The inverse of a function in standard form can be determined by substituting the function in the standard form
Every function does not have an inverse. A function needs to be a one-to-one function, which means there can't be repeated output for distinct inputs, in order to have an inverse. When a quadratic function is in vertex form, if the leading coefficient (a) is not zero, the function has an inverse. The various steps to find the inverse of a function in standard form can be noted as -
The function should first be written in standard form as follows: y = f(x) = a*x + b, where a and b are constants.Substituting y for x and x for y in the equation: x = f(y)(a*y + b)(y)Isolate y on one side of the equation and solve for it: y = (x-b)/aIf required, make the equation simpler.The original function f(x) = a*x + b is now the inverse function f-1(y) = (x-b)/a.Read more about standard form on:
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Carla’s dining room and living room have the same perimeter. The width of her rectangular dining room is 18 feet, and the width of her rectangular living room is 12 feet. The length of her dining room is 1 foot less than 3/4 the length of her living room. What is the length of Carla’s living room?
Answer:
Length of Carla's living room is 20 feet
Step-by-step explanation:
The perimeter of a rectangle is 2(L + W)
In the case of both dining and living rooms, we are given the widths as dining room width = 18, living room width = 12 and asked to calculate living room length from the information given
Let x be the length of the dining room and y the length of the living room
Perimeter of dining room = 2(18 + x)
Perimeter of living room = 2(12 + y)
Since these are equal
2(18 + x) = 2(12 + y)
=> 18 + x = 12 + y
Subtract x from both sides:
18 = 12 + y - x
Subtract 12 from both sides:
18 - 12 = y - x
Simplifying and rearranging terms we get
- x + y = 6 [1]
We are also given that:
The length of her dining room is 1 foot less than 3/4 the length of her living room.
This mathematically translates to:
[tex]x = \dfrac{3}{4}y - 1[/tex]
which can be re-written as
[tex]x - \dfrac{3}{4} = -1 \; \cdots[2][/tex]
So the two equations are
[tex]- x + y = 6 \;\cdots [1][/tex]
[tex]x - \dfrac{3}{4}y = - 1 \; \cdots [2][/tex]
Notice that the coefficients of x are the same(1) but with opposite signs
Add [1] and [2]
[tex](-x + y) +(x - \dfrac{3}{4}y) = 6 - 1\\\\-x + x + y - \dfrac{3}{4}y = 5\\\\\dfrac{1}{4}y = 5\\\\y = 20\\\\[/tex]
So the length of the living room is 20 feet
(If you wanted to find the length of the dining room, substitute 20 into equation 1 to get -x + 20 = 6 which will give x = 14 and is the length of the dining room)
How do you do algebraic expressions easy?.
An algebraic expression (or) a variable expression is a combination of terms by operations such as addition, subtraction, multiplication, and division. For example, let's look at the expression 5x + 7.
Algebraic Expression:
Algebraic expressions are mathematical expressions that result when you perform operations such as addition, subtraction, multiplication, and division on variables and constants. For example, let's say James and Natalie were playing with matches and wanted to create a pattern of numbers in matches. James played four games and formed No. 4. Natalie added three more games and two he formed a pattern of four. They realized they could add 3 matches to each round to get an extra '4'. From this, they concluded that, in general, 4 + 3 (n-1) double crochet stitches are required to create n 4 patterns. Here 4+ 3(n-1) is called an algebraic expression. To solve a linear equation with one variable, use the transpose method. Follow this procedure to perform the same operations on both sides of the equal sign. That is, perform operations to isolate variables.
Example:
5x + 7 = 32
Subtract 7 from both sides
⇒ 5x = 25
Divide both sides by 5
⇒ x = 5
⇒ 5x = 25
Divide both sides by 3
⇒ 2y – 12 = 24
Add 12 to both sides
⇒ 2y = 36
Divide both sides by 2
⇒ y = 18.
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Munir has $115 to spend on shoes and socks. He buys a pair of shoes for $80 and wants to buy socks with the rest of his money. If each pair of socks costs $4.50, what is the maximum number of pairs of socks that Munir can buy?
They combine their money to buy as much ice cream as they can afford. If . each pair of socks costs $4.50, what is the maximum number of pairs of socks .
