There does not exist a constant C > 0 such that p∞(v) ≤ Cp1(v) for all vectors v in V. Therefore, p1-norm < p2-norm < p∞-norm implies there is no c such that p∞-norm < c p1-norm.
Given two norms, p and q, on a vector space V, we say that p is less than q, written p ≤ q, if there exists a constant C > 0 such that for all vectors v in V,
p(v) ≤ Cq(v)
Now, let's consider the three norms:
The 1-norm: p1(v) = ∑|v_i|
The 2-norm: p2(v) = √(∑v_i^2)
The infinity-norm: p∞(v) = max{|v_i|}
We are given that p1(v) ≤ p2(v) for all vectors v in V and that p2(v) ≤ p∞(v) for all vectors v in V. We want to show that there does not exist a constant C > 0 such that p∞(v) ≤ Cp1(v) for all vectors v in V.
Suppose for the sake of contradiction that such a constant C does exist. Then, for any non-zero vector v in V,
p∞(v) ≤ Cp1(v)
max{|v_i|} ≤ C ∑|v_i|
Since v is non-zero, there exists an index i such that |v_i| > 0. We can divide both sides of the inequality by |v_i| to get:
1 ≤ C ∑|v_j| / |v_i|
Now, since p1(v) ≤ p2(v) for all vectors v in V, we have:
p1(v) = ∑|v_j| ≤ √(∑v_j^2) = p2(v)
So, we can substitute this into the inequality above to get:
1 ≤ C √(∑v_j^2) / |v_i| = Cp2(v) / |v_i|
Finally, since p2(v) ≤ p∞(v) for all vectors v in V, we have:
p2(v) / |v_i| ≤ p∞(v) / |v_i| = 1
So, we have reached a contradiction:
1 ≤ Cp2(v) / |v_i| ≤ C
This contradicts the assumption that C > 0, so there does not exist a constant C > 0 such that p∞(v) ≤ Cp1(v) for all vectors v in V. Therefore, p1-norm < p2-norm < p∞-norm implies there is no c such that p∞-norm < c p1-norm.
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For every 50p Max saves, his mum gives him 20p. Max saves £5.50. How much does he have altogether, including the money his mum gives him?⭐⭐⭐⭐⭐
Max saves £5.50, which is 550p.
For every 50p Max saves, his mum gives him 20p, so for 550p, he gets 550/50 * 20p = 440p from his mum.
Altogether, Max has 550p + 440p = 990p, which is equivalent to £9.90.
One base of a trapezoid is 1 in longer than 2 times the other base. find the lengths of the bases if the trapezoid is 4in high and a area of 20in.
The length of the bases for the trapezoid are
shorter base is 3 in and the longer base is 7 in
How to find the length of the basesThe formula for area of trapezoid is
= 1/2 sum of bases * height
base of the trapezoid
20 = 1/2 sum of bases * 4
1/2 sum of bases = 5
sum of bases = 10
The relationship between the bases
longer base = 2 * shorter base + 1
sum of bases = longer base + shorter base
sum of bases = 2 * shorter base + 1 + shorter base
sum of bases = 3 shorter base + 1
substituting will result to
10 = 3 shorter base + 1
10 - 1 = 3 shorter base
9 = 3 shorter base
shorter base = 3 in
longer base = 2 * shorter base + 1
longer base = 2 * 3 + 1
longer base = 7 in
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Cost and revenue functions. Please tysm!!!
