The solution of the given problem of quadratic equation comes out to be K thus has a value of 63/4.
What is quadratic equation?Regression modelling uses the polynomial solutions x = ax² + b + c=0 for one-variable equations. The First Principle of Algebra states that there can only be one solution because it has an extra order. There are both simple and complex solutions available. As the name suggests, a "non-linear formula" has four variables. This implies that there may only be one squared word. In the equation "ax² + bx + c = 0.
Here,
We know that if and are the zeros of the quadratic equation x²-x+k then:
=> α + β = 1
=> αβ = k
Additionally, we are told that 3 + 2 = 20.
We may find as = 1 - by using the equation + = 1.
By replacing this expression for in terms of in the formula k = a, we obtain:
=> (1 - β)β = k
=> β² - β + k = 0
=> 3α + 2(1 - α) = 20
=> α = 6 - 2β/3
=> (6 - 2β/3)²- (6 - 2β/3) + k = 0
=> 4β² - 36β + 72 + 3k = 0
=> 3(6 - 2β/3) + 2β = 20
=> 4β/3 = 2
=> β = 3/2
=> 4(3/2)² - 36(3/2) + 72 + 3k = 0
When we simplify and find k, we obtain:
=>k = 63/4
K thus has a value of 63/4.
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The expected increase (I) of a population of organisms is directly proportional to the current population (n). If a sample of 240 organisms increases by 20, by how many will a population of 600 increase?
A population of 600 organisms is expected to increase by 50.
To solve this problem
We can use the equation I = k * n to determine whether the anticipated growth (I) of an organism population is directly related to the size of the current population (n).
Where k is a proportionality constant that connects the anticipated growth to the current population.
We can use the details provided in the problem to determine the value of k. We can write: 20 = k * 240 to represent an increase of 20 in a sample of 240 organisms.
We can calculate k as follows: k = 20 / 240 = 1 / 12.
Now that we know the value of k, we can use it to calculate the predicted growth for a population of 600:
I = (1 / 12) * 600 = 50
Therefore, a population of 600 organisms is expected to increase by 50.
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Prove that 5 to the power of 7 plus 5 to the power of 6 is divisible by 6
To prove that 5^7 + 5^6 is divisible by 6, we can factor out the common term 5^6 and apply the divisibility rule for 6. Here's the solution:
1. Factor out the common term 5^6:
5^7 + 5^6 = 5^6(5 + 1) = 5^6 * 6
2. Apply the divisibility rule for 6:
A number is divisible by 6 if it is divisible by both 2 and 3. Since 6 is already a multiple of 6 (6*1), the expression 5^6 * 6 is divisible by 6.
Thus, we have proved that 5^7 + 5^6 is divisible by 6.
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What are the zeros of the function y = (x − 4)(x2 − 12x + 36)
The zeros of the function y = (x − 4)(x² − 12x + 36) are 4 and 6.
To find the zeros of the function y = (x - 4)(x² - 12x + 36), we need to set y to zero and solve for x.
0 = (x - 4)(x² - 12x + 36)
Now, solve for each factor separately:
1) x - 4 = 0
x = 4
2) x² - 12x + 36 = 0
This is a quadratic equation, and we can factor it as (x - 6)(x - 6).
So, x - 6 = 0
x = 6
The zeros of the function are x = 4 and x = 6. The zeros of a function are the values of its variables that meet the equation and result in the function's value being equal to 0.
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Solve for X. Round to the nearest tenth, if necessary.
The value of x to the nearest tenth is 1.4
What is trigonometric ratio?Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle.
