To schedule the examination so that no student has two examinations on the same day, we can use graph coloring. Graph coloring is the process of assigning colors to the vertices of a graph in such a way that no two adjacent vertices have the same color.
1. First, we can create a graph with the seven subjects as vertices and the pairs of subjects with common students as edges.
2. Next, we can assign a color to each vertex, starting with S1 and moving clockwise around the graph. We can use the smallest available color for each vertex, making sure that no two adjacent vertices have the same color.
3. Once we have assigned a color to each vertex, we can use the colors to schedule the examinations. Each color represents a different day, and all of the subjects with the same color can be scheduled on the same day.
The resulting schedule would look something like this:
Day 1: S1, S5
Day 2: S2, S4
Day 3: S3, S6
Day 4: S7
This schedule ensures that no student has two examinations on the same day, since no two adjacent vertices in the graph have the same color.
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HELPPPP ASAP PLEASE HURRY 50 POINTS
The interval giving the middle 68% of the lifetime of bulbs in hours is given as follows:
(1350, 1450).
What does the Empirical Rule state?The Empirical Rule states that, for a normally distributed random variable, the symmetric distribution of scores is presented as follows:
The percentage of scores within one standard deviation of the mean of the distribution is of approximately 68%.The percentage of scores within two standard deviations of the mean of the distribution is of approximately 95%.The percentage of scores within three standard deviations of the mean off the distribution is of approximately 99.7%.The middle 68% of scores is within one standard deviation of the mean, hence the bounds of the interval are given as follows:
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when an integer m is divided by 5 the remainder is 3. When m is divided by 7 the remainder is 1. If m is greater than 40 but less than 80, what is one possible value of m.
One possible value of m is 53.
A possible value of m can be found using the Chinese Remainder Theorem. The Chinese Remainder Theorem states that if two numbers a and b are relatively prime, then there exists an integer x such that:
x ≡ a (mod m)
x ≡ b (mod n)
In this case, we have:
m ≡ 3 (mod 5)
m ≡ 1 (mod 7)
Both 5 and 7 are prime so we can start by finding the smallest positive integer that is congruent to 3 (mod 5) and 1 (mod 7). This integer is 18:
18 ≡ 3 (mod 5)
18 ≡ 1 (mod 7)
Now, we can add multiples of 5*7 = 35 to 18 until we find a value of m that is greater than 40 but less than 80:
18 + 35 = 53
53 is greater than 40 but less than 80, so one possible value of m is 53.
Therefore, one possible value of m is 53.
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How do you find the scale factor of a shape. For common core? And what is the sequence of transformations?
The process to find the scale factor of a shape and the concept of sequence of transformations is explained below.
In order to find the scale factor of a shape, we need to compare the corresponding side lengths of the original shape to the corresponding side lengths of the new shape after a dilation.
The scale factor is the ratio of any two corresponding side lengths.
The sequence of transformations is defined as the order in which different transformations are applied to a shape.
For example, if we have a triangle and we first translate it, then rotate it, and then dilate it, the sequence of transformations would be : translate → rotate → dilate.
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The given question is incomplete, the complete question is
How do you find the scale factor of a shape? And What is the sequence of transformations?
Mark is training for a mini triathlon he rode his bike 3/4 Mille ran 2/4 milks and swam 1/4 mile each day how does the distance he biked in 3 days compare to the distance he swam in 3 days in 5 days in 6 days why
The comparison between the distance he biked and swam is 3 : 1.
What is a numerical expression?A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.
From the given information the comparison of the distance he swam to the distance he biked can be represented in the form of ratios.
Given, He rode his bike 3/4 miles and swam for 1/4 mile.
Therefore, The ratio of biking : swimming = 3/4 : 1/4.
Now, Multiplying by 4 to get rid of the denominator we have,
Biking : Swimming = 3 : 1.
As the days in both comparisons increase in the same manner the ratio will remain the same.
