This is a mathematical problem where the student needs to allocate $5000 between two investments, Investment A and B, fitting certain conditions. By developing a series of equations based on the conditions given, it is possible to determine the appropriate allocations.
Explanation:The subject of this question pertains to the allocation of funds in two investments, a process which involves applying principles of mathematics and financial planning. The person wants to invest $5000, with a certain percentage in Investment A (yielding 5%) and the rest in Investment B (yielding 8%), as per the stipulated conditions. To adhere to these requirements, let's denominate the investment in A as 'x' and that in B as 'y'. The restrictions provided, i.e., x needs to be at least 25% of $5000 (i.e., $1250) and y should not be more than 50% of $5000 (i.e., $2500), and x should be half the investment in y, lead us to the equation x = y/2. If you solve this system of equations, the allocations into A and B can be found. For instance, one feasible solution might be $2000 in A and $3000 in B. This ensures that A is at least 25%, B is at most 50%, and A is half of B, which abides by all the stipulations provided.
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The individual should allocate $1250 to Investment A and $3750 to Investment B.
Let's denote the amount invested in Investment A as[tex]\( x[/tex] and the amount invested in Investment B as[tex]\( y \)[/tex] . The individual has a total of $5000 to invest, so we have our first equation:\[ x + y = 5000 \]The individual wants to invest at least 25% of the total amount in Investment A, which gives us the second equation[tex]:\[ x \geq 0.25 \times 5000 \][ x \geq 1250 \][/tex]. The individual also wants to invest at most 50% of the total amount in Investment B, which gives us the third equation:[tex]\[ y \leq 0.50 \times 5000 \][ y \leq 2500 \[/tex]]. Additionally, the investment in A should be at least half the investment in B, leading to the fourth equation:[tex]\[ x \geq \frac{1}{2} y \][/tex] Now, let's solve these equations. From the first equation, we can express[tex]\( y \) in terms of \( x \):[ y = 5000 - x ]Substituting \( y \) into the inequality from the third equation, we get:[ 5000 - x \leq 2500 \][ x \geq 5000 - 2500 \][ x \geq 2500 \]This satisfies the condition from the second equation \( x \geq 1250 \).Now, we substitute \( y \) into the fourth equation:\[ x \geq \frac{1}{2} (5000 - x) \] 2x \geq 5000 - x \] 3x \geq 5000 \][ x \geq \frac{5000}{3} \][ x \geq 1666.\overline{6} Since \( x \)[/tex] must be a whole number of dollars, the smallest whole number that satisfies[tex]\( x \geq 1666.\overline{6} \) is \( x = 1667 \)[/tex] . However, we must also ensure that \( y \) is within the allowed range. Let's calculate [tex]\( y \) using \( x = 1667 \):\[ y = 5000 - x \]\[ y = 5000 - 1667 \]\[ y = 3333 \][/tex]
This allocation does not satisfy the condition that[tex]\( y \)[/tex] must be at most $2500. Therefore, we need to find the maximum value of \[tex]( x[/tex] that satisfies both [tex]\( x \geq 1666.\overline{6} \) and \( y \leq 2500 \).Since \( x \) must be at least half of \( y \), and \( y \) must be at most $2500, we can set \( x \) to half of $2500, which is $1250:\[ x = \frac{1}{2} \times 2500 x = 1250 \]Now, let's check if \( y \) is within the allowed range:[ y = 5000 - x \][ y = 5000 - 1250 \][ y = 3750 \][/tex]This allocation satisfies all the conditions:-[tex]\( x = 1250 \)[/tex] is more than 25% of the total investment.- [tex]\( y = 3750 \[/tex] ) is less than 50% of the total investment.- [tex]\( x \)[/tex] is half o[tex]f \( y \).[/tex] Therefore, the individual should allocate $1250 to Investment A and $3750 to Investment B.
Machine A can produce 435 widget in 3 hours. At this rate how many widget can machine A produce in 7 hours?
Answer:
1015 widgets
Step-by-step explanation:
First investigate how many widgets can machine A produce in 1 hour:
If it produces 435 in 3 hours, then it will produce one third of that in one hour:
In ONE (1) hour [tex]\frac{435}{3} = 145[/tex] widgets
Therefore in seven (7) hours it will produce seven times the amount it does in one hour, that is:
[tex]145 * 7 = 1015[/tex] widgets
Answer:
answer is 1015.
Step-by-step explanation:
Five-year-old students at an elementary school were given a 30-yard head start in a race. The graph shows how far the average student ran in 30 seconds.
Age of Runner
Which statement best describes the domain of the function represented in the graph?