The profit function is P ( x ) = 2x² + 5x - 12 and the number of months during which the company is not making a profit is x = 1.5 years = 18 months
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the profit function be represented as P ( x )
Let the cost function of the company be C ( x ) = 17 - 8x - 2x²
Let the revenue function of the company be R ( x ) = 5 - 3x
Now , Profit = Revenue - Costs
Substituting the values in the equation , we get
P ( x ) = R ( x ) = C ( x )
P ( x ) = 5 - 3x - ( 17 - 8x - 2x² )
On simplifying the equation , we get
P ( x ) = 2x² + 5x - 12
Now , when the profit of the company is 0
2x² + 5x - 12 = 0
On factorizing the equation , we get
2x² + 8x - 3x - 12 = 0
2x ( x + 4 ) - 3 ( x + 4 ) = 0
Taking the common factors of the equation , we get
( 2x - 3 ) ( x + 4 ) = 0
The value of x cannot be negative , so
2x - 3 = 0
Adding 3 on both sides of the equation , we get
2x = 3
Divide by 2 on both sides of the equation , we get
x = 3/2 years
Therefore , the number of months x = 18 months
Hence , the equation is solved
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Over the last three evenings, Elsa received a total of 96 phone calls at the call center. The first evening, she received 8 more calls than the second evening. The third evening, she received 2 times as many calls as the second evening. How many phone calls did she receive each evening?
Answer:30, 22, 44
Step-by-step explanation: 2x+x+(x+8) represents how many calls for each evening. you simplify,
2x+x+(x+8)=4x+8
you know the total is equal to 96, so you do 4x+8=96.
x=22 calls for the SECOND evening. Next, you double and add 8 to get each evening.
I'm giving out 50 points for this riddle: if a homeless eats it, he dies, rich people need this, and some people already have this that are poor
Answer:
nothing
Step-by-step explanation:
Answer:
Step-by-step explanation:
nothing
Which graph represents the equation y equals one half times x minus 1?
Step-by-step explanation:
The graph would represent a straight line that slopes downwards and passes through the point (2,1).
Given P is the incenter of ABC find the length of PX
The length of PX is 1.56.
The Pythagorean Theorem is a mathematical principle that relates the lengths of the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
c² = a² + b²
The Pythagorean Theorem is one of the most famous and widely-used theorems in mathematics, and is a fundamental principle in geometry. It is used to find the length of one side of a right triangle when the lengths of the other two sides are known, and is also used to test whether a triangle is a right triangle.
Since YP is perpendicular to AB, we have right triangle APY with AP as hypotenuse and PY as the height.
Using the Pythagorean theorem, we can find the length of AP:
AP² = AY² + PY²
AP² = 4.7² + 1.4²
AP² = 22.09
AP = 4.7
Similarly, since ZP is perpendicular to BC, we have right triangle CPZ with CP as hypotenuse and PZ as the height.
Using the Pythagorean theorem, we can find the length of CP:
CP² = CZ² + PZ²
CP² = 3.2² + PZ²
CP² = 10.24
CP = 3.2
Since PX is perpendicular to AB, we have right triangle BPX with BP as hypotenuse and PX as the height.
Using the Pythagorean theorem, we can find the length of BP:
BP^2 = AP^2 + CP^2
BP^2 = 4.7^2 + 3.2^2
BP^2 = 26.69
BP = 5.15
Finally, using the Pythagorean theorem, we can find the length of PX:
PX^2 = BP^2 - PY^2
PX^2 = 5.15^2 - 1.4^2
PX^2 = 2.46
PX = 1.56
Therefore, the length of PX is 1.56.
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Hellllp I neeeeddd helllp asap
Answer:
B.80
Step-by-step explanation:
use of pythagoras theorem.
a^2 + b^2 = c^2
where c is hypotenuse
Answer:
D
Step-by-step explanation:
first we can see from question is finding perimeter which means first step we need is
24+24=48
second , we can know it from picture
ABD=[tex]\frac{1}{2}[/tex] times 60= 30, which means it is 1/2 marks
P=DB/BA
which we see DB=BE
so 2 time DB( or BE)=18 which is 18
so we just need use 24+24+18=66
What is the critical F value for a sample of six observations in the numerator and four in the denominator? Use a two-tailed test and the .10 significance level., Mark Nagy has averaged 9 rejects, with a standard deviation of 2 rejects per day. Debbie Richmond averaged 8.5 rejects, with a standard deviation of 1.5 rejects, over the same period. At the .05 significance level, can we conclude that there is more variation in the number of rejects per day attributed to Mark?
