which of the following rational functions is graphed below? A. F(x)=(x+2)/(x-2) B. F(x)=(x-1)/x(x-2) C. F(x)= (x+2)/(x+1) D. F(x)= (x+1)/(x+2)(x-2)
Answers
Answer:
[tex]C. \; \dfrac{x+2}{x+1}[/tex]
Step-by-step explanation:
Since the vertical asymptotes are at x = -1
This means the denominator must have x + 1 in it
The only expression is the one in choice C
Related Questions
Ash spent 50% of his time at the gym lifting weights. If he spent 45 minutes lifting weights, how many minutes was he at the gym? Use any strategy to solve.
Answers
Ash spent 90 minutes at the gym
pre calc
combine radicals/fractional exponents
Answers
[tex]\begin{array}{llll} ~\hspace{7em}\textit{negative exponents} \\\\ a^{-n} \implies \cfrac{1}{a^n} ~\hspace{4.5em} a^n\implies \cfrac{1}{a^{-n}} \end{array} ~\hfill \begin{array}{llll} \textit{rational exponents} \\\\ a^{\frac{ n}{ m}} \implies \sqrt[ m]{a^ n} \end{array} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\cfrac{6x^{-1}y^{-1}}{\sqrt[4]{xy^4}}\implies \cfrac{6x^{-1}y^{-1}}{\left( xy^4 \right)^{\frac{1}{4}} }\implies \cfrac{6x^{-1}y^{-1}}{x^{\frac{1}{4}} y^{4\cdot \frac{1}{4}}}\implies \cfrac{6x^{-1}y^{-1}}{x^{\frac{1}{4}} y^1}\implies \cfrac{6x^{-1}y^{-1}x^{-\frac{1}{4}} y^{-1}}{1}[/tex]
[tex]6x^{-1}x^{-\frac{1}{4}}y^{-1} y^{-1}\implies 6x^{-1-\frac{1}{4}} y^{-1-1}\implies {\Large \begin{array}{llll} \underset{\textit{\small a}}{6} \stackrel{\textit{\small ~~ b}}{x^{-\frac{5}{4}}} \stackrel{\textit{\small ~~ c}}{y^{-2}} \end{array}}[/tex]
A cone with height h and radius r has volume V = 1/3πr^2h. If a certain cone with a height of 6 in. and a volume of V = 8πx^2 + 24πx + 18π, what is its radius r in terms of x?
A. R = 3x + 2
B. 4x^2 + 12x +9
C. 2x + 3
D. (2x + 3) (2x -3)
Which one is the answer? A explanation would be nice too! (Many points!)
Answers
Answer: C. 2x+3
Step-by-step explanation:
[tex]\displaystyle\\V=\frac{\pi r^2h}{3} \\\\V=\frac{\pi r^2(6)}{3}\\\\V=2\pi r^2\\\\\\8\pi x^2+24\pi x+18\pi \\\\V=2\pi (4x^2+12\pi +9)\\\\V=2\pi ((2x)^2+2(2x)(3)+3^2)\\\\V=2\pi (2x+3)^2\\\\THUS,\\[/tex]
[tex]2\pi r^2=2\pi (2x+3)^2[/tex]
Divide both parts of the equation by 2π:
[tex]r^2=(2x+3)^2[/tex]
Extract the square root of both parts of the equation:
[tex]r=2x+3[/tex]
Use the definition of the integral (the long way) to evaluate the integral
Answers
The answer is 10.67. It can be calculated by integrating and then applying the limits.
What is the integration of [tex]x^n[/tex]?
The integration of power of a variable is given by the following formula:
[tex]\int x^ndx=\frac{x^{n+1}}{n+1}[/tex]
If limits are there then limits can be applied after integration.
Consider the given function.
[tex]\int^3_1(x^2-4x+9)dx[/tex]
Now, apply integration on each term as follows:
[tex]\int^3_1x^2dx-4\int^3_1xdx+9\int^3_1dx[/tex]
Now, use the following formula to integrate:
[tex]\int x^ndx=\frac{x^{n+1}}{n+1}[/tex]
Now, the integration will look like as follows:
[tex][x^3/3]^3_1-4[x^2/2]^3_1+9[x/1]^3_1[/tex]
Now, apply limits to find the answer as follows:
[tex][3^3/3-1^3/3]-4[3^2/2-1^2/2]+9[3-1]\\[/tex]
[tex][9-\frac{1}{3}]-4[9/2-1/2]+18\\10.67[/tex]
Hence, the answer is 10.67. It can be calculated by integrating and then applying the limits.