What is costs?In production, research, retail, and accounting, a cost is the value of money that has been used up to produce something or deliver a service, and hence is not available for use anymore. In business, the cost may be one of acquisition, in which case the amount of money expended to acquire it is counted as cost.Cost is the expenditure required to create and sell products and services, or to acquire assets. When sold or consumed, a cost is charged to expense. In the case of an asset, the charge to expense could be significantly deferred. The cost concept underlies the transition of assets from the balance sheet to expenses in the income statement.To learn more about income statement refer to:
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Make into simplest form I. Need Hellp lIKE HELP
Answer:
8/10 = 4/5
6/12 = 1/2
10/20 = 1/2
4/5= 4/5
3/12= 1/4
13/21= 2/3
9/24 = 3/8
7/14 = 1/2
15/30 = 1/2
12/16 = 3/4
Step-by-step explanation
Hope this helps
How many images will form when two mirrors are placed at 120 degree and 30 degree object is placed symmetrically with respect to both the mirrors?.
The number of images formed by using the angle 120° is 3 or 30° is 11.
Angle in math is defined as the image formed when two rays are joined together at a common point and the common point here is called node or vertex and the two rays are called arms of the angle.
Here we have given that 120 degree and 30 degree object is placed symmetrically with respect to both the mirrors.
As we all know that the formula number of images formed between two mirrors kept at some angles is written as,
=> n = (360 / θ) - 1
Here we have n is the number of images and θ is the angle between the mirrors.
The value of θ is 120 or 30.
If the angle is 120, then the number of image is calculated as,
=> n = (360 / 120) - 1
=> n = 3 - 1 = 2
Where as the angle is 30, then the number of image is calculated as,
=> n = (360 / 30) - 1
=> n = 12 - 1 = 11
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Which prefix means 10?
deci
centi
Deca
Milli
For the number 10, the prefix used is 10. The correct option is A.
What is a numeral prefix?Prefixes derived from numerals or other numbers are known as numeral or number prefixes. They are used to coin numerous series of words in English and many other languages.
The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
The given number is 10. The prefix used for the number 10 is deci.
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What is the SSA theorem?.
The SSA theorem tells us that if two angles and a non-included side of one triangle are congruent to two angles and also the corresponding non-included side of another triangle, then it is claimed that the two triangles supplied are congruent to one another.
Similar triangles are triangles that have the same shape, but their sizes may vary. All equilateral triangles and squares of any side length are similar objects. To put it another way, if two triangles are similar, then their respective sides are proportionately equal and their corresponding angles are congruent.
The SSA congruence rule is not a reliable standard for determining if two triangles are congruent to one another.
Hence, there is no such SSA similarity from which you can make predictions that whether the two triangles are similar or not.
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Of the 120 students honored at an academic banquet, 40% won awards for mathematics and 55% for English. Fourteen of these students won awards for both mathematics and English. One of the 120 students is chosen at random to be interviewed for a newspaper article. What is the probability that the student won an award in mathematics or English?
The probability that a pupil would excel in math's or English = 0.691
How are probabilities defined?A probability is a numerical representation of the likelihood or chance that a specific event will take place. Both proportions ranging from 0 to 1 and percentages range from 0% to 100% can be used to describe probabilities.
According to the given information:120 students are enrolled in total. Let A represent the scenario in which a student receives a mathematics award, and B represent the scenario in which a student receives an English award. The odds of these events are as follows:
48 won awards for mathematics
78 won awards for English.
14 students won awards for mathematics and English.
P(A) = 48/120
P(B) = 78/120
P(A ∩ B) = 14/120
Due to the fact that 14 students shared the honors from both events, the following equation should be written for overlapping sets:
P(A or B ) = P(A) + P(B) - P(A ∩ B)
= 48/120 + 78/120 - 14/120
= 56/ 81
= 0.691
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y=2x+1 y=x+3
what is the solution to this system of equations?
Answer:
x = 2
y = 5
Step-by-step explanation:
A system of equations is a set of two or more equations that contain multiple variables. To solve a system of equations, you can use different methods such as substitution, elimination or graphing.
The given system of equations is:
y = 2x + 1
y = x + 3
We are trying to find the x and y values that make both equations true at the same time.
One way to solve this system is to use the substitution method, by replacing one of the variables in one equation with the other equation.
