Answer:
y = -2(x + 3)^2 - 1
Step-by-step explanation:
The vertex equation of a parabola is y = a(x - h)^2 + k. If the vertex here is (-3, 1), then this equation becomes y = a(x + 3)^2 + 1.
Knowing that the graph passes through (-4, -1), we let y = -1 and x = -4, obtaining:
-1 = a(-4 + 3)^2 - 1, which must be solved for a:
-1 = a - 1. Therefore, a = -2, and the exact equation of this parabola is:
y = -2(x + 3)^2 - 1
two numbers which differ by 3 have a product of 88. Find the two numbers.
Answer:
Step-by-step explanation:
Let's call the two numbers x and y, where x is the smaller number and y is the larger number. We are given that x and y differ by 3, which means y - x = 3. We are also given that the product of x and y is 88, which means x * y = 88. We can solve this system of equations by substituting the second equation into the first equation:
y - x = 3
y = x + 3
Substituting this expression for y into the second equation, we get:
x * (x + 3) = 88
x^2 + 3x - 88 = 0
This is a quadratic equation that we can solve using the quadratic formula:
x = (-3 +/- sqrt(3^2 - 41(-88))) / (2*1)
= (-3 +/- sqrt(9 + 352)) / 2
= (-3 +/- sqrt(361)) / 2
= (-3 +/- 19) / 2
= -11/2 or 8/2
Therefore, the two numbers are x = -11/2 and y = 8/2, or x = -5.5 and y = 4. These are the solutions to the original system of equations.
at a local fair, a smartphone company hosted a booth and brought several models of its phones for customers to play with and look at up close. customers were then encouraged to fill out comment cards letting the company know which of the models was their favorite. at the end of the fair, the company would randomly select a card and the owner of that card would win the smartphone of his or her choice. this is an example of
The given content is an example of sweepstakes.
What is a sweepstake?
A sweepstake is a type of contest in which a prize or prizes are given to the winner or winners. Sweepstakes began as a type of lottery tied to the sale of a product. Sweepstakes became strictly "No purchase necessary to enter or win" and "A purchase will not increase your chances of winning" under these laws, especially because many sweepstakes companies skirted the law by stating only "no purchase necessary to enter," removing the consideration (one of the three legally required elements of gambling) to prevent sweepstakes abuse.
Based on the above definition,
we can conclude that the given question is an example of a sweepstake.
Hence, this is an example of sweepstakes.
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What is the base area B?
The space occupied by the flat surface of a cylinder's base, also known as the base area, is quantifiable.
Given,
Base area of cylinder;
The amount of space that a flat surface at the base of a cylinder occupies is known as the base area. Since the cylinder's base is a circle, the base area of the cylinder is actually an area of a circle.
Formula for base area;
The cylinder's base area is calculated by multiplying the square of its radius by π.
Then,
"πr²" is the formula for the cylinder's base area with radius r.
That is,
Base Area of a Cylinder = (π × radius²) square units.
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Complete question is here;
What is the base area B of cylinder?
What is rational root theorem give one example?
The rational root theorem is the required to represents rational root or zeros or the solution of the given polynomial function with example: g(x) = 4x² - 4x + 1.
As given in the question,
Rational root theorem is also called rational zero or the solution theorem.let us take a polynomial function g(x) = b₀ + b₁x + b₂x² + ...+ bₙxⁿRational root or the solution can be derived in the simplest form x/y by using the following formula:r / s = ( One of the prime factor of b₀ ) / ( One of the prime factor of bₙ )The value of r/s is always a representation of a prime number. rational root theorem is known by different names such as root, zeros or the solution of polynomial function.Example of rational root theorem is : consider g(x) = 4x² - 4x + 1.Here b₀ = 1 , bₙ = 4 and prime factor of 4 is 2 then r/s = 1/2 is one of the factor or zero or root.Therefore, the rational root theorem is the required to find the rational root or the solution or the zero of the polynomial function.
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When the sun is at a certain angle in the sky, a 50-foot building will cast a 20-foot shadow. What is the length of the shadow in feet cast by a 20-foot pole at the same time? Enter only the number.