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20. Let V be an Euclidean space and letx,y∈V. (a)†If∥x+y∥=∥x−y∥then determine⟨x,y⟩. (b) * Show that2⟨x,y⟩≤∥x∥2+∥y∥2. (c) If(z,y)=0for allz∈Vthen show thaty=0. (d) †If∥x∥=∥y∥for somex,y∈Vthen⟨x+y,x−y⟩=0. (e)†Pythagorean theorem: Show that⟨x,y⟩=0if and only if∥x+y∥2=∥x∥2+∥y∥2.
Pythagorean theorem: Show that ⟨x, y⟩ = 0
Let V be an Euclidean space and let x, y ∈ V.(a) If ∥x + y∥ = ∥x − y∥ then determine ⟨x, y⟩.We know that ∥x + y∥² = ⟨x + y, x + y⟩ = ⟨x, x⟩ + ⟨x, y⟩ + ⟨y, x⟩ + ⟨y, y⟩ = ∥x∥² + 2⟨x, y⟩ + ∥y∥²And ∥x − y∥² = ⟨x − y, x − y⟩ = ⟨x, x⟩ − ⟨x, y⟩ − ⟨y, x⟩ + ⟨y, y⟩ = ∥x∥² − 2⟨x, y⟩ + ∥y∥²Since ∥x + y∥ = ∥x − y∥, we have ∥x∥² + 2⟨x, y⟩ + ∥y∥² = ∥x∥² − 2⟨x, y⟩ + ∥y∥²⟨x, y⟩ = 0(b) Show that 2⟨x, y⟩ ≤ ∥x∥² + ∥y∥².We know that ∥x + y∥² = ∥x∥² + 2⟨x, y⟩ + ∥y∥²And ∥x + y∥² ≥ 0So ∥x∥² + 2⟨x, y⟩ + ∥y∥² ≥ 0⟨x, y⟩ ≤ (∥x∥² + ∥y∥²)/2(c) If (z, y) = 0 for all z ∈ V then show that y = 0.Let z = y, then (y, y) = 0⟨y, y⟩ = ∥y∥² = 0∥y∥ = 0So y = 0(d) If ∥x∥ = ∥y∥ for some x, y ∈ V then ⟨x + y, x − y⟩ = 0.We know that ∥x + y∥² = ∥x∥² + 2⟨x, y⟩ + ∥y∥²And ∥x − y∥² = ∥x∥² − 2⟨x, y⟩ + ∥y∥²Since ∥x∥ = ∥y∥, we have ∥x∥² = ∥y∥²So ∥x + y∥² − ∥x − y∥² = 4⟨x, y⟩ = 0⟨x, y⟩ = 0(e) Pythagorean theorem: Show that ⟨x, y⟩ = 0 if and only if ∥x + y∥² = ∥x∥² + ∥y∥².We know that ∥x + y∥² = ∥x∥² + 2⟨x, y⟩ + ∥y∥²If ⟨x, y⟩ = 0, then ∥x + y∥² = ∥x∥² + ∥y∥²If ∥x + y∥² = ∥x∥² + ∥y∥², then 2⟨x, y⟩ = 0⟨x, y⟩ = 0
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a company is building square-bottomed boxes with no tops. the material for the bottom costs $2 per cm2 and the material for the sides costs $1 per cm2. the company has budgeted $96 to build one box. what is the box of largest volume the company can build for $96
The box of largest volume the company can build for $96 is the box that costs exactly $96.
The company can build a box of the largest volume for $96 by calculating the area of the bottom and sides of the box. To calculate the area of the bottom, use the formula A=l x w, where l is the length of one side and w is the width of one side. Since the bottom is square, l=w. Thus, A=l^2. The area of the bottom is then multiplied by the cost of the material for the bottom, which is $2 per cm2, to calculate the total cost of the bottom.
Similarly, to calculate the cost of the sides, use the formula A=2lh + 2wh, where h is the height of the box. The area of the sides is then multiplied by the cost of the material for the sides, which is $1 per cm2, to calculate the total cost of the sides.
The total cost of the box is the cost of the bottom plus the cost of the sides. Since the company has budgeted $96 to build one box, the total cost of the box must not exceed $96. Therefore, the box of largest volume the company can build for $96 is the box that costs exactly $96.