Sin(θ) = opp/hyp
cos(θ)= adj/hyp
tan( θ) = opp/adj
x is opposite side to angle F and 32 is adjascent to angle F.
therefore;
tan F = opp/adj
tan23 = x/3.2
x = tan23 × 3.2
x = 1.4
therefore the value of x to the nearest tenth is 1.4
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[tex]4^3\sqrt{-88}[/tex]
Help I don't know where the negative goes.
if you meant simplification
[tex]4^3\sqrt{-88}\implies 64\sqrt{-88}\implies 64\sqrt{(-1)(2^2)(2)(11)}\implies 64(2)\sqrt{(-1)(2)(11)} \\\\\\ 128\sqrt{(2)(11)}\cdot \sqrt{-1}\implies 128\sqrt{22}~i[/tex]
A regular size chocolate bar was 5 4/9 inches long. if the king size bar is 3 2/5 inches longer, what is the length of the king size bar?
please help me!!!!
Answer:
8 38/48
Step-by-step explanation:
You want to estimate the number of people in a grocery store who buy milk. which sample is unbiased?group of answer choices70 shoppers at random as they leave the store8 shoppers as they enter the store60 women with children at random80 shoppers in the dairy section at random
80 shoppers in the dairy section at random is a biased sample (d).
To estimate the number of people who buy milk in a grocery store, we need an unbiased sample that represents the entire population. The options provided are 70 shoppers at random as they leave the store, 8 shoppers as they enter the store, 60 women with children at random, and 80 shoppers in the dairy section at random.
The most unbiased sample is selecting 80 shoppers in the dairy section at random because the dairy section is where people are most likely to buy milk, and selecting a random sample of shoppers in that section ensures that we are not introducing any biases.
The other options may introduce biases since they are not necessarily representative of the entire population of shoppers in the store. Therefore, selecting 80 shoppers in the dairy section at random provides the most unbiased estimate of the number of people who buy milk in the grocery store. So d is correct option.
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A marine biologist is researching the population density of manatees. The marine biologist counts 550 manatees within a radius of 5 miles of a half circle off of Crystal Bay. What is the population density of the manatees? Use 3. 14 for pi, and round to the nearest whole number
The population density of manatees within a radius of 5 miles of a half circle off of Crystal Bay is approximately 14 manatees per square mile
The population density of manatees within a radius of 5 miles of a half circle off of Crystal Bay is approximately 70 manatees per square mile. To calculate this, we first need to find the area of the half circle. Using the formula for the area of a circle, A=πr^2, where r is the radius, we can find the area of the full circle with a radius of 5 miles:
A = 3.14 x 5^2
A = 78.5 square miles
Since we only want to consider the area of the half circle, we divide this by 2:
A = 78.5 / 2
A = 39.25 square miles
Now we can calculate the population density by dividing the number of manatees by the area:
Density = 550 / 39.25
Density ≈ 14
Therefore, the population density of manatees within a radius of 5 miles of a half circle off of Crystal Bay is approximately 14 manatees per square mile. Rounded to the nearest whole number, this is 14.
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"Francine made 11 long distance calls in March. The calls ranged from 11 minutes to 25 minutes in length. If Francine pays 0. 99 per minute for long distance calls, which is the closest to the total cost of her calls?"
The closest value to the total cost of Francine's calls is $191.07
To find the total cost of Francine's calls, we need to know the total number of minutes she spent on the phone. We can calculate this by adding up the lengths of all 11 calls:
Total minutes = 11 + 14 + 12 + 20 + 25 + 16 + 11 + 22 + 18 + 19 + 15 = 193
The total cost of Francine's calls can then be found by multiplying the total number of minutes by the cost per minute:
Total cost = 193 * 0.99
= 191.07
Therefore, the closest value to the total cost of Francine's calls is $191.07.
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Six friends are selling crafts at a flea market. they each need to pay $7. 20 to pay
for the table rental. they each sell 3 items. if every item is the same price, and the
6 friends make a total profit of $25. 20, what was the sale price of each item?
To find the sale price of each item, we need to use some basic algebra. Let's call the sale price of each item "x".
First, we need to find the total cost of the table rental for all six friends. Since each friend needs to pay $7.20, the total cost of the table rental is 6 * $7.20 = $43.20.