Using the concept of proportions, the 20 feet pole will cast an 8ft shadow
ProportionsProportion is a mathematical comparison between two numbers. According to proportion, if two sets of given numbers are increasing or decreasing in the same ratio, then the ratios are said to be directly proportional to each other.
In this problem, we can find the length of the shadow casted if we write a proportional relationship between the two object.
50 / 20 = 20 / x
Cross multiply both sides and solve for x
50 × x = 20 × 20
50x = 400
x = 400 / 50
x = 8
The shadow is 8ft
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Last summer, Sheila decided to measure change in a swimming pool's
water level each week. There was a hot and dry spell that lasted 4
weeks and the water level dropped the following amounts each week
Sheila measured the level: 8 millimeters in the first week, 5 millimeters
in the second week, 12 millimeters in the third week, and 5 millimeters
in the fourth week. Express the total gain/loss of water over the 4
weeks as an integer
Sheila measured the water level in the swimming pool every week for 4 weeks and found that the water level decreased by 8 millimeters in the first week, 5 millimeters in the second week, 12 millimeters in the third week and 5 millimeters in the fourth week. The net result was a loss of 10 millimeters of water over the 4 week period.
Sheila wanted to track the changes in the water level of a swimming pool over time. She measured the level every week for 4 weeks. In the first week, the water level dropped by 8 millimeters. In the second week, the level dropped by 5 millimeters. In the third week, the level dropped by 12 millimeters. Finally, in the fourth week, the level dropped by 5 millimeters. The net result of these measurements was a loss of 10 millimeters of water over the course of the 4 week period. This indicates that the hot and dry spell had a negative effect on the water level in the pool.
Total Water Loss = 8 millimeters + 5 millimeters + 12 millimeters + 5 millimeters = 30 millimeters
Total Water Gain = 0 millimeters
Net Loss = Total Water Loss - Total Water Gain = 30 millimeters - 0 millimeters = -10 millimeters
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Which equations are continuous?
Any equation that does not contain a discontinuity, defined as a point where the graph of the equation has a sudden break or jump, is considered continuous. Examples of continuous equations include linear equations, polynomial equations, and exponential equations.
Continuity is an important concept in mathematics and related disciplines. A continuous equation is defined as any equation that does not contain a discontinuity, which is a point where the graph of the equation has a sudden break or jump. Discontinuous equations are equations with discontinuities that cause the graph of the equation to have sudden jumps or breaks. Linear equations, polynomial equations, and exponential equations are all examples of continuous equations, since they do not contain any discontinuities. Continuous equations are important because they are used in many areas of mathematics, such as calculus, to solve problems and understand the behavior of certain function
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How do you know if a curve is positive or negative?
Positive curve graphs will be upward-sloping.
Negative curve graphs will be downward-sloping.
Both graphs in the first image attached below, show curves sloping upward from left to right. As with upward-sloping straight lines, we can say that generally the slope of the curve is positive. While the slope will differ at each point on the curve, it will always be positive.
In the graphs in the second image attached, both of the curves are downward sloping. Straight lines that are downward sloping have negative slopes; curves that are downward sloping also have negative slopes. The slopes change from point to point on a curve, but all of the slopes along these two curves will be negative.
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Straight lines that are downward sloping have negative slopes; curves that are downward sloping also have negative slopes. The slopes change from point to point on a curve, but all of the slopes along these two curves will be negative.
What is the graph for y = 3x - 1?
Answer: i think c or d
Step-by-step explanation:
Answer:
the answer would be D
Step-by-step explanation:
How do you write quadratic equations in standard form give at least 3 examples?
The quadratic equations are
a) x² + 5x - 6 = 0 ( the roots are x = -6 , x = 1 )
b) x² - 5x + 6 = 0 ( the roots are x = 2 , x = 3 )
c) x² + 2x - 3 = 0 ( the roots are x = -3 , x = 1 )
What is Quadratic Equation?
A quadratic equation is a second-order polynomial equation in a single variable x , ax² + bx + c=0. with a ≠ 0. Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has at least one solution. The solution may be real or complex.