Let's replace y from the second equation with the first equation:
x + 3 = 2x + 1
Subtract x from both sides:
3 = x + 1
Subtract 1 from both sides:
2 = x
Now that we have the value of x, we can substitute it back into either one of the original equations to find the value of y.
Let's substitute it back into the first equation:
y = 2(2) + 1
y = 5
So the solution to this system of equations is x = 2 and y = 5. It means that both equations are true when x = 2 and y = 5
You can also check your solution by substituting x and y back into the equations and see if both of them are true.
How do you find y and y intercepts?.
Substitute y = 0 and solve for x to find the x-intercept, and do the same for the y-intercept.
How do you find an example of the y-intercept?Examples on the Y Intercept Using the y-intercept formula, we can find the y-intercept of a function by substituting x = 0 and solving for y; consequently, the given function lacks a y-intercept. 2nd Example: Find the value of "a" if the y-intercept of a function y = a (x - 1) (x - 2) (x - 3) is (0, 12).
With two points, how do you find the y-intercept?We can solve for the y-intercept in the slope-intercept equation y=mx+b by first determining the slope between two points on a line and then writing an equation for that line.
How are the slope and y-intercept of Y determined?The slope-intercept form is used to write the line's equation, which looks like this: y is equal to mx + b, where m is the slope and b is the y-intercept. We can see that the line has a slope of 6 in our equation, which is y = 6x + 2.
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using trigonometric identities to decode a secret message
Using trigonometric identities to decode, the secret message is
2 20-14-25-12 4-19-17 20-14-11-14-25 23-2-21-14-5 23-26-5-7-2-21-14-5! A Nerd who never makes mistakes!
What is trigonometric identities?Trigonometric Identities are equality statements involving trigonometry functions that hold true for all values of the input variables in the equation.
The side length and angle of a triangle are both subject to a variety of distinctive trigonometric identities. Only for right-angle triangles do the trigonometric identities hold true.
All trigonometric identities are built upon the foundation of the six trigonometric ratios. It consists of sine, cosine, tangent, cosecant, secant, and cotangent. These right triangle sides—including the adjacent side, opposite side, and hypotenuse side—are used to define all of these trigonometric ratios. The six trigonometric ratios are the starting point from which all fundamental trigonometric identities are derived.
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What is the rule for fg )( x?.
The notation "FG)(x" refers to the composition of two functions, "F" and "G", where the output of "G" is used as the input for "F". The notation for this is usually represented as "F(G(x))", where "x" is the input for both functions.
The rule for this notation is that the innermost function is applied first. In this case, the function "G" is applied to the input "x" first, and then the output of that function is used as the input for the function "F". The composition of the functions F and G is denoted by the notation F(G(x)) and the result of this notation is a new function obtained by applying F to the result of G on x.
The function "F" is applied to the output of the function "G", meaning that the output of the function "G" is used as the input for the function "F".. For example, if the function "F" is applied to the input "x" first, followed by the function "G", the notation would be represented as "G(F(x))", and the resulting function would be different.
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what is the gcf of 36 and 69?
Answer:
The GCF of 36 and 69 is 3
Step-by-step explanation:
hope this helps
The ma of the blue copper ulfate crytal i two-third the ma of the red fluorite crytal. Write an equation you can ue to find the ma $m$ (in gram) of the blue copper ulfate crytal
The equation-CuSO4.5H2O(s)→CuSO4.3H2O(s)+2H2O(g) for blue copper sulfate crytal I two-thirds the ma of the red fluorite crytal.
The following steps are used to make copper sulfate crystals:
Add a few drops of diluted sulfuric acid to a beaker of water.
Water heating. Add the powdered copper sulfate gradually as it begins to boil.
Stirring is ongoing during this procedure.until no more copper sulfate powder can be dissolved, keep adding it.Filter the remedy.Allow it to cool.The copper crystals will gradually separate.The ma of the blue copper sulfate crystal I two-thirds the ma of the red fluorite crystal
The equation-
CuSO4.5H2O(s)→CuSO4.3H2O(s)+2H2O(g)
the ma (in grams) of the blue copper sulfate crytal
It is 30 divided by 1/3 or 3.