Next, we need to find the total revenue from selling the items. Each friend sells 3 items, so the total number of items sold is 6 * 3 = 18. The total revenue is the number of items sold multiplied by the sale price, so the total revenue is 18x.
We know that the total profit is $25.20, which is the total revenue minus the total cost of the table rental. So we can set up the equation:
18x - $43.20 = $25.20
Simplifying this equation, we get:
18x = $68.40
Dividing both sides by 18, we get:
x = $3.80
Therefore, the sale price of each item is $3.80.
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PLS HELP! LAST QUESTION!
I WILL MAKE U BRAINLIST AND I NEED THIS!
PLS USE A DESMOS CALCULATOR AND SHOW ALL STEPS! I NEED IT.
Answer:
8.19 units.
Step-by-step explanation:
I didn't use desmos, i completed the question with all steps and working and didn't require desmos. Hope this Helps. Question was solved using trignometric ratios.
The first is kept by the albatross, which first lands in our region. The second occurs every night in shooting, swimming, fighting, singing and others. Separately, they would reveal many to you. Together, they will reveal only one to you. And whose one is it?
Based on the information provided, it is not clear what "they" refer to. However, it seems that there are two distinct events or phenomena being described here.
The first involves the albatross, which is known to be a migratory bird that visits certain regions at certain times of the year. It is unclear what exactly is being "kept" by the albatross, but it could be some sort of information or knowledge that is unique to this bird.
The second event or phenomenon is described as happening every night and involving various activities such as shooting, swimming, fighting, and singing.
It is unclear what this refers to exactly, but it could be some sort of nightly ritual or tradition that takes place in a particular community or region.
The statement that "separately, they would reveal many to you" suggests that each of these phenomena has its own unique insights or knowledge to offer.
However, the statement that "together, they will reveal only one to you" implies that there is some sort of deeper connection or meaning between these two events that is not immediately apparent.
The question of "whose one is it?" suggests that there is some sort of ownership or responsibility associated with this revelation.
It is unclear who this refers to, but it could be interpreted as asking who is responsible for uncovering the deeper meaning or connection between these two phenomena.
In summary, the information provided is somewhat cryptic and open to interpretation. However, it seems that there are two distinct events or phenomena being described, and that there is some sort of deeper connection or meaning between them that is not immediately apparent.
The question of who is responsible for uncovering this connection remains unanswered.
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|x-3|+|x+2|-|x-5| if 3
|x-3|+|x+2|-|x-5| if x<-2
|x-3|+|x+2|-|x-5| if -2
pls help
i will give brainliest
|x-3|+|x+2|-|x-5| can be broken down into three separate cases based on the value of x:
(x-3) + (x+2) - (x-5) = 2x + 4
(x-3) + (x+2) - (5-x) = 2x - 6
-(x-3) - (x+2) - (5-x) = -3x - 4
We break down the expression into three separate cases based on the value of x. This is because the absolute value function creates "turning points" at which the behavior of the expression changes. We analyze the expression for each case and simplify it to obtain the final answer. The answer depends on the value of x, and we must consider the expression separately for each case.
If x ≥ 5, then the expression becomes:
(x-3) + (x+2) - (x-5) = 2x + 4
If -2 ≤ x < 3, then the expression becomes:
(x-3) + (x+2) - (5-x) = 2x - 6
If x < -2, then the expression becomes:
-(x-3) - (x+2) - (5-x) = -3x - 4
Therefore, the final answer depends on the value of x. If x is greater than or equal to 5, then the answer is 2x + 4. If x is between -2 and 3, then the answer is 2x - 6. And if x is less than -2, then the answer is -3x - 4.
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A person sleeps in a tent while camping.
which equation correctly determines the
amount of material used, m, to construct
the fully enclosed tent.
a. =(8∙9)+2(7.5∙8)
b. =(8∙9)+2(7.5∙8)+2(1
2∙9∙6)
c. =2(8∙9)+2(7.5∙8)+(1
2 ∙9∙6)`
d. =3(8∙9)+2(1
2 ∙9∙6)
.m The correct equation to determine the amount of material used, m, to construct the fully enclosed tent is:
c. =2(8∙9)+2(7.5∙8)+(12∙9∙6)
To determine the amount of material used to construct the fully enclosed tent, we need to consider the surface area of the tent. The tent is fully enclosed, so we need to calculate the area of all the sides.