55X 560, orx is from 5 to 60
55x5 30. or x is from 5 to 30
30 SXS 60, or x is from 30 to 60
0 SXS 30 or x is from 0 to 30
The domain of the function represented in the graph is 30 SXS 60 or 30 to 60. Hence option C is correct.
Given that,
Five-year-old students at an elementary school were involved.
They were given a 30-yard head start in a race.
The graph represents the distance run by the average student in 30 seconds.
The graph is related to the age of the runner.
Since we can see that,
The graph starts from 30 yards and ends at 60 yards
Therefore,
The domain of the function represented in the graph is 30 SXS 60 or 30 to 60, which is option C.
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Suppose the augmented matrix for a given system has a pivot in every column. Say as much as you can about the solutions to the corresponding system of equations, with explanation.
Answer:
The system is inconsistent.
Step-by-step explanation:
Consider the provided information.
The augmented matrix for a given system has a pivot in every column.
Let us understand this fact with the help of an example.
Suppose an 8 × 6 augmented matrix has a pivot in every column.
The above matrix is augmented and has a pivot in every column, that means the rightmost column in the above matrix must be a pivot column.
The rightmost column of the above matrix will be something like this:
0 0 0 0 0 | b
Where b≠0 because if b is 0 then it can't be a pivot position.
But from the above 0 = b, which is false.
This means that the system is not consistent,
Hence, the system is inconsistent.
what geometric postulates justifies the statement below
If AB is congruent to CD and CD is congruent to EF, then AB
iscongruent to EF.
Answer: Transitive property of congruence.
Step-by-step explanation:
Transitive property of congruence is similar to transitive property of equality , it says that if a is congruent to b and b is congruent to c , then a is congruent to c.
Given statement : If AB is congruent to CD and CD is congruent to EF, then AB is congruent to EF.
Here a= AB , b= CD and c= EF
Therefore, the geometric postulates justifies given the statement must be Transitive property of congruence.
The final exam of a particular class makes up 40% of the final grade, and Moe is failing the class with an average (arithmetic mean) of 45% just before taking the final exam. What grade does Moe need on his final exam in order to receive the passing grade average of 60% for the class?
Answer:
%82.5
Step-by-step explanation:
The final exam of a particular class makes up 40% of the final gradeMoe is failing the class with an average (arithmetic mean) of 45% just before taking the final exam.From point 1 we know that Moe´s grade just before taking the final exam represents 60% of the final grade. Then, using the information in the point 2 we can compute Moe´s final grade as follows:
[tex]FG=0.40*FE+0.60*0.45[/tex],
where FG is Moe´s Final Grade and FE is Moe´s final exam grade. Then,
[tex]\frac{ FG-0.60*0.45}{0.40}=FE[/tex].
So, in order to receive the passing grade average of 60% for the class Moe needs to obtain in his exam:
[tex]FE=\frac{ 0.60-0.60*0.45}{0.40}=0.825[/tex]
That is, he need al least %82.5 to obtain a passing grade.
Find all solutions of the equation algebraically. Use a graphing utility to verify the solutions graphically. (Enter your answers as a comma-separated list.) 16x^4 - 24x^3 +9x^2 =0
Answer:
The solutions of the equation are 0 and 0.75.
Step-by-step explanation:
Given : Equation [tex]16x^4 - 24x^3 +9x^2 =0[/tex]
To find : All solutions of the equation algebraically. Use a graphing utility to verify the solutions graphically ?
Solution :
Equation [tex]16x^4 - 24x^3 +9x^2 =0[/tex]
[tex]x^2(16x^2-24x+9)=0[/tex]
Either [tex]x^2=0[/tex] or [tex]16x^2-24x+9=0[/tex]
When [tex]x^2=0[/tex]
[tex]x=0[/tex]
When [tex]16x^2-24x+9=0[/tex]
Solve by quadratic formula, [tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
[tex]x=\frac{-(-24)\pm\sqrt{(-24)^2-4(16)(9)}}{2(16)}[/tex]
[tex]x=\frac{24\pm\sqrt{0}}{32}[/tex]
[tex]x=\frac{24}{32}[/tex]
[tex]x=\frac{3}{4}[/tex]
[tex]x=0.75[/tex]
The solutions of the equation are 0 and 0.75.
For verification,
In the graph where the curve cut x-axis is the solution of the equation.
Refer the attached figure below.
A physician prescribes 2.5 million units of penicillin G potassium daily for 1 week. If 1 unit of penicilin G potassium equals 0.6 mcg, how many tablets, each containing 250 mg, will provide the prescribed dosage regimen un
If a physician prescribes 2.5 million units of penicillin G potassium daily for 1 week. To provide the prescribed dosage regimen of 2.5 million units of penicillin G potassium daily for 1 week we would need 42 tablets, each containing 250 mg.