The answer is 10
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the measurements, to the nearest centimeter, of a box are given as 10cm x 7cm x 4cm. calculate the upper and lower bounds for the volume of the box.
In summary, the upper and lower bounds for the volume of the box are:
Upper bound: 382.125cm³
Lower bound: 238.375cm³
The volume of the box is calculated by multiplying the length, width, and height together. In this case, the measurements of the box are given as 10cm x 7cm x 4cm, so the volume of the box is:
volume = 10cm x 7cm x 4cm = 280cm³
However, since the measurements are given to the nearest centimeter, there is a slight degree of uncertainty in the exact measurements of the box. To calculate the upper and lower bounds for the volume of the box, we need to consider the possible variations in the measurements.
The upper bound for the volume would be calculated by taking the highest possible measurements for each dimension. So, if the length could be as much as 10.5cm, the width could be as much as 7.5cm and the height could be as much as 4.5cm. Then the upper bound for the volume would be:
Upper bound = 10.5cm x 7.5cm x 4.5cm = 382.125cm³
The lower bound for the volume would be calculated by taking the lowest possible measurements for each dimension. So, if the length could be as little as 9.5cm, the width could be as little as 6.5cm and the height could be as little as 3.5cm. Then the lower bound for the volume would be:
Lower bound = 9.5cm x 6.5cm x 3.5cm = 238.375cm³
In summary, the upper and lower bounds for the volume of the box are:
Upper bound: 382.125cm³
Lower bound: 238.375cm³
And the exact volume is 280cm³
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19. A rectangular lawn is to be laid with turf. James measures the sides of the rectangle as 30 m by
40 m, with both values being given to the nearest metre. He sends his friend Toby to buy the
turf.
a. Find the area of the maximum area of turf that Toby needs to buy, giving your answers
to 2 decimal places.
b.
Find the area of the minimum area of turf that Toby needs to buy, giving your answers
to 2 decimal places.
A rectangle is a four-sided geometric shape with opposite sides that are parallel and equal in length. The opposite sides are called the length and width of the rectangle. Area = (30m - 0.5m) x (40m - 0.5m) = 29.5m x 39.5m = [tex]1162.75 m^2[/tex]
What is rectangle?a. The maximum area of turf that Toby needs to buy can be found by using the measurements given by James, which are the length and width of the rectangle.
The area of a rectangle is found by multiplying its length and width, so the maximum area of turf needed would be:
Area = length x width = 30m x 40m = [tex]1200 m^2[/tex]
b. The minimum area of turf that Toby needs to buy can also be found by using the measurements given by James, but this time taking into account the margin of error due to the measurements being given to the nearest metre.
If the true length and width of the rectangle are slightly less than the measurements given by James, the mini mum area of turf needed would be:
Area = (30m - 0.5m) x (40m - 0.5m) = 29.5m x 39.5m = [tex]1162.75 m^2[/tex]
So the minimum area of turf that Toby needs to buy would be [tex]1162.75 m^2[/tex]and the maximum area of turf that Toby needs to buy would be [tex]1200 m^2.[/tex]
In general, the measurements given by James could be off by half a meter in both directions, therefore the minimum area of turf that Toby needs to buy would be the area calculated by taking into account the margin of error, in this case, it's [tex]1162.75 m^2.[/tex]
The maximum area of turf that Toby needs to buy would be the area calculated using the measurements given by James, which is [tex]1200 m^2.[/tex]
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If the interior angle of a regular polygon is 108°.The polygon has how many sides?
If the interior angle of a regular polygon is 108°.The polygon has five (5) sides.
Method 1: Each exterior angle of a regular polygon is equal to 180-108°, or 72°, if each internal angle is 108°.Regardless of the number of sides, any figure's outer angles add up to 360 degrees.
The figure has 360/72 or 5 sides since each outer angle is 72 degrees. Therefore, the polygon is a standard pentagon. Answer.
Method 2: Using n isosceles triangles, a regular polygon with n sides is created. 54° is the basis angle, or 108/2. Each vertical angle is equal to 2*36°, or 72°, which is the semi-vertical angle of 90-54°. 360/72 equals 5, the number of isosceles triangles. Therefore, the polygon is a standard pentagon.
Method 3: [n-2]*180 or 108n, which is the sum of the internal angles of a regular polygon.