Option a. =(8∙9)+2(7.5∙8) calculates the area of the top and two sides of the tent. This does not include the front and back of the tent, so it is not the correct equation.
Option b. =(8∙9)+2(7.5∙8)+2(12∙9∙6) calculates the area of the top, two sides, front and back of the tent, but it also includes an extra term of 2(12∙9∙6) which is not necessary for a fully enclosed tent. This option overestimates the amount of material used.
Option c. =2(8∙9)+2(7.5∙8)+(12∙9∙6) calculates the area of the top, bottom, and all four sides of the tent. This is the correct equation to determine the amount of material used in a fully enclosed tent.
Option d. =3(8∙9)+2(12∙9∙6) overestimates the amount of material used because it includes an extra term of 3(8∙9) which is not necessary for a fully enclosed tent.
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Regina writes the expression y + 9 x 3/4. Which expression is equivalent to the one Regina writes?
The expression that is equivalent to the one Regina wrote is y + 27/4
Which expression is equivalent to the one Regina wrote?From the question, we have the following parameters that can be used in our computation:
y + 9 x 3/4
This means that
Expression = y + 9 x 3/4
When expanded, we have
Expression = y + 27/4
Using the above as a guide, we have the following:
The expression that is equivalent to the one Regina wrote is y + 27/4
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25 points help asap
will mark brainiest
select the correct answer from each drop-down menu.
a lake currently has a depth of 30 meters. as sediment builds up in the lake, its depth decreases by 2% per year
this situation represents _____
the rate of growth or decay,r, is equal to _____
so the depth of the lake each
year is ______ times the depth in the previous year,
it will take between _____
years for the depth of the lake to reach 26. 7 meters
It will take approximately 5.28 years for the depth of the lake to reach 26.7 meters.
How to find the lake depth decay?The situation represents decay because the depth of the lake decreases over time.
The rate of growth or decay, r, is equal to -0.02 (negative because it represents a decrease of 2% per year).
So the depth of the lake each year is 0.98 times the depth in the previous year (100% - 2% = 98%).
To find the number of years it will take for the depth of the lake to reach 26.7 meters, we can use the formula for exponential decay:
26.7 = 30 *[tex](0.98)^n[/tex]
Solving for n, we get:
[tex](0.98)^n = \frac{26.7 }{ 30}[/tex]
n =[tex]log\frac{(\frac{26.7}{30}) }{ log(0.98)}[/tex]
Using a calculator, we find n ≈ 5.28 years.
Therefore, it will take approximately 5.28 years for the depth of the lake to reach 26.7 meters.
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use 3.14 to approximate pi the options are
a) 32
b) 49.12
c) 36.56
d)25.12
e) 41.12
f ) 10.21
Answer:
E
Step-by-step explanation:
This figure contains a square and a quarter of a circle
The perimeter is the sum of all side lengths
Let's find the circumference of the quarter of a circle:
[tex]0.25c = 2\pi \times r[/tex]
The diameter is equal to a square's side length:
d = 8
hence, r = 0,5 × 8 = 4
[tex]0.25c = 2 \times 3.14 \times 4 = 25.12[/tex]
Now, we can find the perimeter:
P = 25,12 + 8 + 8 = 41,12
Andrew invests $500 into an account with a 2. 5% interest rate that is compounded quarterly. How much money will he have in this account if he keeps it for 5 years?
Round your answer to the nearest dollar
He will have $566 in this account if he keeps it for 5 years.
