If Anthonia and Zahra compete at the same event, the probability that Anthonia will successfully land her jump and Zahra will not land hers is 18.33%.
What is the probability?We describe probability as the chance or likelihood that an expected event occurs when there are many possible outcomes.
Probability lies between one and zero when it is not 100% certain or uncertain.
The probability that Anthonia lands her triple axel jump = 33.33%
The chance that Anthonia does not land her jump = 66.67% (1 - 33.33%)
The likelihood of Zahra landing her jump successfully = 45%
The odds that Zahra will not successfully land her jump = 55% (1 - 45%)
The probability that Anthonia lands successfully while Zahra does not = 18.33% (33.33% x 55%)
Thus, Anthonia has the chance of winning Zahra at 18.33%.
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Problem 1)
a) What is the possible number of microstates associated with tossing a coin N times and having it come up H times heads and T times tails?
b) For a series of 1000 tosses, what is the total number of microstates associated with 50% heads and 50% tails?
c) How less probable is the outcome that the coin will land 40% heads and 60% tails?
a) The number of microstates associated with tossing a coin N times and having it come up H times heads and T times tails is given by the binomial coefficient "N choose H" (or "N choose T", as H+T=N), which is equal to N!/(H! * T!)
b) For a series of 1000 tosses, with 50% heads and 50% tails, there are a total of "1000 choose 500" microstates. This is approximately 2.07*10²⁵ microstates.
c) The probability of getting a specific outcome when tossing a coin is given by the binomial probability mass function. The probability of getting 40% heads and 60% tails in 1000 tosses is given by (1000 choose 400)(0.4⁴⁰⁰)(0.6⁶⁰⁰) which is very less probable.
Coin Toss Microstates Probabilitya) The number of microstates associated with a coin flip can be determined using combinatorics. The coin flip can have two possible outcomes: heads or tails. To calculate the number of microstates for N coin flips with H heads and T tails, we use the binomial coefficient formula "N choose H" (or "N choose T", as H+T=N), which is equal to N!/(H! * T!) where ! denotes factorial. This formula gives the number of ways to choose H heads out of N coin flips.
b) To calculate the total number of microstates for a series of 1000 tosses with 50% heads and 50% tails, we use the binomial coefficient formula "1000 choose 500". This is equivalent to (1000!/(500!500!)) which is approximately 2.0710²⁵ microstates.
c) The probability of getting a specific outcome when tossing a coin is given by the binomial probability mass function. The probability of getting 40% heads and 60% tails in 1000 tosses is given by (1000 choose 400)(0.4⁴⁰⁰)(0.6⁶⁰⁰) which is very less probable. We can calculate this using the binomial probability mass function which is ( n choose k ) * p^k * (1-p)^(n-k) where n is the number of trials, k is the number of successful outcomes, p is the probability of success in one trial, 1-p is the probability of failure in one trial.
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A coin is flipped 200 times. The table shows the frequency of each event.
Outcome Frequency
Heads 96
Tails 104
Determine the experimental probability of landing on tails.
50%
52%
96%
104%
Answer:
52%
Step-by-step explanation:
Since the question asks about tails, we don't need the information about heads.
So out of 200 flips, 104 were tails. If we put 104 out of 200 in fraction form, you get;
104
-----
200
Now we just divide;
104/200 = 0.52, also known as 52%
which of the following expressions have dimensions of length? assume the standard variables have the following dimensions: t is in units of s, x is in units of m, v is in units of m/s, and a is in units of m/s2.a. atb. 1/2at^2c. x^3/td. √2axe. v^2/a
The following expressions have length dimensions: a. at b. x3/t
d. √2ax
a. at: has a dimension of m because it is the product of acceleration (m/s2) and time(s). b. 1/2at2: has a dimension of m because it is the product of acceleration (m/s2) and time(s) squared. c. x3/t: has a dimension of m because it is the product of distance x (m) cubed and time(s).
e. v2/a: is a product of velocity (m/s) squared and acceleration (m/s2) thus it doesn't have a dimension of length. d. 2ax: is a product of 2, distance x (m), and acceleration (m/s2) so it has dimension of m.
