Use cylindrical coordinates to find the volume of the region in the first octant bounded by the cylinder r and the plane z.
Answers
The volume of the region in first octant is 9.
Given that the region in the first octant bounded by the cylinder r=3 and the plane z=y.
The graph is shown below.
we are given that first octant is bound by r=3 and z=y
We will use the conversion formula, i.e.
x²+y²=r² where x=rcos∅ and y=rsin∅
Firstly, we will find bound octant
0≤∅≤π/2
0≤r≤3
0≤z≤r sin∅
Now, we can set up integral
[tex]V=\int_{0}^{\frac{\pi}{2}}\int_{0}^{3}\int_{0}^{rsin(\theta)}rdzdrd\theta[/tex]
Further, we can solve it.
Firstly, we solve integral for z then we solve for r and after that we will solve for ∅, we get
[tex]\begin{aligned}V&=\int_{0}^{\frac{\pi}{2}}\int_{0}^{3}r^2 \sin (\theta)dr d\theta\\&=\int_{0}^{\frac{\pi}{2}}\frac{1}{3}r^3\sin \theta d\theta\left|_{0}^{3}\\ &=\int_{0}^{\frac{\pi}{2}}\left(\frac{1}{3}\sin\theta d\theta(27-0)\right)\\ &=\int_{0}^{\frac{\pi}{2}}9\sin \theta d\theta\\ &=0-(-9)\\ &=9\end[/tex]
Hence, the volume of the region in the first octant is bounded by the cylinder r=3 and the plane z=y is 9.
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Related Questions
find the median
643258
Answers
Answer:
median = 4.5
Step-by-step explanation:
the median is the middle value of the data set arranged in ascending order. If there is no exact middle value then it is the average of the values either side of the middle
express the values in ascending order ( smallest to largest )
2 , 3 , 4 , 5 , 6 , 8
↑
the middle is between 4 and 5 , then
median = [tex]\frac{4+5}{2}[/tex] = [tex]\frac{9}{2}[/tex] = 4.5
Why do you think banks will try to sell you credit cards or personal loans?
Answers
Answer: liquidity reasons
Step-by-step explanation:
Banks sell credit cards and personal loans to customers to increase their profits, deepen customer relationships, generate more revenue, and compete in the market.
What is Loan?A loan is the lending of money by one or more individuals, organizations, or other entities to other individuals, organizations, etc.
Banks will try to sell credit cards and personal loans to customers for a few reasons:
Profit: Banks earn money by charging interest on loans and credit cards. By selling these products to customers, banks can earn significant profits.
Customer retention: By offering loans and credit cards to customers, banks can deepen their relationships with them and increase customer loyalty.
Increased revenue: Offering loans and credit cards can help banks generate more revenue from customers.
Market competition: Banks face intense competition from other financial institutions, including credit unions and online lenders.
Hence, banks sell credit cards and personal loans to customers to increase their profits, deepen customer relationships, generate more revenue, and compete in the market.
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In circle O, FG=40 RS=37 OP=19 FG is perpendicular OP, and RS is perpendicular to OQ A. 27.2 B. 18.5 C. 19 D. 20.5
Answers
Answer: In order to solve this problem, you should use the Pythagorean theorem. Since FG is perpendicular to OP, and OP and OQ are radii of the same circle, then triangle OFG is a right triangle. Similarly, triangle ORS is also a right triangle. By using the Pythagorean theorem, you can find that:
OF = sqrt(OP^2 + FG^2) = sqrt(19^2 + 40^2) = sqrt(361 + 1600) = sqrt(1961)
and
OS = sqrt(OQ^2 + RS^2) = sqrt(19^2 + 37^2) = sqrt(361 + 1369) = sqrt(1740)
Therefore, the length of the segment connecting the feet of the altitudes is:
OF + OS = sqrt(1961) + sqrt(1740) = 27.2
So the answer is A. 27.2
Step-by-step explanation:
What is quotient rule Class 7?
Answers
The quotient rule in calculus is a method used to find the derivative of any function given in the form of a quotient obtained from the result of the division of two differentiable functions.
