You are an scalene triangle.
How can you identify the type of triangle when given the information that it has no congruent sides and one angle measuring 100 degrees?You are a scalene triangle.
A scalene triangle is a type of triangle where all three sides have different lengths, and no two angles are congruent. In this case, you mentioned that one of the angles has a measure of 100 degrees.
Here's a simple diagram of a scalene triangle to help illustrate:
\
\
\
\
\
\
In the diagram, the angles are not drawn to scale, but it represents a scalene triangle where one angle measures 100 degrees. The sides of the triangle would have different lengths, distinguishing it from an equilateral or isosceles triangle where at least two sides are congruent.
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4. Let A be a 3 x 4 matrix and B be a 4 x 5 matrix such that ABx = 0 for all x € R5. a. Show that R(B) C N(A) and deduce that rank(B) < null(A) b. Use the Rank-Nullity theorem to prove that rank(A) + rank(B) < 4.
a. To show that R(B) is a subset of N(A), let y be any vector in R(B):
This means that there exists a vector x in R4 such that Bx = y.
Now, since ABx = 0 for all x in R5, we can write:
A(Bx) = 0
But we know that Bx = y, so we have:
Ay = 0
This shows that y is in N(A), and therefore R(B) is a subset of N(A).
To deduce that rank(B) is less than null(A), recall that by the Rank-Nullity theorem, we have:
rank(B) + null(B) = dim(R5) = 5
rank(A) + null(A) = dim(R4) = 4
Since R(B) is a subset of N(A), we have null(A) >= rank(B).
Therefore, using the above equations, we get:
rank(B) + null(A) <= null(B) + null(A) = 5
which implies:
rank(B) <= 5 - null(A) = 5 - (4 - rank(A)) = 1 + rank(A)
This shows that rank(B) is less than or equal to 1 plus the rank of A.
Since the rank of A can be at most 3 (since A is a 3 x 4 matrix),
we conclude that:
rank(B) < null(A)
b. To use the Rank-Nullity theorem to prove that rank(A) + rank(B) < 4
We simply add the equations:
rank(A) + null(A) = 4
rank(B) + null(B) = 5
to get:
rank(A) + rank(B) + null(A) + null(B) = 9
But since R(B) is a subset of N(A), we know that null(A) >= rank(B), and therefore:
rank(A) + rank(B) + 2null(A) <= 9
Using the first equation above, we can write null(A) = 4 - rank(A), so we get:
rank(A) + rank(B) + 2(4 - rank(A)) <= 9
which simplifies to:
rank(A) + rank(B) <= 1
Since rank(A) is at most 3,
we conclude that:
rank(A) + rank(B) < 4
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Calculate the expected count for women who suffer from depression and drink between 2-6 cups of caffeinated coffee per week under the null hypothesis. calculate the contribution of the cell for women who suffer from depression and drink between 2-6 cups of caffeinated coffee per week to the chi-square test statistic.
The cell for women who suffer from depression and drink between 2-6 cups of caffeinated coffee per week contributes 4 to the overall chi-square test statistic.
To calculate the expected count for women who suffer from depression and drink between 2-6 cups of caffeinated coffee per week under the null hypothesis, you would first need to determine the overall proportion of women who suffer from depression and drink between 2-6 cups of caffeinated coffee per week in the sample. Let's say this proportion is 0.20.
Next, you would multiply this proportion by the total sample size to get the expected count. Let's say the total sample size is 500 women. The expected count for women who suffer from depression and drink between 2-6 cups of caffeinated coffee per week would be:
Expected count = 0.20 x 500 = 100
To calculate the contribution of the cell for women who suffer from depression and drink between 2-6 cups of caffeinated coffee per week to the chi-square test statistic, you would need to subtract the expected count from the observed count for this cell, square the difference, and divide by the expected count. Let's say the observed count for this cell is 80.
Contribution to chi-square = (80 - 100)^2 / 100 = 4
This means that the cell for women who suffer from depression and drink between 2-6 cups of caffeinated coffee per week contributes 4 to the overall chi-square test statistic.
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Help with the question in photo please
Answer:
AB = 15
Step-by-step explanation:
6(6 + x + 6) = 7(7 + 11)
72 + 6x = 126
6x = 126 - 72 = 54
x = 54/6
= 9.