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the solid with a semicicular base of radius 5 whose cross sections perpendicular to the base and parallel to the diameter are squares
The volume of the solid with the given semi-circular base with radius 5 units is equal to 333.33 cubic units.
As given in the question,
Radius of the semicircular base = 5 units
Equation of the circle is given by :
x² + y² = r²
⇒ x² + y² = 5²
⇒ x² + y² = 25
⇒ x = √25 - y²
Cross section is perpendicular to the base and it is parallel to the diameter are squares:
Diameter is double of the radius
s = 2x
= 2√25 - y²
Volume of the given solid is equal to :
V = [tex]\int\limits^5_0 {s^{2} } \, dy[/tex]
= [tex]\int\limits^5_0 {( 2\sqrt{25 - y^{2} }) ^{2} } \, dy[/tex]
= [tex]\int\limits^5_0 {( 4({25 - y^{2} }) } \, dy[/tex]
= 100y - 4y³/3 (for limit 0 to 5)
= ( 500 - 500/3 ) - 0
= 500 ( 1 - 1/3 )
= 500( 2/3 )
= 1,000/3
= 333.33 cubic units.
Therefore, the volume of the given solid with the given measures is equal to 333.33.
The above question is incomplete, the complete question is :
Use the general slicing method to find the volume of the following solid.
The solid with a semicircular base of radius 5 whose cross sections perpendicular to the base and parallel to the diameter are squares. Place the semicircle on the xy-plane so that its diameter is on the x-axis and it is centered on the y-axis. Set up the integral that gives the volume of the solid. Use increasing limits of integration.
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For each of the following subsets of R3 determine whether it is a vector subspace of R3 or not. Justify your answer.
So, b) and e) are subspaces of R3, and a), c), and d) are not.
a) {(x,y,z) ∈R3 : x = 0} is not a subspace of R3, because it doesn't contain the zero vector (0,0,0) which is a requirement for a subspace. It's also not closed under scalar multiplication.
b) {(x,y,z) ∈R3: x + y = 0} is a subspace of R3, because it's closed under scalar multiplication and addition and it contains the zero vector (0,0,0).
c) {(x,y,z) ∈R3 : xz = 0} is not a subspace of R3, because it's not closed under scalar multiplication and it's not closed under addition
d) {(x,y,z) ∈ R3: y ≥0} is not a subspace of R3, because it's not closed under addition, it doesn't contain the zero vector (0,0,0) and it's not closed under scalar multiplication.
e) {(x,y,z) ∈ R3: x = y = z} is a subspace of R3 because it satisfies all the properties of a subspace, it is closed under addition, scalar multiplication, it contains the zero vector (0,0,0) and the negative of any vector in the set is also in the set. This set contains all the vectors that are multiples of (1,1,1) which forms a subspace.
So, b) and e) are subspaces of R3, and a), c), and d) are not.
Complete Question: For each of the following subsets of R3 determine whether it is a vector subspace of R3 or not. Justify your answer.
a) {(x,y,z) ∈R3 : x = 0}
b) {(x,y,z) ∈R3: x + y = 0}
c) {(x,y,z) ∈R3 : xz = 0}
d) {(x,y,z) ∈ R3: y ≥0}
e) {(x,y,z) ∈ R3: x = y = z}
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What is the area, in square meters, of the shaded part of the rectangle below? 9 m 20 m 4 m
49 is the shaded part of the rectangle.
What are the names of the shaded areas?Therefore, the rule for its area is base1+base2/2xh
Base1=9
Base2=20
Height = 4
Therefore, 9+20/2 x 4= 49.
The denominator is the total number of pieces that make up the whole, which is eight. That is the figure beneath the fraction bar. The numerator is the number of shaded portions, which is 7. That is the value shown above the fraction bar.
The shaded region’s area is the difference between the area of the complete polygon and the area of the polygon’s unshaded portion. In polygons, the area of the shaded component can arise in two ways. The darkened zone might be found in the middle of a polygon or on its sides.
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The area of the shaded region of the rectangle is 130 sq. meter
What is Rectangle ?A quadrilateral with four right angles is a rectangle. It may alternatively be described as a parallelogram with a right angle or an equiangular quadrilateral, where equiangular denotes that all of its angles are equal. A square is a rectangle with four equally long sides.