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12. WRITE The sketch shows the side view of a sculpture that is
being designed by an artist. Determine whether AABC=
ADCA. If yes, then provide a paragraph proof. If no, then
explain your reasoning.
B
Note that in the above sketch, ΔABC ≇ ΔDCA. That is they are NOT congruent. The proof is that ∠ADC which is supposed to be = ∠BAC are not equal. Thus, line AC which is supposed to be equal to Line BC are not equal.
When two triangles are congruent, it means they are exactly the same in terms of their shape and size, so all corresponding sides and angles are equal.
In ΔABC,
Line AB = 8ft
Line BC = 11ft
Line AC = 13.6ft
∠ABC = 90°
∠BAC = 53.97°
∠ACB = 36.03°
As you can see,
Where as, ∠ADC = 62° - given
∠BAC = 53.97°
Also
Where as:
AC (The Hypotenuse ΔABC which is also the Adjacent Side of ΔACD) = 13.6ft
BC (The adjacent side of ΔABC) = 11ft.
Note that in order to prove congruence, at least two angles and one side from both Triangles must be equal (Angle Angle Side Theorem). Or
Two sides and one angle from both Triangles must be equal (Side - Angle - Side).
Or All three angles (Angle - Angle - Angle);
Or All three Sides (Side - Side Side).
In this case, only one side from both Triangles AC is common to both Triangles.
On the basis of the above, therefore, ΔABC ≇ ΔDCA.
The calculations showing how we arrived at the missing sides and angles are given below:
ΔABC:
AC = √(AB² + BC²)
AC = √(8² + 11²)
AC = √(64 + 121)
AC = √(185)
AC = 13.6014705087
AC [tex]\approx[/tex] 13.60
∠ ACB = arcsin (AB/AC)
∠ ACB = arcsin (8/13.6014705087)
= arcsin(0.58817169767505)
∠ ACB = 0.6288 rad converted to degrees
∠ ACB = 36.03°
Thus, since the sum of Angles in a Triangle is 180°
∠BAC = 180-36.03° -90°
∠BAC = 53.97°
ΔACD
y (AD) = AC / sin(β)
y (AD) = 13.6/ sin(62°)
y (AD) = 13.6/0.88294759285
AD = 15.40°
CD = √(AD² - AC²)
CD = √(15.4029526893712 - 13.62)
CD = √52.290951550999
CD = 7.23
The outstanding angle in ΔACD is ∠CAD
Since the sum of Angles in a Triangle is 180°
∠CAD = 180 - 90 - 62
∠CAD = 28°
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Full Question:
Although part of your question is missing, you might be referring to this full question: See attached Image.
(Please answer quickly!!!)Which graph represents the solution to one seventh m is less than or equal to negative one twenty second?
number line with closed circle on point negative 14 over 11 and arrow shaded to the right
number line with closed circle on point negative 7 over 22 and arrow shaded to the right
number line with closed circle on point negative 14 over 11 and arrow shaded to the left
number line with closed circle on point negative 7 over 22 and arrow shaded to the left
Option D i.e. number line with closed circle on point negative 7 over 22 and arrow shaded to the left represents the solution to the inequality.
What is inequality?A comparison between two or more mathematical expressions is known as an inequality.
We are given an inequality as one seventh m is less than or equal to negative one twenty second.
So, from the above information, we get the inequality as
m/7 ≤ −1/22
Now, on solving the above inequality, we get the following
⇒m/7 ≤ −1/22
⇒m ≤ −7/22
Hence, Option D i.e. number line with closed circle on point negative 7 over 22 and arrow shaded to the left represents the solution to the inequality.
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A firm manufactures a product that sells for $13 per unit. Variable cost per unit is $2 and fixed cost per period is $1540. Capacity per period is 2500 units. (a) Develop an algebraic statement for the revenue function and the cost function. (b) Determine the number of units required to be sold to break even. (c) Compute the break-even point as a percent of capacity. (d) Compute the break-even point in sales dollars.
