The rate of change of the volume at that particular instant is - 62.8 m^3/h.
How to get the rate of change of the volume?
First, we can write the dimensions as:
radius = R = (5m + 1m/h*t)
height = H = (8m - 4 m/h*t)
Where t is the time in hours.
Then the volume of the cylinder will be:
V = pi*R^2*H = 3.14*(5m + 1m/h*t)^2*(8m - 4 m/h*t)
To get the rate of change, we need to differentiate it with respect to t, we will get:
V' = 3.14*(2*(5m + 1m/h*t)*(1m/h)*(8m - 4 m/h*t) + (5m + 1m/h*t)^2*(-4m/h))
V' = 3.14*( (2 m/h)*(5m + 1m/h*t)*(8m - 4 m/h*t) - (4m/h)*(5m + 1m/h*t)^2)
V' = 3.14*(2m/h)*( (5m + 1m/h*t)*(8m - 4 m/h*t) - 2*(5m + 1m/h*t)^2)
As you can see the rate of change depends on t, but we want the rate of change at this instant, then we use t = 0, replacing that on the above equation we get:
V'(0) = 3.14*(2m/h)*( (5m + 1m/h*0)*(8m - 4 m/h*0) - 2*(5m + 1m/h*0)^2)
V'(0) = 3.14*(2m/h)*( (5m)*(8m ) - 2*(5m)^2)= -62.8 m^3/h
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Find the volume of a cone with a base radius of 4 cm and a height of 6 cm. Write the exact volume in terms of it, and be sure to include the correct unit in your answer.
Answer:
Step-by-step explanation:
The volume of a cone with a base radius of 4 cm and a height of 6 cm is equal to 33.51 cm³. This is calculated by using the formula V = (1/3)πr²h, where V is the volume, π is the constant Pi, r is the radius of the base, and h is the height of the cone. Plugging in the given numbers, we have V = (1/3)(3.14)(4²)(6) = 33.51 cm³.
Sarah uses a counterexample in her proof. How is this different from a direct proof? In using a counterexample, Sarah used a single example to show that a statement was . This technique is unlike a direct proof, in which evidence is used to support a series of statements to show why a statement is . A single example is to prove a statement true.
In using a counterexample, Sarah used a single example to show that a statement was False. This technique is unlike a direct proof, in which evidence is used to support a series of statements to show why a statement is True. A single example is enough to prove a statement true.
What is a counterexample ?A counterexample is a specific example that contradicts a general statement or claim. In mathematics, a counterexample is used to disprove a conjecture, hypothesis or theory by providing an example that does not follow the rule or pattern that is being proposed. It is a method of proof by contradiction.
Counterexamples can also be used to show that a conjecture or assumption is not universally true. They are important tools in mathematics and logic, as they allow us to test the validity of statements and identify any exceptions or limitations.
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university of texas students were fitted with belt-worn tape recorders for up to four days so that researchers could sample their daily activities. the researchers employed a scientific method known as
The researchers employed a scientific method known as Ethnography.
Ethnography to study the recorded data. Ethnography is a method of research that involves the systematic study of people and cultures, often through the use of participant observation and the collection of detailed field notes. In this case, the researchers used the recordings to gain an understanding of the student's daily activities and how they were affected by their environment.
The researcher also often conducts in-depth interviews, focus groups, or surveys with group members to comprehensively understand their perspectives and experiences.
Ethnography can be applied in various fields, including anthropology, sociology, education, and business. It is commonly used to study communities, organizations, and other social groups.
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a giraffe was 1 feet tall at birth,7 feet tall at the age of 4, and 11 and half feet at the age of 7
Which expressions(s) is equivalent to
(7^2)^5 • 7^4?
A. 7^2•^5+^4
B. 7^3(^5+^4)
C. 7^2+^5+^4
D. 7^2+^5•7^4
(7²)⁵ * 7⁴ is equal to the expression 7²ˣ⁵ ⁺ ⁴.
What is expression?Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.
Given an expression (7²)⁵ * 7⁴
Since as per the rule of power we can add the power is base is same and also there is a rule if Power is the power of another power they will be multiplied
From the second rule
(7²)⁵ * 7⁴ = 7²ˣ⁵ * 7⁴
From the first rule of power
7²ˣ⁵ * 7⁴ = 7²ˣ⁵ ⁺ ⁴
therefore, (7²)⁵ * 7⁴ is equal to the expression 7²ˣ⁵ ⁺ ⁴.