That means if we are given a function of the form: f(x) = u(x)/v(x),
we can find the derivative of this function using the quotient rule derivative:
f'(x) = [u(x)/v(x)]' = [v(x) × u'(x) - u(x) × v'(x)]/[v(x)]2
Where:
f(x) = The function of the form u(x)/v(x) for which the derivative is to be calculated.u(x) = A differentiable function that makes numerator of the function f(x).u'(x) = Derivative of function u(x).v(x) = A differentiable function that makes the denominator of the given function f(x).v'(x) = Derivative of the function v(x).To know more about the quotient rule:
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I need help with my math homework
Answers
The number of hours until the city is locked down is given as follows:
229 hours.
How to obtain the number of hours until the city is locked down?The function for this problem is defined as follows:
p(t) = 0.004t³ + 0.001t + 1.
For which the variables are given as follows:
t is the time in days.p(t) is the number of people infected.The rate of change of the infection in 72 hours = 3 days is given by:
p'(3).
The derivative of the function is given as follows:
p'(t) = 0.012t² + 0.001.
After the instantaneous rate after 3 days is given as follows:
p'(3) = 0.012(3)² + 0.001
p'(3) = 0.109.
The city will lock down when p'(t) = 10 x p'(3) = 1.09 that is:
0.012t² + 0.001 = 1.09.
0.012t² + 0.001 - 1.09 = 0.
Hence the time in days is obtained as follows:
0.012t² = 1.089
t² = 1.089/0.012
t² = 90.75
[tex]t = \sqrt{90.75}[/tex]
t = 9.53 days.
Each day is composed by 24 hours, hence the time in hours is obtained as follows:
9.53 x 24 = 229 hours. (rounding to the nearest whole integer).
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A spinner is divided into 8 equal sections, and each section contains a number from 1 to 8. What is the probability of the spinner landing on 5?.
Answers
The probability the spinner landing on 5 is 0.125. The result is obtained by dividing the number of favorable outcomes by the number of events.
What is probability?Probability is a numerical description of how possible an event to happen. Tha value of probability is between 0 to 1. Zero probability means something is impossible to happen. Probability 1 shows certainty.
Probability of an event can be expressed as
P(A) = n(A) / n(S)
Where
P(A) is the probability of an event An(A) is the number of favorable outcomesn(S) is the total number of events in the sample spaceA spinner is divided into 8 equal sections numbered 1 to 8. Find the probability that the spinner lands on number 5!
From the information, we have:
Total number of events, n(S) = 8.The favorable outcomes that the spinner lands on 5, n(A) = 1.P(A) = n(A) / n(S)
P(A) = 1/8
P(A) = 0.125
Hence, the probability that the spinner lands on number 5 is 0.125.
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Simplify the expression by combining like
terms:
-3c+ c² - 2 + c² + 5c +3
Enter the number that belongs in the green box.
[?] c² + c + ]
Answers
The expression -3c+ c² - 2 + c² + 5c +3 is simplified to 2c² + 2c - 1
What are algebraic expressions?Algebraic expressions are described as mathematical expressions that are comprised of coefficients, terms, factors, constants and variables.
These expressions are known to be made up of mathematical operations, which incudes;
AdditionFloor divisionBracketParenthesesSubtractionMultiplicationDivisionFrom the information given, we have that;
-3c+ c² - 2 + c² + 5c +3
Now, collect like terms, we get;
c² + c² - 3c + 5c - 2 + 3
Now, add or subtract the like terms, we get;
2c² + 2c - 1
Hence, the expression is 2c² + 2c - 1
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A:B=B:C=C:D if C:D is 1:4, what is A:D?
Answers
The ratio A:D = 1:64
What is a ratio?A ratio is the comparison of two amounts of the same unit which shows the amount of one present in the other.
How to find the ratio A:D?Given that A:B = B:C = C:D if C:D is 1:4, we need to find A:D.
So, we proceed as follws
A:B = B:C = C:D
⇒ A/B = B/C = C/D = 1/4
So, we have that the equation
A/B = 1/4
This implies that
A = B/4
Also, we have that the equation
B/C = 1/4
This implies that
B = C/4
Finally, we have that
C/D = 1/4
This implies that
C = D/4
Since A = B/4, B = C/4 and C = D/4,
Substituting C into B, we have that
B = C/4
= (D/4)/4
B = D/16
Since A = B/4, substituting this B into A, we have that
A = B/4
= (D/16)/4
A = D/64
So, A/D = 1/64
⇒ A:D = 1:64
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The pressure P of a compressed gas is inversely proportional to the volume V. If there is a pressure of 25 pounds per square inch when the volume of gas is 400 cubic inches, write the equation that can be used to find the pressure when the gas is compressed to 200 cubic inches?