The roots of the quadratic equations are
x = [ -b ± √ ( b² - 4ac ) ] / ( 2a )
where ( b² - 4ac ) is the discriminant
when ( b² - 4ac ) is positive, we get two real solutions
when discriminant is zero we get just one real solution (both answers are the same)
when discriminant is negative we get a pair of complex solutions
Given data ,
Let the quadratic equation in the standard form is ax² + bx + c = 0
Now , the value of A is
a)
Let the quadratic equation be x² + 5x - 6 = 0
Now , on simplifying the equation , we get
x² + 5x - 6 = 0
x² - x + 6x - 6 = 0
Taking the common factors in the equation , we get
x ( x - 1 ) + 6 ( x - 1 ) = 0
On factorizing the equation , we get
( x + 6 ) ( x - 1 ) = 0
So , ( x + 6 ) = 0 or ( x - 1 ) = 0
And , x = -6 or x = 1
So , the roots of the quadratic equations are x = -6 or x = 1
b)
Let the quadratic equation be x² - 5x + 6 = 0
Now , on simplifying the equation , we get
x² + 5x + 6 = 0
x² + 2x + 3x - 6 = 0
Taking the common factors in the equation , we get
x ( x + 2 ) + 3 ( x + 2 ) = 0
On factorizing the equation , we get
( x + 3 ) ( x + 2 ) = 0
So , ( x + 3 ) = 0 or ( x + 2 ) = 0
And , x = -3 or x = -2
So , the roots of the quadratic equations are x = -2 or x = -3
c)
Let the quadratic equation be x² + 2x - 3 = 0
Now , on simplifying the equation , we get
x² + 2x - 3 = 0
x² - x + 3x - 3 = 0
Taking the common factors in the equation , we get
x ( x - 1 ) + 3 ( x - 1 ) = 0
On factorizing the equation , we get
( x + 3 ) ( x - 1 ) = 0
So , ( x + 3 ) = 0 or ( x - 1 ) = 0
And , x = -3 or x = 1
So , the roots of the quadratic equations are x = 1 or x = -3
Hence , the quadratic equations are solved
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pls answer need help.
Answer:
No
Step-by-step explanation:
5+10<17
What is the exact value of x?
5·215x=16
Answer:
X=log 3.2/5 log 21
Step-by-step explanation:
X=0.0764094
How do you solve for y in a triangle?
Either Pythagorean Theorem or the Law of Cosines is used to solve for y in a triangle, you can use either of these equations.
The Pythagorean Theorem states that the sum of the squares of the two shorter sides of the triangle is equal to the square of the longest side. This is written as:
a2 + b2 = c2,
where a and b are the lengths of the two shorter sides, and c is the length of the longest side. In this equation, y is the length of the longest side.
The Law of Cosines states that:
a2 = b2 + c2 - 2bc cos A
where A is the angle opposite side a. In this equation, y is side a.
Therefore, to solve for y in a triangle, you can use either the Pythagorean Theorem or the Law of Cosines, depending on the information given.
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What is the formula in solving rectangular prism?
The formula used to calculate the solving the volume of the rectangular prism is equal to ( length × width × height ).
As given in the question,
Given base of the prism is rectangular in shape ,
Area of the base with rectangular base is given by :
area of the rectangle = length × width
Volume of the rectangular prism = base area × height
⇒Volume of the rectangular prism = ( length × width ) × height
Therefore, the formula required to calculate the volume of the rectangular prism is given by ( length × width × height ).
The above question is incomplete , the complete question is:
What is the formula in solving volume of the rectangular prism?
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A game is played by randomly throwing a dart at the hexagon. If it lands on the dotted sector the player wins $60. Otherwise, the player loses $10. What is the value of the game?
The value of the game would be 13.33 dollars.
What is probability?
The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a proposition is true. An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty.
Value of the game = E(x)
[tex]= [(60\times\frac{5}{6}) + (-10\times\frac{1}{6})]\\\\=50 -1.66\\\\= 48.33[/tex]
P(Plain surface) = 2/6 and
P(Not plain surface) = 1 - 2/6 = 2/3
Let x: Value won/lost in the game
Value of the game
[tex]= [(60\times\frac{2}{6}) + (-10\times\frac{2}{3})]\\\\=20 -6.66\\\\= 13.33[/tex]
Hence, the value of the game would be 13.33 dollars.