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On any given day, the proportion of workers at a factory who are more than 5 minutes late to work is 0.11. A random sample of 20 workers will be selected. Which of the following is the best interpretation of the mean of the sampling distribution of the sample proportion of workers in the sample who are more than 5 minutes late to work for samples of size 20
The sampling distribution's mean will be interpreted as meaning A. The mean of all potential sample proportions for all samples of size 20 is equal to 0.11.
What is sampling distribution ?The sample proportion's sampling distribution is roughly regularly distributed. It is merely the population's mean drawn from the sampled population. This is the sample fraction of employees who arrive more than five minutes late, according to the sampling distribution's mean.
For samples of size 20, the mean of all potential sample proportions is equal to 0.11, which is the sampling distribution's mean for the sample proportion of workers who arrive at work more than 5 minutes late.
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In the accompany diagram, the width of the inner rectangle is represented by x-3 and the length by x+3. The width of the outer rectangle is represented by 3x-4 and the length by 3x+4. Express the area of the region inside the larger rectangle but outside the smaller rectangle as a polynomial in terms of x.
The inner triangle's area is equal to x2 - 9.The larger triangle has a surface area of (9x2 - 16) square units.8x2 - 7 is the necessary polynomial in terms of x.
Express the area of the region ?The following is the formula for determining a rectangle's area:
A = LW
The length is L.
The width is W.
For the inner triangle, see (A)
Size = x - 3
Height = x +3
(x-3)(x+3) is the inner triangle's area.
x2-3x+3x+9 is the inner triangle's area.
Inner triangle's area is equal to x2 - 9.
B) Regarding the bigger triangle
Size = 3x-4
Height 3x + 4
A = (3x-4)(3x+4)
A = 9x²+12x-12x-16
A = (9 x 2 – 16) square units
Consequently, the larger triangle's area is (9x2 - 16) square units.
C) We will make the difference in the areas in order to describe the size of the region inside the bigger rectangle by outside the smaller rectangle as a polynomial in terms of x:
P(x) is equal to the smaller triangle's area minus the larger triangle's area.
P(x) = (9x² - 16) - (x² - 9)
P(x) = 9x² - 16 - x² + 9
P(x) = 9x² - x² - 16 + 9
P(x) = 8x² - 7
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Is it possible to construct a triangle with its sides as 9.1 cm 8.3 cm and 17.5 cm?
No, it is not possible to construct a triangle with the sides 9.1cm ,8.3cm and 17.5cm because sum of the any two sides is not greater than the third side .
The side lengths are given as : 9.1cm ,8.3cm and 17.5cm ;
we know that , the triangle is formed only when the sum of two sides of triangle is greater than third side .
applying this property , we get that ;
⇒ 9.1 + 8.3 = 17.4 > 17.5 ; False
⇒ 8.3 + 17.5 = 25.8 > 9.1 ; True
⇒ 9.1 + 17.5 = 26.6 > 8.3 ; True .
we can see that the sum of the sides 9.1 and 8.3 is 17.4 which is not greater than the third side (17.5) ,
Therefore , the triangle cannot be constructed .
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f(x)=x^2-4
g(x)=2x+1
solve fg(x)>0
SHOW WORKING
Answer:
To find the values of x that make fg(x) > 0, we first need to find fg(x). We can do this by composing the functions, which means replacing x in g(x) with the expression for f(x):
fg(x) = f(g(x)) = f(2x + 1) = (2x + 1)^2 - 4
Now we can solve the inequality fg(x) > 0:
(2x + 1)^2 - 4 > 0
(2x + 1)^2 > 4
2x + 1 > 2 or 2x + 1 < -2
x > -1.5 or x < -2.5
So the values of x that make fg(x) > 0 are x > -1.5 or x < -2.5.
11+4z+z-6
how would I solve this
Answer:
5 + 5z
Step-by-step explanation:
You just combine the like terms:
11 - 5 = 5
4z + 1z = 5z
=
5 + 5z
A company has learned that by pricing a newly released toy at $5, sales will reach 2500 per day. raising the price to $7 will cause the sales to fall to
1500 per day. assume that the ratio of change in price to change in daily sales is constant and let x be the price of the toy and y be the number of
sales.
a. find theinear equation that models the price-sales relationship for this toy. [hint: the line must pass through (5,2500) and (7,1500).]
b. use this equation to predict the daily sales of the toy if the price is set at $6.50.