How to determine how much money he will have in this account?To determine how much money he will have in this account if he keeps it for 5 years, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
P = the principal, $500
r = the interest rate, 2.5% = 0.025
n = the number of times the interest is compounded per year, in this case quarterly (n = 4)
t = the time period in years, 5
Substituting the values :
A = 500(1 + 0.025/4)^(4 * 5)
A = 500(1 + 0.00625)²⁰
A = 500(1 .00625)²⁰
A = $566
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For each problem, determine what will happen to the first factor.
10x1/2
15 x 7/2
Answer:
52.5
Step-by-step explanation:
15×7=105÷2
=52.5 ans it means fifteen times seven divided by two
Identify the parent function for g(x) = (x + 3)^2 and describe the transformation.
The resulting graph of g(x) will resemble the graph of f(x), but it will be stretched vertically and pushed leftward by three units.
What is function?Each element of a set (referred to as the domain) is mapped by a rule known as a function to a particular element of this other set (called the range). A function is, in other words, a connection between two subsets in which every member of the domain has a unique relationship to every element of the range. One popular approach to write a function is via function notation, which entails writing the function name surrounded by the incoming signal in parentheses, as in the following example: f (x). For instance, the formula f(x) = 2x + 1 takes the input x and produces the result 2x + 1.
given,
The fundamental quadratic function f(x) = x² serves as the parent function for g(x) = (x + 3)².
Since the argument of the function (x + 3) is x shifted left by 3 units, the graph of f(x) is horizontally displaced to the left by 3 units.
Given that the coefficient of the squared term is 1, the graph of the resulting function is shifted up by a factor of 1.
As a result, the transformation can be represented as a 3 unit horizontal shift and a 1 unit vertical stretch.
The resulting graph of g(x) will resemble the graph of f(x), but it will be stretched vertically and pushed leftward by three units.
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Haz la ecuación y la verificación según los datos que dan. 5. Un padre tiene 35 años y su hijo 5. ¿Al cabo de cuántos años será la edad del padre tres veces mayor que la edad del hijo?
6. Si al doble de un número se le resta su mitad resulta 54. ¿Cuál es el número?
7. En una reunión hay doble número de mujeres que de hombres y triple número de niños que de hombres y mujeres juntos. ¿Cuántos hombres, mujeres y niños hay si la reunión la componen 96 personas?
8. Una granja tiene cerdos y pavos, en total hay 35 cabezas y 116 patas. ¿Cuántos cerdos y pavos hay?
ccabo de 25 años será la edad del padre tres veces mayor que la edad del hijo.El número es 36.Hay 8 hombres, 16 mujeres y 72 niños.Hay 27 cerdos y 8 pavos
How many years will it take for the father's age to be three times the age of his son?Sea x el número de años transcurridos. La edad del padre será 35 + x y la edad del hijo será 5 + x. La ecuación que representa la situación es: 35 + x = 3(5 + x). Resolviendo la ecuación, tenemos:
35 + x = 15 + 3x
2x = 20
x = 10
Por lo tanto, después de 10 años, la edad del padre será tres veces mayor que la edad del hijo.
Sea x el número desconocido. La ecuación que representa la situación es: 2x - (1/2)x = 54. Resolviendo la ecuación, tenemos:
(4/2)x - (1/2)x = 54
(3/2)x = 54
x = 36
Por lo tanto, el número desconocido es 36.
Sea x el número de hombres. Según la información dada, el número de mujeres es el doble, es decir, 2x, y el número de niños es el triple, es decir, 3(x + 2x) = 9x. La ecuación que representa la situación es: x + 2x + 9x = 96. Resolviendo la ecuación, tenemos:
12x = 96
x = 8
Por lo tanto, hay 8 hombres, 16 mujeres y 72 niños en la reunión.