(a) Revenue function: R = 13x
Cost function: C = 2x + 1540
(b) Break-even point: 2x + 1540 = 13x
x = 1540/11
x = 140 units
(c) Break-even point as a percent of capacity: 140/2500 x 100% = 5.6%
(d) Break-even point in sales dollars: 13 x 140 = $1820
(a) The revenue function is R(x) = 13x, where x is the number of units sold. The cost function is C(x) = 2x + 1540, where x is the number of units sold.
(b) To find the break-even point, we set R(x) = C(x):
13x = 2x + 1540
11x = 1540
x = 140
So the firm needs to sell 140 units to break even.
(c) The break-even point as a percent of capacity is 140/2500 = 0.056 or 5.6%.
(d) The break-even point in sales dollars is 140 units * $13/unit = $1820.
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The pyramid and prism above have the
the prism?
OA. 84 cm
OB &3 cm³
OC. 42 cm³
OD. 7cm³
width, length, and height. The volume of the pyramid is 21 cm. What is the volume of
According to the information we can infer that the volume of the prism would be 63cm³
How to find the volume of the prism?To find the volume of the prism we must perform the following mathematical operation:
First of all we must find the cube root of each of the options.
[tex]\sqrt[3]{84} = 4,3795191398878892657\\\sqrt[3]{63} = 3,9790572078963918596\\\sqrt[3]{42} = 3,4760266448864497867\\\sqrt[3]{7} = 1,9129311827723891012[/tex]
Once we find these values we must check them with the formula for the volume of a pyramid. In this case, it would be the area of the base times the height. Then we must take into account that the cube root of these numbers would be the equivalent to the side length of the figures.
In the case of 63, the procedure would be as follows:
3.9790572078963918596 * 3.9790572078963918596 = 15.832896315.8328963 / 3 = 5.27763215.2776321 * 3.9790572078963918596 = 21Then we can establish that this value corresponds to the length of the sides and the height of the pyramid because as a result it gives us the volume of the pyramid. So to find the volume of the prism we must do the following operation:
3.9790572078963918596 * 3.9790572078963918596 * 3.9790572078963918596 = 63Learn more about area in: https://brainly.com/question/27683633
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what value of x makes the equation 3x(x-6) -8x=-(2x+1)
The value of x is (12 ± √141) / 3. The solution has been obtained by solving the algebraic equation.
What is an algebraic equation?A mathematical expression is said to be "algebraic" if it includes variables, constants, and algebraic operations (addition, subtraction, etc.). The equals sign is a requirement for the expression to satisfy the algebraic equation.
We are given an equation as 3x (x - 6) - 8x = -(2x + 1)
On solving the given equation, we get
⇒3x (x - 6) - 8x = -(2x + 1)
⇒3x² - 18x - 8x = -2x - 1
⇒3x² - 26x = -2x - 1
⇒3x² - 24x = - 1
⇒3x² - 24x + 1 = 0
Using quadratic formula, we get
x = (12 ± √141) / 3
Hence, the value of x is (12 ± √141) / 3.
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I am needing some help with this
The volume of the cone is 1004.8 cm³.
How to find the volume of a cone?Let's calculate the volume of the cone with radius 8 cm and the slant height is 17 cm. Therefore, let's find the volume of the cone as follows:
volume of a cone = 1 / 3 πr²h
where
r = radiush = height of the coneTherefore,
h² = 17² - 8²
h = √289 - 64
h = √225
h = 15 cm
Therefore,
volume of a cone = 1 / 3 × 3.14 × 8² × 15
volume of a cone = 1 / 3 × 3.14 × 64 × 15
volume of a cone = 3014.4 / 3
volume of a cone = 1004.8 cm³
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Shaun goes to a fast food restaurant and orders some tacos and burritos. He sees on the nutrition menu that tacos are 160160 calories and burritos are 320320 calories. If he ordered 1515 items and consumed a total of 36803680 calories, how many tacos and how many burritos did Shaun order and eat?
Tacos eaten:
Burritos eaten:
Shaun ordered 7 tacos and 8 burritos, for a total of 160160 calories from tacos and 320320 calories from burritos.