Using linear functions, we have that:
a) The equation for the line is: y = -500x + 5000
b) The prediction is 2250 sales with a price of $6.50.
The equation of a line is given by:
y = mx+ b
In which
m is the slope, which is the rate of change.b is the y-intercept, which is the value of y when x = 0.Item a:
In this problem:
There are two points (5, 2500) and (7, 1500).The slope is given by the change in y divided by the change in x, then:[tex]m = \frac{1500-2500}{7-5} = - \frac{1000}{2} = -500[/tex]
Thus:
y = -500x + b
Point (5,2500) means that when x=5, y = 2500, which we use to find b.
y = -500x + b
⇒2500 = -500* (5) +b
⇒2500 = -2500 +b
⇒b = 2500 +2500
⇒b = 5000
Thus
y = -500x + 5000
∴The equation for the line is: y = -500x + 5000
Item b:
The sales is y when x = 6.5, thus:y = -500*(6.5) + 5000
y = 2250
The prediction is 2250 sales with a price of $6.50.
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Complete the following probability distribution table: Probability Distribution Table X P(X)
20 0.2 49 0.4
53 ? 80 0.3 Get help: Written Example
0.1 is Probability in Distribution Table X P(X).
What is Probability ?
The simple definition of probability is the likelihood that something will occur. We can discuss the probabilities of various outcomes, or how likely they are, whenever we are unsure of how an event will turn out.
Statistics describes the examination of events subject to probability.
Here X is a random variable. P(X) is its probabilities corresponding to the different values of X.
now according to the properties of probability theory we know that sum of probabilities is 1.
so here to find the value of the probability of X coresponding to the value 53 we should use the following steps:
sum of given probabilities =(0.2+0.4+0.3) = 0.9
required probability = 1 - sum of given probabilities = 1 - 0.9 = 0.1
hence the complete probability table is
X P(X)
20 0.2
49 0.4
53 0.1
80 0.3
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In mrs.thomas’ class, 16 of the students received A’s and 20 received B’s. In mr.carls class 12 students received a’s and 15 recived b’s which class has a lower ratio to a’s and b’s
Mrs Thomas' and Mr Carl's classes both have same lower ratio of A's to B's which expressed in the lowest term as 4:5 or fraction as 4/5
Expression of ratio as fractionsRatio is used to compare values, and can be expressed as fractions.
Mrs Thomas' class have the ratio of students her students A's to B's as 16:20. And can be expressed as fraction in its lowest term as follows:
16:20 = 16/20
16:20 = 4/5
Mr Carl's class have the ratio of students her students A's to B's as 12:15. And can be expressed as fraction in its lowest term as follows:
12:15 = 12/15
12:15 = 4/5
Therefore, since the lowest form of Mrs Thomas' class and Mr Carl's class ratios of A's to B's are equal, then both class have equivalent lower ratio of A's to B's.
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⅕(x + 2 ) -¼(2-x) ≥ ⅓ (x + 4) - ¼ ( 5-x)
Answer:
x > 11/-8
Step-by-step explanation:
1/5(x + 2) -1/4(2-x)> 1/3(x + 4) -1/4(5-x)
=60×1/5(x +2)-60×1/4(2-x)>60×1/3 (x+4)-60×1/4(5-x)
=12(x +2)-15(2- x)> 20(x+4)-15(5-x)
=12x + 24 - 30 + 15x > 20x + 80 - 75 + 15x
=12x + 15x - 20x -15x > 80 -75 -24 +30
=27x -35x > 80 - 99 + 30
=-8x > 11
=-8x/-8 > 11/-8
=x > 11/-8.
Hence the truth set of the inequality is said that (x : x > 11/-8).
Two companies offer monthly cell phone plans, both of which include free text messaging. Company A charges a $25 monthly fee plus five cents per minute of phone conversation, while Company B charges a $50 monthly fee and offers unlimited calling. Both companies charge the same amount when the total duration of monthly calls is 8 hours and 33 minutes 8 hours and 20 minutes 5 hours 500 hours
The charge of the total duration of monthly calls for company A and company B will be same at 8 hours and 20 minutes.
What is a function?Function: a mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.
Given that the cost function of the two companies is:
Company A: $25 monthly charges, and 5 cents of fee per minute.
Let us suppose the duration per minute = m. The cost function is:
A(x) = 25 + 0.05m
Company B: $50 monthly charges.
B(x) = 50
Both companies will charge the same amount when:
A(x) = B(x)
25 + 0.05x = 50
x = 500 minutes.