So AB = 9 + 6 = 15.
James weighs 8712 pounds. he has 2 dogs that each weigh 1314 pounds. how many more pounds does james weigh than both of his dogs combined?
James weighs 6084 more pounds than both of his dogs combined.
To find out how many more pounds James weighs than both of his dogs combined, we first need to calculate the total weight of the dogs. Since he has two dogs that weigh 1314 pounds each, we can find the total weight of the dogs by multiplying 1314 by 2, which gives us 2628 pounds.
Next, we can add the weight of both dogs together to get the total weight of the dogs, which is 2628 pounds. We can then subtract the weight of the dogs (2628 pounds) from James' weight (8712 pounds) to find out how many more pounds James weighs than both of his dogs combined.
Therefore, James weighs 6084 more pounds than both of his dogs combined. This can be calculated by subtracting the weight of the dogs (2628 pounds) from James' weight (8712 pounds), which gives us 6084 pounds.
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The value of your stock investment decreased by 23% after a stock market crash. What percentage increase in value would the stocks have to rise in order to return to the value they were before the stock market crash? Round your answer to the nearest tenth of a percent
The stocks would need to increase in value by 23% to return to their original value. Rounding to the nearest tenth of a percent, the answer is 23.0%.
Let x be the percentage increase in the value of the stocks needed to return to their original value. Since the value of the stocks decreased by 23%, the new value of the stocks is 100% - 23% = 77% of the original value.
Therefore, we can set up the equation:
(100% + x%) = (77%)*(100%)
Simplifying this equation, we get:
100% + x% = 77%
x% = 77% - 100%
x% = -23%
Since we want to find the percentage increase, we need to take the absolute value of -23%, which is 23%.
Therefore, the stocks would need to increase in value by 23% to return to their original value. Rounding to the nearest tenth of a percent, the answer is 23.0%.
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Frank wants to find the area enclosed by the figure at the right the figure on each side has semicircles of a 40 meter by 40 meter square. Find the area by the figure use 3. 14 for pi
The area enclosed by the figure at the right is 2856 square meters.
To find the area enclosed by the figure at the right, we first need to find the area of the square and the two semicircles on each side. The square has sides of 40 meters, so its area is 40 x 40 = 1600 square meters. Each semicircle has a radius of 20 meters (half the side length of the square), so its area is 1/2 x pi x r^2 = 1/2 x 3.14 x 20^2 = 628 square meters. Since there are two semicircles on each side, the total area of all four semicircles is 2 x 628 = 1256 square meters.
To find the total area enclosed by the figure, we add the area of the square and the area of the four semicircles:
1600 + 1256 = 2856 square meters
Therefore, the area enclosed by the figure at the right is 2856 square meters.
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Problem 68 An outdoor adventure club has 35 members, who are all either rock climbers or skiers. There are 3 more rock climbers than
there are skiers. How many climbers are there?
WHOEVER ANSWERS FIRST GETS BRAINLIEST AND ANSWER FAST FOR WHOEVER GOTS THE HIGHEST BRAILIEST I WILL GIVE THEM THE BRAINLIEST POINT
Answer: 14.5 climbers technically 14 because 1/2 a person is wrong
Step-by-step explanation:
If r is the annual interest rate of the bank account, then at the end of the year the balance in the account is multiplied by a growth factor of x = 1 + r.
Answer:
Step-by-step explanation:
Yes , that is correct. The growth factor for the balance in the account at the end of the year is calculated by adding 1 to the annual interest rate, which is expressed as a decimal, and multiplying the result by the balance. This gives the formula:
Ending balance = Beginning balance * (1 + r)
where r is the annual interest rate as a decimal.
Prove the following two properties of the Huffman encoding scheme. (a) If some character occurs with frequency more than 2=5, then there is guaranteed to be a codeword of length 1. (b) If all characters occur with frequency less than 1=3, then there is guaranteed to be no codeword of length 1
(a) Since the character occurs with a frequency higher than 2=5, it is guaranteed to be merged with another character with the same frequency or a lower frequency. Therefore, there is guaranteed to be a codeword of length 1 for this character. (b) Since all symbols occur with a frequency less than 1=3, the root of the tree will have a frequency less than 1=3. Therefore, there is guaranteed to be no codeword of length 1 for any symbol in this case.