Sea x el número de cerdos y y el número de pavos. Según la información dada, tenemos las siguientes ecuaciones: x + y = 35 (por el total de cabezas) y 4x + 2y = 116 (por el total de patas). Resolviendo este sistema de ecuaciones, obtenemos:
x + y = 35
4x + 2y = 116
Multiplicamos la primera ecuación por 2:
2x + 2y = 70
Restamos la segunda ecuación de la primera:
2x + 2y - (4x + 2y) = 70 - 116
-2x = -46
x = 23
Sustituyendo el valor de x en la primera ecuación, tenemos:
23 + y = 35
y = 12
Por lo tanto, hay 23 cerdos y 12 pavos en la granja.
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What is the amount of carrying charges for a $10,000 for college if there is a 5% down payment, apr of 10%, and a 36-month repayment period?
The amount of carrying charges for a $10,000 college loan with a 5% down payment, a 10% APR, and a 36-month repayment period is approximately $1,571.44.
To calculate the amount of carrying charges for a $10,000 college loan with a 5% down payment, a 10% annual percentage rate (APR), and a 36-month repayment period, follow these steps:
1. Determine the down payment: 5% of $10,000 = $500.
2. Subtract the down payment from the loan amount: $10,000 - $500 = $9,500. This is the principal loan amount.
3. Calculate the monthly interest rate: 10% APR / 12 months = 0.833% or 0.00833 as a decimal.
4. Calculate the monthly payment using the loan payment formula: P = r * PV / (1 - (1 + r)⁻ⁿ), where P is the monthly payment, r is the monthly interest rate, PV is the present value (principal loan amount), and n is the number of monthly payments. P = 0.00833 * $9,500 / (1 - (1 + 0.00833)⁻³⁶) = $307.54.
5. Determine the total amount paid over the loan term: Monthly payment * Number of monthly payments = $307.54 * 36 = $11,071.44.
6. Calculate the carrying charges: Total amount paid - Principal loan amount = $11,071.44 - $9,500 = $1,571.44.
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22
A statistical question is one where you expect to get a variety of answers. Determine whether each question can be classified as a statistical question. Select Yes or No for each question. Yes
No
How many hours a week do people exercise?
How many hours are there in a day?
How many rainbows have students seen this month?
To answer this question, determine the quantity asked for:
Answers are:
Yes - How many hours a week do people exercise?
No - How many hours are there in a day?
Yes - How many rainbows have students seen this month?
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Kyra bought a ""trick"" coin that lands on heads with a probability of 0. 4. Which frequency table would Kyra most likely generate by flipping the coin many times?
Don't mind the answer c because I don't know if that's right
Kyra's trick coin is more likely to generate a frequency table with a higher count of tails than heads, reflecting the 0.4 probability of landing on heads and the 0.6 probability of landing on tails.
Kyra's "trick" coin has a probability of 0.4 for landing on heads, meaning that it is more likely to land on tails. If Kyra flips the coin many times, she will likely generate a frequency table that shows a higher frequency of tails than heads.
A possible frequency table might look like this:
| Outcome | Frequency |
|---------|-----------|
| Heads | 40 |
| Tails | 60 |
In this table, the coin landed on heads 40 times and tails 60 times, representing the probability of 0.4 for heads and 0.6 for tails (since the total probability must equal 1). The more times Kyra flips the coin, the closer the relative frequencies will get to the actual probabilities. It's important to note that this is just an example, and the exact frequency counts may vary in practice due to randomness.
To summarize, Kyra's trick coin is more likely to generate a frequency table with a higher count of tails than heads.
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for each of these relations on the set {1,2,3,4}, decide whether it is reflexive, whether it is symmetric, and whether it is transitive. Which of these relations are equivalence relations?(a) {(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)}. (b) {(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)}. (c) {(2,4),(4,2)}.
The relation is not an equivalence relation. (a) {(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)}:
Reflexive: (2,2), (3,3) are present, but (1,1) and (4,4) are not present. Hence, not reflexive.
Symmetric: (2,3) is present, but (3,2) is also present. Hence, not symmetric.
Transitive: No counterexample to transitivity exists. Hence, it is transitive.
Therefore, the relation is not an equivalence relation.