What is the Number of tablets needed ?
Total amount of penicillin G potassium required
Since there are 7 days in a week the total amount required is:
Total amount required =2.5 million units/day * 7 days
Total amount required = 17.5 million units
Convert the units of penicillin G potassium to micrograms (mcg)
Total amount required in micrograms:
Total amount required = 17.5 million units * 0.6 mcg/unit
Total amount required = 10.5 million mcg
Convert micrograms to milligrams (mg)
Convert the total amount required to milligrams:
10.5 million mcg / 1000
= 10,500 mg
The number of tablets needed:
Number of tablets needed:
Number of tablets needed = 10,500 mg / 250 mg/tablet
Number of tablets needed = 42 tablets
Therefore to provide the prescribed dosage regimen of 2.5 million units of penicillin G potassium daily for 1 week we would need 42 tablets, each containing 250 mg.
The patient needs to take 42 tablets of penicillin G potassium over one week. This is determined by converting the prescribed units to milligrams and dividing by the tablet dosage.
Calculating Dosage of Penicillin G Potassium
To determine how many tablets of penicillin G potassium are needed, we need to go through several steps of unit conversion.
Convert units to micrograms (mcg):Therefore, the patient needs to take 42 tablets over the course of one week.
The probability is 1% that an electrical connector that is kept dry fails during the warranty period. If the connector is ever wet, the probability of a failure during the warranty period is 5%. If 90% of the connectors are kept dry and 10% are wet, what proportion of connectors fail during the warranty period?
Answer:
A 1.4% of the total connectors are expected to fail during the warranty period.
Step-by-step explanation:
Let's assume a population of 1000 connectors (to make the math easiest) and let's analize the dry connectors.
Of the 1000 connectors, 900 are kept dry. and of that number, 9 are the ones that fails during the warranty period. (90% of 1000 is 900. 1% of 900 is 9)
Of the 1000 connectors, 100 are wet. And of that number, 5 are connectors that will fail during the warranty period. (10% of 1000 is 100, and 5% of 100 is 5)
So overall we have 14 connectors that will fail from 1000 connectors.
That is a 1.4% of the total samples.
Let p, q, and r represent the following statements"
p : Sam has pizza last night.
q : Chris finished her homework
. r : Pat watched the news this morning.
Give a formula (using appropriate symbols) for each of these statements:
a) Sam had pizza last night if and only if Chris finished her homework.
b) Pat watched the news this morning iff Sam did not have pizza last night.
c) Pat watched the news this morning if and only if Chris finished her homework and Sam did not have pizza last night.
d) In order for Pat to watch the news this morning, it is necessary and sufficient that Sam had pizza last night and Chris finished her homework. Express in words the statements represented by the following fomulas.
e) q ⇔ r
f) p ⇔ (q ∧ r)
g) (¬p) ⇔ (q ∨ r)
h) r ⇔ (p ∨ q)
In logic, own symbols are used in order to be able to represent the relations between propositions in a general and independent way to the proposition, in order to be able to find the relationship process that operates in the communicated message, the propositional logic.
For this purpose there are, among others, the following logical operators: conjunction (and) ∧, disjunction (or) ∨, denial (not) ¬, conditional (if - then) ⇒ and double conditional (if and only if, iff) ⇔.
So for this case we have:
Answer
a) Sam had pizza last night if and only if Chris finished her homework.
p⇔q
b) Pat watched the news this morning iff Sam did not have pizza last night.
r⇔¬p
c) Pat watched the news this morning if and only if Chris finished her homework and Sam did not have pizza last night.
r⇔(q∧¬p)
d) In order for Pat to watch the news this morning, it is necessary and sufficient that Sam had pizza last night and Chris finished her homework.
r⇔(p∧q)
e) q ⇔ r
Chris finished his homework if and only if Pat watched the news this morning
f) p ⇔ (q ∧ r)
Sam had pizza last night if and only if Chris finished his homework and Pat watched the news this morning
g) (¬p) ⇔ (q ∨ r)
Sam didn't have pizza last night if and only if Chris finished his homework or Pat watched the news this morning
h) r ⇔ (p ∨ q)
Pat watched the news this morning if Sam had pizza last night or Chris finished his homework
find the area of a rectangle with the given base and height
5ft,6 in
Answer:
360 inch
Step-by-step explanation:
Step 1: Converting
6inch = 6inch
5ft = 60inch
Step 2: Finding the area
Area of a rectangle = L x B
A = 6 x 60
A = 360inch
Find the inverse of 12 modulo 19
Answer:
8 is the modular inverse of 12 mod 19 since 12*8 mod 19 ≡ 1.
