YX . . . . . . . makes option 3 false
X'Z' = (1/2)XZ . . . . . . . makes option 4 true
Since no angles are changed, we have ...
Angle XZY ≅ Angle Y'Z'X' (another name for angle X'Z'Y')
... makes option 2 true
Of course, the resulting triangle is similar to the original, so ...
△XYZ ~ △X'Y'Z' . . . . . option 1 is true
A credit card company charges a late fee of 10% on all late payments. last month, jamal was charged a late fee. the total amount paid, including the late fee, was $275. what was the original amount on the credit card before the late fee?
Answer:
265
Step-by-step explanation:
You subtract 275 minus 10
$247.5 was the original amount on the credit card
Lets simplify the problem,
Total amount after late fee = $275
Late fee deduction = 10%
Original amount = 275 - 10% of 275
= 275 - 27.5
= $247.5
So $247.5 was the original amount on the credit card before the late fee.
Percentage:
In mathematics, a percentage is a number or ratio that represents a fraction of 100. It is often denoted by the symbol "%" or simply as "percent" or "pct." For example, 35% is equivalent to the decimal 0.35, or the fraction.
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What is a Turned triangle?
Please tell me QUICKLY.
like the place where you are applying TURNED TRIANGLE.
coz literrally it means that if the trianlge has for example angle a on its right side then we turn the triangle as a whole and make the turn such that angle a is anywhere else but not on our right side....
I believe the explanation is clear, if not do contact me
Identify which of the following is not equivalent to 2 3/4.
The figure that is not equivalent to 2¾ is 2¼ × 2½. That is option B.
Simplification of equivalent fractionsSimplification is a process that is used to make a fraction appear in its simplest forms.
The fraction given is = 2¾
The simplification of the fraction= 2^3/4 = 2
But simplification of 2¼ × 2½ is equal to;
2¼= 0.5
2½= 1
Therefore 2¾ is not equivalent to 2¼ × 2½ when simplified.
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Find the value of x in 1:x=6 1/4
Answer:
x = [tex]\frac{4}{25}[/tex]
Step-by-step explanation:
1 : x = 6 [tex]\frac{1}{4}[/tex] ← change to improper fraction
1 : x = [tex]\frac{25}{4}[/tex] , then
[tex]\frac{1}{x}[/tex] = [tex]\frac{25}{4}[/tex] ( cross- multiply )
25x = 4 ( divide both sides by 25 )
x = [tex]\frac{4}{25}[/tex]
please do this asap with the correct answer!!
The percentage of population has a resting heart rate is 15.7%.
Given that the mean of the data is 70 beats per minute and standard deviation is 4 beats per minute.
A probability distribution is a mathematical function that describes the probability of several possible values of a variable.
Mean, μ=70
Standard deviation, σ=4
Given that the heart rate is resting between 74 and 82.
Now, we will use the standard normal formula
P(x₁<x<x₂)=P((x₁-μ)/σ<x<(x₂-μ)/σ)
Here x₁=74 and x₂=82.
Substitute these values in the formula, we get
P(74<x<82)=P((74-70)/4<x<(82-70)/4)
P(74<x<82)=P(4/4<x<12/4)
P(74<x<82)=P(1<x<3)
P(74<x<82)=P(x<3)-P(x<1)
Now, we will find the values by using the z-table, we get
P(74<x<82)=0.998-0.841
P(74<x<82)=0.157
P(74<x<82)=15.7%
Hence, the percentage of the population has a resting rate between 74 and 82 with the mean data is 70 beats per minute and the standard deviation data is 4 beats per minute is 15.7%.
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Write the expanded expression in expanded form using the definition of an exponent.
2wz^2?
The expanded form of the given expression is 2 × w × z × z
Writing an expressionFrom the question, we are to determine the expanded form of the given expression
The given expression is
2wz²
To write an expression in an expanded form, we will write the expression as a repeated multiplication in the simplest way possible.
To do this, we will first expand the index in the given expression.
The index in the given expression is z²
Thus, we get
z² = z × z
Now, we will include the remaining individual terms in the expression and write them as product.
That is,
2wz² = 2 × w × z × z
Hence, the expanded form of the given expression is 2 × w × z × z
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A margin of sampling _______ is a statistical calculation for how accurate a poll's results are. quizlet
A margin of sampling "error" is a statistical calculation for how accurate a poll's results are.
What is sampling error?A population parameter's sampling error is the discrepancy between the sample statistic used to estimate it and the parameter itself. For instance, sampling error is the variance between a population mean and a sample mean.
The sampling errors can be placed into four categories:
population-specific error: The researcher makes a population-specific error when they are unsure of the appropriate population to sample. When respondents choose to participate in the study on their own, a selection error arises. (This limits responses to those who are interested, which spoils the results.)selection error: The sampling error for a sample chosen using a non-probability method is known as selection error. Selection error can result when participants opt to self-participate in a study and only those who are interested reply, as there may already be an underlying bias.sample frame error: When the incorrect sub-population is utilized to select a sample, a sample frame error happens. Last but not least, a non-response problem happens when potential responders cannot be reached or refuse to respond.non-response error: Non-response error is a mistake that happens when a survey doesn't receive a response to one or more questions.To know more about the sampling error, here
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The table represents a linear function.