Converting minutes into hours we get the answer as 8 hours and 20 minutes.
Hence, the time for which both the companies charge the same amount is 8 hours and 20 minutes.
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Jorge wants to order tickets online so that he and 4 of his friends can go together to a movie. The cost of the tickets is $12.00 per person. The website also charges a transaction fee of $2.75 for the tickets online. Write an expression in terms of m that represents the cost of ordering m tickets online. Now use your expression to find the total cost of ordering 5 movie tickets show your work.
Answer:
Step-by-step explanation:
The expression in terms of m is 12m + 2.75.
To find the total cost of ordering 5 movie tickets, we plug in 5 for m: 12(5) + 2.75 = 62.75.
the rescue center has budgeted a total of 2581.8 for this 30-day month. how many dogs are at the rescue center math problem
11423 be the number of dogs are at the rescue center.
What is Expression?An expression is combination of variables, numbers and operators.
A pet rescue center is creating a monthly food budget.
The center currently has two-thirds as many dogs as cats.
Each animal is fed 2 cans of food a day.
Dog food costs $0.08 per can, cat food costs $0.05 per can
The rescue center has budgeted a total 2581.8 of for food for this .
2(0.08)x+2(0.05)y= 2581.8
Let x be the number of dogs.
cats be y.
2/3x =y
2(0.08)x+2(0.05)2/3x= 2581.8
0.16x+0.066x=2581.8
0.226x=2581.8
x=11423
Hence, 11423 be the number of dogs are at the rescue center.
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Stella has 5 1/3 yards of ribbon to make bows. Each bow is made from a piece of ribbon that is 1/3 yard long. How many bows can Stella make?
Therefore , the solution of the given problem of fraction comes out to be Stella makes 6 bows.
What is fraction?The definition of a fraction is a portion of a whole. A single object or a collection of objects might be the entire. When we cut a piece of cake in real life from the entire cake, the part represents the percent of the cake. The word "fraction" is derived from Latin. "Fractus" means "broken" in Latin. The fraction was expressed verbally in earlier times. It was afterwards presented in numerical form.
Here,
Given :
Take the amount of ribbon and divide by the amount needed per bow
5 1/3÷ 1/3
Change the mixed number to an improper fraction
(5*1 +1)/3 ÷1/3
6/3 ÷1/3
Copy dot flip
6/3 * 3/1
6
Stella makes 6 bows.
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Test whether 712.008 is divisible by 11
A
D
Consider these two claims:
•
ABC is similar to both AACD and ACBD.
• (ACI2 + IBC12 = (ABI2
Explain why these claims are correct or incorrect. Provide valid
mathematical reasoning to support your responses.
As the corresponding angles are same in each of the right triangles, they are similar to each other (AAA rule).
AC² + BC² = AB² is true according to the Pythagorean theorem.
What is Pythagorean theorem?One of the earliest known theorems to ancient civilizations was the Pythagorean Theorem. Pythagoras, a Greek mathematician and philosopher, inspired the name of this famous theorem. He is credited with numerous mathematical contributions, some of which may have been the work of his students.
They were adamant that any two lengths had to be integral multiples of some unit length. There have been numerous attempts to conceal the fact that the square root of 2 is irrational. According to legend, the man who revealed the secret drowned at sea.
The Pythagorean Theorem describes triangles with a right angle. According to Pythagorean Theorem:
"The area of the square constructed on the hypotenuse of a right triangle equals the sum of the areas of the squares constructed on the remaining sides."
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A store sees 120 customers in 8hours. What is the unit rate (in customer per hour) ?
Answer: 128 hours
Step-by-step explanation:
if there are only four recent college graduates with math majors, what is the chance that the sample will consist of two who intend to teach high school?
The chance of the sample consisting of two college graduates with math majors who intend to teach high school is zero. There are only four people in the sample, so the sample must consist of all four people.
Since the sample size is fixed, it is impossible to have two out of the four people have the same trait (intending to teach high school). Furthermore, the sample size is too small to yield a meaningful result; the sample size would need to be much larger in order to accurately represent the population of college graduates with math majors. Therefore, the chance of the sample consisting of two college graduates with math majors who intend to teach high school is zero.
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at a hot air balloon festival. mohamed's balloon is at an altitude of 40m and rises 10m/min dana's balloon is at an altitude of 165m and descends 15m/min. a) in how many minutes will both balloons be at the same altitude?
After 5 minutes, the two balloons' height will be 40 + 10 + 5 = 90 ft.