To solve this problem, we need to use a system of equations. Let's call the number of tacos Shaun ordered x and the number of burritos he ordered y. We can then write the following equations:
160x + 320y = 3680 (the total number of calories consumed)
x + y = 15 (the total number of items ordered)
We can rearrange the second equation to get y = 15 - x and then substitute that into the first equation to solve for x:
160x + 320(15 - x) = 3680
160x + 4800 - 320x = 3680
-160x = -1120
x = 7
Now that we know x, we can plug it back into the second equation to solve for y:
7 + y = 15
y = 8
So, Shaun ordered and ate 7 tacos and 8 burritos.
Tacos eaten (x) : 7
Burritos eaten (y) : 8
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A student used factor by grouping to factor the expression below. How can you tell that the student made an error when factoring? Explain the error.
Student's work: 6x² – x – 12
6x² – 9x + 8x – 12
3x(2x – 3) – 4(–2x +3)
The error made by the student is instead of taking +4 as common he has taken - 4 as common.
Factoring by grouping terms:To factor an expression by grouping, we need to look for groups of terms that have common factors. Then, we factor out those common factors from each group and see if there is a common factor that can be further factored out.
Here we have
6x² – x – 12
The above expression can be factorized as follows
=> 6x² – x – 12
= 6x² – 9x + 8x – 12 [ Splitting the middle term ]
= 3x(2x – 3) + 4(2x – 3) [ Grouping the terms ]
= (3x + 4)(2x – 3)
Hence, the factors of given expression are (3x + 4) and (2x – 3)
The error made by the student is in step 3 instead of taking +4 as common he has taken - 4 as common which doesn't give the common factors in the next step taken.
Therefore,
The error made by the student is instead of taking +4 as common he has taken - 4 as common.
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In 2011, there were approximately 2.8 million nurses in a large country, with demand for nurses expected to increase by 0.044 million each year. a. Assuming the number of nurses in 2011 represents the demand for that year, express the demand for nurses in the country, D, in millions, as a function of the number of years after 2011,x. b. If the demand increases as expected, during which year will demand for nurses in the country reach 3.5 million? a. D(x)=
The demand for nurses in the country can be expressed as the function D(x) = 2.8 + 0.044x. The year in which the demand for nurses in the country will reach 3.5 million is in 2027.
a. The demand for nurses in the country can be expressed as a function of the number of years after 2011, x, by using the equation D(x) = 2.8 + 0.044x. This equation represents the initial demand in 2011 (2.8 million) plus the expected increase in demand each year (0.044 million) multiplied by the number of years after 2011 (x).
b. To find the year in which the demand for nurses will reach 3.5 million, we can set D(x) equal to 3.5 and solve for x:
3.5 = 2.8 + 0.044x
0.7 = 0.044x
x = 0.7 / 0.044
x = 15.9
Since x represents the number of years after 2011, we can add 15.9 to 2011 to find the year in which the demand will reach 3.5 million:
2011 + 15.9 = 2026.9
Therefore, the demand for nurses in the country will reach 3.5 million during the year 2027.
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What is the surface area for a box with 12 as length, 10 as height and 3 as width.
Answer: 372 units²
Step-by-step explanation:
We can use this formula to solve for the surface area:
Surface Area = 2lw + 2lh + 2hw
We will substitute the known values and solve:
Surface Area = 2lw + 2lh + 2hw
Surface Area = 2(12)(3) + 2(12)(10) + 2(10)(3)
Surface Area = 372 units²
Debra has two different square baking dishes.one has a side length of 8 inches, and the other has a side length of 9 inches. what is the difference in the area of Debra's two square baking dishes?
Debra's two area baking plates are 17 square inches different in size.
What distinguishes an area from a perimeter?The area surrounding a shape forms its perimeter. Area is a unit of measurement for interior space. Surface area is a measurement of a solid shape's exposed surface, whereas area is a two-dimensional measurement of the size of a flat surface (three-dimensional).
