To convert the given third-order linear equation into a system of first-order equations, we introduce three new variables:
x₁ = y
x₂ = y'
x₃ = y''
Now, let's differentiate these new variables to express the derivatives in terms of the original equation:
x₁' = y' = x₂
x₂' = y'' = x₃
x₃' = y''' = (1/3)(t³ - t⁴)y' - ycos(t) + 3t⁴/3
Now we have a system of first-order equations:
x₁' = x₂
x₂' = x₃
x₃' = (1/3)(t³ - t⁴)x₂ - x₁cos(t) + t⁴
To determine the initial values, we substitute the given initial conditions:
x₁(-1) = y(-1) = -1
x₂(-1) = y'(-1) = 0
x₃(-1) = y''(-1) = 2
Hence, the equivalent system of first-order equations with initial values is:
x₁' = x₂, x₁(-1) = -1
x₂' = x₃, x₂(-1) = 0
x₃' = (1/3)(t³ - t⁴)x₂ - x₁cos(t) + t⁴, x₃(-1) = 2
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At a Cambridge College, 65% of the math tudent are from England. If there are 240 math tudent, how many are not from England?
84 math students are not from England
Given that
In mathematics, a percentage (from Latin: per centum, "by a hundred") is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%",although the abbreviations "pct.", "pct" and sometimes "pc" are also used. A percentage is a dimensionless number (pure number); it has no unit of measurement.
The term "percent" is derived from the Latin per centum, meaning "hundred" or "by the hundred". The sign for "percent" evolved by gradual contraction of the Italian term per cento, meaning "for a hundred". The "per" was often abbreviated as "p."—eventually disappeared entirely. The "cento" was contracted to two circles separated by a horizontal line, from which the modern "%" symbol is derived.
At a Cambridge College,65% of the math student are from England
If there are 240 math student
now we need to find how many are not from England
= (100 - 65)% = 35% percentage
= [tex]\frac{35}{100}[/tex] * 240
= 84
84 math students are not from England
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9 divided by 20.25 help L
Answer:
0.444
9/20.25=0.444
please help me find the scale factor of this dialation.
Considering the graph the scale factor of the dilation is calculated to be
1/3What is dilation?Dilation is a method of transformation that magnify or shrink the preimage depending on the scale factor
The transformation rule for dilation is as follows
(x, y) for a scale factor of r → (rx, ry)
In solving to get the scale factor of an image reference point is chosen and the proportion of the dimensions are taken
Let the reference point be TV and T'V'
TV = 6 units
T'V' = 2 units
scale factor = T'V' / TV
= 2 / 6
= 1 / 3
the scale factor is 1/3
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I am thinking of two numbers. When i add my numbers, the answer is 1 when i multiply my numbers, the answer is 0. 09 what are my numbers?.
The two numbers are either 0 and 1 or 1 and 0. This is a method of solving a system of equations called substitution method
To solve this problem you can use a mathematical one, called a system of equations. A system of equations is a set of two or more equations that contain multiple variables. In this case, we have two equations with two variables, x and y.
We know that when you add the two numbers, the result is 1. We also know that when you multiply the two numbers, the result is 0.
Given these two equations:
x + y = 1 and x*y = 0
We know that one of the numbers is 0, because when we multiply any number by 0 the result is 0.
So x=0 or y=0
Now we can use the first equation, x + y = 1, and substitute in one of the values we know:
x = 0 and y = 1
or
y = 0 and x = 1
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express the absolute value f(x)=6|x| function as a piecewise-defined function.
The absolute value f(x)=6|x| function as a piecewise-defined function is:
f(x) = { 6x, if x ≥ 0
{ -6x, if x < 0
How to express an absolute value function as a piecewise-defined function?A piecewise-defined function is a function that is defined by multiple sub-functions, each of which applies to a certain range of the input (or domain) values.
Each sub-function is defined by a separate piece of the overall function, and together they define the entire domain of the function. The sub-functions are typically separated by breakpoints, which are values of the input variable at which the sub-functions change.
|x| can either be -x or x. Thus, the absolute value function f(x) = 6|x| can be written as:
f(x)= 6x or f(x)= -6x
Therefore, the piecewise-defined function of f(x) = 6|x| will be:
f(x) = { 6x, if x ≥ 0
{ -6x, if x < 0
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Please answer this question with decent explanation. Thank you.