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A boat traveled on a river 10 km upstream and then comes back to its starting point. Given that the trip takes 2 hours and the river has a current of 4 km an hour, write a quadratic equation that gives the boat's speed for the trip.
Responses
A 2x2 − 20x − 32 =0
B 2x2 − 8x − 32 =0
C 2x2 − 8x − 16 =0
D 2x2 − 16x − 32 =0
Answer:
2x² - 20x - 32 = 0
Step-by-step explanation:
Trip upstream:
distance = 10 km
time = t
current = 4 km/h
speed of boat in still water = x
speed upstream: x - 4
speed = distance/time
distance = speed × time
10 = (x - 4) × t
10 = tx - 4t
t(x - 4) = 10
t = 10/(x - 4)
Trip downstream:
Distance = 10 km
time = 2 - t
current = 4 km/h
speed downstream = x + 4
speed = distance/time
distance = speed × time
10 = (x + 4) × (2 - t)
10 = 2x - tx + 8 - 4t
10 = 2x - 10x/(x - 4) + 8 - 40/(x - 4)
10(x - 4) = 2x(x - 4) - 10x + 8(x - 4) - 40
10x - 40 = 2x² - 8x - 10x + 8x - 32 - 40
2x² - 20x - 32 = 0
What is the difference between negative infinity and positive infinity?
Numbers approach negative infinity as they move left on the number line and positive infinity as they move right on the number line. Negative infinity is always less than any number, positive infinity is always greater than any number.
Main Difference:
There is no positive ∞ nor negative ∞.Infinity is not a number not on the number line. Do not use in arithmetic expressions. Relearn the number line.Infinity is one size, one multiplicity. It's a size that fits a lot.The size of all odd-only sets is infinite.The set of all integers evenly divided by 7 has the same size as infinity.It is the union of the sets described above and the size of the resulting set is the same size of infinity.If we remove all negative integers from this set, the size of the resulting set will be the same size as infinity.The size of the set of all integers is infinite.His five sets above are all the same size infinity.This can happen because infinity is not a number. relearn the number line.Learn more about Negative Infinity:
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Are all equilateral triangles similar and congruent?
All equilateral triangles are similar.
In an equilateral triangle, all the lengths of the sides are equal.
In such a case, each of the interior angles will have a measure of 60°
Since the angles of an equilateral triangle are the same, it is also known as an equiangular triangle.
A property of equilateral triangles includes that all of their angles are equal to 60°
The AAA Triangle Similarity Postulate asserts that if all of the angles in a triangle are equal to the corresponding angles in another triangle, then those two triangles are equal.
Since every equilateral triangle’s angles are, 60°every equilateral triangle is similar to one another due to this AAA Postulate.
=> For congruence of triangles, we need equal sides for two equilateral triangles, but here it has said nothing about sides of both triangles in Congruent. So, this is incorrect. Here, it can be said that all equilateral triangles are similar.
Hence, all equilateral triangles are similar.
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State all integer values of x in the interval 1 ≤ x ≤ 8 that satisfy the following
inequality:
5x − 8 > 6
Answer:
3,4,5,6,7.
Step-by-step explanation:
if yk, yk
Determine the Domain & Range.
Answer:
64, 2
Step-by-step explanation:
`-6+g=11 solve the equation
The function , (1,4) (2,5) (n,3n) has a domain 1
Considering the given data which include the domain and the range the possible value of n is
1 ≤ n ≤ 7How to find the possible value of nThe possible value of n will be n such a way that both the domain and the range are not altered
for a domain of 1, 2, 3, 4, 5, 6, 7, 8 using n, 3n will have a range of
1, 6, 9, 12, 15, 18, 21, 22, 24,
The number 24 is outside the range which is 3, 4, 5, 6, 7, 8,,,,,,,20, 22
n = 8 gave the value of 24 hence 8 cannot be the value of n
n is 1, 2, 3, 4, 5, 6, and 7
written in the form: 1 ≤ n ≤ 7
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The volume of a box, a rectangular prism, is represented by the function f(x) = x3 5x2 – 8x – 12. the length of the box is (x 6), and the width is (x – 2). find the expression representing the height of the box. x – 1 x – 5 x 1 x – 12
The expression representing the height of the box is the difference between the length and width of the box, which is equal to x subtracting two, or x – 1.