[n-2]*5=3n, or
5n-10 = 3n, or
5n-3n = 2n = 10, or
n = 5.
The polygon is a regular pentagon as a result.
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How do you define the range Mcq?.
The correct answer is B. The set of output values (y-values) of a function.
The range of a function is the set of all possible output values (y-values) of the function. It is the set of values that the dependent variable (y) can take on, given the domain (set of input values or x-values) of the function.
It can also be represented graphically by plotting the function on a coordinate plane and observing the set of y-values that the function takes on.
We have the following options:
A. The set of input values (x-values) of a function
B. The set of output values (y-values) of a function
C. The set of all possible x-y pairs of a function
D. The set of all possible x-values of a function
Therefore, The correct answer is B. The set of output values (y-values) of a function.
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sqrt 8 + sqrt 32 - sqrt 2, find the exact value of the expression
i got 6 sqrt 2, but it was wrong.
Answer:sqrt 38
Step-by-step explanation:32+8-2 is 38
For what value of k does the pair of equations 3x y − 8 and 6x KY 16 0 have infinitely many solutions?.
The value of k for which the pair of equations 3xy - 8 = 0 and 6xky - 16 = 0 have infinitely many solutions is 4.
To solve this question, we need to find the value of k for which the two equations are equivalent. Equivalence means that the coefficients of x and y are the same and the constant terms are the same.
From the given equations, we can see that the coefficients of x and y are the same in both equations, which is 6x. The constant terms, however, are different. The first equation has a constant term of -8, while the second equation has a constant term of -16.
To make the equations equivalent, we need to make the constant terms the same. We can do this by dividing the second equation by 2. This gives us 6 ky - 8 = 0. We can see that the constant term is now the same as the first equation. Therefore, the value of k that makes the equations equivalent is k = 4.
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The complete question is:
For what value of k do the pair of equations 3xy - 8 = 0 and 6xky - 16 = 0 has infinitely many solutions?
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Find the value of a in the triangle shown below.
·
102
48°
The value of third angle is 30 and we can find using angle sum property.
What is angle sum property?The triangle's angles can be added up to equal 180 degrees, according to the angle sum property of triangles. A triangle has three sides, one at each vertex, and three angles. The total of the interior angles in any triangle, whether acute, obtuse, or right, is 180 degrees.
One of the most commonly employed properties in geometry is the angle sum property of a triangle. Calculating the unknown angles is primarily done using this characteristic. When the values of the other two angles are known, the measure of an unknown interior angle can be determined using the angle sum property.
Given
The two angles are given as 102 and 48
we need to find the third angle
Let the third angle be a then
using the angle sum property we get
102 + 48 + a = 180
a = 30
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The number of bacteria in a lab experiment can be modeled by the function � ( � ) = 45 ( 3 ) � + 1 . S(t)=45(3) t+1 . Write an equivalent function of the form � ( � ) = � � � . S(t)=ab t .
The equivalent function when the number of bacteria in a lab experiment can be modeled by the function is P(t) = 1161(9)^t
How to calculate the function?A function illustrates the variables given. In this case, it is vital to note that two or more components are considered in order to be able to describe the situation.
Here, we are given that the number of bacteria in a lab experiment can be modeled by the function given as P(t) = 43(39^(2t + 3)
This will be:
= 43 × 3³ × 3(^2t)
= 1161 × 3²^t.
P(t) = 1161(9)^t
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Can the graph of an absolute value be negative?.
The graph of an absolute value is always equal to or greater than zero and can never be negative.
We define the absolute value function f: R→R as:
f(x) = |x|
The absolute value function is also named the modulus function.
An alternative form to express the function is
|x| = -x if x ≤ 0 and x if x ≥ 0
The absolute value function is defined for all real numbers.
For x≥0, f(x) returns the same value, that is, x.
For x<0, f(x) returns the negative of the value of x.
So, we arrive at the conclusion that the domain and range of absolute value function are R and [0, ∞).
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Amelia need to buy ome dog food. At the nearet tore, 2 bag of dog food cot $11. 50. How much would Amelia pend on 5 bag of dog food?
What is the percent of decrease from 95 to 19?