Answer:
15.966
Step-by-step explanation:
a/sinA=b/sinB
12/sin37⁰=x/sin53⁰
cross multiply ;
12sin53⁰=xsin37⁰
9.58=0.60x
x=15.966
What are the 3 ways to calculate slope?
There are several ways to calculate the slope of a line, but the most common ways are:
Rise over Run: This is the most basic and intuitive way to calculate slope. It involves taking the difference in the y-coordinates (rise) and dividing it by the difference in the x-coordinates (run) between two points on the line. The slope is represented by the letter "m" and can be calculated using the following formula: m = (y2 - y1) / (x2 - x1)
For example, if two points on a line are (3,6) and (6,9) the slope can be calculated as: m = (9-6) / (6-3) = 3/3 = 1
Point-Slope Form: This method uses the coordinates of a single point on the line and the slope of the line to find the equation of the line. The slope is represented by the letter "m" and the point by the coordinates (x1, y1). The equation of the line in point-slope form is: y - y1 = m(x - x1)
Slope-Intercept Form: This method expresses the equation of a line in the form of y = mx + b, where m is the slope and b is the y-intercept. To find the slope-intercept form, you need to know the slope and the y-intercept (the point where the line crosses the y-axis) of the line.
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How do you write 1st in exponential form?
The number 1 in exponential form can be written as 1 itself , because 1 is the base and the "number 1" to the exponent of any number will be 1 only .
What is Exponential Form?
The exponential Form of representation is a way of representing the repeated multiplication which involves base and exponents.
For Example : if 3 is multiplied 2 times ,then the exponential form will be [tex]3^{2}[/tex] ;
we have to write the exponential form of the number 1 ;
So , the base of the number is 1 , let x be the exponent ,
So , the exponential form will be ⇒ [tex]1^{x}[/tex] , on substituting any value in x , we will always get the result as 1 .
Therefore , the exponential form of the number 1 is 1 .
The given question is incomplete , the complete question is
How do you write the number 1 in exponential form ?
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May ha 4/15 meter of lace while lovie ha 2/7 meter longer Than may lace. How meter of lace do girl have altogether?
May ha 4/15 meter of lace while lovie ha 2/7 meter longer Than may lace then 86/105 meter of lace do girl have altogether
Meter Definition
The meter is the standard unit of measuring length in the International System of Units (SI). Its symbol is “m” and it is one of the seven base units of the SI system.
May has 4/15 meters of lace
Lovie has a lace that is 2/7 meters longer
4/5 + 2/7
= 58/105
Therefore the lace they own together can be calculated as follows
= 4/15 + 58/105
= 86/105
Hence together they have 86/105 meters of lace
that is May ha 4/15 meter of lace while lovie ha 2/7 meter longer Than may lace then 86/105 meter of lace do girl have altogether
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(171/77 simplify the expression
Answer: 2 17/77
Step-by-step explanation:
I assume that simplify means convert to a fraction because it is a fraction
Hope this helps !
The operation is defined on thet set of R by by a*b-1 for all abER Is R Closed under the operation?
When the operation is defined on the set of R by a*b-1, it is commutative rather than associative.
what is equation ?In algebra, an equation is a mathematical statement that establishes the equality of two mathematical expressions. Take a look at the formula 3x + 5 = 14, where 3x + 5 and 14 are two expressions that are separated by the word "equal." The standard form, the point-slope form, and the slope-intercept form are the three primary kinds of linear equations. Algebra is used in many equations. When it comes to arithmetic, algebra is used when it's difficult to compute an exact amount.
Given,
a∗b=1+ab,∀a,b∈R
Commutativity a∗b=1+ab
=1+b.a
=b∗a
∵ Set of real numbers is commutative.
So, a∗b=b∗a
∴ is commutative Associativity (a∗b)∗c=(1+ab)∗c
=(1+ab)c
=2+abc
a∗(b∗c)=a∗(1+bc)=1+a∗(a+bc)
=1+a+abc
It is clear that (a∗b)∗c
=a∗(b∗c)
So, * operation is not associative.
So, option (a) is correct.
When the operation is defined on the set of R by a*b-1, it is commutative rather than associative.