Step-by-step explanation:
We need to find the inverse of 12 modulo 19.
[tex]12^{-1}(\text{ mod 19})[/tex]
If a is an integer and m is modulo, then the modular multiplicative inverse of a modulo m is an integer b such that
[tex]a\times b\equiv 1(\text{ mod m})[/tex]
Substitute different values of b and check whether that remainder is 1 after modulo 19.
At b=1,
[tex]12\times 1\equiv 12(\text{ mod 19})[/tex]
At b=2,
[tex]12\times 2\equiv 5(\text{ mod 19})[/tex]
At b=3,
[tex]12\times 3\equiv 17(\text{ mod 19})[/tex]
At b=4,
[tex]12\times 4\equiv 10(\text{ mod 19})[/tex]
At b=5,
[tex]12\times 5\equiv 3(\text{ mod 19})[/tex]
At b=6,
[tex]12\times 6\equiv 15(\text{ mod 19})[/tex]
At b=7,
[tex]12\times 7\equiv 8(\text{ mod 19})[/tex]
At b=8,
[tex]12\times 8\equiv 1(\text{ mod 19})[/tex]
Therefore, 8 is the modular inverse of 12 mod 19 since 12*8 mod 19 ≡ 1.
An elementary school class polled 198 people at a shopping center to determine how many read the Daily News and how many read the Sun Gazette. They found the following information: 171 read the Daily News, 40 read both, and 18 read neither. How many read the Sun Gazette?
Answer: 49 people
Step-by-step explanation:
First you need to separate the people that read both from the people that read the daily news. Do this by subtracting 171 - 40 = 131.
Now you can subtract the total amount of people that only read the Daily Mews from the total amount of people polled (198-131=67)
From here you need to subtract the amount of people that read neither from the remaining total. (67-18=49)
this is where you get the 49 people out of 198 that read the Sun Gazette
The number of people who read the Sun Gazette is calculated by subtracting the number of people who only read the Daily News and those who read neither from the total polled. The result is 49 Sun Gazette readers.
Explanation:This problem involves the mathematics concept of set theory, specifically union and intersection of sets. Here, the total number of people polled are considered as the 'Universal Set'. The 'Daily News readers' and the 'Sun Gazette readers' are the two subsets of this Universal set.
The data given can be interpreted as follows:
Total number of people surveyed (universal set) = 198 Number of people who read the Daily News = 171 Number of people who read both newspapers = 40 Number of people who read neither = 18
Since 40 people read both, they are being counted twice in the 171 (Daily News readers) figure. Hence, we subtract 40 from 171.
The number of people who only read the Daily News = 171 - 40 = 131
We subtract this number and those who read neither from the total polled to find the number of Sun Gazette readers.
The number of Sun Gazette readers = 198 - 131 (only Daily News readers) - 18 (neither) = 49 Sun Gazette readers
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Consider all 5 letter "words" made from the full English alphabet. (a) How many of these words are there total? (b) How many of these words contain no repeated letters? (c) How many of these words start with an a or end with a z or both (repeated letters are allowed)?
Answer:
a) There are 11,881,376 of these words in total.
b) 7,893,600 of these words contain no repeated letters.
c) 896,376 of these words start with an a or end with a z or both.
Step-by-step explanation:
The English alphabet has 26 letters.
Our word has the following format
C1 - C2 - C3 - C4 - C5
C1 is the first character, C2 the second, etc...
(a) How many of these words are there total?
Each of C1, C2,... can be 26. So the total is
[tex]T = (26)^{5} = 11,881,376[/tex]
There are 11,881,376 of these words in total.
(b) How many of these words contain no repeated letters?
C1 can be any letter. C2 can be any letter bar the letter at C1. C3 any other than C2, C1... So
26-25-24-23-22
[tex]T = 26*25*24*23*22 = 7,893,600[/tex]
7,893,600 of these words contain no repeated letters.
(c) How many of these words start with an a or end with a z or both (repeated letters are allowed)?
[tex]T = T_{1} + T_{2} + T_{3}[/tex]
[tex]T_{1}:[/tex] Start with a, end with any letter other than z.