(b) {(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)}:
Reflexive: (1,1), (2,2), (3,3), (4,4) are present. Hence, reflexive.
Symmetric: (1,2) is present, but (2,1) is also present. Hence, symmetric.
Transitive: No counterexample to transitivity exists. Hence, it is transitive.
Therefore, the relation is an equivalence relation.
(c) {(2,4),(4,2)}:
Reflexive: (2,2) and (4,4) are not present. Hence, not reflexive.
Symmetric: (2,4) is present, but (4,2) is not present. Hence, not symmetric.
Transitive: No counterexample to transitivity exists. Hence, it is transitive.
Therefore, the relation is not an equivalence relation.
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Solve for f(-2).
f(x) = -3x + 3
f(-2) = [?]
Answer:
f(-2) = 9
Step-by-step explanation:
f(x) = -3x + 3 Solve for f(-2).
f(-2) = -3(-2) + 3
f(-2) = 6 + 3
f(-2) = 9
Find two vectors in opposite directions that are orthogonal to the vector u.
u = 1/4 i - 4/5j
Two vectors in opposite directions that are orthogonal to u are v = 5i + 4j and w = -4i + 5j.
To find two vectors in opposite directions that are orthogonal to u, we need to use the cross product. The cross product of two vectors is a vector that is perpendicular to both of them. We can choose any two non-collinear vectors as long as they are orthogonal to each other and the given vector.
Let's find the cross product of u and a vector v. The cross product of two vectors a and b is given by:
a x b = |a| |b| sinθ n
where |a| and |b| are the magnitudes of the vectors, θ is the angle between them, and n is a unit vector perpendicular to both a and b in the direction given by the right-hand rule.
Since we want v to be orthogonal to u, we need to choose v such that u x v = 0. This means that the angle between u and v is either 0 or 180 degrees, and |v| is arbitrary.
Let v = 5i + 4j. Then, we have:
u x v = (1/4 i - 4/5j) x (5i + 4j)
= (-16/20)i - (5/20)j + (1/20)k
= (-4/5)i - (1/4)j + (1/20)k
Since u x v is not equal to zero, v is not orthogonal to u. To find another vector that is orthogonal to u, we can take the cross product of u and w, where w = -4i + 5j. Then, we have:
u x w = (1/4 i - 4/5j) x (-4i + 5j)
= (-5/20)i + (16/20)j + (1/20)k
= (-1/4)i + (4/5)j + (1/20)k
Since u x w is also not equal to zero, we need to adjust the signs of v and w to make them orthogonal to u. We can do this by taking the opposite of v and w. Therefore, two vectors in opposite directions that are orthogonal to u are v = 5i + 4j and w = -4i + 5j.
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Your local dry cleaners is running a special. The dry cleaner is offering a 13% discount for any drop off that is over $40. Also, they are running a neighborhood special of an additional 5% off. Given this information, determine the decay factor corresponding to each percent decrease. A. 34. 8; 33. 06 c. 0. 87; 0. 95 b. 0. 384; 0. 3306 d. 87; 95
The decay factor corresponding to each percent decrease is 0.87. The correct option is c. The customer would save a total of $17.35, or 17.35% off the original price.
To determine the decay factor corresponding to each percent decrease, we need to use the formula:
decay factor = (100% - percent decrease)/100%
For the first discount of 13%, the decay factor would be:
decay factor = (100% - 13%)/100% = 0.87
For the additional neighborhood discount of 5%, the decay factor would be:
decay factor = (100% - 5%)/100% = 0.95
So the correct answer is option C, with a decay factor of 0.87 for the 13% discount and 0.95 for the neighborhood discount.
These decay factors represent how much of the original price is left after the discount has been applied. For example, if the original price was $100, after the 13% discount, the price would be:
price after first discount = $100 x 0.87 = $87
Then, after the additional 5% discount, the final price would be:
final price = $87 x 0.95 = $82.65
So the customer would save a total of $17.35, or 17.35% off the original price.