Answers
The expression for pressure is P = k/V and pressure is 50 pounds per square inch.
What is the ratio?Comparing two amounts of the same units and determining the ratio tells us how much of one quantity is in the other. Two categories can be used to categorize ratios. Part to whole ratio is one, and part to part ratio is the other. The part-to-part ratio shows the relationship between two separate entities or groupings.
Given pressure and volume are inversely proportional to each other
P ∝ 1/V
P = k/V
given when pressure p = 25 pounds per square inch
then volume V = 400 cubic inches
25 = k/400
k = 10,000
and when V = 200 cubic inches
then P = k/V
P = 10000/200
P = 50 pounds per square inch
Hence pressure is 50 pounds per square inch.
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Can residual value be less than 5%?
Answers
Residual Value should not be more than 5% of Cost of Asset i.e Rs. 5000 and accordingly the amount to be adjusted shall be Rs.
What Exactly Is Residual Value?
The estimated value of an asset that is fixed at the end of its useful life or lease term is its residual value, which is often referred to as salvage value. In lease agreements, the residual value is one of the lessor's main methodologies for calculating the lessee's periodic lease payments.
A good residual value is what?
Check out the Car Lease Ratings section of our Lease Kit for a complete list of high-residual vehicle brands and models.
A car is probably not a good lease deal if the lease-end residual value is less than 50% of the MSRP (for a 36-month lease). 55% to 65% of MSRP would be a nice residual.
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Suppoe you have fewer than 10 place value block. How many different way could you how 1214? Explain
Answers
Since 1214 is a four digit number and you have fewer than 10 blocks it is not possible to represent 1214 in any other way.
With fewer than 10 place value blocks, it would not be possible to show the number 1214 using standard place value notation, as you would need at least 4 blocks to represent the ones, tens, hundreds, and thousands place. Each block would represent a different place value, such as ones, tens, hundreds, and thousands. Each block would have a different numerical value, such as 1, 2, 1, and 4, respectively. Since 1214 is a four digit number and you have fewer than 10 blocks it is not possible to represent 1214 in any other way.
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What is ordered pair and unordered pair?
Answers
If the elements in the pairs are related to each other then the pair is called as Ordered Pair an if the elements in the pair are not related then the pair is called as Unordered Pair .
The Ordered Pair is defined as a pair of elements having specific importance for order of their placements.
Also, the ordered pairs are used to represent elements that have a relation between them .
The Ordered pairs are usually used to represent a point on a coordinate plane.
The Unordered Pair is a set of the form (a,b) that have two elements "a and b" that have no particular relation between them,
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The price of Stock A at 9 A.M. was $12.81. Since then, the price has been increasing at the rate of $0.11 each hour. At noon the price of Stock B was $13.31. It begins to decrease at the rate of $0.16 each hour. If the two rates continue, in how many hours will the prices of the two stocks be the same?
Answers
If the two rates continue, the number of hours when the prices of the two stocks would be the same is equal to 1.85 hours.
How to determine the required number of hours?In order to write a system of algebraic equation to describe this situation, we would assign variables to the price of Stock A and the price of Stock B after a certain number of hours respectively, and then translate the word problem into algebraic equation as follows:
Let the variable x represent the number of hours for the price of Stock A and Stock B to become equal.Let the variable T represent the price of each stockFor the price of Stock A increasing at the rate of $0.11 each hour, an algebraic equation which models it is given by:
T = 12.81 + 0.11x
For the price of Stock A decreasing at the rate of $0.16 each hour, an algebraic equation which models it is given by:
T = 13.31 - 0.16x
Equating the two algebraic equations, we have:
12.81 + 0.11x = 13.31 - 0.16x
0.11x + 0.16x = 13.31 - 12.81
0.27x = 0.5
x = 0.5/0.27
x = 1.85 hours.
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How do you write a function in vertex form from a graph?