The denominator is the total number of pieces that make up the whole, which is eight. That is the figure beneath the fraction bar. The numerator is the number of shaded portions, which is 7. That is the value shown above the fraction bar.
The shaded region’s area is the difference between the area of the complete polygon and the area of the polygon’s unshaded portion. In polygons, the area of the shaded component can arise in two ways. The darkened zone might be found in the middle of a polygon or on its sides.
The base length = 9 m
The base width = 20 m
The height = 4m
The area of the shaded region = 1/2 *(9 + 4)*20 = 130 sq. meter
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The path of a firework can be represented by F9t=16^+92t+160
What is the maximum height of the firework?
How long did it take to reach its maximum height?
a) The maximum height of a firework is 292.25 feet.
b) The time required by a firework to reach its maximum height is 2.875 second.
How to determine the maximum height reached by a firework and its time needed
Herein we find the case of a firework being launched under free fall motion, that is, a vertical uniformly accelerated motion due to gravity, represented by following quadratic equation:
f(t) = - 16 · t² + 92 · t + 160
Where:
f(t) - Height, in feett - Time, in secondsPart A - The maximum height can be found by modifying the quadratic equation into vertex form. This can be done by completing the square:
f(t) = - 16 · t² + 92 · t + 160
f(t) = - 16 · [t² - (23 / 4) · t - 10]
f(t) - 16 · (529 / 64) = - 16 · [t² - (23 / 4) · t + 529 / 64] + 160
f(t) - 292.25 = 16 · (t - 2.875)²
The firework reaches a maximum height of 292.25 feet.
Part B - The firework takes a time of 2.875 seconds to reach its maximum height.
RemarkFree fall equation has several mistakes and was corrected according to principles of free fall motion.
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Sam has 9 1/3 cups of cereal. Each serving is 2/4 cup. How many servings does Sam have?
The number of servings that Sam has would be = 18.67 servings.
What is a fraction?A fraction is part of a whole value that is being represented in a form of numerator and denominator.
The number of cups that Sam has = 9 1/3 = 28/3
The number of cups used for each serving = 2/4
That is ;
1 serving = 2/4 cups
X servings = 28/3 cups
Make X the subject of formula;
X servings = 28/3 ÷ 2/4
= 28/3 × 4/2
= 112/ 6
= 18.67 servings.
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according to the law of large numbers
Answer:
In probability and statistics, the law of big numbers holds that as a sample size increases, its mean gets closer to the average of the entire population. This is because as the sample size increases, the sample becomes more representative of the population.
According to the law of large numbers, a large entity that is expanding quickly cannot continue doing so indefinitely. Often used as instances of this phenomenon are the largest blue chips, with market values in the hundreds of billions.
Two distinct subjects can be covered by the law of huge numbers. First, there are many topics to which the law of large numbers can be applied in statistical analysis. In order to acquire the necessary amount of data, it may not be practical to survey every member of a given population, but every additional data point has the potential to raise the probability that the result is an accurate representation of the mean.
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suppose that you have a photograph and you want to crop it to cut off 2 inches from the right side of the picture and then resize it using various scale factors (for our purposes here we will focus on a scale factor of 1.5). the transition from the dimensions of the original photograph to the dimensions of any scaled version is a two-step process.
Yes, cropping a picture and then using a scale factor to resize it is a two-step process.
The image is first cropped by subtracting 2 inches from the right side, altering the image's proportions.
The image is then enlarged using a scale factor of 1.5, which increases the image's dimensions by 1.5.
The final measurements of the image would be its original measurements less 2 inches (for cropping), then multiplied by 1.5. (for the resizing).
It's vital to keep in mind that both picture cropping and image scaling change the size of the final product.
Yes, cropping a photo, then resizing it with a scale factor, is a two-step process.