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F and H are sets of real numbers defined as follows.
F = { v l v < 4}
H = { v l v ≥ 7}
Write F ∩ H and F ∩ H using interval notation. If the set is empty, write ∅.
Answers
The final interval notation is F ∩ H = [7, 4] & F ∪ H = [-∞, ∞].
What is the interval notation?
A set of real numbers known as an interval in mathematics contains all real numbers falling inside any two of the set's numbers. For instance, the interval containing 0, 1, and all integers in between is the set of values x satisfying 0 x 1.
We have,
F and H are sets of real numbers defined as follows.
F = { v l v < 4}
H = { v l v ≥ 7}
F is the set of all real numbers that are less than or equal to 4. H is the set of real numbers greater than 7.
F U H is the union (or combination) of the two sets, so it is the set of all real numbers that are less than equal to 3 or greater than 6:
F ∩ H is the set of real numbers that are in both sets; but since F is less than or equal to 4 and H is greater than 7, there are no numbers that are in both sets. Hence the answer is the empty set, Ø:
F ∩ H = Ø
F ∩ H = [7, 4]
F ∪ H = [-∞, ∞]
Hence, the final interval notation is F ∩ H = [7, 4] & F ∪ H = [-∞, ∞].
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Malcolm calculated how many liters of milk each team would get if 6 teams shared the milk equally. His work is shown above, but he forgot to place the decimal point in the quotient.Where should he place the decimal point? Use number sense to explain
Answers
Malcolm should place the decimal point immediately after one (1) or before three (3) as follows: 8.25/6 = 1.375.
What is a quotient?In Mathematics, a quotient can be defined as a mathematical expression that is simply used to represent the division of a number by another number.
As a general rule, the smaller the divisor (denominator) of a mathematical expression, the larger would be the quotient. This ultimately implies that, a quotient would be large when the divisor (denominator) of a mathematical expression is small.
On the other hand (conversely), a quotient would be small when the divisor (denominator) of a mathematical expression is large.
Based on Malcom's calculations, he forgot to place the decimal point in the following expression;
8.25/6 = 1375
Next, we would determine the maximum value of this quotient;
8.25/6 = 12/6
8.25/6 = 2
Therefore, the whole number part for this decimal number would be one (1) and as such, the decimal point must be placed immediately after 1 as follows;
8.25/6 = 1.375
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Complete Question;
Malcolm calculated how many liters of milk each team would get if 6 teams shared the milk equally. His work is shown above, but he forgot to place the decimal point in the quotient. Where should he place the decimal point? Use number sense to explain.
8.25/6 = 1375
Help please quick!!!!!!
Answers
first one
2x+43 = 180
.....
Super confused! Please help! Explanation would be amazing, thanks! :)
Answers
Answer:
A = 29°
Step-by-step explanation:
set
(3x-4) + (9x+17) + (6x-31) = 180
Solve for x
18x - 18= 180
18x = 198
x = 11
So A will be
A = 3x - 4
A = 3(11) - 4
A = 29°
Answer:
116°Step-by-step explanation:
total angle of a triangle: 180°thus, (3x-4)+(9x+17)+(6x-31)= 180°.3x-4+9x+17+6x-31= 180° 18x-18= 180° 18x= 198° x= 198°/18 x= 11°angle of A: (9x+17)°9(11)+17= 99+17 = 116°Suppose we flipped a coin and rolled a die, what is the probability of getting a tail on the coin or a 3 on the die?
Answers
Answer:
The probability of a head of a die is 1/6 and that of a side of a coin
The vertices of triangle ABC are A(−4, 3), B(4, 3), and C(−4, −6). The triangle is dilated with a scale factor of 3. Estimate the perimeter of the dilated figure to the nearest whole number.
Answers
The perimeter of the image of the triangle is approximately equal to 87 units.