Answers
By applying the formula x=-b/2a to find the quadratic's vertex, the result can then be used to replace y in the original equation.
What is vertex form from a graph?By applying the formula x=-b/2a to find the quadratic's vertex, the result can then be used to replace y in the original equation. Substitute the vertex into the equation y=a(x-h)2+k in the vertex form. (A will not change; h is x; and K is y.) It is referred to as being in standard form when the quadratic function f(x) = a(x - h)2 + k is not equal to zero. The graph opens either upward or downward depending on whether an is positive or negative. The vertex is the point, while the vertical line x = h is the line of symmetry (h,k).To convert standard form to vertex form,
Convert y = ax2 + bx + c into the form y = a (x - h)2 + k by completing the square.Then y = a (x - h)2 + k is the vertex form.To learn more about vertex form refer to:
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Do sides 7 cm 24cm 25cm form a right angle triangle?
Answers
Answer:
Yes, the sides 7cm, 24cm and 25cm form a right angle triangle.
This is because the sides of a right angle triangle must follow the Pythagorean theorem, which states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, 7cm and 24cm are the two legs and 25cm is the hypotenuse.
Step-by-step explanation:
Heather has a very important exam to take in he morning. Since she wants to be sure to that she will wake up in time, she sets two alarm clocks. One has a. 95 probability that it will ring, and the other has a. 98 probability that it will ring. She sets both clocks. What is the probability that at least one of the alarm clocks will wake her up?.
Answers
Answer: "C. 0.9604.
Step-by-step explanation: The correct answer is The probability that at least one of the alarm clocks will wake her up is 0.9604. Heather has a very important test to take in the morning. So she wants to be sure that she will wake up in time, so she sets two alarm clocks.
a sports ticketing company offers two ticket plans. one plane cost $75 per ticket. the other plan cost $180 plus $15 per ticket. Ig the total cost for the two options is the same. how many tickets did you buy, and how much did you spend?
Answers
You buy 3 tickets. You spent $225 to buy the tickets.
What is an equation?
A mathematical statement made up of two expressions joined by an equal sign is known as an equation. 3x - 5 = 16 is an example of an equation. We get the value of the variable x as x = 7 after solving this equation.
Given that there are two planes to buy tickets.
First plane offers $75 per ticket.
The other plan offers $180 plus $15 per ticket.
Assume that you buy x number of tickets.
According to the first plane, the total cost of x number of tickets is y = $75x.
According to the second plane, the total cost of x number of tickets is y = $180 + 15x
The system of equations is
y = $75x .....(i)
y = $180 + 15x ....(ii)
Using the substitution method solve the equations.
The solution of the equations represents the total cost for the two options is the same.
Putting y = 75x in equation (ii)
75x = 180 + 15x
Subtract 15x from both sides:
75x - 15x = 180
60x = 180
Divide both sides by 60:
x = 3
Putting x = 3 in equation (i)
y = 75 × 3
y = 225
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Solve for y.
8+y=15
Enter your answer in the box.
y =
Answers
Answer:
y=7
Step-by-step explanation:
Answer: y=7
Step-by-step explanation:
15-8=7
y=7
Without using the formula, work out the gradient of the graph shown.
Give your answer in its simplest form.
Answers
The gradient or slope of the line is given by the equation m = -3
What is an Equation of a line?The equation of a line is expressed as y = mx + b where m is the slope and b is the y-intercept
And y - y₁ = m ( x - x₁ )
y = y-coordinate of second point
y₁ = y-coordinate of point one
m = slope
x = x-coordinate of second point
x₁ = x-coordinate of point one
The slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )
Given data ,
Let the equation of line be represented as A
Now , the value of A is
Let the first point be represented as P = P ( -1 , 0 )
Let the second point be represented as Q = Q ( 0 , -3 )
Now , the slope of the line passing through the points is given by
Slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )
Substituting the values in the equation , we get
Slope m = ( -3 - 0 ) / ( 0 - ( -1 ) )
On simplifying the equation , we get
Slope m = -3/1
Slope m = -3
Hence , the slope of the line is m = -3
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6x+8y=13 in inverse form and how you got the answer
Answers
Answer:
6x + 8y = 13. 8y = 13 - 6x.
y = (13 - 6x) / 8.
f(x) = (13 - 6x) / 8.
x = (13 - 6f-1(x)) / 8.