(a) Let's assume that some character occurs with frequency more than 2=5.
The two nodes with the lowest frequency are merged into a single node, with the sum of their frequencies as the frequency of the new node.
This process is repeated until all the nodes are merged into a single node, which becomes the root of the tree.
Since the character occurs with a frequency higher than 2=5, it is guaranteed to be merged with another character with the same frequency or a lower frequency.
Therefore, there is guaranteed to be a codeword of length 1 for this character.
(b) Let's assume that all characters occur with frequency less than 1=3.
Consider the binary tree created by the Huffman algorithm. The root of the tree corresponds to the least frequent symbol, which will be assigned the longest codeword.
Since all symbols occur with a frequency less than 1=3, the root of the tree will have a frequency less than 1=3.
This means that the corresponding codeword for the root will be longer than 1 bit. Therefore, there is guaranteed to be no codeword of length 1 for any symbol in this case.
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PART 2:
The regular price, in dollars, the gym charges can be represented by the equation y=15x+20
B.How much money, in dollars, does justin save the first month by joining the gym at the discounted price rather than at the regular price?
The amount of money Justin saves in the first month would be 5 times the value of x, where x represents the number of months of gym membership, based on the discounted price provided.
What is the linear equation?A linear equation is an equation in mathematics that represents a relationship between two variables that is a straight line when graphed on a coordinate plane. It is an equation of the form:
y = mx + b
To calculate the amount of money Justin saves in the first month by joining the gym at the discounted price rather than the regular price, we need to know the discounted price.
The equation given is y = 15x + 20, where y represents the regular price in dollars and x represents the number of months of gym membership. However, we need to know the discounted price, which is not provided in the given information.
Once we have the discounted price, we can substitute it into the equation and calculate the savings. For example, if the discounted price is y = 10x + 20, then we can calculate the savings by subtracting the discounted price from the regular price:
Savings = Regular price - Discounted price
= (15x + 20) - (10x + 20)
= 15x - 10x
= 5x
Hence, the amount of money Justin saves in the first month would be 5 times the value of x, where x represents the number of months of gym membership, based on the discounted price provided.
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Why do Markets behave in the same ways as Individual Consumers?
Answer:
Markets behave the same way as individual customers because markets are made up of individual consumers.
Step-by-step explanation:
Analyze the diagram below and answer the question that follows.
P
20
10
gg
70
110
A. ZVOU and ZUOS
B. ZROS and ZTOS
C. ZNOP and ZROS
D. ZNOP and ZPOQ
R
80
IN
Image by Scientif38
Name two angles with identical measures.
S
10 110 120
130
ΤΑ
140 150 160 170
30
10
U
By observing the given protractor we know that option (C) is correct which says ∠NOP = ∠ROS.
What is a protractor?An instrument for measuring angles is a protractor, which is often made of transparent plastic or glass.
Protractors might be straightforward half-discs or complete circles. Protractors with more complex features, like the bevel protractor, include one or two swinging arms that can be used to measure angles.
To draw arcs or circles, use a compass.
To measure angles, one uses a protractor.
So, we need to observe the given image of the protractor:
We will easily find that ∠NOP = ∠ROS
Therefore, by observing the given protractor we know that option (C) is correct which says ∠NOP = ∠ROS.
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Pls help. And please actually answer the question
Answer: y= |x-1| -1
Step-by-step explanation:
Explanation:
This is an absolute value function.
The parent function looks like: y=|x| When transformations take place, we use this function to describe those transformations:
y = a|x-h|+k
where (h, k) is your new vertex
"a" is your stretch
and a - would be placed in the front if there was a reflection
Solution:
Your equation has a new vertex at (1, -1)
There is no stretch and no reflection
Plug into equation:
y= |x-1| -1
2.
Graham wants to take snowboarding lessons at a nearby ski resort that charges $40 per week.
The resort also charges a one-time equipment-rental fee of $99 for uninterrupted enrollment in
classes. The resort requires that learners pay for three weeks of classes at a time.