How to determine the perimeter of the image of a triangle
In this problem we find the coordinates of the vertices of a triangle: A(x, y) = (- 4, 3), B(x, y) = (4, 3), C(x, y) = (- 4, - 6), of which we must determine the perimeter of its image, which is the result of a scale factor of 3. The perimeter is the sum of the lengths of the triangle. The length of each side of the image of the triangle can be found by Pythagorean theorem and definition of dilation:
l = √[(Δx)² + (Δy)²]
Where:
l - Side lengthΔx - Change in the x-direction.Δy - Change in the y-direction.And the perimeter of the image of the triangle is described below:
p = k · (AB + BC + AC)
Where k is the scale factor.
First, determine the lengths of sides AB, BC and AC:
Side AB
AB = √[[4 - (- 4)]² + (3 - 3)²]
AB = 8
Side BC
BC = √[(- 4 - 4)² + (- 6 - 3)²]
BC = √[(- 8)² + (- 9)²]
BC = √145
Side AC
AC = √[(- 4 - 4)² + (- 6 - 3)²]
AC = 9
Second, determine the perimeter of the image:
p = 3 · (8 + √145 + 9)
p = 3 · (17 + √145)
p = 51 + 3√145
p ≈ 87.125
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Your employer pays you 1 1/2 times your normal wage on holidays. If your normal hourly wage is $12/hour. how much will you make per hour on the holiday?
Answers
Answer: $18/hour
12 x 1.5 = 18
The solution must be complete, with explanations that are based on already studied facts, formulas, definitions, axioms, theorems and consequences from them.
All assignments require drawing.
Exercise 1.
Rectangle ABCD is given. The line a is parallel to AD and does not lie in the plane of the rectangle.
a) Prove that a||BC (7 points).
b) Prove that lines a and BD are intersecting (7 points).
c) Determine the cosine of the angle between the lines a and BD, if AB = 18 cm, BC = 24 cm. Justify your answer (15 points).
Task 2.
Given a rectangular box MNKLM1N1K1L1. The point E lies on the edge KK1, the point G lies on the edge NK, and the point F lies on the bottom face.
a) Construct a section of a parallelepiped by three given points E, F, G. Explain the construction of each of the segments (22 points).
b) Indicate the name (type) of the resulting polygon and shade its inner part (8 points).
1
Task 3.
An isosceles triangle ABC with base AB is given. From a point S lying outside the plane ABC, a perpendicular is dropped to point C.
a) Draw the linear angle of the dihedral angle SABC. Explain why this particular angle is the linear angle of the dihedral angle (16 points).
b) Find SC if AC = BC = 15 cm, AB = 18 cm, dihedral angle SABC is 30° (25 points).
Answers
Answer:
and the point F lies on the bottom face.
a) Construct a section of a parallelepiped by three given points E, F, G. Explain the construction of each of the segments (22 points).
b) Indicate the name (type) of the resulting polygon and shade its inner part (8 points).
1
Task 3.
An isosceles triangle ABC with base AB is given. From a point S lying outside the plane ABC, a perpendicular is dropped to point C.
a) Draw the linear angle of the dihedral angle SABC. Explain why this particular
Find the sum (9 + 6) (mod 10)
Answers
Answer:
15d10mo
Step-by-step explanation:
Simplify:
(9+6)(mod10)
=15d10mo
how to solve for reflection under transformation
Answers
Similarly, when a point is reflected across the line y = -x, the x-coordinates and y-coordinates change their place and are negated. Therefore, The reflection of the point (x, y) across the line y = x is (y, x). The reflection of the point (x, y) across the line y = – x is (-y, -x).
Hope this helps!
Two dice are rolled, find the probability that the total showing is less than 10.
Answers
The probability that the total showing is less than 10 is 5/6.
What is a expression? What is a mathematical equation? What is Probability?A mathematical expression is made up of terms (constants and variables) separated by mathematical operators. A mathematical equation is used to equate two expressions. Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem. The formula to calculate the probability of occurrence of an event 'A' can be written as -
P(A) = n(A)/n(S)
where -
n(A) = Number of outcomes favorable to event A.
n(S) = Total number of outcomes
Given are two dices rolled.