A two column table with six rows. The first column, x, has the entries, negative 2, negative 1, 0, 1, 2. The second column, y, has the entries, negative 2, 1, 4, 7, 10.
What is the slope of the function?
–3
–2
3
4
The slope of given data is a -3.
According to the statement
we have given a some data and we have to find the slope of the data.
So, data given is:
x, has the entries, negative 2, negative 1, 0, 1, 2.
y, has the entries, negative 2, 1, 4, 7, 10.
and we have to find the slope of the data
So, we know that it is a data represent in the graph then
We know that a there is a graphical representation of data.
A graph is the representation of the data on the vertical and horizontal coordinates so we can see the trend of the data.
And when we observe this data then we see that the there is some increase the y axis value as compare to the x axis values.
So,
For each unit of run in the x-values, there is an increase of 3 in the y-values. The slope is rising over Run so 3/1 = 3
So, The slope of given data is a -3.
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Answer: the answer is a -3
Step-by-step explanation:
Jina needs to solve a problem that involves using a different type of units for angle measure. The problem given to her to calculate 6/5 * 240
The value of the expression 6/5 * 240 is 288
How to calculate the expression?The expression is given as:
6/5 * 240
Divide 240 by 5
6 * 48
Multiply 6 and 48
288
Hence, the value of the expression 6/5 * 240 is 288
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URGENT WHOEVER ANSWERS CORRECTLY AND FAST GETS BRAINLIEST
What is the equation that can be used to find a percent of a number?
A .part = percent + whole
B. part = percent/whole
C. part = percent x whole
D. part = whole/ percent
Answer:
C
Step-by-step explanation:
Example
Find 55% of 200.
200 times 0.55 = 110
Answer: the answer is B
Step-by-step explanation:
if we want to find the percent of 700 people out of 4400 people, we would do 700/4400, which is equal to 0.16, or 16 percent.
Rewrite the ratio 15: 39 as an equivalent ratio of the form 1 : n.
Give any decimals in your answer to 1 d.p.
Answer:
n = 2.6
Step-by-step explanation:
Hello!
We can create an equation with equivalent fractions. Remember that a ratio can also be represented as a fraction.
[tex]a:b = \frac ab[/tex]We can create two fractions that are equivalent to each other given the ratios.
[tex]\frac{15}{39} = \frac 1n[/tex]Solve for n[tex]\frac{15}{39} = \frac 1n[/tex][tex]\frac{5}{13} = \frac 1n[/tex][tex]5n = 13[/tex][tex]n = \frac{13}{5}[/tex][tex]n = 2.6[/tex]The value of n is 2.6.
Which set of numbers may represent the lengths of the sides of a triangle?
A. (2,2,6)
B. (5,12,13)
C. (5,5,10)
D. (1,8,10)
Answer:
B. (5,12,13)
Step-by-step explanation:
The only answer with two side lengths that are greater than the largest side length.
How can you use the converse of the Pythagorean Theorem
to tell if a triangle is a right triangle?
Answer:
Yes. If the squares of the 2 short sides add up to the square of the hypotenuse, the triangle contains a right angle. Hope this helped.
The mean of three numbers is 10 more than the least of the numbers and 15 less than the greatest. The median of the three numbers is 5. What is their sum
Answer:
Sum is
Step-by-step explanation:
Study the following data set.
{8,15,9,18,9,17,22,10,11,9,13}
What is the quarterfinal range of the data set?
The interquartile range of the data set is 8
How to solve for the interquartile rangeTo do this we have to arrange in ascending order
= 8,9,9,9,10,11,13,15,17,18,22
Q2 = The median of the data is 11.
We have to find the median of the half before 11 and the median of the half after 11.
We would have
8,9,9,9,10
Q1 = 9
And 13,15,17,18,22
Q3 = 17
The interquartile range is
Q3 - Q1
= 17 - 9
= 8
The interquartile range of the data set is 8
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Correct question
Study the following data set.
{8,15,9,18,9,17,22,10,11,9,13}
What is the interquartile range of the data set?
32. Which of the following would be in the solution set to the system of inequalities given below?
y< 2x - 4
2-y-3
(0,-5)
Both (0,-5) and (0, -2) are solutions
(0,-2)
No solution
Answer:
no
Step-by-step explanation:
The diameters of bolts produced in a machine shop are normally distributed with a mean of 5.34 millimeters and a standard deviation of 0.03 millimeters. Find the two diameters that separate the top 8% and the bottom 8%. These diameters could serve as limits used to identify which bolts should be rejected. Round your answer to the nearest hundredth, if necessary.
Using the normal distribution, we have that:
The diameter that separates the top 8% is of 5.38 mm.The diameter that separates the bottom 8% is of 5.30 mm.Normal Probability DistributionThe z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score measures how many standard deviations the measure is above or below the mean. Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.The mean and the standard deviation for this problem are given, respectively, by:
[tex]\mu = 5.34, \sigma = 0.03[/tex]
The 8th percentile separates the bottom 8%, that is, X when Z = -1.405, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.405 = \frac{X - 5.34}{0.03}[/tex]
X - 5.34 = -1.405 x 0.03
X = 5.30.