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What number should be added to the polynomial 2x³ − 3x² − 8x so that the resulting polynomial leaves the remainder 12 when divided by 2x 1?
The missing number to be added to the polynomial is 16. 2x³ − 3x² − 8x + 16 leaves a remainder of 12 when divided by 2x.
The polynomial 2x³ − 3x² − 8x + ? leaves a remainder of 12 when divided by 2x. We can solve this by using the Remainder Theorem. The Remainder Theorem states that if a polynomial p(x) is divided by (x-a), the remainder is p(a).
We can use this theorem to solve the problem by setting x=2 and p(x)=2x³ − 3x² − 8x + ?. We can then solve for ? to find the missing number.
p(2) = 2(2³) − 3(2²) − 8(2) + ? = 16 - 12 - 16 + ? = ?
Therefore, the missing number to be added to the polynomial is 16. 2x³ − 3x² − 8x + 16 leaves a remainder of 12 when divided by 2x.
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Ron and his friends like to send messages to one another. One evening, Ron estimated how many messages he'd sent that day. When he looked at his phone, he saw he'd sent 25 messages. His estimate had been 40% more than that. How many messages did Ron estimate he'd sent?
We know that Ron's estimate for the number of messages he sent that day was 40% more than the actual number of messages he sent. To find out how many messages he estimated he sent, we need to undo this percentage increase.
If the actual number of messages Ron sent was 25, and his estimate was 40% more than that, we can use the following equation:
Estimated number of messages = Actual number of messages + (Actual number of messages * 40%)
We can then substitute in the known values and solve for the estimated number of messages:
Estimated number of messages = 25 + (25 * 40%)
Estimated number of messages = 25 + (25 * 0.4)
Estimated number of messages = 25 + 10
Estimated number of messages = 35
Therefore, Ron estimated that he sent 35 messages that day.
What is right triangle class 7?
A "right triangle" in 7th-grade math class refers to a triangle with one angle measuring exactly 90 degrees, and students learn about the properties of right triangles, the Pythagorean theorem, and how to use it to solve problems.
In 7th-grade math, students typically learn about the basic properties of triangles, including their angle measures and side lengths. They learn how to classify triangles based on their angle measures, and the Pythagorean theorem, which relates the side lengths of a right triangle. They also learn how to use the theorem to solve real-world problems, such as finding the length of the missing side of a right triangle.
In a 7th-grade math class, students will likely also learn about the special right triangles, such as the 30-60-90 triangle and the 45-45-90 triangle. These triangles have special ratios of their side lengths that make them useful for solving mathematical problems. They also will learn about solving problems with the Pythagorean theorem.
In a 7th grade math class, students are expected to have a good understanding of triangles, and be able to classify them according to their angle measures, solve problems with the Pythagorean theorem, and be familiar with the special right triangles.
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Consider the equation `q=4+0.8p`.
What value of `q` would make the equation true when `p` is 100?
q = 84
q = 4 + 0.8p (where "p" = 100)
q = 4 + 0.8 × 100
q = 4 + 80
q = 84
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q = 84 for the equation q = 4 + 0.8p to be true, when `p` is 100.
q=4+0.8p (given)
If 'p' is equal to 100
Putting p = 100 into the above equation,
q = 4 + 0.8p
q = 4 + 0.8 × 100
q = 4 + 80
q = 84
Hence, when 'p' is 100, q would be 84 to make the equation true.
At 10:00 a. M. , Han and Tyler both tarted running toward each other from oppoite end of a 10-mile path along a river. Han run at a pace of 12 minute per mile. Tyler run at a pace of 15 minute per mile. How far doe Han run after a half hour? After an hour?
2.5 miles Han run after a half hour
it will take 1[tex]\frac{1}{9}[/tex] to meet both
Given that
At 10:00 a. M. , Han and Tyler both tarted running toward each other from opposite end of a 10-mile path along a river.
Han run at a pace of 12 minute per mile.
Tyler run at a pace of 15 minute per mile.