The critical F value for a sample of 6 and 4 degrees of freedom at a .10 two-tailed significance level can be found using a F-distribution table. The exact value depends on the specific table used.
How do we compare Mark and Debbie?For the comparison between Mark and Debbie, we would need to perform an F-test for equality of variances to determine if there is more variation in Mark's data. The null hypothesis is that the variances are equal, and the alternative hypothesis is that the variance of Mark's data is greater. The test statistic would be the ratio of the larger variance to the smaller variance, with the degrees of freedom being the smaller of the two sample sizes minus 1.
A .05 significance level means that we reject the null hypothesis if the p-value is less than .05. The p-value can be found using an F-distribution table or a statistical software.
The conclusion would depend on whether the p-value is less than .05 or not.
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Which of the following variables use the ratio scale of measurement? Income Temperature Gender Social security number
Income normally uses a ratio scale of measurement.
Hence, option (a) is correct choice.
A ratio scale is a numerical scale with a genuine zero and equal gaps between adjacent points. A zero on a ratio scale, unlike an interval scale, indicates the complete absence of the variable being measured. Ratio scales include length, area, and population.
A ratio scale may be used to label variables, arrange them in order, and calculate the equal distance between variables. The ratio Scale enables unit conversion. British thermal units, such as Joules, kilograms, and grams, are ratio scale units.
As several elements of a given unit, interval scale may quantify size and magnitude. A ratio scale can be used to quantify size and magnitude by dividing one designated unit by another.
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A baseball diamond measures 28 meters along each side. If a batter hit 1 single in a game, how many kilometers did the batter run?
The batter ran a distance of 0.112 kilometers or approximately 112 meters.
What is the distance?A mathematical number known as distance measures "how much ground an object has traveled" while moving. Distance is defined as the product of speed and time.
A baseball diamond is made up of four bases, and the distance a batter runs for a single is from home plate to first base, then first base to second base, then second base to third base, and finally third base back to home plate.
So the total distance the batter runs is 4 times the distance from home plate to first base, which is 28 meters.
28 meters x 4 = 112 meters
To convert meters to kilometers, divide by 1000:
112 meters / 1000 = 0.112 kilometers
Thus, the batter ran 0.112 kilometers or approximately 112 meters.
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Six less than six times a number is the same as eight times the number. Find the number.
The number is
Answer:
below
Step-by-step explanation:
6x -6 = 8 x
-6 = 2x
x = -3
Answer:
X = -3
Step-by-step explanation:
Let's slowly create an equation
Six less than six times a number = eight times the number
if x = that number...
Six less than six times of x = eight times of x
6 less than 6 times of x is 6x-6
8 times x is simply 8x
So 6x - 6 = 8x
We can rearrange this equation by adding 6 to one side and subtracting 8x from the other
6x - 8x = 6
Combining like terms makes it -2x = 6
Last step is dividing by -2, giving us x = -3
A=1/2h (b1 + b2) for h
Select the correct answer. The sum of the digits of a three-digit number is 13. The tens digit, t, is 1 more than the hundreds digit, h. The units digit, u, is 3 more than the sum of the tens and hundreds digits. Which system of equations can be used to find each digit?
Could someone help me understand? im a but confused
Using the system of equations we get the values of 3 digit number as:
238
Given:
The sum of the digits of a three-digit number is 13.
The tens digit, t, is 1 more than the hundreds digit, h. The units digit, u, is 3 more than the sum of the tens and hundreds digits.
we are asked to determine each digit of the given number:
we frame the equation as:
tens digit (t) = 1+h
hundreds digit (h) = h
units digit (u) = 3+(t+h)
hence,
h + (1+h) + (3+(t+h)) = 13
open the brackets.
h+1+h+3+(t+h)=13
substitute t value from the above mentioned state.
h+1+h+3+((1+h)+h)=13
h+1+h+3+1+h+h=13
arrange the like terms and add.