The total number of favorable outcomes can be calculated as -
[a] = (1,1) to (1,6) = 6
[b] = (2,1) to (2,6) = 6
[c] = (3,1) to (3,6) = 6
[d] = (4,1) to (4,5) = 5
[e] = (5,1) to (5,4) = 4
[f] = (6,1) to (6,3) = 3
n(A) = [a] + [b] + [c] + [d] + [e] + [f] = 6+6+6+5+4+3 = 30
n(S) = 36
So, the probability of occurrence of an event 'A' -
P(A) = n(A)/n(S)
P(A) = 30/36
P(A) = 5/6
Therefore, the probability that the total showing is less than 10 is 5/6.
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The red figure is similar to the blue figure. Which best describes a sequence of transformations in which the blue figure is the image of the red figure?
A.Rotate 90° clockwise about the origin and then dilate with respect to the origin using a scale factor of 12.
B.Rotate 90° counterclockwise about the origin and then dilate with respect to the origin using a scale factor of 12.
C.Rotate 90° counterclockwise about the origin and then dilate with respect to the origin using a scale factor of 2.
D.Rotate 180° counterclockwise about the origin and then dilate with respect to the origin using a scale factor of 3.
Answers
The statement that best describes a sequence of transformations in which the blue figure is the image of the red figure is given as follows:
A. Rotate 90° clockwise about the origin and then dilate with respect to the origin using a scale factor of 1/2.
What are the transformations?The horizontal side lengths on the blue figure are given as follows:
8 - 4 = 6 - 2 = 4.
The vertical side lengths on the transformed red figure is of:
4 - 2 = 3 - 1 = 2.
Hence the scale factor of the dilation is given as follows:
2/4 = 1/2.
Which means that options C and D are not correct.
The rules for each rotation are given as follows:
90º degrees clockwise: (x,y) -> (y,-x).90º degrees counter-clockwise: (x,y) -> (-y,x).On the second quadrant, of the blue figure, we have that:
x is negative.y is positive.On the first quadrant, of the red figure, we have that:
x is positive.y is positive.Hence the rotation was clockwise, and option A is correct.
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please see the attached the answer is wrong that i have
Answers
what is a compressed graph
Answers
Answer: A transformation in which all distances on the coordinate plane are shortened by multiplying either all x-coordinates (horizontal compression) or all y-coordinates (vertical compression) of a graph by a common factor less than 1.
Step-by-step explanation:
If a person travels at a speed of 28 m/s for 10 seconds, how far have they traveled?
Answers
It is a uniform rectilinear movement which is one in which an object moves in a straight line, in one only direction, with a constant speed.
When we spoke of constant speed we mean that the movement retains the same speed, that is; that the object does not move faster, or slower and always at the same speed.
If a person travels at a speed of 28 m/s for 10 seconds, how far have they traveled?We obtain the data, as specified by the exercise.
Data:
V = 28 m/s
T = 10 s
D = ?
We have that the uniform motion formula is:
[tex]\large\displaystyle\text{$\begin{gathered}\sf V=\dfrac{d}{t}, \to where \end{gathered}$}[/tex]
[tex]\large\displaystyle\text{$\begin{gathered}\sf V=Speed \end{gathered}$}[/tex][tex]\large\displaystyle\text{$\begin{gathered}\sf D=distance \end{gathered}$}[/tex][tex]\large\displaystyle\text{$\begin{gathered}\sf T=Time \end{gathered}$}[/tex]We solve the formula, to calculate the distance and we obtain.And we also substitute our data.
[tex]\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{d=v*t} \end{gathered}$} }[/tex]
[tex]\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{d=28 \ \frac{m}{\not{s}}*10 \not{s} } \end{gathered}$}}[/tex]
[tex]\boxed{\boxed{\large\displaystyle\text{$\begin{gathered}\sf \bf{d=280 \ m } \end{gathered}$}}}[/tex]
The person travels a distance of 280 meters.The high school a athletic director is asked if football players are doing as well as academically asthe other students athletes. We know from a previous study that the average GPA for the student athletes is 3.10. after an initiative to help improve the GPA of students athletes, the athletic director randomly samples 20 football players and finds that the average GPA of the sample is 3.18 with a sample standard deviation of 0.54 is there a significant improvement use a 0.05 significance level
Answers
The athletic director can therefore draw the conclusion that the average academic standing of football players is comparable to the average standing of other student athletes.