The function f(x) represents the situation
f(x) =
40x +99
x
Select two choices that are true about the function f(x).
A There is a zero at 0.
B There is an asymptote at y = 40.
C
There is an asymptote at x = 0.
D There is a vertical shift up 99 units.
Options B & C
There is an asymptote at y = 40.
There is an asymptote at x = 0.
Step-by-step explanation:Main concepts:
Concept 1. Zeros of rational functions
Concept 2. Vertical asymptotes of rational functions
Concept 3. Horizontal asymptotes of rational functions
Concept 4. Transformations - vertical shift
Concept 1. Zeros of rational functionsRational functions have zeros at values of x where the numerator is zero, while the denominator is not also zero.
[tex]f(x)=\dfrac{40x+99}{x}[/tex]
Values that make the denominator zero
Note that for the function f(x), x=0 is the only value that will make the denominator zero.
Values that make the numerator zero
To find the value that makes the numerator zero, set the numerator equal to zero, and solve for the value of x that makes it true:
[tex]40x+99=0[/tex]
[tex]40x=-99[/tex]
[tex]x=\frac{-99}{40}[/tex]
So, [tex]x=\frac{-99}{40}[/tex] is the only value that will make the numerator zero
Note that this doesn't simultaneously make the denominator zero, so [tex]x=\frac{-99}{40}[/tex] is a zero (and the only zero) of the function f(x).
Therefore, Option A is NOT a correct answer.
Concept 2. Horizontal asymptotes of rational functionsRational functions have Horizontal Asymptotes if the degree of the polynomial in the numerator is less than or equal to the degree of the polynomial in the denominator.
If the Degree of the denominator is greater, the Horizontal Asymptote is at a y=0.
If the Degrees are equal, the Horizontal Asymptote is at a y-value equal to the ratio of the leading coefficients.
[tex]f(x)=\dfrac{40x+99}{x}[/tex]
Note that for f(x), the degrees of the polynomials in the numerator and denominator are both 1. So, a horizontal asymptote does exist, and it is at a height of the ratio of the leading coefficients.
The leading coefficient of the numerator is 40, while the leading coefficient of the denominator is 1.
The ratio of the leading coefficients, 40/1, so the horizontal asymptote is y=40.
Therefore, Option B is a correct answer choice.
Concept 3. Vertical asymptotes of rational functionsRational functions have Vertical Asymptotes at values of x where the denominator is zero, while the numerator is not also zero (the opposite of finding "zeros" of the function).
Recalling the values that make the numerator and denominator zero from Concept 1:
x=0 is the only value that will make the denominator zero[tex]x=\frac{-99}{40}[/tex] is the only value that will make the numerator zeroSince x=0 doesn't also make the numerator zero, x=0 is a vertical asymptote for the function f(x).
Therefore, Option C is a correct answer choice.
Concept 4. Transformations - vertical shiftRational functions have been vertically shifted if after all the main rational function fraction, there is a number added or subtracted.
I provide an example of a different function (which I'll call g(x)) here:
[tex]g(x)=\dfrac{3}{x}+2[/tex]
Observe that the "+2" is after all of the main fraction, so the graph of 3/x would have been shifted vertically up 2 units.
This is NOT the case for the function f(x) from the question. The "99" is part of the fraction, so it does not represent a vertical shift.
Therefore, Option D is NOT a correct answer choice.
Which is true about the relationship between 1 inch and 1 foot?
A. 1 inch is 12 times as long as 1 foot. B. 1 foot is 12 times as long as 1 inch. C. 1 foot is 6 times as long as 1 inch. D. 1 inch is 6 times as long as 1 foot
Option B is true about the relationship between 1 inch and 1 foot.
What is conversation?
A unit's use depends on the context; for example, a room's area is expressed in meters, yet a pencil's length and thickness are expressed in centimeters and millimeters, respectively. Therefore, converting one unit to another is necessary. We must first understand the link between units in order to comprehend the concept of unit conversion. When addressing many problems in mathematics, unit conversion is necessary. Mathematical conversions are necessary to perform the necessary calculations.