1-26-26-26-25
[tex]T_{1} = 26^{3}*25 = 439,400[/tex]
[tex]T_{2}:[/tex]End with z, start with any other letter than A
25-26-26-26-1
[tex]T_{2} = 26^{3}*25 = 439,400[/tex]
[tex]T_{3}:[/tex]Start with A, end with Z
1-26-26-26-1
[tex]T_{3} = 17,576[/tex]
[tex]T = T_{1} + T_{2} + T_{3} = 439,400 + 439,400 + 17,576 = 896,376[/tex]
896,376 of these words start with an a or end with a z or both.
The vector y = ai + bj is perpendicular to the line ax + by = c. Use this fact to find an equation of the line through P perpendicular to v. Then draw a sketch of the line including v as a vector starting at the origin. P(-1,-9)v = -4i + j The equation of the line is _____.
Answer:
[tex]-4x+y=-5[/tex]
Step-by-step explanation:
It is given that he vector y = ai + bj is perpendicular to the line ax + by = c.
Given given point is P(-1,-9) and given vector is v = -4i + j.
Here, a = -4 and b= 1.
We need to find the equation of the line through P perpendicular to v.
Using the above fact, the vector v = -4i + j is perpendicular to the line
[tex](-4)x+(1)y=c[/tex]
[tex]-4x+y=c[/tex] ... (1)
The line passes through the point (-1,-9).
Substitute x=-1 and y=-9 in the above equation to find the value of c.
[tex]-4(-1)+(-9)=c[/tex]
[tex]4-9=c[/tex]
[tex]-5=c[/tex]
The value of c is -5.
Substitute c=-5 in equation (1).
[tex]-4x+y=-5[/tex]
Therefore the equation of line which passes through P and perpendicular to v is -4x+y=-5.
A survey of 255 adults found that during the last year, 40 traveled by plane but not by train, 75 traveled by train but not by plane, 50 traveled by bus but not by plane or by train, 55 traveled by bus and plane, 35 traveled by all three, and 170 traveled by plane or train. How many did not travel by any of these modes of transportation?
Answer:
35 people did no travel by any of those modes of transportation.
Step-by-step explanation:
Reviewing the groups described, we can see there are 170 that traveled by either plane or train and 50 traveled by bus, but not plane or train.
This means that the 50 people that traveled by bus are not included in the group of 170 that used either plane or train.
Since those two groups include people that used bus (50 people) and people that used either plane or train (170 people), the union of these two sets include all the people that used at least one of the tree modes of transportation mentioned (bus, plane or train).
The result of the union between these two sets is 50+170=220 people which travel by at least one of those modes of transportation.
Since the total number of surveyed adults is 255, the number of people that did not travel by any of these modes of transportations is 255-220=35 people.
let f(x) be a func. satisfying f(-x)=f(x) for all real x.if f"(x) exist, find its value.
Answer:
[tex]f''(x)=f''(-x)[/tex]
Step-by-step explanation:
A function satisfying the equation [tex]f(x)=f(-x)[/tex] is said to be an even function. This denomination comes from the fact that the same relation is satisfied for functions of the form [tex]x^{n}[/tex] with [tex]n[/tex] even. Observe that if [tex]f[/tex] is twice differentiable we can derivate using the chaing rule as follows:
[tex]f(x)=f(-x)[/tex] implies [tex]f'(x)=f'(-x)\cdot (-1)=-f'(-x)[/tex]
Applying the chain rule again we have:
[tex]f'(x)=-f'(-x)[/tex] implies [tex]f''(x)=-f''(-x)\cdot (-1)=f''(-x)[/tex]
So we have that function [tex]f''(x)[/tex] is also an even function.
The Social Security Administration increased the taxable wage base from $106,800 to $110,100. The 6.2% tax rate is unchanged. Joe Burns earned over $120,000 each of the past two years.
a.
What is the percent increase in the base?(Round your answer to the nearest hundredth percent.)
Percent increase
%
b.
What is Joe’s increase in Social Security tax for the new year?(Round your answerto the nearest cent.)
Increase in social security tax
$
Answer:
The Social Security Administration increased the taxable wage base from $106,800 to $110,100.
A:
The percent increase in the base =
[tex]\frac{110100-106800}{106800}\times100=3.09[/tex]%
B:
Previous year tax = [tex]0.062\times106800=6621.60[/tex] dollars
This year tax = [tex]0.062\times110100=6826.20[/tex] dollars
Increase in tax = [tex]6826.20-6621.60=204.60[/tex] dollars
Hence, Joe’s increase in Social Security tax for the new year is $204.60.
The taxable wage base increased by 3.09%. The additional amount that Joe will pay in Social Security tax due to this increase is $204.60.