The expression representing the height of the box can be determined by subtracting the width of the box from the length. The length of the box is x 6, and the width is x – 2. To calculate the height, we subtract the width from the length: x – (x – 2) = x – x + 2 = 2. Therefore, the expression representing the height of the box is x – 1. This indicates that the height of the box is equal to the length of the box minus one. This calculation can be applied to any rectangular prism to calculate the height.The volume of the box is represented by the function f(x) = x3 5x2 – 8x – 12. To calculate the volume, we can substitute the length and width into the function:
f(x) = (x – 6)(x – 2)(x – 1) = x3 – 8x2 + 11x – 12.
Therefore, the volume of the box is x3 – 8x2 + 11x – 12.
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What is the graph of a quadratic equation and inequality called?
The graph of quadratic inequality when plotted represents the parabola.
The quadratic inequality will have a range of solutions, unlike the quadratic equation.
The quadratic inequality could be solved using factorization method for example
The quadratic inequality x²+6x+5 ≤0 can be solved by factorizing but instead of two solutions, there are a range of solutions.
For given equation we need to calculate value of x such that the graph is less than zero.
Hence solution to this inequality can be written as
−5≤x≤−1.
And the graph of given quadratic inequality represents the parabola.
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a farmer wishes to enclose two identical adjoining rectangular pens against the side of a large barn. she wants each pen to have an area of 750 square feet and will use the barn as one side of the enclosure. what is the least amount of fencing needed (in feet) to create the two pens?
Using basic geometry and area formula for calculation of length, Apply fencing on only 3 side , The answer is 109.54 feet.
What do you mean by geometry?A subfield of mathematics called geometry examines the dimensions, placements, angles, and sizes of objects.
What is a rectangle and it's formula for area?A plane shape, as opposed to a square, with four straight sides and four right angles, particularly one with uneven neighboring sides.
Area of rectangle is:
A= l x b
area=750 square feet.
for attaining minimum area , these area must be fitted in square,
so, [tex]l = \sqrt {750}[/tex]
=27.38 feet
since, one side is taken by barn, l should be extended 4 times, 2 on 1 side and 2 on sides each.
Total length of fence = 4 * 27.38
= 109.54 feet
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Show that x(x - 1)(x + 1) = x³ - x
x (x - 1) (x + 1) = [tex]x^{3}[/tex]- x
Distribute
1. (x + 1) [tex]x^{2}[/tex] - x (x + 1) = [tex]x^{3}[/tex] - x
2. Distribute
(x + 1) [tex]x^{2}[/tex] - x (x + 1) = [tex]x^{3}[/tex] - x
[tex]x^{3 }[/tex] + 1 [tex]x^{2}[/tex] - x (x + 1) = [tex]x^{3 }[/tex] - x
3. Multiply by 1
[tex]x^{3 }[/tex] + 1 [tex]x^{2}[/tex] - x (x + 1) = [tex]x^{3 }[/tex] - x
[tex]x^{3 }[/tex] + [tex]x^{2}[/tex] - x (x + 1) = [tex]x^{3 }[/tex] - x
4. Distribute
[tex]x^{3 }[/tex] + [tex]x^{2}[/tex] - x (x + 1) = [tex]x^{3 }[/tex] - x
[tex]x^{3 }[/tex] + [tex]x^{2}[/tex] - ([tex]x^{2}[/tex] + x) = [tex]x^{3 }[/tex] - x
5. Distribute
[tex]x^{3 }[/tex] + [tex]x^{2}[/tex] - ([tex]x^{2}[/tex] + x) = [tex]x^{3 }[/tex] - x
[tex]x^{3 }[/tex] + [tex]x^{2}[/tex] - [tex]x^{2}[/tex] - x = [tex]x^{3 }[/tex] - x
6. Combine like terms
[tex]x^{3 }[/tex] + [tex]x^{2}[/tex] - [tex]x^{2}[/tex] - x = [tex]x^{3 }[/tex]- x
[tex]x^{3 }[/tex] - x = [tex]x^{3 }[/tex] - x
7. Move terms to the left side
[tex]x^{3 }[/tex] - x = [tex]x^{3 }[/tex] - x
[tex]x^{3 }[/tex] - x - ([tex]x^{3 }[/tex] - x) = 0
8. Distribute
[tex]x^{3 }[/tex] - x - ([tex]x^{3 }[/tex] - x) = 0
[tex]x^{3 }[/tex] - x - [tex]x^{3 }[/tex] + x = 0
9. Combine like terms
[tex]x^{3 }[/tex] - x - [tex]x^{3 }[/tex] + x = 0
- x + x = 0
10. Combine like terms
- x + x = 0
0 = 0
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How do you prove AAS congruence rule?