We can say that he is capable of running at a pace of 5 mph. (Since speed is equal to 1/pace, and 12 minutes equal 1 mile and 60 minutes equal 5 miles),
Similar to this, Tyler would run at a speed of 4 miles per hour if he ran a mile at a pace of 15 minutes.
distance = speed * time
The distance covered by Han in half an hour = 1/2 × 5 = 2.5 miles
the time at which Han and tyler will meet,
T= Total distance/ ( the sum of their speed)
T = 10/5+4 = 10/9
T = 1[tex]\frac{1}{9}[/tex] hours is greater than 1 hour
so, they will not meet within 1 hour
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Determine the equation of a straight line passing through (-1,3) and parallel to the line whose equation is 3x-5y=10.
Answer:
y = (3/5)x+3.6
Step-by-step explanation:
Let's look for an equation of the standard form: y=mx+b, where m is the slope and b is the y-intercept (the value of y when x is 0). Parallel lines have the same slope, m. Next, rearrange the reference line to standard format:
3x-5y=10
-5y = -3x + 10
y = (3/5)x + 2
Now we can easily find the slope. The slope of the reference line is (3/5). That will also be the slope of the parallel line, so we can write:
y = (3/5)x + b
We need a value of b that will shift the line so that it touches point (-1,3) . ["passes through"]. Enter that point into the above equation and solve for b:
y = (3/5)x + b
(3) = (3/5)*(-1) + b for point (-1,3)
3 = -(3/5) + b
-(3/5) = 3-b
-b = -3 - (3/5)
b = 3+(3/5)
b = (18/5), or 3.6
The equation becomes y = (3/5)x+3.6
See the attached graph.
[Parallel lines have a lot in common. Too bad they'll never meet.]
14. The sum of money of $3.500 is divided among Adrian, Sean and James. Scan received half. Adrian received $850 and James received the remainder. Calculate:
(a) Sean's share
(b) James' share
c) the ratio in which the $3 500 was divided among the three persons
D) the percentage of the total amounts that Adrian received
a. 1750; b. 900; c. Sean 50%, Adrian = 850/3500= 24.3%, James =25.7%
How to calculate the problem?step1
The sum of money of $3500 is divided among Adrian, Sean and James. Sean received half. Adrian received $850 and James received the remainder, calculate:
(a) Sean's share:
Sean received = 3500/2 = $ 1750
= (1750/3500)*100 = 50%
(b) James' share:
James received = 3500 - 1750 - 850 = $ 900
= (900/3500)*100 = 25.7 %
step2
(c) the ratio in which the $3500 is divided Amon the three persons.
Adrian :Sean: Ames = 850:1750:900 = 17:35:18
(d) the percentage of the total that Adrian received.
Adrian received = (850/3500)*100 = 24.3 %
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The cost y (in dollars) to rent a kayak is proportional to the number x of hours that you rent the kayak. It costs $27 to rent the kayak for 3 hours. Write an equation that represents the situation.
Answer:
y = 9x
Step-by-step explanation:
Let y = the cost
Let x = the number of hours
It costs $9 and hour to rent 27/3 = 9
Which method would be the simplest way to solve the system?7x + 5y = 19–7x – 2y = –16a. graphingb. substitutionc. eliminationd. distributive
The simplest way to solve the system of equations 7x + 5y = 19 and -7x - 2y = -16 would be using the method of c i.e Elimination.
Elimination, also known as the method of adding or subtracting equations. This method involves adding or subtracting the equations in such a way that one of the variables is eliminated, making it easier to solve for the other variable.
Here is one way to solve the system of equations 7x + 5y = 19 and -7x - 2y = -16 using the elimination method:
Add the two equations together:
(7x + 5y) + (-7x - 2y) = 19 + (-16)
This eliminates the x variable and gives you:
3y = 3
Divide both sides of the equation by 3 to get:
y = 1
Substitute this value of y back into one of the original equations (either 7x + 5y = 19 or -7x - 2y = -16) to find the value of x. For example, using the first equation:
7x + 5(1) = 19
7x + 5 = 19
7x = 14
x = 2
So the solution to the system of equations is x = 2 and y = 1.
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How do you find the length of the third side of a triangle for Class 7?
Answer:
To find the length of the third side of a triangle for class 7, you can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, 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.
The theorem can be written as: c^2 = a^2 + b^2, where c is the hypotenuse and a,b are the other two sides.