The square baking dish has the following surface area with an 8-inch side length:
Area of first square = (side length)² = 8² = 64 square inches
Similarly, the surface area of a square baking dish with 9-inch sides is:
Area of second square = (side length)² = 9² = 81 square inches
Difference in area = |Area of second square - Area of first square|
Difference in area = |81 - 64|
Difference in area = 17 square inches
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A European company is selling a new brand of headphones. For h headphones sold, in thousands, a profit of E(h) = -4h^(4) + 7h^(3) - 7h + 20, in ten thousands of Euros, will be earned. How much will be earned in profit for selling 1,500 headphones?
The European company will earn 128,750 Euros in profit for selling 1,500 headphones.
How to find sale profitTo find the profit for selling 1,500 headphones, we need to plug in the value of h into the profit function E(h) and simplify.
Since h is measured in thousands of headphones, we need to divide 1,500 by 1,000 to get h = 1.5.
Now we can plug in h = 1.5 into the profit function and simplify:
E(1.5) = -4(1.5)^(4) + 7(1.5)^(3) - 7(1.5) + 20
E(1.5) = -4(5.0625) + 7(3.375) - 10.5 + 20
E(1.5) = -20.25 + 23.625 - 10.5 + 20
E(1.5) = 12.875
So the profit for selling 1,500 headphones is 12.875 ten thousands of Euros, or 128,750 Euros.
Therefore, the European company will earn 128,750 Euros in profit for selling 1,500 headphones.
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Julian makes and sells wallets. He estimates that his income can be modeled by y = 18x - 140, where x is the number of wallets he sells. He estimates that his costs to make the wallets can be modeled by y = 7x+160. How many wallets does Julian need to make in order to break even?
Therefore , the solution of the given problem of equation comes out to be julian must sell at least 28 wallets in order to make even.
How do equations operate?Mathematical formulas frequently use the same variable letter to try to impose unity between two assertions. Many academic numbers are shown to be equal using mathematical fraction equations, also known as assertions. Using y + 6 as an illustration, the normalise does not divide 12 into two parts, but instead b + 6. It is possible to determine the connection between each sign part and the number of lines. The significance of a symbol usually contradicts itself.
Here,
For Julian to break even, his revenue must match his expenses. Therefore, we can equalise the two equations and find x:
=> 18x - 140 = 7x + 160
7x is subtracted from both lines to yield:
=> 11x - 140 = 160
140 added to both ends results in:
=> 11x = 300
When we multiply both parts by 11, we get:
=> x = 27.27 (rounded to two integer places) (rounded to two decimal places)
We can round up to the nearest whole amount because Julian cannot sell a fraction of a wallet. Julian must therefore sell at least 28 wallets in order to make even.
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If 2x-1 is a factor of 24x^4 - 2x^2 +2x +f find f
If 2x-1 is a factor of 24x⁴ - 2x² +2x +f , the value of f is 1 - 2x.
Describe Long division?Synthetic division is a method of polynomial division used to divide a polynomial by a linear expression of the form x - a, where a is a constant. It is a shorthand way of performing long division for polynomials and is particularly useful when the divisor is of the form x - a, as it eliminates the need for writing out the full divisor in each step of the division process.
Since 2x-1 is a factor of the given polynomial, we know that it will divide exactly into the polynomial, leaving no remainder. Therefore, we can use polynomial long division to find the quotient when 24x⁴ - 2x² + 2x + f is divided by 2x-1:
12x³ + 5x² - 3x - f/2 + 1/4
____________________________________
2x - 1 | 24x⁴ + 0x³ - 2x² + 2x + f
- (24x⁴- 12x³)
----------------------
12x^3
- (12x³ - 6x^2)
----------------------
5x^2
- (5x² - 2.5x)
----------------------
2.5x
- (2.5x - f/4)
-----------------------
f/4 + 2x/4 - 1/4
Since there is no remainder, the last term in the quotient must be zero. Therefore:
f/4 + 2x/4 - 1/4 = 0
Simplifying and solving for f, we get:
f/4 = 1/4 - 2x/4
f = 1 - 2x
Therefore, f = 1 - 2x.