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At Lauren's school library, students are allowed to have a balance of no less than -$12 and still check out books. Five days ago, Lauren's balance was $0, but now she owes for her books that are five days late. If the total charge on her checked out books is $2 per day and she turns them in today, can she still check out books? No Yes
Answer:
No
Step-by-step explanation:
2 * 5 = 10
12 does not equal 10
Modeling Problems with Equations Dean ran 2.3 fewer kilometers than Sam. If Dean ran 6.8 km, how far did Sam run? A 2-column table with 4 rows. Column 1 is labeled Situation with entries increasing, difference, finding part of a total, sharing or grouping. Column 2 is labeled Operation with entries +, minus, times, divided by. Select all that apply. You know the difference in the distances the boys ran, so this is a subtraction problem. You are finding the total distance the boys ran, so this is an addition problem. Dean ran part of the distance Sam ran, so this is a multiplication problem. The correct equation is s + 2.3 = 6.8. The correct equation is s – 2.3 = 6.8. The correct equation is 2.3s = 6.8.
Since Dean ran 2.3 fewer kilometers than Sam. If Dean ran 6.8 km, the true statements about the distance which Sam ran are:
A. You know the difference in the distances the boys ran, so this is a subtraction problem.
E. The correct equation is s – 2.3 = 6.8.
What is distance?In Mathematics, distance can be defined as the total amount of ground that is covered or travelled by a physical object (body) over a particular period of time and speed, irrespective of its direction, starting point or ending point.
Since Dean ran 6.8 km and 2.3 fewer kilometers than Sam, we can logically deduce that this is a subtraction problem. Assuming the variable s represent the distance which Sam ran, Sam's distance can be calculated as follows;
s – 2.3 = 6.8
s = 6.8 + 2.3
s = 9.1 kilometers.
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Which expression is equivalent to 5f+4f?
Hey there!
Answer:
An equivalent to the given expression [tex]5f+4f[/tex] would be [tex]9f[/tex].
Hope this helps!
Answer:I hope this helps
Step-by-step explanation:
5f+4f
Collect like terms by adding their coefficients
which is like this
(5f+4f)f and now we
Add the numbers in the parentheses
=9f
Left F(x) be the number, in thousands, of vehicles a manufacturer estimates will be sold when x million dollars are spent on advertising. If f(1.8)=240, and f(2.5)=325, compute the average rate of change for 1.8 < x< 2.5
The average rate of change for 1.8 < x < 2.5 is given as follows:
121.43.
How to obtain the average rate of change?The average rate of change of a function is given by the change in the output divided by the change in the input.
The change in the output for this problem is given as follows:
325 - 240 = 85.
The change in the input for this problem is given as follows:
2.5 - 1.8 = 0.7.
Hence the average rate of change is given as follows:
85/0.7 = 121.43.
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write an equation of the line that passes through (-6,0) and (0,-24)
Answer:
y=-4x-24
Step-by-step explanation:
draw the line on a grid paper so that you know the slope must be negative. then plug in the numbers.
I NEED HELP ASAP WHOEVER ANSWERS GETS BRAINLIST
The side length of the cube is 1/6 and the volume is 1/216
How to determine the side length of the cubeFrom the question, we have the following parameters that can be used in our computation:
Face area = 1/36
This means that
Side length = √(1/36)
Evaluate
Side length = 1/6
How to determine the volue of the cubeFrom the question, we have the following parameters that can be used in our computation:
Side length = 1/6
This means that
Volume = (1/6)^3
Evaluate
Volume = 1/216
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Select the correct answer.
Which expression is equivalent to the given expression? Assume the denominator does not equal zero. a^3b^35/a^4b
By Assuming the denominator does not equal zero. a³b³/a⁴b equivalent expression is b⁴/a.
What is an equivalent expression?
Equivalent expressions are expressions that have the same result when they are simplified.
Two expressions are considered equivalent if they can be transformed into each other by applying mathematical operations such as simplification, combining like terms, or applying the properties of arithmetic and algebra.
Equivalent expressions can be used to solve problems more efficiently or to help understand complex mathematical concepts.
For example, simplifying an expression to its simplest form can make it easier to understand or perform further calculations with it.
Equivalent expressions can also be used to check the correctness of a solution to a problem, by verifying that the answer can be transformed into the expected result.
The given expression can be simplified as follows:
a³b³ / a⁴b = (a³/a⁴) * (b³/b)
a³/a⁴ = 1/a
b³/b = b²
Hence, By Assuming the denominator does not equal zero. a³b³/a⁴b equivalent expression is b⁴/a.