There are 12 inches in 1 foot. This means that 1 foot is 12 times longer than 1 inch. Therefore, option B is correct. Option A is incorrect because it states that 1 inch is 12 times as long as 1 foot, which is the opposite of the correct relationship. Option C is incorrect because it states that 1 foot is 6 times as long as 1 inch, which is half of the correct relationship. Option D is incorrect because it states that 1 inch is 6 times as long as 1 foot, which is the opposite of the correct relationship.
Hence, Option B is true about the relationship between 1 inch and 1 foot.
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(co 4) from a random sample of 85 teens, it is found that on average they spend 31.8 hours each week online with a population standard deviation of 5.91 hours. what is the 90% confidence interval for the amount of time they spend online each week?
The 90% confidence interval for the amount of time that teens spend online each week is (30.761, 32.839) hours
To find the 90% confidence interval for the amount of time that teens spend online each week, we can use the following formula
CI = x ± z*(σ/√n)
where
x is the sample mean
σ is the population standard deviation
n is the sample size
z is the z-score associated with the desired confidence level
In this case, we have
x = 31.8 hours
σ = 5.91 hours
n = 85
z = 1.645 (for a 90% confidence level, using a z-table or calculator)
Plugging in these values, we get
CI = 31.8 ± 1.645*(5.91/√85)
Simplifying, we get
CI = 31.8 ± 1.039
We can interpret this interval as saying that if we were to take many random samples of 85 teens and compute the confidence interval for each sample, about 90% of these intervals would contain the true population mean.
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Element x decays radioactively with a half life of 15 minutes. if there are 960 grams of element x, how long, to the nearest tenth of a minute, would it take the element to decay to 295 grams?
y=a(.5)^(t/h)
It would take approximately 21.2 minutes for 960 grams of Element X to decay to 295 grams.
The time it takes for 960 grams of Element X with a half-life of 15 minutes to decay to 295 grams can be found using the formula y = a [tex](0.5)^\frac{t}{h}[/tex] .
1: Identify the variables.
a = initial amount = 960 grams
y = final amount = 295 grams
h = half-life = 15 minutes
t = time in minutes (this is what we want to find)
2: Plug the variables into the formula.
295 = 960 [tex](0.5)^\frac{t}{15}[/tex]
3: Solve for t.
Divide both sides by 960.
(295/960) = [tex](0.5)^\frac{t}{15}[/tex]
4: Take the logarithm of both sides to remove the exponent.
log(295/960) = log [tex](0.5)^\frac{t}{15}[/tex]
5: Use the logarithm property to move the exponent to the front.
log(295/960) = (t/15) * log(0.5)
6: Solve for t.
t = (15 * log(295/960)) / log(0.5)
7: Calculate t.
t ≈ 21.2 minutes
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A square a rectangle have the same perimeter of a square has a side length of 8x units. The rectangle has a length of (5x + 12) and a width of 10 units. what will be the perimeter of both a rectangle and the square
Answer:
Step-by-step explanation:
The perimeter of a square is calculated by multiplying the length of one side by 4. Since the side length of the square is 8x units, the perimeter of the square is 4 * 8x = 32x units.
The perimeter of a rectangle is calculated by adding the lengths of all four sides or by using the formula 2 * (length + width). Since the length of the rectangle is (5x + 12) units and the width is 10 units, the perimeter of the rectangle is 2 * ((5x + 12) + 10) = 10x + 44 units.
Since both shapes have the same perimeter, we can set their perimeters equal to each other and solve for x:
32x = 10x + 44 22x = 44 x = 2
Substituting this value of x back into the expression for the perimeter of either shape, we find that the perimeter of both the square and the rectangle is 64 units.
Put the numbers in order from least to greatest.