Explanation:The percent increase in the base can be found by subtracting the old base from the new base, dividing the difference by the old base, and finally multiplying by 100 to convert to a percentage. So, the percent increase is: ((110,100 - 106,800) / 106,800 ) * 100 = 3.09%.
To calculate Joe's increase in Social Security tax, you need to determine the taxable increase, which is the difference between the new and old taxable wage bases, and then multiply this amount by the tax rate. For Joe, the taxable increase is $3,300 (110,100 - 106,800). Therefore, Joe's increase in Social Security tax is: 3,300 * 6.2% = $204.60.
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Let f and g be decreasingfunctions for all real numbers. Prove
that f o g isincreasing.
Answer with Step-by-step explanation:
We are given that two functions f and g are decreasing function for all real numbers .
We have to prove that fog is increasing.
Increasing Function: If [tex]x_1 \leq x_2[/tex]
Then, [tex]f(x_1)\leq f(x_2)[/tex]
Decreasing function: if [tex]x_1\leq x_2[/tex]
Then, [tex]f(x_1)\geq f(x_2)[/tex]
Suppose [tex]f(x)=-x[/tex]
[tex]g(x)=-2x[/tex]
fog(x)=f(g(x))=f(-2x)=-(-2x)=2x
Therefore, f and g are decreasing function for all real numbers then fog is increasing.
Hence, proved
Mark the statement either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If the statement is true, give a justification. If v1,…,v4 are in R4 and {v1,v2,v3} is linearly dependent then {v1,v2,v3,v4} is also linearly dependent.
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
(A) True. Because v3 = 2v1 + v2, v4 must be the zero vector. Thus, the set of vectors is linearly dependent.
(B) True. The vector v3 is a linear combination of v1 and v2, so at least one of the vectors in the set is a linear combination of the others and set is linearly dependent.
(C) True. If c1 =2, c2 = 1, c3 = 1, and c4 = 0, then c1v1 + ........... + c4v4 =0. The set of vectors is linearly dependent.
(D) False. If v1 = __, v2 =___, v3 =___, and v4 = [1 2 1 2], then v3 = 2v1 + v2 and {v1, v2, v3, v4} is linearly independent.
Answer with Step-by-step explanation:
We are given that [tex]v_1,v_2,..,v_4[/tex] are in [tex]R^4[/tex] and [tex]v_1,v_2,v_3[/tex] is linearly dependent then {v_1,v_2,v_3,v_4}[/tex] is also linearly dependent.
We have to find that given statement is true or false.
Dependent vectors:Dependent vectors are those vectors in which atleast one vector is a linear combination of other given vectors.
Or If we have vectors [tex]x_1,x_2,....x_n[/tex]
Then their linear combination
[tex]a_1x_1+a_2x_2+.....+a_nx_n=0[/tex]
There exist at least one scalar which is not zero.
If [tex]v_1,v_2,v_3[/tex] are dependent vectors then
[tex]a_1v_1+a_2v_2+a_3v_3=0[/tex] for scalars [tex]a_1,a_2,a_3[/tex]
Then , by definition of dependent vectors
There exist a vector which is not equal to zero
If vector [tex]v_3[/tex] is a linear combination of [tex]v_1\;and \;v_2[/tex], So at least one of vectors in the set is a linear combination of others and the set is linearly dependent.
Hence, by definition of dependent vectors
{[tex]v_1,v_2,v_3,v_4[/tex]} is linearly dependent.
Option B is true.
Draw the top view of this
figure.
Counting 5-card hands from a deck of standard playing cards. A 5-card hand is drawn from a deck of standard playing cards. How many 5-card hands have at least one club? (b) Hown -card hands have at least two cards with the same rank?
Answer:
5-card hands with at least one club: [tex] {52 \choose 5}-{39 ]choose 5}[/tex]
5-card hands with at least two cards of the same rank: [tex]{52 \choose 5}-{13 \choose 5}4^5[/tex]
Step-by-step explanation:
To determine how many 5-card hands have at least one club, we can count how many do NOT have at least one club, and then subtract that from the total amount of 5-card hands that there are.
A 5-card hand that doesn't have at least one club, is one whose 5 cards are from spades,hearts or diamonds. Since a standard deck of cards has 13 clubs, 39 of the cards are spades, hearts of diamonds. Getting a 5-card hand out of those cards, is choosing 5 cards out of those 39 cards. So there are [tex] {39 \choose 5}[/tex] 5-card hands without any clubs.
The total amount of 5-card hands is [tex]{52 \choose 5}[/tex], since a 5-card hand is simply a group of 5 cards out of the full deck, which has 52 cards.
Therefore the number of 5-card hands that have at least one club is [tex] {52 \choose 5}-{39 \choose 5}[/tex].