So, if you know the lengths of the other two sides, you can use this theorem to find the length of the third side. For example, if you know the lengths of sides a and b, you can use the following formula to find the length of side c: c = √(a^2 + b^2)
You can also use the converse of the Pythagorean theorem, which states that if the square of the length of one side is equal to the sum of the squares of the other two sides, the triangle is a right triangle.
It is always a good idea to draw a diagram to help you understand the problem and to double check the answer.
Step-by-step explanation:
if y is the number of yellow cars which variable expression represents the phrase below? the sum of yellow cars and 9 red cars
The y + 9 is the variable expression represents the phrase below.
What is an expression?One mathematical expression makes up a term. It might be a single variable (a letter), a single number (positive or negative), or a number of variables multiplied but never added or subtracted. Variables in certain words have a number in front of them. A coefficient is the number used before a phrase.
Given:
y is the number of yellow cars.
And the expression,
"the sum of yellow cars and 9 red cars."
The expression,
= the number of yellow cars + the number of red cars
= y + 9
Therefore, y + 9 is the required expression.
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Can anyone help me with this ASAP I have done this but it ain’t working thanks
Answer:
No picture is shown?
Step-by-step explanation:
Answer: i think i did it
Step-by-step explanation:
HURRY WILL MARK BRAINLIEST An egg carton of a dozen eggs consists of 1 randomly selected empty cardboard container and 12 randomly selected eggs. The weights of the empty cardboard containers have a mean of 20
grams and a standard deviation of 1.7 grams. The weight of a single egg has a mean of 68.33 grams and standard deviation of 2.23 grams. It is reasonable to assume independence between the weights of the empty cardboard containers and the weights of the eggs. It is also reasonable to assume independence among the weights of the 12 eggs that are randomly selected for a full carton. Let the random variable X be the weight of a randomly selected carton of a dozen eggs.
i) The mean of x is 68.33 grams.
ii) The standard deviation of x is 2.23 grams.
What is standard deviation?
A standard deviation or (σ ) is a measurement of the data's dispersion from the mean. A low standard deviation indicates that data are concentrated around the mean, whereas a high standard deviation shows that data are more dispersed.
If the standard deviation is close to zero, the data points are close to the mean; otherwise, if the standard deviation is high or low, the data points are, respectively, above or below the mean.
i) Properties of expected values establish that
[tex]$\mathrm{E}(W)=\mathrm{E}(P)+\mathrm{E}\left(X_1\right)+\ldots+\mathrm{E}\left(X_{12}\right)$[/tex]
Because all 12 eggs have the same mean weight, this becomes
[tex]$\mathrm{E}(W)=\mathrm{E}(P)+12 \times \mathrm{E}\left(X_i\right)$[/tex]
We were told that
[tex]$\mathrm{E}(W)=840$ and $\mathrm{E}(P)=20$, so we can solve[/tex]
[tex]$840=20+12 \times \mathrm{E}\left(X_i\right)$ to find $\mathrm{E}\left(X_i\right)=\frac{840-20}{12} \approx 68.33$ grams.[/tex]
ii) Because of independence, properties of variance establish that
[tex]$${Var}(W)={Var}(P)+{Var}\left(X_1\right)+{Var}\left(X_2\right)+\ldots+{Var}\left(X_{12}\right)$$[/tex]
Because all 12 eggs have the same variance of their weights, this becomes
[tex]${Var}(W)={Var}(P)+12 \times {Var}\left(X_i\right)$[/tex]
We were told that
[tex]${SD}(W)=7.9\ \text {and}\ {SD}(P)=1.7$ Therefore, ${Var}(W)=(7.9)^2=62.41$ and ${Var}(P)=(1.7)^2=2.89$[/tex]
We can solve
[tex]$ 62.41=2.89+12 \times {Var}\left(X_i\right)$[/tex]
To find
[tex]${Var}\left(X_i\right)=\frac{62.41-2.89}{12}=4.96$[/tex]
Thus, SD[tex](X_i\right))=\sqrt{(4.96)} \approx[/tex] 2.23 grams.
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Which statement about the function fis false?
A) -2 and 1 are zeros of the function.
B) The function has a local minimum at x = 1 and a local maximum at x = -1.
C) The function is increasing when x < -1 and x > 1, and decreasing when -1 < x < 1.