4h + 5 = 13
4h = 13-5
4h=8
h = 8/4
h = 2
hence we get the hundreds digit as 2.
now substitute it to get t and u value.
t = 1+h
t = 1+2
t = 3
tens digit = 3
u = 3+(t+h)
u = 3+(3+2)
u = 3+5
u = 8
units digit = 8
Consider the graph please
Age of first heart attack - The horizontal axis is age. The vertical axis is frequency (number of people who had their first heart attack at that age). The population is people in the US who have had at least one heart attack. Shape of distribution:
Answer: The shape of the distribution of the age of first heart attack in the US population who have had at least one heart attack is likely to be positively skewed, with the majority of cases occurring in older age groups and a relatively small number of cases occurring in younger age groups. The frequency of heart attacks generally increases with age, reaching a peak in the older age groups.
Step-by-step explanation:
Please I need help , thanks
The statements that are true on the graph include:
The point ( -8, 9 ) lies on the line The point ( 0, 4 ) lies on the line The point ( 4, 0 ) lies on the line What points lie on the line ?At the point where the line hits x = 4, the y coordinates would be 0. This means that the point ( 4, 0 ) lies on the line. The point where the line crosses x = 0, y is the value of 4 which means that ( 0, 4 ) lies on the line.
The point ( - 8, 9 ) also lies on the line because as the line extends upwards, it would intersect the point ( - 8, 9 ).
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Solve with full solution
Q1. A ladder 9 m long reaches to a point below the top of building. From the foot of the ladder, the angle of olevation of the building is 60°. Find the height of the building. Here the ladder makes the angle 45° with ground. (Ans=11.02m)
Q2. A vertical pole is divided in the ratio 9:1 .If both segments of pole subtend equal angles to each other at a distance of 20m away from the foot of pole, find height of pole. (Ans=178.8m)
Answer:
Q 1 : Solution:
Let the height of the building be h.
We know that tan(60) = h/9 (angle of elevation formula)
h = 9tan(60) = 9sqrt(3)/3 = 3*sqrt(3) = 3.464101615 m (approx)
Q 2 : Solution:
Let the height of the pole be h.
Let the shorter segment be x, and the longer segment be 9x.
We know that tan(theta) = x/(20) = 9x/(20+h) (angle subtended formula)
By cross multiplying and simplifying, we get:
x = (20h)/(h+180)
The ratio of the segments is x:9x = 1:9, so x = (20h)/(h+180) = 1/10 * h
h = 10x = (20h)/(h+180)
h^2 + 180h - 20h^2 = 0
h(20-h) = 0
h = 0 or h = 20
Since the pole has a height, h = 20m is not possible, so the height of the pole is h = 178.8m
a child tosses a baseball up into the air. On its way down, it gets caught in a tree for several seconds before falling back down into the ground. Select the best description of the range of this function.
The range includes all numbers from 0 to 12.5. So correct option is D.
What do you mean by probability?Probability is a branch of mathematics that deals with the likelihood or chance of an event happening. It is a number between 0 and 1 that indicates the likelihood of an event occurring, with 0 meaning that an event is impossible and 1 meaning that an event is certain to occur.
Probability is used to model and make predictions about real-world situations such as the outcome of a coin flip, the results of a survey, or the success of a marketing campaign. It is also used in many fields including statistics, finance, engineering, and sciences to make decisions based on uncertain data.
Probabilities can be calculated using various methods, such as counting, Bayes' theorem, and simulation. In general, the probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
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Find all values of x where f(x)=-7.
Answer: A line parallel to the x-axis and at a distance of 7 units above the x-axis.
Farthest Point. Which of the following points is farthest from P (–1, 3)? Show the necessary solutions.
a. R(2, 4)
b. S(-4, 1)
c. T(0, 5)
d. U(3, -2)
The farthest point from P is point U with a distance of 6.4 units
How to determine the farthest pointFrom the question, we have the following parameters that can be used in our computation:
P = (-1, 3)
To determine the farthest point, we make use of the distance formula represented as
distance = √[(y₂ - y₁)² + (x₂ - x₁)²]
So, we have
PR = √[(4 - 3)² + (2 + 1)²] = 3.2
PS = √[(1 - 3)² + (-4 + 1)²] = 3.6
PT = √[(5 - 3)² + (0 + 1)²] = 2.2
PU = √[(-2 - 3)² + (3 + 1)²] = 6.4
This means that the fartherst point is point U
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What is the zero in the equation y=2x+8?