8x = 13 - 6f-1(x).
6f-1(x) + 8x = 13 S
the question is in the picture below
Answers
Answer:-2
Step-by-step explanation:
2(x-4) + 3x -x^2
2(2-4)+ 3(2) - 2^2
2(-2)+ 3(2)-2^2
-4+6-4=
-2
Which function has a vertex on the y-axis?
f(x) = (x – 2)2
f(x) = x(x + 2)
f(x) = (x – 2)(x + 2)
f(x) = (x + 1)(x – 2)
Answers
The function whose vertex is on the y-axis is f(x) = (x – 2)(x + 2).
Option C is the correct answer.
What is a function?A function is a relationship between inputs where each input is related to exactly one output.
Example:
f(x) = 2x + 1
f(1) = 2 + 1 = 3
f(2) = 2 x 2 + 1 = 4 + 1 = 5
The outputs of the functions are 3 and 5
The inputs of the function are 1 and 2.
We have,
Write each of the functions in the form of ax² + bx + c
Vertex of the function.
= ( -b/2a, f(-b/2a) )
Now,
f(x) = (x – 2)²
x² - 2x + 4
a = 1, b = -2, c = 4
-b/2a = 2/2 = 1
f(1) = -2
Vertex = (1, -2)
f(x) = x(x + 2)
x² + 2x
a = 1, b = 2, c = 0
-b/2a = -2/2 = -1
f(-1) = -1(-1 + 2) = -1
Vertext = (-1, -1)
f(x) = (x – 2)(x + 2)
x² + 2x - 2x + 4
x² + 4
a = 1, b = 0, c = 4
-b/2a = 0
f(0) = (0 -2)(0 + 2) = -4
Vertex = (0, -4)
f(x) = (x + 1)(x – 2)
x² - 2x + x - 2
x² - x + 2
a = 1, b = -1, c = 2
-b/2a = 1/2
f(1/2) = (1/2 + 1)(1/2 - 2) = 3/2 x -3/2 = -9/4
Vertex = (1/2, -9/4)
Thus,
The vertex (0, -4) is on the y-axis.
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What is the area of a triangle with the base 4.8 cm and height 3.6 cm *?
Answers
The area of a triangle with the base 4.8 cm and height 3.6 cm is 8.64 cm².
Therefore the answer is 8.64 cm².
The area of a triangle can be calculated using the formula:
Area = (1/2) x base x height
In this case, the base of the triangle is 4.8 cm and the height is 3.6 cm. Plugging these values into the formula, we get:
Area = (1/2) x 4.8 x 3.6
Area = (1/2) x 17.28
Area = 8.64 cm²
So the area of the triangle is 8.64 square centimeterS.
It's important to note that the formula for area of triangle uses the concept of base and height. A base is a side of the triangle and height is the perpendicular distance from the base to the opposite vertex.
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The larger triangle is the image of the smaller triangle after a dilation whose center is at (-2,1).
What scale factor was used in the dilation?
Answers
Considering that the larger triangle is the image of the smaller triangle after a dilation whose center is at (-2,1), the scale factor that was used in the dilation is 2.
What is the dilation of a triangle?Dilation is a technique for creating comparable figures that have their size reduced or increased by certain scaling factors.
Hence, the dilation of a triangle involves decreasing or increasing their sizes by a scale factor.
Considering the larger triangle which is the image of the smaller triangle after a dilation with center at (-2,1);
Scale factor = 5 - 1 / 1 - (-1)
Scale factor = 4 / 2
Scale factor = 2
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Mike asked 200 students to select their favorite sport and then recorded the results in the bar graph below. Mike will ask another 140 students to select their favorite sport.
Based on the information in the bar graph, about how many more students of the next 140 asked will select football rather than track and field as their favorite sport?
Answers
Based on the information in the bar graph, the number of students of the next 140 asked will select football rather than track and field as their favorite sport is, 35
What is the percentage?The percentage is a mathematical term, which means a number or a ratio which is represented in fractions of 100. We use the '%' symbol to represent the percentage.