The claim about the function that is false is At x = 1 and x = -1, respectively, the function has a local minimum and a local maximum.
what is function ?
The study of mathematics comprises the study of quantities and their variations, equations and associated structures, shapes and their positions, and locations where they can be found. A combination of inputs and corresponding outputs are referred to as a "function," which describes the relationship between them. The term "function" refers to an association between inputs and outputs where each input produces a single, unique result. There are two domains, or scopes, assigned to each function. Usually, the symbol f is used to signify functions (x). input is an x. There are four basic categories of functions that are available: on functions, one-to-one functions, many-to-one functions, within functions, and on functions.
given
Let y=x^x
⇒ dx/dy =x ^x (1+logx)
For increasing function, dx/dy >0
⇒x ^x (1+logx)>0
⇒1+logx>0
⇒log e
x>log e/e
⇒x> 1/e
Therefore, the function is increasing, when x> 1/e
The claim about the function that is false is At x = 1 and x = -1, respectively, the function has a local minimum and a local maximum.
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3 cars are washed every 27 minutes at a car wash
Express the unit rate as a shaded fraction of the circle below
Answer: The unit rate is the number of cars washed per minute. To find the unit rate, we need to divide the number of cars washed by the number of minutes.
3 cars / 27 minutes = 1/9 car/minute
To express the unit rate as a shaded fraction of a circle, we can divide the circle into 9 equal parts and shade 1 of those parts to represent the unit rate of 1/9 car/minute.
It is not possible to provide a shaded fraction of a circle here as it is a text based interface.
Step-by-step explanation:
The total weight of red balls and yellow balls in a basket is
4,300 g. Each red ball weighs 35 g, and each yellow ball
weighs 40 g. How many yellow balls are there if there are 60
red balls in the basket?
yellow balls
According to the condition given in the question that the basket contains red balls and yellow balls weighing 4300 g, the total number of yellow balls in the basket would be 55.
What is algebra?Algebra is a branch of mathematics that deals with mathematical symbols and the rules for manipulating these symbols, including solving equations and analyzing graphs
What is the difference between an equation and an expression?An equation is a statement of equality between two expressions, while an expression is a combination of numbers, variables, and mathematical operations. An equation is something that can be solved to find the value of the unknown, while an expression is a combination of numbers and operations that stands on its own.
Let there are R red balls and Y yellow balls in the basket,
As given in the question, each red ball weighs 35 g and each yellow ball weighs 40g, and the total weight of basket is 4300 g,
so, 35 * R + 40 * Y = 4300
Also, there are 60 red balls in the basket , so R =60,
35 * 60 + 40 * Y = 4300
therefore, Y = 55
Hence there are 55 yellow balls in the basket.
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What are altitudes of a triangle class 7?
Answer:
Step-by-step explanation:
The altitude of a triangle is the perpendicular drawn from the vertex of the triangle to the opposite side.
How do you translate an image on a coordinate plane?
Therefore , the solution of the given problem of coordinate plane is we can translate an image on a coordinate plane .
What is a coordinate plane?The term "coordinate plane" refers to a two-dimensional region that consists of two number lines. It emerges when the X-axis & Y-axis intersect at a location known as the origin. Using the numbers on a coordinate grid, one can locate points.
Here,
If we are aware of the direction and magnitude of the figure's movement, we may draw the translation in the coordinate plane.
Use P'(x+a,y+b) to translate the point P(x,y), a unit to the right, and b unit up.
So, translating an image onto a coordinate plane is the answer to the provided coordinate plane problem.
Therefore , the solution of the given problem of coordinate plane is we can translate an image on a coordinate plane .
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What is a scale factor of 4?
The scale factor is the ratio of the length of a side of one figure to the length of the corresponding side of the other figure
What is a scale factor of 4?Scale factors in geometry are defined as a conversion factor for distances in similar shapes, so the scale factor is the factor you would have to multiply all of the dimensions of one shape by in order for it to have the same dimensions as the other, so a scale factor of 4 means that if you multiplied all of the dimensions of one shape by 4, they would be the same dimensions as the dimensions of the other.
Alternatively, if this is not involving another shape, a scale factor of 4 is merely a factor for dilation, the transformation that makes a shape larger, therefore a scale factor of 4 would simply need you to multiply all of the original’s dimensions.
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