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Consider the three points P (3,0,0), Q (0,0,-9), and R (0, -6,0). (a) Find a non-zero vector orthogonal to the plane through the points P, Q, and R. (b) Find the area of the parallelogram with sides PQ and PR.
(a) To find a non-zero vector orthogonal to the plane through the points P, Q, and R, we need to take the cross product of two vectors in the plane. One way to do this is to subtract one point from another to get a vector, and then take the cross product of the two resulting vectors. For example, we could subtract point Q from point P to get the vector PQ, and subtract point R from point P to get the vector PR. Then, taking the cross product of PQ and PR will give us a vector orthogonal to the plane:PQ = <-3, 0, -9>PR = <-3, -6, 0>PQ x PR = <54, 27, 18>Therefore, the vector <54, 27, 18> is orthogonal to the plane through the points P, Q, and R.(b) To find the area of the parallelogram with sides PQ and PR, we need to find the length of the projection of PQ onto PR, and then multiply by the length of PR. The projection of PQ onto PR is given by:proj_PR(PQ) = (PQ · u) uwhere u is a unit vector in the direction of PR, and · denotes the dot product. Since PR = <-3, -6, 0>, we can take u = <-1/sqrt(10), -3/sqrt(10), 0>, which is a unit vector in the direction of PR. Then:proj_PR(PQ) = (PQ · u) u = (-3/sqrt(10)) <-1/sqrt(10), -3/sqrt(10), 0> = <9/10, 27/10, 0>The length of this vector is sqrt((9/10)^2 + (27/10)^2 + 0^2) = 3sqrt(10), so the area of the parallelogram is:A = |PQ| |proj_PR(PQ)| = sqrt((-3)^2 + 0^2 + (-9)^2) * 3sqrt(10) = 27sqrt(10)
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A. Orthogonal vector = (0, -27, 18)
B. The area of the parallelogram with sides PQ and PR is 1053 square units.
(a) To find a non-zero vector orthogonal to the plane through points P, Q, and R, we need to compute the cross product of vectors PQ and PR.
Vector PQ = Q - P = (-3, 0, -9)
Vector PR = R - P = (-3, -6, 0)
Cross product PQ x PR = (i, j, k) × ((-3, 0, -9), (-3, -6, 0))
= i(0 * 0 - (-9) * (-6)) - j(-3 * 0 - (-3) * (-9)) + k(-3 * -6 - 0 * (-3))
= i(0) - j(27) + k(18)
Orthogonal vector = (0, -27, 18)
(b) To find the area of the parallelogram with sides PQ and PR, we can use the magnitude of the cross product of PQ and PR.
Magnitude of PQ x PR = ||(0, -27, 18)||
= √(0^2 + (-27)^2 + 18^2)
= √(729 + 324)
= √1053
The area of the parallelogram with sides PQ and PR is 1053 square units.
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In a shop where all items cost a whole number of dollars, I bought 3 packets of plain biscuits and 5 packets of chocolate biscuits. The total cost was $34. Harold says, ‘The packets of chocolate biscuits must have cost $2 each. Show that Harold is wrong
To show that Harold is wrong about the cost of chocolate biscuits being $2 each, we can use the given information:
You bought 3 packets of plain biscuits and 5 packets of chocolate biscuits, and the total cost was $34.
Let's use the variables P for the cost of plain biscuits and C for the cost of chocolate biscuits.
We can write the equation:
3P + 5C = $34
Harold claims that the cost of chocolate biscuits is $2 each. So, let's substitute C = $2 into the equation:
3P + 5($2) = $34
Now, we can solve for P:
3P + $10 = $34
3P = $24
P = $8
So, the cost of plain biscuits is $8 each. This means you bought 3 packets of plain biscuits for $24 and 5 packets of chocolate biscuits for $10, which adds up to the total cost of $34.
Since the cost of plain biscuits came out to be a whole number, Harold's claim that chocolate biscuits cost $2 each is not necessarily wrong.
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