Answer:
The zero in the equation y = 2x + 8 is when x = -4. When x = -4, y = 0.
Categorize the type of sampling (simple random sample, stratified sample; systematic sample; cluster sample; or convenience sample) used: To judge the appeal of a proposed television sitcom, a random sample of 10 people from each of three different age categories was selected and those chosen were asked to rate a pilot show: simple random sample stratified sample systematic sample cluster sample
The type of sampling used to judge the appeal of the television sitcom is a b) stratified sample.
A stratified sample is a type of sampling where the population is divided into subgroups or strata based on certain characteristics, and then a random sample is taken from each of these strata. In this case, the population is divided into three age categories and a random sample of 10 people is selected from each category.
Stratified sampling is useful when the characteristics of each stratum are different and you want to make sure that the sample accurately reflects the entire population. By dividing the population into age categories, this method ensures that the sample is representative of the different age groups and their opinions on the pilot show.
Overall, stratified sampling is a useful method for accurately estimating the characteristics of a population by taking into account any relevant differences within it.
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Determine the value of h such that the matrix is the augmented matrix of a consistent linear system.
6 -3 h
-12 6 4
The value of h such that the matrix is the augmented matrix of a consistent linear system is -2
How to determine the value of hFrom the question, we have the following parameters that can be used in our computation:
6 -3 h
-12 6 4
For a consistent system of linear equation, the equations are the equivalent
So, we have
6x - 3y = h
-12x + 6y = 4
Divide the second equation by -2
6x - 3y = -2
By comparing the equations, we have
h = -2
Hence, the solution of h is -2
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Help! I will be very grateful if you help, 25 points
Therefore , the solution of the given problem of linear equation comes out to be total amount collected is $4956.
What is a linear equation, exactly?The foundation for a linear regression curve is the equation y=mx+b. B is the slope, and m is the y-intercept. The previous sentence is sometimes referred as a "mathematical equation involving two variables," even though that y or y are separate components. Two variables make up bivariate linear equations. There are no known solutions to application issues involving linear equations. Y=mx+b is the equation for a single set of equations.
Here,
Given :
Pupil : price per ticket is $8
teacher : price per ticket is $20
Thus ,
Let the x be the total amount of teacher is :
So,
Pupil collected 416$ more than teachers total amount is :
Pupil total amount = 416 + x
So,
Total tickets sold is 1/4 of the ticket sold :
=> 1/4y * 20 = x
=> 5 = x/y
=>x = 5y
So,
=> 416 + 5y = 3/4 * 8 y
=> 416 + 5y = 6y
=> y = 516 tickets were sold
Thus ,
=> 1/4 * 516 = 129
=> 129 * 20 = 2580
For pupil is :
=> (516 -219) * 8
=> 2376
Total amount collected is => 2580 + 2376 = 4956
Therefore , the solution of the given problem of linear equation comes out to be total amount collected is $4956.
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(b) If the sum of two numbers is 10 and difference is 4. Find the number.
Answer:6
Step-by-step explanation:
The two numbers are 3 and 7.
Let the two numbers be x and y.
Considering the first statement :
x + y = 10 ...........................(1)
Considering the second statement:
x - y = 4 ............................(2)
Adding equations (1) and (2):
x + y + x - y = 10 + 4
2x = 14
x = 7
Substituting the value of x in equation (1):
7 + y = 10
y = 10 - 7
y = 3
Therefore, the value of the two numbers is 3 and 7.