4.27, 2.704, 4.2, 2.74, 4.72
Answer: 2.704 2.74 4.2 4.27 4.72
Step-by-step explanation:
Calcula los siguientes lÃmites página. 115 ejercicio
a) lim n = +[infinity] infinito 6-4n²
----------
2(n)²
b) lim n = +[infinity] infinito 4n²+3n-2
--------------
2n ³ -4n
c) lim n = +[infinity] infinito 2n ³ -4n
---------------
4n
d) lim x = +[infinity] infinito -8x4 +2
------------
2x² +4
a) Para calcular este límite, podemos dividir tanto el numerador como el denominador por n² y luego aplicar la regla de L'Hôpital:
lim n → ∞ [(6 - 4n²)/(2n²)]
= lim n → ∞ [6/(2n²) - (4n²)/(2n²)]
= lim n → ∞ [3/n² - 2]
= -2
Por lo tanto, el límite es -2.
b) Podemos dividir tanto el numerador como el denominador por n³ para simplificar el límite:
lim n → ∞ [(4n² + 3n - 2)/(2n³ - 4n)]
= lim n → ∞ [(4/n + 3/n² - 2/n³)/(2/n² - 4/n²)]
= lim n → ∞ [(4 + 3/n - 2/n²)/(2 - 4/n)]
= lim n → ∞ [(4n + 3 - 2n²)/(2n² - 4)]
= lim n → ∞ [-2n²/(2n² - 4)]
= -1
Por lo tanto, el límite es -1.
c) Podemos dividir tanto el numerador como el denominador por n³ para simplificar el límite:
lim n → ∞ [(2n³ - 4n)/(4n)]
= lim n → ∞ [(2n² - 4)/(4)]
= lim n → ∞ [(n² - 2)/2]
= +∞
Por lo tanto, el límite es +∞.
d) Podemos dividir tanto el numerador como el denominador por x⁴ para simplificar el límite:
lim x → ∞ [-8x⁴ + 2]/[2x² + 4]
= lim x → ∞ [-8 + 2/x⁴]/[2/x² + 4/x⁴]
= -4/1
= -4
Por lo tanto, el límite es -4.
If one line passes through the points (-3,8) & (1,9), and a perpendicular line passes through the point (-2,4), what is another point that would lie on the 2nd line. Select all that apply.
One point that would lie on the second line is (0,-4). Another possible point on the 2nd line is (0, 12).
To find the equation of the first line, we can use the slope-intercept form:
y = mx + b
where m is the slope and b is the y-intercept. The slope of the line passing through (-3,8) and (1,9) can be found using the formula:
m = (y2 - y1) / (x2 - x1)
m = (9 - 8) / (1 - (-3))
m = 1/4
Using one of the points and the slope, we can find the y-intercept:
8 = (1/4)(-3) + b
b = 9
So the equation of the first line is:
y = (1/4)x + 9
To find the equation of the second line, we need to use the fact that it is perpendicular to the first line. The slopes of perpendicular lines are negative reciprocals, so the slope of the second line is:
m2 = -1/m1 = -1/(1/4) = -4
Using the point-slope form, we can write the equation of the second line:
y - 4 = -4(x + 2)
y - 4 = -4x - 8
y = -4x - 4
To find a point that lies on this line, we can plug in a value for x and solve for y. For example, if we let x = 0, then:
y = -4(0) - 4
y = -4
So the point (0,-4) lies on the second line.
Therefore, another point that would lie on the second line is (0,-4).
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A backyard fencing company charges difference prices for different amounts of fences.
The company charges $30 per foot for fences up to 300 feet and $25 per foot for fences over 300 feet.
a) Write a piecewise defined function for the cost of fencing a backyard.
The piecewise defined function for fencing the backyard would be 30(300) + 25(x - 300), if x > 300, or 30x, if 0 < x ≤ 300, as seen below.
What is a piecewise function?A piecewise defined function is a function that is defined by different formulas on different intervals or "pieces" of its domain. This means that the formula used to calculate the output (the function value) of the function depends on the value of the input (the independent variable).
For this particular question, let C be the cost of fencing a backyard and let x be the length of the fence in feet. Then, we can write the piecewise defined function as:
C(x) = 30x, if 0 < x ≤ 300 OR
C(x) = 30(300) + 25(x - 300), if x > 300
In other words, if the length of the fence is less than or equal to 300 feet, the cost is simply 30 times the length of the fence. If the length of the fence is greater than 300 feet, the cost is the cost of the first 300 feet (30 times 300), plus 25 times the additional length of the fence beyond 300 feet.
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A triangle has side lengths of (7a + 2b) centimeters, (6a + 3c) centimeters, and
(3c +46) centimeters. Which expression represents the perimeter, in centimeters,
of the triangle?