The two triangles are congruent if two angles and a non-included side of one triangle are congruent with two angles and the corresponding non-included side of a second triangle. So it was proved.
The two triangles are congruent if two angles and a non-included side of one triangle are congruent with two angles and the corresponding non-included side of a second triangle.
If angle ad, side ac-df, and angle cf are present, then ABCDEF is true.
To support: By the angle sum property, ABC DEF A+B+C=180° ______ (1)
In DEF D+E+F=180°___________ (2) (By angle sum property)
From (1) and (2), A+B+C= D+E+F A+E+F= D+E+F
(Given B=E and C=F) A=D ____________________ (3)
Currently, in ABC and DEF A=D (from (3))
AC = DF (Given)
C=F (Give (AAS congruency)
So it was proved.
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What is the easiest way to find the domain?
We simply solve the equation y = f(x) to determine the values of the independent variable x and obtain the domain.
What is the domain?The collection of all potential inputs for a function is its domain.
For instance, the domain of f(x)=x² and g(x)=1/x are all real integers with the exception of x=0.
The set of numbers x for which a function has a valid definition is known as its domain.
A closed interval is referred to as a function's domain if it is specified over the interval a ≤ x ≤ b.
In order to identify the values of the independent variable x and acquire the domain, we simply solve the equation y = f(x).
Simply put, x=g(y) will calculate the function's range after we identify g(y).
Therefore, we simply solve the equation y = f(x) to determine the values of the independent variable x and obtain the domain.
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Why does the square root of 16 have two answers?
When we multiply 4 with 4 we get 16 again
when we multiply (-4) with (-4) we get 16. so 16 has 2 square root.
What are Square and Square root ?There are always two solutions to the square root of a number: a positive response and a negative answer. Because the square of a negative and positive integer is the same, there are two possible solutions.
Squares are the numbers that are produced when a value is multiplied by itself. In contrast, a number's square root is a value that, when multiplied by itself, returns the original value.
The values that result from multiplying an integer by itself are known as square numbers. Among the attributes are:
The square of one is one.Positive squares have positive characteristics.The square of a negative integer has a positive aspect as well.Square of 0 equals 0The value under the root is equal to a number's square root.Only even numbers will be present in every even number's unit location of square.A number's square ends in 1 if either 1 or 9 were substituted for the unit.If a number has a 4 or 6 in the place of the unit, then the square it forms ends in 6.When we multiply 4 with 4 we get 16 again
when we multiply (-4) with (-4) we get 16. so 16 has 2 square root.
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The table shows the favourite type of film of 20 people.
Action
Horror
Comedy
The complete table for the pie chart is;
Type of Film Frequency Angle in °
Action 4 72°
Horror 11 198
Comedy 5 90
How to interpret Pie chart?
From the given table used to form the pie chart, we see that;
Type of Film Frequency Angle in °
Action 4 72°
Horror 11
Comedy 5
We know that the angles in a circle is equal to 360 degrees. Thus;
Angle composed of horror and comedy = 360 - 72 = 288°
Thus;
Angle for horror film = (11/(4 + 11 + 5)) * 360°
= (11/20) * 360°
= 198°
Thus;
Angle for comedy = 288° - 198° = 90°
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