Explain the term t-distribution?Whenever the population's standard deviation is unknown and the data come from such a normally distributed population, the t-distribution characterizes the standardized distances between the sample means and the population mean.The following formula is used to determine the test statistic:
SRS, s = 0.54, n = 40 > 30,
x = 3.18
H0: μ = 3.10
H1: μ > 3.10
α = 0.05
The t-distribution table and our alpha level of.05 are then used to determine our crucial values.
Our critical values are 2.093 for a two specimens with 19 levels of freedom and α = 0.05 level of significance.
t = x - μ / s/√n
t = 3.18 - 3.10/ 54/√20
t = 0.66
This indicates that the measured sample mean of football players, 3.18, is 0.66 standard errors higher than the 3.10 number that was predicted. We are unable to disprove the null hypothesis because the test statistic's value is below the threshold of 2.093.As a result, It is clear that sampling error is the only possible explanation for the discrepancy between both the sample mean as well as the hypothesized value.
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what is the slope of the line perpendicular to x-3y=4
Answers
Answer:
-3
Step-by-step explanation:
x - 3y = 4 Write in the slope intercept form. Subtract x from both sides
-3y = -x + 4 Divide all the way through by -3
y = [tex]\frac{1}{3}[/tex] x - [tex]\frac{4}{3}[/tex]
The slope is the number before the x ( 1/3 )
The perpendicular slope is the opposite reciprocal or -3
Donna buys 4 hardcover books for h dollars each and 5 paperback books
for p dollars each. Sales tax is 8%. Which expression represents the total
cost of Donna's purchase?
A (4h + 5p) (1.08)
B. (4h + 5p)(0.08)
C. (4h + 5p) + 1.08
D. (4h + 5p) + 0.08
Answers
The 1.08 represents the sales tax of 8%
Please help, I will give brainliests to the person to answer correctly.
Answers
The animal which travel at a rate of 1.5 meters per minutes is ants.
The correct answer option is option D
Which animal travel at 1.5 meters per minutes?Based on the given task content, it can be solved using fraction.
Speed refers to the rate of change of distance at a given time period.
Average speed = Total distance / Time taken
Turtle = Total distance / Time taken
= 6/5
= 1.2 meters per minutes
Snail = Total distance / Time taken
= 4/8
= 0.5 meters per minutes
Caterpillar = Total distance / Time taken
= 7/7
= 1 meters per minutes
Ant = Total distance / Time taken
= 12/8
= 1.5 meters per minutes
In conclusion, ant is the animal which has an average speed of 1.5 meters per minutes
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PS Unit 02: Prove These Triangles are Congruent
please help I need this within the next few days and I don't understand how to do it
Answers
Answer:
Step-by-step explanation:
We can use the Side-Side-Side (SSS) Theorem – Three pairs of corresponding sides are congruent in two triangles.
Band students are tested on, and required to pass, a certain number of scales during the year. As of today, Becky has passed 4 scales, whereas her friend Charlotte has passed 8 of them. Going forward, Becky has committed to passing 4 scales per week, and Charlotte has committed to passing 2 per week. At some point soon, the two friends will have passed the same number of scales. How long will that take? How many scales will that be?
Answers
They each will pass 12 scales.
What do you mean by algebraic expression?
The concept of algebraic expressions is the use of letters or alphabets to represent numbers without providing their precise values. We learned how to express an unknown value using letters like x, y, and z in the fundamentals of algebra. Here, we refer to these letters as variables. Variables and constants can both be used in an algebraic expression. A coefficient is any value that is added before a variable and then multiplied by it.
It is given that Becky has passed 4 scales, whereas her friend Charlotte has passed 8 of them.
Going forward, Becky has committed to passing 4 scales per week, and Charlotte has committed to passing 2 per week.
Let number of weeks be x
Becky has passed 4x + 4
Charlotte has passed 2x + 8
According to question:
4x + 4 = 2x + 8
4x - 2x = 8 - 4
2x = 4
x = 4/2 = 2
Becky has passed 4x + 4 = 4(2) + 4 = 8 + 4 = 12
Charlotte has passed 2x + 8 = 2(2) + 8 = 4 + 8 = 12
Therefore, They each will pass 12 scales.
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The width of a rectangle is 6 yards less than the length. The perimeter is 132 yards. Find the length and width.