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I'm so confused on thins can someone PLEASE help me??? It's Algebra by the way
Answer:
Degree: The degree of the function is 3.
X-Intercepts with multiplicity greater than 1: x = -2 and x = 0
How many distinct x-intercepts?: There are two distinct x-intercepts.
How many zeros are there?: There are three zeros, one at x = -2 and two at x = 0 (with a multiplicity of 2).
pls help asap will give brainiest
Which is closest to the surface area of the figure below?
Responses
1463.8 m2
578.1 m2
923.2 m2
3692.6 m2
The surface area of the cone is 578.1 square feet.
How to get the surface area of the cone?We know that for a cone of radius R and slant height H the surface area is given by the formula:
S = pi*R^2 + pi*R*(H)
with pi = 3.14
On the diagram we can see that H = 19.3ft, and the diameter of the base is 14 ft, then the radius is:
R = 14ft/2 = 7ft
REplacing these values in the formula above we will get:
S = 3.14*(7ft)^2 + 3.14*7ft*(19.3ft) = 578.1 ft²
So the correct option is the second one.
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12. For this problem, consider the permutationsα=(1223344551677886),β=(1123384756657284),γ=(1235)(24568)(a) Write each of the permutations as a product of disjoint cycles. (b) Find the order of each permutation. (c) Write each permutation as a product of 2-cycles and determine which are even and which are odd. (d) Write the inverse of each of the permutations as a product of disjoint cycles. (e) Writeαβandβαas products of disjoint cycles, and find their orders.
(a) The permutations can be written as a product of disjoint cycles as follows:
α = (1 5 4 3 2)(6 7 8)
β = (1)(2 3 8 4 7)(5 6)
γ = (1 2 3 5)(4 5 6 8)
(b) The order of each permutation is the least common multiple of the lengths of the disjoint cycles. Therefore:
Order of α = lcm(5, 3) = 15
Order of β = lcm(1, 5, 2) = 10
Order of γ = lcm(4, 4) = 4
(c) The permutations can be written as a product of 2-cycles as follows:
α = (1 2)(2 3)(3 4)(4 5)(5 1)(6 7)(7 8)(8 6)
β = (1 1)(2 3)(3 8)(8 4)(4 7)(7 1)(5 6)(6 5)
γ = (1 2)(2 3)(3 5)(5 1)(4 5)(5 6)(6 8)(8 4)
Since α and β have an even number of 2-cycles, they are even permutations. γ has an odd number of 2-cycles, so it is an odd permutation.
(d) The inverse of each permutation can be found by reversing the order of the elements in each cycle. The inverses can be written as a product of disjoint cycles as follows:
[tex]α^-1 = (1 2 3 4 5)(6 8 7)β^-1 = (1)(2 7 4 8 3)(5 6)γ^-1 = (1 5 3 2)(4 8 6 5)[/tex]
(e) The products αβ and βα can be found by performing the permutations in the given order. The products can be written as a product of disjoint cycles as follows:
αβ = (1 6 8 4 7 2 3 5)(5 1)
βα = (1 7 4 8 6 5 3 2)(2 1)
The order of αβ is lcm(8, 2) = 8, and the order of βα is lcm(8, 2) = 8.
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bvement of the progress bor moy be uneven becouse questions can be worth more or less Rewrite ((1)/(3))^(-4)=81 as a logarithmic equation.
The logarithmic equation that is equivalent to the given exponential equation
The logarithmic equation that is equivalent to the exponential equation [tex]((1)/(3))^-4=81 (is) log(1/3) 81 = -4[/tex]
To rewrite an exponential equation as a logarithmic equation,
we can use the following formula:
bx = y → logb y = x
In this case, b = (1/3), x = -4, and y = 81. So, we can plug these values into the formula to get the logarithmic equation:
[tex]log(1/3) 81 = -4[/tex]
This is the logarithmic equation that is equivalent to the given exponential equation.
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population 31,200 now... increase 10% per year... what ws the population 2 years ago?
The population two years ago was 28,080.
To calculate this, we need to use the following formula: Population two years ago = Population now - (Population increase rate x Number of years). Therefore, the population two years ago was 31,200 - (10% x 2) = 28,080.