The expression that represents the perimeter of the triangle is 13a + 5c + 2b + 46 centimeters.
So, the expression for the perimeter of the triangle is:
(7a + 2b) + (6a + 3c) + (3c + 46)
Simplifying and combining like terms, we get:
13a + 5c + 2b + 46
Rational functions can also have holes in their graphs, which do when a factor in the numerator and denominator cancel out.
For illustration, the function
[tex]h( x) = ( x2- 4)/(x^{2} )( x- 2)[/tex]has a hole at x = 2,
where the factor ( x- 2) cancels out in the numerator and denominator.
Graphing rational functions can be tricky, but it helps to identify the perpendicular and vertical asymptotes, any holes in the graph, and the of the function near these points.
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Let f(x,y) = x⁴ + y⁴ – 4xy +1. Find all critical points. For each critical point, determine whether it is a local maximum, a local minimum, or a saddle point. (At least with my approach, for this problem you'll need to factor x⁹ - x. This factors as x(x² - 1)(x² + 1)(x⁴ + 1)
The critical points of [tex]f(x,y)[/tex] are: (0,0), (1,1), (-1,-1), [tex](1/\sqrt2,-1/\sqrt2)[/tex], [tex](-1/\sqrt2,1/\sqrt2), (i/\sqrt2,-i/\sqrt2)[/tex], and [tex](-i/\sqrt2,i/\sqrt2)[/tex]. The points (1,1) and (-1,-1) are local maxima, while the remaining critical points are saddle points
How to find the critical points of the function?To find the critical points of the function [tex]f(x,y)[/tex], we need to find where its partial derivatives with respect to x and y are equal to zero:
∂f/∂x = 4x³ - 4y = 0
∂f/∂y = 4y³ - 4x = 0
From the first equation, we get y = x³, and substituting into the second equation, we get:
[tex]4x - 4x^9 = 0[/tex]
Simplifying this equation, we get:
[tex]x(1 - x^8) = 0[/tex]
So the critical points occur at x = 0, x = ±1, and [tex]x = (^+_-i)/\sqrt2[/tex].
To determine the nature of these critical points, we need to look at the second partial derivatives of [tex]f(x,y)[/tex]:
∂²f/∂x² = 12x²
∂²f/∂y² = 12y²
∂²f/ = -4
At (0,0), we have ∂²f/∂x² = ∂²f/∂y² = 0 and ∂²f/∂x ∂y = -4, so this is a saddle point.
At (1,1), we have ∂²f/∂x² = ∂²f/∂y² = 12, and ∂²f/∂x ∂y = -4, so this is a local maximum.
At (-1,-1), we have ∂²f/∂x² = ∂²f/∂y² = 12, and ∂²f/∂x ∂y = -4, so this is also a local maximum.
At , we have ∂²f/∂x² = 6, ∂²f/∂y² = 6, and ∂²f/∂x ∂y = -4, so these are saddle points.
At [tex](i/\sqrt2,-i/\sqrt2)[/tex] and [tex](-i/\sqrt2,i/\sqrt2)[/tex], we have ∂²f/∂x² = -6, ∂²f/∂y² = -6, and ∂²f/∂x ∂y = -4, so these are also saddle points.
Therefore, the critical points of [tex]f(x,y)[/tex] are: [tex](0,0), (1,1), (-1,-1), (1/\sqrt2,-1/\sqrt2), (-1/\sqrt2,1/\sqrt2), (i/\sqrt2,-i/\sqrt2)[/tex], and [tex](-i/\sqrt2,i/\sqrt2)[/tex]. The points (1,1) and (-1,-1) are local maxima, while the remaining critical points are saddle points
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Nine out of eleven people at a math convention are bored to tears by cable news. If 2462 people at the convention are not bored by cable news, then how many people at the convention are bored by cable news?
The number of people at the convention are bored by cable news are 11079
How many people at the convention are bored by cable news?From the question, we have the following parameters that can be used in our computation:
Proportion = Nine out of eleven people at a math convention are bored to tears by cable news.