Width =
Length =
Answers
The length and width of a rectangle are 36 and 30 yard respectively.
What is perimeter?Its the sum of length of the sides used to made the given figure.
We are given that width of a rectangle is 6 yards less than the length. The perimeter is 132 yards.
Therefore, the perimeter = (2 x length)+(2 x width)
The length = 6 + w
Hence, we have;
132 = (2(6 + w))+(2w)
132 = 12 + 2w + 2w
132 = 12 + 4w
120 = 4w
w = 30
Length = 36 yard.
Thus, the length and width of a rectangle are 36 and 30 yard respectively.
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Find the Maclaurin series of the following functions. Step by step please
Answers
The Maclaurin series of ln (1 + x³) is
[tex]= x^3-\dfrac{1}{2}x^6+\frac{1}{3}x^9+\ldots \:[/tex]
The function is given in the question as
h(x) = ln (1 + x³)
Taylor (Maclaurin) series ofln (1 + x³) up to n = 3
A Maclaurin series is given by f(x),
[tex]f\left(x\right)=f\left(a\right)+\frac{f^'\left(a\right)}{1!}\left(x-a\right)+\frac{f^{''}\left(a\right)}{2!}\left(x-a\right)^2+\frac{f^{'''}\left(a\right)}{3!}\left(x-a\right)^3+\ldots[/tex]
[tex]f\left(x\right)=f\left(0\right)+\frac{f^'\left(0\right)}{1!}\left(x\right)+\frac{f^{''}\left(0\right)}{2!}\left(x\right)^2+\frac{f^{'''}\left(0\right)}{3!}\left(x\right)^3+\ldots[/tex]
We need to do to get the desired polynomial is to calculate the derivatives, evaluate them at the given point, and plug the results into the given formula.
f⁽⁰⁾(x) = f(x) = ln (1 + x³)
Evaluate the function at the point: f(0)=0
1st derivative: f(1)(x)=(ln(x3+1))′=3x²/(x³+1)
1st derivative at the given point: (f(0))′=0
2nd derivative: f(2)(x)=(f(1)(x))′=(3x²/(x³+1) )′=3x(2−x³)/(x⁶+2x³+1)
Evaluate the 2nd derivative at the given point: (f(0))′′=0
3rd derivative: f(3)(x)=(f(2)(x))′= (3x(2−x³)/(x⁶+2x³+1) )′= 6(x⁶−7x³+1)x9+3x⁶+3x³+1
Evaluate the 3rd derivative at the given point: (f(0))′′′=6
Now, use the calculated values to get a polynomial:
f(x)≈0/0!x⁰+0/1!x¹+0/2!x²+6/3!x³+0/4/!x⁴+0/5!x⁵
Thus, the Maclaurin series of ln (1 + x³) is
[tex]= x^3-\dfrac{1}{2}x^6+\frac{1}{3}x^9+\ldots \:[/tex]
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Question
DIG DEEPER A group of friends visits a movie theater.
A menu of Royal Cinemas shows cost of tickets and refreshments. The cost of daytime and evening tickets are 5 dollars and 7.50 dollars respectively. The cost of small, medium, and large drinks are 1.75 dollars, 2.75 dollars, and 3.50 dollars respectively. The cost of small and large popcorns are 3 dollars and 4 dollars respectively.
Each person buys a daytime ticket and a small drink. The group shares 2 large popcorn. What is the average cost per person when there are 4 people in the group?
Answers
The average cost per person when there are 4 people in the group is of:
$8.75.
How to obtain the average cost per person?The average cost per person represents the mean cost per person in the data-set.
The mean of a data-set is given by the sum of all observations in the data-set divided by the number of observations.
Hence, the average cost per person in this problem is obtained as the division of the total spending by four, as there are four people.
The total spending is obtained as follows:
Four daytime tickets -> 4 tickets of 5 dollars.Four small drinks -> 4 drinks of 1.75 dollars.Two large popcorns -> 2 popcorns of 4 dollars.Hence the numeric value of the total spending is of:
4 x (5 + 1.75) + 2 x 4 = $35.
Thus the average spending per person is of:
$35/4 = $8.75.
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f(x) =x^3 + 2x^2 -2x
Answers
Divide (5x^4-33x^3+13x^2+25x+20) by (x-6) using long polynomial divison.