To determine how many 5-card hands have at least two cards with the same rank we can follow the same approach. We determine how many 5-card hands have NO cards with the same rank, and the subtract that out of the total amount of 5-card hands.
A 5-card hand that doesn't have cards of the same rank, is a group of 5 cards all from different ranks. Such hand can be made then by choosing first which 5 different ranks are going to be present in the hand, out of the 13 available ranks. So there are [tex] {13 \choose 5}[/tex] possible combinations of ranks. Then, choosing which card from each of the chosen ranks is the one that is going to be in the hand, is choosing which of 4 cards from EACH rank is going to be in the hand. So for each rank there are 4 availble choices, and so there are [tex]4^5[/tex] possible ways to choose the specific cards from each rank that will be in the hand. So the amount of 5-card hands with all ranks different is [tex] {13 \choose 5}\cdot{4^5}[/tex]
Therefore the amount of 5-card hands with at least two cards with the same rank is [tex]{52 \choose 5}-{13 \choose 5}\cdot4^5[/tex]
Answer:
b
Step-by-step explanation:
A wind turbine is located at the top of a hill where the wind blows steadily at 12 m/s, and stands 37 m tall. The air then exits the turbine at 9 m/s and the same elevation. Find the power generated by the wind if the mass flow rate is 137 kg/s. Report your answer in kW and to 2 decimal places.
Answer:
4.3155 kW
Step-by-step explanation:
Given,
speed of wind when enters into the turbine, V = 12 m/s
speed of wind when exits from the turbine, U = 9 m/s
mass flow rate of the wind = 137 kg/s
According to the law of conservation of energy
Energy generated = change in kinetic energy
Hence,energy generated in 1 sec can be given by
[tex]E\ =\ \dfrac{1}{2}.m.V^2\ -\ \dfrac{1}{2}.m.U^2[/tex]
[tex]=\ \dfrac{1}{2}\times 137\times 12^2\ -\ \dfrac{1}{2}\times 137\times 9^2[/tex]
[tex]=\ 9864\ -\ 5548.5[/tex]
= 4315.5 J
So, the power generated in 1 sec will be given by
[tex]P\ =\ \dfrac{energy\ generated}{time}[/tex]
[tex]=\ \dfrac{4315.5}{1}[/tex]
= 4315.5 W
= 4.13 kW
So, the power generated will be 4.13 kW.
Use the graph below to determine the number of solutions the system has.
Answer:
one
Step-by-step explanation:
I think, there were 4 lines and the two lines are reaching each other
There are 100 centimeters in a meter, 2.54 centimeters in an inch, 12 inches in a foot, 3 feet in a yard, and 1760 yards in a mile 1) A race is 6.2 miles long. Use conversion factors and dimensional analysis to determine the length of the race measured in: a) yards. b) meters.
Answer:
a) 10912 Yards
b) 9977.93 meters
Step-by-step explanation:
We are given the following information in the question.
1 Meter = 100 Centimeter
1 Inch = 2.54 Centimeter
1 Foot = 12 Inch
1 Yard = 3 Foot
1 Mile = 1760 Yards
The length of race is 6.2 miles.
a) We have to find length of race in yards.
Since we know 1 Mile = 1760 Yards
Thus, 6.2 Miles = 6.2 × 1760 Yards = 10912 Yards
Thus , the race is 10912 yards in length.
b) We have to calculate the length of the race in meters.
Since we know 1 Mile = 1760 Yards
Thus. 6.2 miles = [tex]\frac{(6.2)(1760)(3)(12)(2.54)}{100}[/tex] = 9977.93 meters
Triphasil-28 birth control tablets are taken sequentially, 1 tablet per day for 28 days, with the tablets containing the following: Phase 1: 6 tablets, each containing 0.050 mg of levonorgestrel and 0.030 mg ethinyl estradiol Phase 2: 5 tablets, each containing 0.075 mg of levonorgestrel and 0.040 mg ethinyl estradiol Phase 3: 10 tablets, each containing 0.125 mg of levonorgestrel and 0.030 mg ethinyl estradiol Then 7 inert tablets (no drug). How many total milligrams each of levonorgestrel and ethinyl estradiol are taken during the 28-day period?
Over the 28-day cycle of taking Triphasil-28 birth control tablets, a total of 1.925 mg of levonorgestrel and 0.680 mg of ethinyl estradiol are consumed.
Explanation:The Triphasil-28 birth control tablets are taken in a sequence of 21 active pills containing hormones and 7 inactive pills (placebos). To calculate the total amount of levonorgestrel and ethinyl estradiol consumed over the 28-day period, each phase needs to be taken into account.