For reference:
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If X and Y are independent exponential random variables with pdf f(x) = 0.5e-as, x 0, find p(X > 1,Y > 4)
The joint probability distribution function for exponential random variables with pdf f(x) = 0.5e^(-0.5x), x ≥ 0, find P(X > 1,Y > 4) is, e^(-5/2).
The X and Y are independent variables, so probability distribution function of X is same as that of Y.
The joint probability distribution function for independent variables is,
[tex]\int \int f(x)f(y) dx dy[/tex]
where f(x) and (y) are the probability distribution function in x and y respectively.
In this question, X>1 and Y>4, so the limits of integration for X goes from 1 to infinity and for Y it goes from 4 to infinity.
[tex]\int_4^\infty \int_1^\infty 0.5 \times 0.5 \times e^{-0.5x} e^{-0.5y} dx dy\\= 0.25\int_4^\infty e^{-0.5y} dy \int_1^\infty e^{-0.5x} dx\\= 0.25 \times [-2e^{-0.5y}]_4^\infty \times [-2e^{-0.5x}]_1^\infty \\= 0.25 \times \dfrac{2}{{e}^2} \times \dfrac{2}{\sqrt{e}}[/tex]
The integral is, e^(-5/2).
The joint probability distribution function is, e^(-5/2).
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--The complete question is, If X and Y are independent exponential random variables with pdf f(x) = 0.5e^(-0.5x), x ≥ 0, find P(X > 1,Y > 4).--
Carmen is starting a small gardening and lawn care business in her neighborhood for the summer. To get the business started she spends some money on advertising. The table shows her weekly profit for the first few weeks of the summer.
Week Profit ($)
1 −44.50
2 −26.25
3 80.50
4 88.75
5 95.50
Based on the average profit per week, how much total profit can she expect to earn over the course of the 12-week summer break?
On average Carmen earned $38.8 per week of profit and for 12 weeks the profit earned is $465.6.
What is average?
In everyday terms, an average is one number chosen to represent a list of numbers; it is often the sum of the numbers divided by the number of numbers in the list (the arithmetic mean).
The profit earned by Carmen in her first week = -$44.50
The profit earned by Carmen in her second week = -$26.25
The profit earned by Carmen in her third week = $80.50
The profit earned by Carmen in her fourth week = $88.75
The profit earned by Carmen in her fifth week = $95.50
The average profit earned by Carmen is = sum of profit earned / number of weeks profit earned
Substitute the values into the equation -
Average = [(-44.50) + (-26.25) + 80.50 + 88.75 + 95.50] / 5
Average = 194/5
Average = 38.8
The avearge profit per week is $38.8
To find the profit Earned by Carmen in 12 weeks use the arithmetic operation of multiplication -
Profit Earned = 12 × Average profit per week
Substitute the values in the equation -
Profit Earned = 12 × 38.8
Profit Earned = 465.6
Therefore, the profit earned is $465.6.
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Answer:465.6
Step-by-step explanation:
Please help me with this question.
The solution of the given differential initial value problem is
y(x) = 1/25 e⁻⁵ˣ(6e⁵ˣ - 5x - 6).
What is initial value problem?In multivariable calculus, an initial value problem is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain.Given is the initial value problem -
y''' + 10y'' + 25y' = 0
We can write the -
[tex]$25 \frac{d}{d x} y{\left(x \right)} + 10 \frac{d^{2}}{d x^{2}} y{\left(x \right)} + \frac{d^{3}}{d x^{3}} y{\left(x \right)} = 0$[/tex]
The given differential equation can be solved and written as -
y(x) = 1/25 e⁻⁵ˣ(6e⁵ˣ - 5x - 6)
Therefore, the solution of the given differential initial value problem is
y(x) = 1/25 e⁻⁵ˣ(6e⁵ˣ - 5x - 6).
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the diffrence of 54 and 32 mutlipted by the diffrence of 8 and 5
Answer:
66
Step-by-step explanation:
the difference of 54 and 32 is 54 - 32 = 22
the difference of 8 and 5 is 8 - 5 = 3
then
22 × 3 = 66