What is unitary method?The unitary technique involves first determining the value of a single unit, followed by the value of the necessary number of units.
Let's say you go to the store to buy six apples.
You are informed by the shopkeeper that he is offering 10 apples for Rs 100. In this instance, the value and the units are the price of the apples.
Recognizing the units and values is crucial when using the unitary technique to a problem.
Always write the items that need to be computed on the right side and the things that are known on the left side to simplify things.
We are aware of the quantity of apples and the amount of money in the aforesaid problem.
According to our question-
Mohammed's altitude will be after a certain amount of time.
40 +10m
Dana's elevation will be...
165 -15m
When..., the elevations will be the same.
40 +10m = 165 -15m
25m = 125
add 15m-40
M = 5 is divided by 25 using the formula:
After five minutes, the two balloons' height will be 40 + 10 + five feet, or 90 feet.
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urgent, rlly need help w/ this !!
Answer:
We can solve the simultaneous equations by using substitution or elimination method. Here, we will use the substitution method:
x - 2y = 3 and x ^ 2 - y ^ 2 + 2x = 10
Solving the first equation for x, we get
x = 2y + 3
Now, substitute this value of x in the second equation
(2y + 3)^2 - y^2 + 2(2y + 3) = 10
Expanding the square on the left-hand side
4y^2 + 12y + 9 - y^2 + 4y + 6 = 10
Combining like terms
3y^2 + 16y + 3 = 0
Now, we can solve this equation by factoring or using the quadratic formula. Here, we will use factoring method:
3y^2 + 16y + 3 = 0
(3y + 3)(y + 1) = 0
So, y = -1 or y = -1
Now, we can substitute either value of y back into x = 2y + 3
For y = -1, x = 2(-1) + 3 = 1
For y = -1, x = 2(-1) + 3 = 1
So, the solutions of the simultaneous equations are x = 1 and y = -1 or x = 1 and y = -1
Answer:
If y = -5, x = -7
If y = -3, x = 2 1/3 or 7/3
Step-by-step explanation:
Alr alr,
x - 2y = 3
x = 2y + 3
Now, lets substitute that into the other equation,
x² - y² + 2x = 10
(2y + 3)² - y² + 2(2y+ 3) - 10 = 0
4y² + 9 + 12y - y² + 4y + 6 - 10 = 0
3y² + 16y + 5 = 0
3y² + 15y + y + 5 = 0
3y(y + 5) + y + 5 = 0
(3y + 1)(y + 5) = 0
y = -5, -1/3
Put both these values in the first equation, u will get,
x = 2y + 3
x = -10 + 3
x = -7
or
x = -2/3 + 3
x = 2 1/3
sorry for taking this long,
My internet gave out after I was about to add the answer.
pls mark as brainliest nevertheless. I don't know why i need it, but i need it. cheers
1. A firefighter is rescuing a cat in a tree. If the branch that the cat is on is 15 feet above the ground and the ladder makes an angle of 63 with the ground, how long is the ladder?
Answer:
13.24 ft
Step-by-step explanation:
To solve this problem you must apply the proccedure shown below:
1. You have:
Sinα=opposite/hypotenuse
α=62°
opposite=x
hypotenuse=15 ft
2. When you substitute the values into Sinα=opposite/hypotenuse, you obtain:
Sinα=opposite/hypotenuse
Sin(62°)=x/15
3. You must clear "x", as below:
x=(15)(Sin(62°))
x=13.24 ft
The answer is. 13.24 ft
Consider the point (-4, -2). What will be the coordinates of the new point if we
-Reflect it across the line y=x
-Translate it 2 units to the right and 3 units up?
The coordinates of the new point after applying the given set of transformation is given by the ordered pair (0, -1).
What is a reflection across the x-axis?In Mathematics, a reflection across the x-axis would maintain the same x-coordinate while the sign of the y-coordinate changes from positive to negative.
Furthermore, a reflection across the line y = x would interchange (switch) the x-coordinate and y-coordinate and this is given by this transformation rule:
(x, y) → (y, x)
Ordered pair A = (-4, -2) → Ordered pair A' = (-2, -4).
Next, we would apply a translation of 2 units to the right and 3 units up:
(x, y) → (x + 2, x + 3)
Ordered pair A' = (-2, -4) → (-2 + 2, -4 + 3) → Ordered pair A' = (0, -1).