In phase 1, there are 6 tablets each with 0.050 mg of levonorgestrel and 0.030 mg ethinyl estradiol. So, 6 tablets * 0.050 mg = 0.300 mg of levonorgestrel and 6 tablets * 0.030 mg = 0.180 mg of ethinyl estradiol.
In phase 2, there are 5 tablets each with 0.075 mg of levonorgestrel and 0.040 mg ethinyl estradiol. So, 5 tablets * 0.075 mg = 0.375 mg of levonorgestrel and 5 tablets * 0.040 mg = 0.200 mg of ethinyl estradiol.
In phase 3, there are 10 tablets each with 0.125 mg of levonorgestrel and 0.030 mg ethinyl estradiol. So, 10 tablets * 0.125 mg = 1.250 mg of levonorgestrel and 10 tablets * 0.030 mg = 0.300 mg of ethinyl estradiol.
Adding up all the quantities, we have a total of 0.300 mg + 0.375 mg + 1.250 mg = 1.925 mg of levonorgestrel and 0.180 mg + 0.200 mg + 0.300 mg = 0.680 mg of ethinyl estradiol consumed over the 21 active days of the 28-day cycle.
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According to Newton’s law of cooling, the temperature u(t) of an object satisfies the differential equationdu/dt= −k(u−T)where T is the constant ambient temperature and k is a positive constant. Suppose that the initial temperature of the object is u(0)= u0.(a) Find the temperature of the object at any time. (I know how to resolve this)
Answer:
[tex]u(t)=T+(u_{0}-T})e^{-kt}[/tex]
Step-by-step explanation:
We know:
[tex]\frac{du}{dt} = -k(u-T)[/tex]
We integrate in order to find u(t):
[tex]\int\limits^u_{u_{0}} {\frac{1}{-k(u-T)} \, du } = \int\limits^t_0 \, dt[/tex]
[tex]ln(\frac{u-T}{u_{0}-T} )=-kt\\[/tex]
[tex]u(t)=T+(u_{0}-T})e^{-kt}[/tex]
Suppose A is a 3 x 3 matrix such that det (A) = 9. Prove det (3 (A-!') is equal to 3
Answer: The proof is done below.
Step-by-step explanation: Given that A is a 3 x 3 matrix such that det (A) = 9.
We are to prove the following :
[tex]det(3A^{-1})=3.[/tex]
For a non-singular matrix B of order n, we have two two properties of its determinant :
[tex](i)~det(B^{-1})=\dfrac{1}{det(B)},\\\\\\(ii)~det(kB)=k^ndet(B),~\textup{k is a scalar.}[/tex]
Therefore, we get
[tex]det(A^{-1})=\dfrac{1}{det(A)}=\dfrac{1}{9},[/tex]
and so,
[tex]det(3A^{-1})~~~~~~~[\textup{since A is of order 3}]\\\\=3^3det(A^{-1})\\\\=27\times\dfrac{1}{9}\\\\=3.[/tex]
Hence proved.
Estimate the sum. Use benchmarks with decimal parts of 0, 0.25, 0.50, or 0.75.
2.52+4.03
A. 6.50
B. 6.75
C. 7
An astronaut with a mass of 80.70 kg is standing still on the surface of the Mars (Mars has an acceleration due to gravity of 3.80 m / s^2). How many pounds force is she exerting on the martian surface?
Answer:
Force, F = 68.93 pounds
Step-by-step explanation:
Given that,
Mass of the astronaut, m = 80.70 kg
Acceleration due to gravity on Mars, [tex]a=3.8\ m/s^2[/tex]
The force acting on him is given by using second law of motion as :
[tex]F=m\times a[/tex]
[tex]F=80.70\ kg\times 3.8\ m/s^2[/tex]
F = 306.66 N
Since, 1 newton = 0.224 pound
F = 306.66 N = 68.93 pounds
So, 68.93 pounds force she exerts on the martian surface. Hence, this is the required solution.
In what type quadrilateral are the diagonals NOT
alwayscongruent to each other?
Answer:
Parallelogram.Rhombus.Trapezoid.Kite.Step-by-step explanation:
There are six basic types of quadrilaterals:
Rectangle. Square. Parallelogram. Rhombus. Trapezoid and isosceles trapezoid. Kite.Then we have:
Quadrilaterals with NOT ALWAYS congruent diagonals:
Parallelogram.Rhombus.Trapezoid.Kite.Quadrilaterals with ALWAYS congruent diagonals:
Rectangle.Square.Isosceles trapezoid.