Formula for percentage;
Percentage = (present quantity/whole quantity)×100
Given that,
Mike asked 200 students to select their favorite sport and then recorded the results in the bar graph
the percentage of students who selects football is,
Percentage = (50/200)×100
Percentage = 25%
If we consider in same percentage students will select football in next 140 students,
then, number of those students will be,
25 = (a/140)×100
a = 35
Therefore, the number of students is, 35
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HELP ITS DUE BY 9:00 I PUT THE PICTURE DOWN BELOW
Answers
2 x 4.5 : 9 yd sq
The length of the hypothesis of an isosceles right triangle is 8. Find the perimeter of the triangle with a picture.
Answers
On solving the provided question, we can say that The perimeter of the Isosceles triangle is 8 + 4v'5+ 4[tex]\sqrt{2}[/tex]= 8 + 8[tex]\sqrt{2}[/tex] = +[tex]\sqrt{2}[/tex])
What is Isosceles Triangle ?A triangle with two sides that have the same length is said to be isosceles. It can be stated as having exactly two equal-length sides or at least two equal-length sides, with the latter definition containing the equilateral triangle as an exception.
A right triangle, also known as a right-angled triangle, right-perpendicular triangle, orthogonal triangle, or previously rectangled triangle, is a triangle with one right angle, or two perpendicular sides. The foundation of trigonometry is the relationship between the sides and various angles of the right triangle.
A right triangle's three sides are related in Euclidean geometry by the Pythagorean theorem, also known as Pythagoras' theorem. According to this statement, the areas of the squares on the other two sides add up to the size of the square whose side is the hypotenuse.
The hypotenuse is 8 so the equal sides are x lets say.
Now, by Pythagoras Theorem
x^2 + x^2 + = 8^2
x^2 = 64/2
x = [tex]4\sqrt{2}[/tex]
The perimeter of the triangle is 8 + 4v'5+ 4[tex]\sqrt{2}[/tex]= 8 + 8[tex]\sqrt{2}[/tex] = +[tex]\sqrt{2}[/tex])
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How do you know if an expression has infinitely many solutions?
Answers
When you try to solve an equation and receive a variable or a number equal to itself, you know it has an endless number of solutions.
What is equation?In its most basic form, an equation is a mathematical statement that indicates that two mathematical expressions are equal. 3x + 5 = 14, for example, is an equation in which 3x + 5 and 14 are two expressions separated by a 'equal' sign. A mathematical statement made up of two expressions joined by an equal sign is known as an equation. 3x - 5 = 16 is an example of an equation. We get the value of the variable x as x = 7 after solving this equation.
Here,
If you try to solve an equation and receive a variable or a number equal to itself, you know it has an endless number of solutions.
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help pls i have already answered but not sure if my answer is right
Answers
Answer:
China = 6018 / 1339 = 4.49
US = 5093 / 307 = 16.59
Russia = 1704 / 140 = 12.17
India = 1293 / 1157 = 1.12
Japan = 1246 / 127 = 9.81
Germany = 858 / 82 = 10.46
Canada = 614 / 33 = 18.60
Find the greatest common factor of 48xy^2 and 54x^3y^2z
Answers
The common factors for 48,36,−12 48 , 36 , - 12 are 1,2,3,4,6,12 1 , 2 , 3 , 4 , 6 , 12 . The GCF for the numerical part is 12.
What is common factors?"Factors" are numbers we can multiply together to get another number.When we find the factors of two or more numbers, and then find some factors are the same ("common"), then they are the "common factors"."Factors" are numbers we can multiply together to get another number.When we find the factors of two or more numbers, and then find some factors are the same ("common"), then they are the "common factors".A factor of a number is an exact divisor of the given number or when a number is exactly divided by a divisor that divisor is known as a factor of that number. Factors of a number cannot be greater than the given number, they are either less than or equal to the given number. 1 is a common factor of all numbers and every number is a factor of itself. When we find the factors of two or more numbers we notice that some factors are common, this is because a number can be exactly divided by many divisors if the number is a multiple of that divisor. For example, the factors of 35 are {1,5,7,35} and factors of 45 are {1,3,5,9,15,45}. 1 and 5 are the numbers that are exactly dividing both the numbers. Thus the common factors of 35 and 45 are 1 and 5.To learn more about divisor refer to:
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