The 92th percentile separates the top 8%, that is, X when Z = 1.405, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.405 = \frac{X - 5.34}{0.03}[/tex]
X - 5.34 = 1.405 x 0.03
X = 5.38.
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5. You are a game manufacturer. You and your team are trying to
find the best pair of dice for a new board game you have made. You
are examining the possible results of using one 6-sided die and
one 10-sided die and rolling them together. If the random variable
is defined as the sum of the two dice when rolled, which of the following
values are included in the random variable?
0
16
1
14
10
17
All of the above
The values that are included in the random variable of the experiment that include rolling one 6-sided die and one 10-sided die are 10, 14 and 16
How to determine the values in the random variable?The experiment elements are given as:
One 6-sided dieOne 10-sided dieThe sample spaces of both dice are:
6-sided = {1, 2, 3...., 6}
10-sided = {1, 2, 3...., 10}
When the outcomes of both dice are added, we have:
Sample space = {2, 3, 4, 5, 6, 7, 8, 9, .............. 16}
Minimum = 2
Maximum = 16
This means that the values that are included in the random variable are in the range 2 - 16 i.e. starting from 2 and ending in 16
Hence, the values that are included in the random variable of using one 6-sided die and one 10-sided die and rolling them together are 10, 14 and 16
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Use Newton's method to approximate the zero of the function f(x)=x^3-4x+33 to five decimal places. Round any intermediate calculations, if needed, to no less
than six decimal places, and round your final answer to five decimal places.
If B P ⊥ P C , what is m B C ^ ? 180 90 60
The measure of angle BC (m ∠BC) is 90°
Perpendicular linesFrom the question, we are to determine the measure of angle BC
From the given information, we have that
B P ⊥ P C
This means line BP is perpendicular to line PC
The angle between two perpendicular lines is 90°
Hence, the measure of angle BC (m ∠BC) is 90°.
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which statement best compares the rate of change of the two functions?
Answer:
For every x value, the y value would be double of the x value. (ex, x=2, y=4)
Step-by-step explanation:
If you look closely at every x value and see their y-values for the ordered pair, you will see that the x-value in question was doubled in order to find the y-value. This will result in the linear equation: [tex]y=2x[/tex].
Answer:
6288388288288828ffjfjjrjir
help please its my last one!
Using compound interest, it is found that they must deposit $9,143.47 in order to have the desired amount.
What is compound interest?The amount of money earned, in compound interest, after t years, is given by:
[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]
In which:
A(t) is the amount of money after t years. P is the principal(the initial sum of money). r is the interest rate(as a decimal value). n is the number of times that interest is compounded per year.For this problem, the parameters are:
A(15) = 30000, r = 0.08, n = 4
Then we solve for P to find the initial deposit:
[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]
[tex]30000 = P\left(1 + \frac{0.08}{4}\right)^{4 \times 15}[/tex]
[tex]P = \frac{30000}{(1.02)^{60}}[/tex]
P = $9,143.47.
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The art club had an election to select a president nine out of the 12 members of the our club voted in the election what percentage of the members voted
Answer:
75%
Step-by-step explanation:
9 divided by 12 is 0.75, so 9 is 75 percent of 12.
I hope this helps, have a good day!
How many times does the quadratic function below the intersect the x-axis?
What is an appropriate display to show the spread of 150 test scores for Mrs. Hansen's last science test?
(ignore my answer, i didn't mean to click it)
The appropriate display to show the spread of 150 test scores for Mrs. Hansen's last science test is: b. box and whisker plot.
What is box and whisker plot?Box and whisker plot which is also called box plot can be defined as the plot that show a graphical presentation of data set by helping to display the data set or data score .
Hence a graphical presentation display to show the spread of 150 test scores for Mrs. Hansen's last science test is option B which is Box and whisker plot .
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Write a function of the geometric sequence with a starting term of 8 and a common ratio of 2. Find the fourth term.
The expression that represents well the geometric sequence is [tex]f(n) = 8 \cdot 2^{n-1}[/tex] and the value of the fourth term is 64. (Correct choice: B)
How to analyze geometric sequencesGeometric sequences are exponential expressions with discrete domain, whose form is presented and explained below:
[tex]f(n) = a \cdot r^{n-1}[/tex], where a, r are real numbers and n is a natural number. (1)
Where:
Value of the starting term.Common ratio of the series.According to the statement, we know that the first term of the geometric sequence is 8 and between any two consecutive terms there is a common ratio of 2. If we know that a = 8, r = 2 and n = 4, then the fourth term of the series by means of (1) is:
[tex]f(4) = 8 \cdot 2^{4-1}[/tex]
f(4) = 8 · 8
f(4) = 64
The expression that represents the geometric sequence is [tex]f(n) = 8 \cdot 2^{n-1}[/tex] and the value of the fourth term is 64.
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