The percent of decrease from 95 to 19 is 80%
What is percentage?A percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate percent of a number, divide the number by the whole and multiply by 100. Hence, the percentage means, a part per hundred. The word per cent means per 100.
Given that, what is the percent of decrease from 95 to 19,
Percent of decrease = final value - initial value / initial value x 100
Here, final value = 95
initial value = 19
Therefore,
Percent of decrease = 19-95 / 95 x 100
= -76 / 95 x 100
= -0.8 x 100
= -80%
Here, - sign is representing the decrease,
Hence, the percent of decrease from 95 to 19 is 80%
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Laura cell phone service costs $65 per month, plus an additional $0.10 per text message sent. the table below shows the cost for zach’s cell phone service based on the number of text messages he sends.
how much cheaper is Zach cell phone service than Laura’s when no text messages are sent?
First, find out how much each text will cost, which is always $ 0.25. When there are no text messagers, collukate the fee to $2 per hour .
How much cheaper is Zacha's cell phone sevices ?When 50 texts are added, the price goes up for each text when there are two of them (57,5-50=7.5). [tex](7.5 \div 30=0.25)[/tex] Zach would incur costs if there were no texts. [tex](50-20 \times 0.25=45)$[/tex] , And Laura's price without a text is $65 Thus, (65-45=20) would apply. [First, find out how much each text will cost, which is always $ 0.25. When there are no text messagers, collukate the fee to $2 per hour.]
Laura's monthly cell phone service fee is $65 plus $0.10 for each text message sent. Based on how many texts Zach sends, the following table displays the cost of his cell phone service. When no text messages are sent, Zach's cell phone service is $2 per hour less expensive than Laura's.
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How much of a circle is a 60 degree angle turn?.
A circle has a total degree of 360 so 1/6 is a circle is a 60 degree angle turn.
An irregular plane figure is a circle. Every point on the circle is equally distant from the circle's fixed center. It has a radius of 1, making it a 2D shape. The Latin word "circulus," which means "little ring," is the source of the English word "circle.
A total angle of circle is 360 degrees so if a turn is of a 60 degree so then 6 turns are possible to make a whole circle giving us 1/6 part as the total of which the only 60 degree turn is taken.
Angles can be measured using one of two standard units. Degrees are the most widely used unit of measurement. A straight angle is 90° since a circle has 360 equal degrees.
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PLSSS HELP IF YOU TURLY KNOW
Answer:
14
Step-by-step explanation:
If d is 5
2(5) + 4(5) - 9 - 7
10 + 20 - 9 - 7
30 - 16
= 14
Given a polynomial and one of its factors, find the remaining factors of the polynomial.
x^3-9x^2+27x-27; x-3
The factors of the polynomial x^3-9x^2+27x-27 are (x -3), (x - 3) and (x - 3)
What is Factor of the PolynomialThe polynomial you provided is x^3-9x^2+27x-27. If you know one of the factors of this polynomial, we can use that to find the remaining factors.
One factor of this polynomial is (x-3) as it gives remainder zero when divided by (x-3)
quotient is x² - 6x + 9
remainder is zero
So the remaining factors are (x-3) and x² - 6x + 9
Alternatively, we can factor the remaining quadratic polynomial x² - 6x + 9
by using Factor theorem or by using difference of squares
x² - 6x + 9 = (x - 3)²
So the remaining factors are (x-3) and (x - 3)
So the original polynomial can be factored as (x-3)(x-3)(x-3) = (x-3)³
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Tasha incorrectly draws this graph to represent the balance in her saving account over time.
A. What is the y intercept of the graph and what does it represent in the situation?
B. Does the y intercept make sense in this situation? Explain.
C. Explain Tasha possible error.
Answer: A. The y-intercept of a graph is the point at which the graph crosses the y-axis. In the graph provided, the y-intercept appears to be at (0, -500). This means that Tasha's account balance starts at -500 dollars.
B. The y-intercept does not make sense in this situation, as a savings account balance cannot be negative. It is not possible to have negative dollars in a savings account.
C. Tasha's possible error is that she may have misinterpreted the y-axis to represent a negative balance, when in reality it represents zero balance. The y-intercept represents the initial balance of the account, which should be zero or a positive number. This error can be corrected by starting the y-axis from zero and making sure the y-intercept is positioned at a positive value.
Step-by-step explanation:
