Answer:
The set of solutions is [tex]\{\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}12\\-7-r\\r\end{array}\right]: \text{r is a real number} \}[/tex]
Step-by-step explanation:
The augmented matrix of the system is [tex]\left[\begin{array}{ccccc}3&6&6&-9\\-2&-3&-3&3\end{array}\right][/tex].
We will use rows operations for find the echelon form of the matrix.
In row 2 we subtract [tex]\frac{2}{3}[/tex] from row 1. (R2- 2/3R1) and we obtain the matrix [tex]\left[\begin{array}{cccc}3&6&6&-9\\0&1&1&-7\end{array}\right][/tex]We multiply the row 1 by [tex]\frac{1}{3}[/tex].Now we solve for the unknown variables:
[tex]x_2+x_3=-7[/tex], [tex]x_2=-7-x_3[/tex][tex]x_1+2x_2+2x_3=-2[/tex], [tex]x_1+2(-7-x_3)+2x_3=-2[/tex] then [tex]x_1=12[/tex]The system has a free variable, the the system has infinite solutions and the set of solutions is [tex]\{\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}12\\-7-r\\r\end{array}\right]: \text{r is a real number} \}[/tex]
Find the number of 3-digit numbers formed using the digits 1 to 9, without repetition, such the numbers either have all digits less than 5 or all digits greater than 4.
Answer: 120
Step-by-step explanation:
The total number of digits from 1 to 9 = 10
The number of digits from less than 5 (0,1,2,3,4)=5
Since repetition is not allowed so we use Permutations , then the number of 3-digit different codes will be formed :-
[tex]^5P_3=\dfrac{5!}{(5-3)!}=\dfrac{5\times4\times3\times2!}{2!}=5\times4\times3=60[/tex]
The number of digits from greater than 4 (5,6,7,8,9)=5
Similarly, Number of 3-digit different codes will be formed :-
[tex]^5P_3=60[/tex]
Hence, the required number of 3-digit different codes = 60+60=120
What weight of magnesium chloride (MgCl2, formula weight 95.3; Mg2, atomic weight = 243; Cl, atomic weight-35.5) is required to prepare 1,000 mL of a solution that contains 5.0 mEq of magnesium?
Answer:
238.25 mg
Step-by-step explanation:
Given:
Molar mass of MgCl₂ = 95.3
atomic weight of Mg₂ = 243
Atomic weight of Cl = 35.5
Volume of solution required = 5.0 mEq of magnesium
Now,
mEq = [tex]\frac{\textup{Weight in mg\timesValency}}{\textup{Atomic mass}}[/tex]
on substituting the values, we get
5 = [tex]\frac{\textup{Weight in mg\times2}}{\textup{95.3}}[/tex]
or
weight of magnesium chloride = 238.25 mg
Therefore,
the required mass of MgCl₂ is 238.25 mg
The "Double-R-7" Ranch has a new owner. The 20 animals, all hummingbirds and mice, are dismayed, as they have heard that he is both foolish and inexperienced. Not being quite sure what he was looking for, however, he checked on the health of his animals by inspecting all their feet. There were 64 feet in all. How many hummingbirds and how many mice are on this ranch?
Please help immediately I'm so confused!!! :(
A survey of 1,107 tourists visiting Orlando was taken. Of those surveyed:
268 tourists had visited the Magic Kingdom
258 tourists had visited Universal Studios
68 tourists had visited both the Magic Kingdom and LEGOLAND
79 tourists had visited both the Magic Kingdom and Universal Studios
72 tourists had visited both LEGOLAND and Universal Studios
36 tourists had visited all three theme parks
58 tourists did not visit any of these theme parks
How many tourists only visited the LEGOLAND (of these three)?
Answer:
602 tourists visited only the LEGOLAND.
Step-by-step explanation:
To solve this problem, we must build the Venn's Diagram of this set.
I am going to say that:
-The set A represents the tourists that visited LEGOLAND
-The set B represents the tourists that visited Universal Studios
-The set C represents the tourists that visited Magic Kingdown.
-The value d is the number of tourists that did not visit any of these parks, so: [tex]d = 58[/tex]
We have that:
[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]
In which a is the number of tourists that only visited LEGOLAND, [tex]A \cap B[/tex] is the number of tourists that visited both LEGOLAND and Universal Studies, [tex]A \cap C[/tex] is the number of tourists that visited both LEGOLAND and the Magic Kingdom. and [tex]A \cap B \cap C[/tex] is the number of students that visited all these parks.
By the same logic, we have:
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
This diagram has the following subsets:
[tex]a,b,c,d,(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C)[/tex]
There were 1,107 tourists suveyed. This means that:
[tex]a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1,107[/tex]
We start finding the values from the intersection of three sets.
The problem states that:
36 tourists had visited all three theme parks. So:
[tex](A \cap B \cap C) = 36[/tex]
72 tourists had visited both LEGOLAND and Universal Studios. So:
[tex](A \cap B) + (A \cap B \cap C) = 72[/tex]
[tex](A \cap B) = 72 - 36[/tex]
[tex](A \cap B) = 36[/tex]
79 tourists had visited both the Magic Kingdom and Universal Studios
[tex](B \cap C) + (A \cap B \cap C) = 79[/tex]
[tex](B \cap C) = 79 - 36[/tex]
[tex](B \cap C) = 43[/tex]
68 tourists had visited both the Magic Kingdom and LEGOLAND
[tex](A \cap C) + (A \cap B \cap C) = 68[/tex]
[tex](A \cap C) = 68 - 36[/tex]
[tex](A \cap C) = 32[/tex]
258 tourists had visited Universal Studios:
[tex]B = 258[/tex]
[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]
[tex]258 = b + 43 + 36 + 36[/tex]
[tex]b = 143[/tex]
268 tourists had visited the Magic Kingdom:
[tex]C = 268[/tex]
[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]
[tex]268 = c + 32 + 43 + 36[/tex]
[tex]c = 157[/tex]
How many tourists only visited the LEGOLAND (of these three)?
We have to find the value of a, and we can do this by the following equation:
[tex]a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1,107[/tex]
[tex]a + 143 + 157 + 58 + 36 + 32 + 43 + 36 = 1,107[/tex]
[tex]a = 602[/tex]
602 tourists visited only the LEGOLAND.
A simple random sample of men over age 18 is taken to estimate the mean weight of all adult males.
Is this study
A. REPRESENTATIVE?
B. NON-REPRESENTATIVE?
Answer:
A. Representative
Step-by-step explanation:
The sample is a subset of the population. We take samples because it is easy to analyze samples as compared to population and it is less time taken. Further, Sample is said to be good if the sample is the representative of the population.
Simple Random Sampling is the sampling where samples are chosen randomly, where each unit has an equal chance of being selected in a sample.
Since, observer is taken the weight of men over age 18. Thus, it is good sample which represent whole population.
One of the events at a swim meet is the 500 meter freestyle. Use conversion factors and dimensional analysis to determine the length of this race in: 2) a) feet. b) miles.
Answer:
1) Distance in feet equals 1640.4 feet
2) Distance in miles equals 0.3107 miles.
Step-by-step explanation:
From the basic conversion factors we know that
1 meter = 3.2808 feet
Thus by proportion we conclude that
[tex]500meters=500\times 3.2808feet\\\\\therefore 500meters=1640.4feet[/tex]
Similarly using the basic conversion factors we know that
1 mile = 1609.34 meters
thus we conclude that 1 meter =[tex]\frac{1}{1609.34}[/tex] miles
Hence by proportion 500 meters = [tex]500\times \frac{1}{1609.34}=0.3107[/tex] miles
A = {a, b, c, d}
B = {c, d, e, f, g, h}
LaTeX: A\cup BA∪B is the set of all elements that belong to A or B.
When given this information Johnny stated that union of A and B is {a, b, e, f, g, h}.
POST (by Thursday): Is Johnny correct? If not, correct his answer. Either way give a every day example to justify your answer. After posting your response, please REPLY to the responses of your group members explaining whyyou agree or disagree with their response, or asking clarifying questions.
Answer:
Johnny is wrong.
[tex]A \cup B = \left\{a,b,c,d,e,f,g,h\right\}[/tex]
Step-by-step explanation:
Johnny is wrong.
A better definition would be: [tex]A \cup B[/tex] is the set of all elements that belong to at least one A or B.
So, the elements that belong to both A and B, like c and d in this exercise, also belong to [tex]A \cup B[/tex].
So:
[tex]A \cup B=\left\{a,b,c,d,e,f,g,h\right\}[/tex]
Mrs. Brown needs $5,000 in three years. If the interest rate on her investment account is 8.4% compounded monthly, how much should she put into her account at the end of each month to have $5,000 in three years?
Give answer in US dollars and cents rounded to the nearest cent. Do NOT enter "$" sign
Answer:
ohhh you have the right time and I am so glad to see it in a better time and if it was not to do you have no doubt
Step-by-step explanation:
your body has to have your body to have the best results you need for kids in a group that can help with a lot to learn about it as you do so much of the time it works out to you to get the job you need for the next few years or 8īijjhguu is not just the best person but the meaning and skills of a person is the meaning of life and the meaning is the only 3AM in a school that you need a lot of a job 3to to do with you need to get your mind off to get your job done with your mind and then you have to do it right now and.
Calculate the Laplace transforms of the following from the definition. 1. y = t^2. y = t^3
Answer:
1) [tex]L(y)=\frac{2}{s^{3}}[/tex]
2) [tex]L(y)=\frac{6}{s^{4}}[/tex]
Step-by-step explanation:
To find : Calculate the Laplace transforms of the following from the definition ?
Solution :
We know that,
Laplace transforms of [tex]t^n[/tex] is given by,
[tex]L(t^n)=\frac{n!}{s^{n+1}}[/tex]
1) [tex]y=t^2[/tex]
Laplace of y,
[tex]L(y)=L(t^2)[/tex] here n=2
[tex]L(y)=\frac{2!}{s^{2+1}}[/tex]
[tex]L(y)=\frac{2}{s^{3}}[/tex]
2) [tex]y=t^3[/tex]
Laplace of y,
[tex]L(y)=L(t^3)[/tex] here n=3
[tex]L(y)=\frac{3!}{s^{3+1}}[/tex]
[tex]L(y)=\frac{3\times 2}{s^{4}}[/tex]
[tex]L(y)=\frac{6}{s^{4}}[/tex]
Use De Morgan's Law to negate the following statements. a. Vx(x >5) b. 3.(x2+2x + 1 = 0)
Answer:
a) [tex]\neg \forall x :x>5 \equiv \forall x:x\leq 5[/tex]
b) [tex]\neg [x^2+2x+1=0]\equiv x^2+2x+1\neq0[/tex] or the set [tex]\{x:x\neq-1\}[/tex]
Step-by-step explanation:
First, notice that in both cases we have to sets:
a) is the set of all real numbers which are higher than 5 and in
b) the set is the solution of the equation [tex]x^2+2x+1=0[/tex] which is the set [tex]x=-1[/tex]
De Morgan's Law for set states:
[tex]\overline{\rm{A\cup B}} = \overline{\rm{A}} \cap \overline{\rm{B}}\\[/tex], being [tex]\overline{\rm{A}}[/tex] and [tex]\overline{\rm{B}}[/tex] are the complements of the sets [tex]A[/tex] and [tex]B[/tex]. [tex]\cup[/tex] is the union operation and [tex]\cap[/tex] the intersection.
Thus for:
a) [tex]\neg \forall x :x>5 \equiv \forall x:x\leq 5[/tex]. Notice that [tex]\forall x:x\leq 5[/tex] is the complement of the given set.
b) [tex]\neg [x^2+2x+1=0]\equiv x^2+2x+1\neq0[/tex] which is the set [tex]B = \{x:x\neq-1\}[/tex]
Suppose that A and B are square matrices and that ABC is invertible. Show that each of A, B, and C is invertible.
Answer:
Step-by-step explanation:
Let A, B and C be square matrices, let [tex]D = ABC[/tex]. Suppose also that D is an invertible square matrix. Since D is an invertible matrix, then [tex]det (D) \neq 0[/tex]. Now, [tex]det (D) = det (ABC) = det (A) det (B) det (C) \neq 0[/tex]. Therefore,
[tex]det (A) \neq 0[/tex]
[tex]det (B) \neq 0 [/tex]
[tex]det (C) \neq 0[/tex]
which proves that A, B and C are invertible square matrices.
The cost of stainless steel tubing varies jointly as the length and the diameter of the tubing. If a 5ft length with diameter 2 inches costs $48.00, how much will a 17tf length with diameter 5in. cost?
Answer:
The cost for 17 ft length with diameter 5 in is $408.
Step-by-step explanation:
Consider the provided information.
It is given that the cost of stainless steel tubing varies jointly as the length and the diameter of the tubing. If a 5 foot length with a diameter 2 inches costs $48.00.
let [tex]C\propto L\cdot D[/tex]
[tex]C=k\cdot L\cdot D[/tex]
[tex]48=k\cdot 5\cdot 2[/tex]
[tex]\frac{48}{10}=k[/tex]
[tex]k=4.8[/tex]
For 17 ft length with diameter 5in the cost will be:
[tex]C=k\cdot L\cdot D[/tex]
Substitute the respective values in the above formula.
[tex]C=4.8\cdot 17\cdot 5[/tex]
[tex]C=408[/tex]
Cost = $408
Hence, the cost for 17 ft length with diameter 5in is $408.
What horsepower is required to lift an 8,000 pound aircraft six feet in two minutes?
Answer:
The horsepower required is 235440 watt.
Step-by-step explanation:
To find : What horsepower is required to lift an 8,000 pound aircraft six feet in two minutes?
Solution :
The horsepower formula is given by,
[tex]W=\frac{mgh}{t}[/tex]
Where, W is the horsepower
m is the mass m=8000 pound
g is the gravitational constant g=9.81
t is the time t= 2 minutes
h is the height h=6 feet
Substitute all values in the formula,
[tex]W=\frac{8000\times 9.81\times 6}{2}[/tex]
[tex]W=\frac{470880}{2}[/tex]
[tex]W=235440[/tex]
Therefore, The horsepower required is 235440 watt.
You know of a relative who is addicted to pain killers. Due to his history, 2.3g of Tylenol per day is dangerous. He takes 200 mg scored Tylenol tablets every six hours. For each six hour period (he takes the same number of tablets every six hours), how many scored tablets can the relative take without the amount being dangerous
Answer:
relative can take 2 tablets every six hours or 8 tablets a day without the amount being dangerous
Step-by-step explanation:
Given:
Amount of Tylenol per day that is dangerous = 2.3 g
1 g = 1000 mg
thus,
= 2.3 × 1000 mg
= 2300 mg
Amount of scored Tylenol tablets every six hours = 200 mg
Now,
for 6 hours intervals, total intervals in a day = [tex]\frac{\textup{24}}{\textup{6}}[/tex] = 4
thus,
He takes Tylenol tablets 4 times a day
Now,
let x be the number of tablets taken in every interval
thus,
4x × 200 mg ≤ 2300
or
800x ≤ 2300
or
x ≤ 2.875
hence, relative can take 2 tablets every six hours or 8 tablets a day without the amount being dangerous
A person should take 2 tablets of 200mg Tylenol every six hours to keep the daily dose under the dangerous level of 2.3g.
Explanation:The question is asking how many 200 mg tablets of Tylenol can a person consume every six hours (which is four times a day), while keeping the daily dose under 2.3g, to avoid a risk level that can be dangerous. The first step is to convert the maximum safe dose from grams to milligrams, because the dose per tablet is given in milligrams. 2.3 g equals 2300 milligrams.
Then, to find out the number of safe tablets per dose, you divide the total safe amount by the number of doses per day. So, 2300 milligrams divided by 4 equals 575 milligrams per dose. Last but not least, to find out the number of tablets, you should divide the amount per dose by the amount in each tablet: 575 divided by 200 equals about 2.875.
Since you cannot take a fraction of a tablet, the safest number of tablets to take every six hours would be 2.
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A commercial package contains thirty-six 200-mg tablets of ibuprofen. How many kilograms of ibuprofen were
Final answer:
To find the total kilograms of ibuprofen, multiply 36 tablets by 200 mg each to get 7200 mg, and then divide by 1,000,000 to convert to kilograms, resulting in 0.0072 kg.
Explanation:
To calculate the total amount of ibuprofen in kilograms, you start by finding the total amount in milligrams. Multiply the number of tablets (36) by the amount of ibuprofen per tablet (200 mg).
36 tablets × 200 mg/tablet = 7200 mg
Now, since there are 1,000,000 milligrams in a kilogram, you'll need to divide the total milligrams by 1,000,000 to convert it to kilograms.
7200 mg ÷ 1,000,000 mg/kg = 0.0072 kg
Therefore, the package contains 0.0072 kilograms of ibuprofen.
-. Mario walks 7 blocks from his home to
a restaurant. He then walks back toward
home for five blocks, where he stops at
a bookstore. How many blocks is Mario
from his home?
Answer:
the correct answer is 2 blocks
Mario is 2 blocks away from his home after walking 7 blocks to a restaurant and then 5 blocks back towards home with help of subtraction.
The number of blocks that Mario is from his home can be calculated by finding the difference between the number of blocks he walked away from home and the number of blocks he walked back home. If Mario starts by walking 7 blocks from his home to a restaurant, he is initially 7 blocks away from home. If he then walks 5 blocks back toward home, he is effectively reducing the distance from his home by 5 blocks. You can calculate the new distance to his home by subtracting 5 from 7, which equals 2 blocks. Therefore, Mario is 2 blocks away from his home when he stops at the bookstore.
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18. Naruto practices his harmonica 1 hour, spends hour working on
homework, and takes another hour to do chores every day. How much
longer does Naruto spend practicing his harmonica and doing his chores
than working on homework?
Answer:
If he uses 1 hour to practice harmonica, one our on homework, and another hour doing chores. He will be spending twice as much time practicing harmonica and doing homework than he would spend on chores.
Answer:
2
Step-by-step explanation:
UP
An office has 30 computers. Seventeen of the 30 are Macintosh, and the remaining thirteen are windows. Two computers are randomly selected without replacement. What is the probability that the sample contains exactly one windows machine and exactly one Macintosh? If needed, round to FOUR decimal places. Pr(One Widows and One Macintosh) = ___________
To find the probability, you need to calculate the number of ways to choose one Macintosh and one Windows computer from the given options and divide it by the total number of ways to choose two computers. The probability is 0.507.