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A rental car company offers two rental plans, Plan A and Plan B, for the same economy size car. For both plans, the total rental cost is a function of the number of miles that the car is driven. Plan A: Plan B: I. In complete sentences, translate each function into a verbal model describing the total cost of the rental in terms of the number of miles that the car is driven. II. For each function, determine how the rate of change will affect the total cost of a car rental. III. For a car rental that will include a maximum of 250 miles for the duration of the rental, which plan is the most cost effective
So, on solving the provided question we can say that in the function we have m = 500, after 500 miles of driving, the expenses of the 2 plans are equal.
what is function?The subject of mathematics includes quantities and their variations, equations and related structures, shapes and their locations, and places where they can be found. The term "function" refers to the relationship between a set of inputs, each of which has an associated output. A connection between inputs and outputs in which each input leads to a single, distinct result is known as a function. Each function is given a domain and a codomain, or scope. Usually, f is used to denote functions (x). input is an x. There are four main types of functions accessible. based on the following factors: on functions, one-to-one functions, many-to-one functions, inside functions, and on functions.
f(m) = .2m + 75
f(m) = .35m
.2m + 75 = ,35m
.15m = 75
m = 500
After 500 miles of driving, the expenses of the 2 plans are equal.
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PLEASE HELP!!
I just can’t seem to get this at all.
The equation solved is 1+-2/x²=-x-1 is 2x²+x³-2=0
Find the value of p and q ?the value of p is p=-x-q/x²the value of q is q=-x³-px²Coordinates are written as (x, y) meaning the point on the x axis is written first, followed by the point on the y axisThe coordinates of a point are a pair of numbers that define its exact location on a two-dimensional plane. Recall that the coordinate plane has two axes at right angles to each other, called the x and y axis. The coordinates of a given point represent how far along each axis the point is located.The point where the axes intersect is called the origin. Let's plot a point on the coordinate plane. The location of Point A is given as a horizontal component (x) and a vertical component (y) and written as (x,y). From the origin, to get to Point A, we would count two units to the right and three units upTo learn more about value of p and q refers to:
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1. please help me. 3 divide 255 2. 16 divide 368 3. 65 divide 2 210
The answer of the given conditions is 255/3 =83, 368/16 = 23 and 2 210/65 = 34.
What are the numerator and denominator?
For example, 4/5 is a fraction, and the line separating the numbers 4 and 5 is the fraction bar. Here the number above the fraction bar is the numerator, and the one below the fraction bar is the denominator. A numerator represents the number of parts out of the whole, which is the denominator.
3 divided by 255 is equal to 3/255.
16 divided by 368 is equal to 16/368.
65 divided by 2 210 is equal to 65/2210.
Note that for 2 and 3 the numerator and denominator have been swapped and the division is not correct, it should be 368/16 = 23 and 2 210/65 = 34
Hence, the answer of the given conditions are 255/3 =83, 368/16 = 23, and 2 210/65 = 34.
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Use the Laplace transform to solve the given initial-value problem.
dy/dt -y=1, y(0)=0
to solve the given initial-value problem: dy/dt -y=1, y(0)=0 is, u(t) = e^(t) - 1.
This can be solved using Laplace transform.
What is Laplace transform?In order to translate a given derivative function with a real variable t into a complex function with a variable s, the Laplace transform is used. Let f(t) be supplied for t ≥ 0 and suppose that the function meets some later-explained requirements.
As we just saw, the Laplace transform turns partial differentials to regular differentials, making it one of the most crucial tools for solving ODEs and, more especially, PDEs. In general, applications in the time-domain for t ≥ 0 employ the Laplace transform.
Laplace transformation X is transformed linearly to a new, typically complex variable called s. Differential equations are transformed into only algebraic equations using this method.
Given that,
dy/dt -y=1
y' - y = 1
⇒ L(y') - L(y) = L(1) .....(i)
⇒L(yⁿ) = sⁿ Fy - sⁿ⁻¹y(0) - sⁿ⁻²-2y' (0) — ..... yⁿ⁻¹(0)
yⁿ means nth derivative and F(y) stands Laplace transformation of y so 1 becomes
⇒sF - y(0) - F = 1/c as L(tⁿ) = (sⁿ⁺¹) / (n!)
⇒ F(s - 1) = 1/s
⇒F = 1/(s(s - 1)) = 1/(s - 1) - 1/s
⇒L⁻¹(F) = L⁻¹ [1/(s - 1)] - L⁻¹ (1/s)
=> u(t) = e^(t) - 1
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