We can also look at this another way. To calculate the 10% increase per year, we can use the following formula: Population increase rate = (Population now - Population two years ago) / Number of years. Therefore, 10% = (31,200 - 28,080) / 2 = 3,120.
We can use this same formula to calculate the population three years ago. Using the same formula, the population three years ago was 25,960.
To summarize, the population two years ago was 28,080 and the population three years ago was 25,960. The population has been increasing by 10% each year.
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The diameter of a cylindrical construction pipe is 3ft. If the pipe is 17ft long, what is its volume?
Use the value 3.14 for /pi, and round your answer to the nearest whole number.
Be sure to include the correct unit in your answer.
Answer:
First, we need to find the radius of the pipe by dividing the diameter by 2:
radius = 3ft / 2 = 1.5ft
Next, we can use the formula for the volume of a cylinder:
V = πr²h
where V is volume, π (pi) is approximately 3.14, r is the radius, and h is the height (or length) of the cylinder.
Plugging in the values we have:
V = 3.14 x (1.5ft)² x 17ft
V = 3.14 x 2.25ft² x 17ft
V = 3.14 x 38.25ft³
V ≈ 120ft³
Rounding to the nearest whole number and including the unit:
The volume of the pipe is about 120 cubic feet.
[tex]\bold{Work \ Shown:}[/tex]
[tex]d = 3 = diameter[/tex]
[tex]r = d\div2 = 3\div2 = 1.5 = radius[/tex]
[tex]h = 17 = height \ or\ length \ of \ the\ pipe[/tex]
[tex]\pi = 3.14\ approximately[/tex]
[tex]V = volume \ of \ the \ cylinder \ pipe[/tex]
[tex]V = \pi \times r^2 \times h[/tex]
[tex]V = 3.14 \times (1.5)^2\times 17[/tex]
[tex]V = 120.105[/tex]
[tex]V=120[/tex]
The cylindrical pipe's volume is about 120 cubic feet.
PLS HELP OML IM STRUGGLINGGG !!!!!! best answer = brainliest !!!
Answer:
Area = 15.6
Perimeter = 20.6
Step-by-step explanation:
Both smaller triangles are right triangles
The left triangle is isosceles right triangle since 2 angles are 45o so AB is hypotenuse and the other 2 sides are equal
so c^2 = a^2 + b^2
c^2 = 2a^2
(3√2)^2 = 2a^2
2a^2 = 18
a^2 = 9
a = 3
For the right triangle the angle at B is 60o, hypotenuse is BC
so cos(60) = adjacent/hypotenuse
1/2 = 3/BC
=> BC = 6
sin(60) = opposite/hypotenuse
√3/2 = opp/6
opp = 3√3
so AC = 3 + 3√3 = 6√3
Area of triangle = A = 1/2bh = 1/2(6√3)(3) = 9√3 = 15.6
Perimeter = 3√2 + 6√3 + 6 = 20.6
Write an equation of a circle with center at (4,-1) and a point at (2,-4)
Explain please
Answer:
y=3/2x-7
Step-by-step explanation:
7. You have a 8-kilogram bag of dog food. You give your dog 100 grams of food
oqeach day. How many days does the bag of food last?
teris fiel even uoy ob enjur to equo
pje ever vo
We can start by converting the mass of the bag of dog food from kilograms to grams:
8 kilograms = 8,000 grams
Then, we can divide the total mass of the bag of food by the daily amount given to the dog:
8,000 grams ÷ 100 grams/day = 80 days
Therefore, the bag of dog food will last for 80 days.
Write the expression as a single power using only positive exponents, if possible. Assume no denominator equals zero.
PLS ANSWER ASAP, I need to know in at least under 3 minutes, your help would be very much appreciated.
d^5/d^-3
Possible answers:
A.) d^2
B.) 1/d^2
C.) 1/d^8
D.) d^8
Answer:
Step-by-step explanation:
d^2
When dividing subtract indices OR
dxdxdxdxd /dxdxd
d's cancel out leaving dxd=d^s