Explanation:To find the probability that the sample contains exactly one Windows machine and exactly one Macintosh machine, we can use the concept of combinations. There are a total of 30 computers, out of which 17 are Macintosh and 13 are Windows. The number of ways to choose one Macintosh and one Windows computer can be calculated by multiplying the number of ways to choose one Macintosh from 17 and the number of ways to choose one Windows from 13.
The number of ways to choose one Macintosh from 17 is C(17, 1) = 17 and the number of ways to choose one Windows from 13 is C(13, 1) = 13. Therefore, the total number of ways to choose one Macintosh and one Windows computer is 17 * 13 = 221.
The sample space is the total number of ways to choose two computers from 30, which is C(30, 2) = 435. So the probability of selecting exactly one Windows machine and exactly one Macintosh machine is 221 / 435 = 0.507.
A movie theater manager wants to determine whether popcorn sales have increased since the theater switched from using "butter-flavored topping" to real butter. Historically the average popcorn revenue per weekend day was approximately $3,500. After the theater started using real butter, the manager randomly sampled 12 weekend days and calculated the sample’s summary statistics. The average revenue per weekend day in the sample was approximately $4,200 with a standard deviation of $140. Select the function that would correctly calculate the 90% range of likely sample means.A. 3,500±CONFIDENCE.T(0.10,140,12)B. 4,200±CONFIDENCE.T(0.10,140,12)C. 3,500±CONFIDENCE.NORM(0.10,140,12)D. 4,200±CONFIDENCE.NORM(0.10,140,12)
Answer:
B. 4,200±CONFIDENCE.T(0.10,140,12)
Step-by-step explanation:
We are in posession's of the sample standard deviation, so the t-distribution is used.
The confidence interval is a function of the sample mean and the margin of error.
That is:
[tex]C.I = S_{M} \pm M_{E}[/tex]
In which [tex]S_{M}[/tex] is the sample mean, and the [tex]M_{E}[/tex] is the margin of error, related to the confidence level, the sample's standard deviation and the sample size.
So the correct answer is:
B. 4,200±CONFIDENCE.T(0.10,140,12)
Rewrite the set T = { 5a + 2b; a,b belong Z} as a list of its elements
Answer:
T={...,-3,-2,-1,0,1,2,3,4,....}
Step-by-step explanation:
We are given that T={5a+2b; a,b belongs Z}
We have to rewrite the given set T as a list of its elements
Substitute a=0 and b=0
Then we get 5(0)+2(0)=0
Substitute a=-1 and b=2 then we get
5(-1)+2(2)=-1
Substitute a=2 and b=0
Then , 5(2)+2(0)=10
If a=0 and b=1
Then, 5(0)+2(1)=2
Substitute a=0 and b=2
Then, 5(0)+2(2)=4
If substitute a=1 and b=1
5(1)+2(1)=7
If substitute a=-1 and b=3
Then, we get 5(-1)+2(3)=1
Then, T={...,-3,-2,-1,0,1,2,3,4,....}
If 30 participants each completed a memory task three times – once each after having no, moderate, or high levels of caffeine, calculate the total degrees of freedom.
Answer: 29
Step-by-step explanation:
In statistics, the degrees of freedom gives the number of values that have the freedom to vary.
If the sample size is 'n' then the degree of freedom is given by:-
df= n-1
Given: The number of participants for the sample =30
Then the degree of freedom for this sample=30-1=29
Hence, The total degree of freedom= 29
In the context of the experimental design, the total degrees of freedom would be calculated based on the between-group and within-group variations. With three groups and three repeats, the total degrees of freedom would be 62.
Explanation:When calculating the total degrees of freedom in an experiment where 30 participants each completed a memory task three times under different levels of caffeine, we can determine the degrees of freedom by considering the number of groups and the number of repeats within each group. Since there are three groups (no, moderate, and high levels of caffeine), and each participant completes the task three times, this setup hints at an Analysis of Variance (ANOVA) scenario with repeated measures. However, without the detailed design for the within-subject factors, we can only calculate the between-group degrees of freedom. For three groups, the between-group degrees of freedom would be the number of groups minus one, which would give us 2 degrees of freedom (dfbetween = number of groups - 1).
For within-group or repeated measures, you typically have dfwithin = (number of repeats - 1) × number of subjects. Since each participant repeats the task three times, and there are 30 participants, the calculation would be dfwithin = (3 - 1) × 30 = 2 × 30 = 60 degrees of freedom. Hence, the total degrees of freedom would be the sum of between-group and within-group degrees of freedom, which is 2 + 60 = 62.
Joe's annual income has been increasing each year by the same dollar amount. The first year his income was $17 comma 90017,900, and the 44th year his income was $20 comma 30020,300. In which year was his income $ 30 comma 700 question mark
Answer:
In 17th year, his income was $30,700.
Step-by-step explanation:
It is given that the income has been increasing each year by the same dollar amount. It means it is linear function.
Income in first year = $17,900
Income in 4th year = $20,300
Let y be the income at x year.
It means the line passes through the point (1,17900) and (4,20300).
If a line passes through two points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex], then the equation of line is
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]
The equation of line is
[tex]y-17900=\frac{20300-17900}{4-1}(x-1)[/tex]
[tex]y-17900=\frac{2400}{3}(x-1)[/tex]
[tex]y-17900=800(x-1)[/tex]
[tex]y-17900=800x-800[/tex]
Add 17900 on both sides.
[tex]y=800x-800+17900[/tex]
[tex]y=800x+17100[/tex]
The income equation is y=800x+17100.
Substitute y=30,700 in the above equation.
[tex]30700=800x+17100[/tex]
Subtract 17100 from both sides.
[tex]30700-17100=800x[/tex]
[tex]13600=800x[/tex]
Divide both sides by 800.
[tex]\frac{13600}{800}=x[/tex]
[tex]17=x[/tex]
Therefore, in 17th year his income was $30,700.
Final answer:
Joe's annual income increases by approximately $55.81 each year, starting at $17,900 in the first year. By dividing the desired income minus the starting income by the annual increase, we find that Joe will reach an income of $30,700 in his 230th year.
Explanation:
To determine in which year Joe's income reached $30,700, we need to establish a pattern of how his income has increased over the years. Given that Joe's income started at $17,900 in the first year and was $20,300 in the 44th year, we can calculate the annual increase in his income.
The total increase over 43 years (from year 1 to year 44) is $20,300 - $17,900 = $2,400. To find the annual increase, divide this by the number of years the increase occurred over, which is one less than the total number of years, because the increase starts after the first year. That is:
Annual Income Increase = Total Increase / Number of Years
Annual Income Increase = $2,400 / 43
Annual Income Increase = approximately $55.81 (rounded to two decimal places)
Now we need to calculate the number of years it would take for his income to reach $30,700, starting from $17,900 and increasing at a rate of approximately $55.81 per year.
Years Needed = (Desired Income - Starting Income) / Annual Increase
Years Needed = ($30,700 - $17,900) / $55.81
Years Needed = approximately 229.84, which we round up to 230, because you can't have a partial year in this context.
Therefore, Joe will reach an income of $30,700 in his 230th year of work (adding 230 to the first year).
The length of the hypotenuse of an isosceles right triangle
is30 meters. Find the area of the triangle. Round to thenearest
tenth, if necessary.
Answer: [tex]224.9\ m^2[/tex]
Step-by-step explanation:
An isosceles right triangle is a right triangle having two legs (other than hypotenuse ) of same length .
Given : The length of the hypotenuse of an isosceles right triangle is 30 meters.
Let x be the side length of the other two legs, then by using the Pythagoras theorem for right triangle , we have
[tex](30)^2=x^2+x^2\\\\\Rightarrow\ 900=2x^2\\\\\Rightarrow\ x^2=\dfrac{900}{2}\\\\\Rightarrow\ x^2=450\\\\\Rightarrow\ x=\sqrt{450}=\sqrt{9\times25\times2}=\sqrt{3^2\times5^2\times2}\\\Rightarrow\ x=3\times5\sqrt{2}=15(1.414)=21.21[/tex]
Thus, the other two legs have side length of 21.21 m each.
Now, the area of a right triangle is given by :-
[tex]A=\dfrac{1}{2}\times base\times height\\\\\Rightarrow\ A=\dfrac{1}{2}(21.21)\times(21.21)=224.93205\approx224.9\ m^2[/tex]
Hence, the area of the given isosceles right triangle= [tex]224.9\ m^2[/tex]
Calculate how many Fluid ounces (fl oz) of water is needed if a recipe requires 2 cups of water.
Answer:
Is needed 16 fluid oz of water to fill in 2 cups of water
Step-by-step explanation:
1 cup have 8 fluid oz
2 cups = 2 * 8 fluid oz = 16 fluid oz
Final answer:
The recipe requires 2 cups of water, which is equivalent to 16 fluid ounces since there are 8 fluid ounces in one cup.
Explanation:
To calculate how many fluid ounces of water is needed if a recipe requires 2 cups of water, you need to understand the unit equivalence between cups and fluid ounces. One cup is equivalent to 8 fluid ounces. Since the recipe requires 2 cups, you will multiply the number of cups by the unit equivalence to find the total number of fluid ounces.
Here is the calculation:
2 cups × 8 fluid ounces/cup = 16 fluid ounces
Therefore, the recipe requires 16 fluid ounces of water.
The cost, in dollars, of making x items is given by the function C(C) = 250 + 300. Find C(1500), the cost of making 1500 items. a) $4050 Ob) $25,300 Oc) $30,300 Od) $37,800 O e) none
Answer:
$37,800.
Step-by-step explanation:
We have been given that the cost, in dollars, of making x items is given by the function [tex]C(x)=25x+300[/tex]. We are asked to find [tex]C(1500)[/tex].
To find [tex]C(1500)[/tex], we will substitute [tex]x=1500[/tex] in our given function as:
[tex]C(1500)=25(1500)+300[/tex]
[tex]C(1500)=37,500+300[/tex]
[tex]C(1500)=37,800[/tex]
Therefore, the cost of making 1500 items would be $37,800 and option D is the correct choice.
Find the derivative of the following functions:
(i) f(x) = −x 2 + 10x + 4
(ii) f(x) = 20 − 1 x+2
(iii) f(x) = x 4 e −2x 2
(iv) f(x) = ln(x 2 + 2x + 2).
using the differentiation rules :Derivative definition and operation rules (Sum, subtraction, multiplication, quotient), Derivative of many basic functions (power, quadratic, exponential and natural logarithm functions), Function of Functions: (Chain Rule).
Explain carefully what rule(s) you have used and where you have applied it(them)
I will only do the first-three. You can repost number 4.
Question 1
f(x) = -x^2 + 10x + 4
dy/dx = -2x + 10
Question 2
f(x) = 20 - x + 2
dy/dx = -1
Question 3
f(x) = ex^4 - 2x^2
dy/dx = e•4x^3 + x^4 - 4x
dy/dx = 4ex^3 + x^4 - 4x
Let A-10, 1, 2, 3, 4, 5,6), let B = 10, 1, 2, 3, 4, 5,6, 7,8), and let R be the relation from A to B given by "the greatest common divisor of a and b is 2." [Note: "greatest common divisor" is sometimes called "highest common factor"] List the elements of R.
Answer:
R = {(10,2), (10,4), (10,6), (10,8), (2,10), (2,2), (2,4), (2,6), (2,4), (4,10), (4,2), (4,6), (6,10), (6,2), (6,4), (6,8)}
Step-by-step explanation:
A = (10, 1, 2, 3, 4, 5, 6)
B = (10, 1, 2, 3, 4, 5, 6, 7, 8)
R is a relation and defined as
R: A→B
R: The greatest common divisor of a and b is 2
To find the elements of R we need to find pair from A and B respectively.
R = {(10,2), (10,4), (10,6), (10,8), (2,10), (2,2), (2,4), (2,6), (2,4), (4,10), (4,2), (4,6), (6,10), (6,2), (6,4), (6,8)}
Now, these pairs were found by picking one element from A and checking with elements of B if they have a gcd of 2. If yes, they fall under the defined relation.
use Gaussian elimination to write each system in triangular form
x+ y+z+ w= 1
x+y −w=−1
−x+ y+z+2w= 2
x+2y−z+ w= 0
Answer:
To see the steps to the diagonal form see the step-by-step explanation. The solution to the system is [tex]x = -\frac{1}{9}[/tex], [tex]y= -\frac{1}{9}[/tex], [tex]z= \frac{4}{9}[/tex] and [tex]w = \frac{7}{9}[/tex]
Step-by-step explanation:
Gauss elimination method consists in reducing the matrix to a upper triangular one by using three different types of row operations (this is why the method is also called row reduction method). The three elementary row operations are:
Swapping two rowsMultiplying a row by a nonzero numberAdding a multiple of one row to another rowTo solve the system using the Gauss elimination method we need to write the augmented matrix of the system. For the given system, this matrix is:
[tex]\left[\begin{array}{cccc|c}1 & 1 & 1 & 1 & 1 \\1 & 1 & 0 & -1 & -1 \\-1 & 1 & 1 & 2 & 2 \\1 & 2 & -1 & 1 & 0\end{array}\right][/tex]
For this matrix we need to perform the following row operations:
[tex]R_2 - 1 R_1 \rightarrow R_2[/tex] (multiply 1 row by 1 and subtract it from 2 row)[tex]R_3 + 1 R_1 \rightarrow R_3[/tex] (multiply 1 row by 1 and add it to 3 row)[tex]R_4 - 1 R_1 \rightarrow R_4[/tex] (multiply 1 row by 1 and subtract it from 4 row)[tex]R_2 \leftrightarrow R_3[/tex] (interchange the 2 and 3 rows)[tex]R_2 / 2 \rightarrow R_2[/tex] (divide the 2 row by 2)[tex]R_1 - 1 R_2 \rightarrow R_1[/tex] (multiply 2 row by 1 and subtract it from 1 row)[tex]R_4 - 1 R_2 \rightarrow R_4[/tex] (multiply 2 row by 1 and subtract it from 4 row)[tex]R_3 \cdot ( -1) \rightarrow R_3[/tex] (multiply the 3 row by -1)[tex]R_2 - 1 R_3 \rightarrow R_2[/tex] (multiply 3 row by 1 and subtract it from 2 row)[tex]R_4 + 3 R_3 \rightarrow R_4[/tex] (multiply 3 row by 3 and add it to 4 row)[tex]R_4 / 4.5 \rightarrow R_4[/tex] (divide the 4 row by 4.5)After this step, the system has an upper triangular form
The triangular matrix looks like:
[tex]\left[\begin{array}{cccc|c}1 & 0 & 0 & -0.5 & -0.5 \\0 & 1 & 0 & -0.5 & -0.5\\0 & 0 & 1 & 2 & 2 \\0 & 0 & 0 & 1 & \frac{7}{9}\end{array}\right][/tex]
If you later perform the following operations you can find the solution to the system.
[tex]R_1 + 0.5 R_4 \rightarrow R_1[/tex] (multiply 4 row by 0.5 and add it to 1 row)[tex]R_2 + 0.5 R_4 \rightarrow R_2[/tex] (multiply 4 row by 0.5 and add it to 2 row)[tex]R_3 - 2 R_4 \rightarrow R_3[/tex](multiply 4 row by 2 and subtract it from 3 row)After this operations, the matrix should look like:
[tex]\left[\begin{array}{cccc|c}1 & 0 & 0 & 0 & -\frac{1}{9} \\0 & 1 & 0 & 0 & -\frac{1}{9}\\0 & 0 & 1 & 0 & \frac{4}{9} \\0 & 0 & 0 & 1 & \frac{7}{9}\end{array}\right][/tex]
Thus, the solution is:
[tex]x = -\frac{1}{9}[/tex], [tex]y= -\frac{1}{9}[/tex], [tex]z= \frac{4}{9}[/tex] and [tex]w = \frac{7}{9}[/tex]
A firm produces a product that has the production cost function
C(x)equals=195195xplus+88408840
and the revenue function
R(x)equals=260260x.
No more than
229229
units can be sold. Find and analyze the break-even quantity, then find the profit function.
Answer:
136 units
65x - 8840
Step-by-step explanation:
Given,
The production cost function is,
[tex]C(x) = 195x + 8840[/tex]
Revenue function,
[tex]R(x)=260x[/tex]
So, profit would be,
P(x) = Revenue - cost
= 260x - 195x - 8840
= 65x - 8840
In break even condition,
Profit, P(x) = 0
65x - 8840 = 0
65x = 8840
⇒ x = 136.
Hence, the break even quantity is 136 units.
The break-even quantity is 1360 units, where the revenue equals the cost. The profit function is given by P(x) = R(x) - C(x).
Explanation:To find the break-even quantity, we need to find the point where the revenue function equals the production cost function. So we set R(x) equal to C(x) and solve for x:
260x = 195x + 88408
Simplifying the equation, we get:
65x = 88408
x = 1360
Now we analyze the break-even quantity. The break-even quantity is the point at which the firm's revenue equals its cost. In this case, it occurs at 1360 units. At this quantity, the firm's total revenue will equal its total cost, resulting in zero profit.
To find the profit function, we subtract the cost function C(x) from the revenue function R(x). The profit function can be expressed as:
P(x) = R(x) - C(x)
How can this be proved to be a tautology using laws of logical equivalence?
((x ∨ y) ∧ (x → z) ∧ (¬z)) → y
Answer:
Step-by-step explanation:
If we assume that [tex][(x \vee y) \wedge (x \rightarrow z) \wedge (\neg z)][/tex] is true, then:
[tex](x \vee y)[/tex] is true
[tex](x \rightarrow z)[/tex] is true
[tex](\neg z)[/tex] is true
If [tex](\neg z)[/tex] is true, then [tex]z[/tex] is false.
[tex](x \rightarrow z) \equiv (\neg x \vee z)[/tex], since [tex](x \rightarrow z)[/tex] is true, then [tex](\neg x \vee z)[/tex] is true
If [tex]z[/tex] is false and [tex](\neg x \vee z)[/tex] is true, then [tex]\neg x[/tex] is true.
If [tex]\neg x[/tex] is true, then [tex]x[/tex] is false, as [tex](x \vee y)[/tex] is true and [tex]x[/tex] is false, then [tex]y[/tex] is true.
Conclusion [tex]y[/tex] it's true.