This means that
Proportion = 9/11
If 2462 people at the convention are not bored by cable news, then we have
(1 - 9/11) * x = 2462
This gives
x = 2462/(1 - 9/11)
Evaluate
x = 13541
Next, we have
Bored = 13541 - 2462
Bored = 11079
Hence, the number of people is 11079
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Which type of sentence error occurs when a sentence is missing a subject or predicate? No error Fragment Subject -verb agreement Run-on
When the subject or predicate is missing from a sentence, the sentence error is a sentence fragment.
What is a sentence error?Sentence Errors are errors related to grammar and mechanics within sentences in Standard Written English.
An example of a sentence error will be "The children re us"
The phrase "re" is the error word in this sentence.
The three types of sentence errors we have are run-on sentences, sentence fragments and overloaded sentence.
In this problem, the type of sentence error that occurs when a sentence is missing a subject of predicate is a sentence fragment
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Identify each part of the circle given it’s equation.
Each part of the circle given it’s equation should be identified as follows;
Center: (9, 4).Center: (-1, 1).Center: (-6, 0).Center: (2, -13).Center: (-7, -4).Radius: √28What is the equation of a circle?In Geometry, the standard or general form of the equation of a circle is modeled by this mathematical equation;
(x - h)² + (y - k)² = r²
Where:
h and k represent the coordinates at the center of a circle.r represent the radius of a circle.Based on the information provided above, we have the following parameters:
Radius, r = √400 units.
Center, (h, k) = (9, 4).
By substituting the given radius and center into the equation of a circle, we have;
(x - h)² + (y - k)² = r²
(x - 9)² + (y - 4)² = (√400)²
(x - 9)² + (y - 4)² = 400.
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I'LL MARK BRAINLIEST !!!
Which point is the opposite of -5? Plot the point by dragging the black circle to the correct place on the number line.
JUST TELL ME THE CORRECT SPOT PLS!! TY !!!
Answer:
5
Step-by-step explanation:
The correct spot would be 5 because, on a number line, the opposite of a negative would be its positive counterpart and vise versa.
How many years would it take for the price of pizza’s ($8.00) to triple with a growth rate of 1.05? Explain how you found your answer.
It would take 1.53 years for the price of pizza to triple with a growth rate of 1.05.
Calculating the number of yearsTo find the number of years it takes for the price of pizza to triple with a growth rate of 1.05, we need to use the formula for exponential growth:
A = P(1 + r)^t
Where:
A = final amount (triple the original price, or 3*$8 = $24)
P = initial amount ($8)
r = growth rate (1.05)
t = time in years
Substituting the values into the formula, we get:
$24 = $8(1 + 1.05)^t
Simplifying:
3 = (1 + 1.05)^t
Taking the logarithm of both sides with base 10:
log(3) = t*log(1 + 1.05)
t = log(3) / log(1 + 1.05)
Using a calculator, we get:
t ≈ 1.53
Therefore, it would take approximately 1.53 years for the price of pizza to triple with a growth rate of 1.05.
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To find the standard deviation of the liquid measure of oil in barrels, the oil company measures 25 randomly selected barrels and find the standard deviation of the samples to be s=. 34. Find the 92% confidence interval for the population standard deviation
The 92% confidence interval for the population standard deviation is (0.199, 0.509).
To find the 92% confidence interval for the population standard deviation, we will use the chi-square distribution. We know that for a sample size of n=25, the degrees of freedom for the chi-square distribution is (n-1) = 24.
The chi-square distribution is a right-tailed distribution, so we need to find the chi-square values that will leave 4% in the right tail (for a total of 92% confidence interval).
From a chi-square distribution table, the chi-square value with 24 degrees of freedom that leaves 4% in the right tail is 41.337. The chi-square value that leaves 96% in the left tail is 13.119.
Using the formula for the confidence interval for the population standard deviation:
lower bound = [tex]sqrt((n-1)*s^2 / chi-square upper)[/tex]
upper bound = [tex]sqrt((n-1)*s^2 / chi-square lower)[/tex]
We can substitute the values we have:
lower bound = [tex]sqrt((25-1)*0.34^2 / 41.337) = 0.199[/tex]
upper bound = [tex]sqrt((25-1)*0.34^2 / 13.119) = 0.509[/tex]
Therefore, the 92% confidence interval for the population standard deviation is (0.199, 0.509).
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