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I'm not sure how to answer part b, an explanation would be appreciated.
The true statement is that the sample mean is not biased
How to determine the true statementFrom the question, we have the following parameters that can be used in our computation:
Sample selection of 2
The sample is selected such that all 4 samples are equally likely
This means that
P(Each selection) = 1/4
An unbiased sample is such that reflects all members of the population and in this case, this sample reflects the population
Hence, the sample is unbiased
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42. HOW DO YOU SEE IT?
Write an expression in rational
exponent form that represents
the side length of the square.
Area:
x in.²
The expression for the side of the square will be s=x^(1/2) in
where s=side.
what are squares?
A square is a two-dimensional plane figure with four equal sides and all four angles are equal to 90 degrees. The properties of the rectangle are somewhat similar to a square, but the difference between the two is, a rectangle has only its opposite sides equal. Therefore, a rectangle is called a square only if all its four sides are of equal length.
The most important properties of a square are listed below:
All four interior angles are equal to 90°.All four sides of the square are congruent or equal to each other.The opposite sides of the square are parallel to each other.The diagonals of the square bisect each other at 90°.The two diagonals of the square are equal to each other.The square has 4 vertices and 4 sides.The diagonal of the square divides into two similar isosceles triangles.The length of diagonals is greater than the sides of the square.Area of square=(side)²
Perimeter of square=4*side
Now,
Area of square=(side)²
and given that
A=x in^2
s^2=x
s=√x
Therefore, expression in rational exponent form will be
s=x^(1/2)
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enter the correct 4 digit code
Answer:
DCBA
Step-by-step explanation:
It's just reverse look closely and you'll notice
knewton.com a
First up, let's review the assignment's learning
objectives
Get familiar with this topic by reviewing instruction and answering a couple of
questions.
Provide your answer below:
LTE 36%
Question
Montana has campgrounds over 41 counties. Nevada has 637 campgrounds over 18 counties. If
Montana was proportional to Nevada in the number of campgrounds to counties, how many
campgrounds would Montana expect to have? Round to the nearest whole number.
Answer: In order to find the number of campgrounds that Montana would expect to have if it was proportional to Nevada in the number of campgrounds to counties, we need to determine the ratio of campgrounds to counties in Nevada and then apply that same ratio to Montana.
First, we'll find the ratio of campgrounds to counties in Nevada by dividing the number of campgrounds (637) by the number of counties (18): 637 / 18 = 35.389
Now we'll apply that same ratio to Montana by multiplying the number of counties in Montana (41) by the ratio of campgrounds to counties in Nevada: 41 x 35.389 = 1458.369
Rounding to the nearest whole number, we get 1458. So if Montana was proportional to Nevada in the number of campgrounds to counties, we would expect to find 1458 campgrounds in Montana.
Step-by-step explanation:
The venn diagram shows sports played by 10 students let event A= The student play basketball let event B= the student plays soccer
The value of P(A|B) from the given data is 1/6 ≅ 0.17
Conditional Probability:The term P(A/B) is known as a conditional probability which indicates the probability of an event A that depends on another event B.
The probability of an event or outcome occurring due to the existence of a prior event or outcome is known as conditional probability.
P(A/B) is used to find this conditional probability.
The formula for P(A/B) is given by
P(B/A) = P(A∩B) / P(A)Here we have
A ven diagram which shows sports played by 10 students
From given figure
Number of students who play basketball, p(A) = 3
Number of students who play soccer, p(B) = 2
Number of students who play both games, A∩B = 3
As we know, P(A/B) = P(A∩B)/P(B)
= 1/3 [2]
= 1/6 ≅ 0.17
Therefore,
The value of P(A/B) from the given data is 1/6 ≅ 0.17
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Recall that the side length of the square was 14 units long. What is the base area of the prism?
Diameter of cylinder base - 19.8
base area of cylinder - 307.8
volume of cylinder - 9.234
Answer:
56
Step-by-step explanation:
The base area will be 4 times the side length
56 units
Read the following prompt and type your response in the space provided.
Describe a real-life representation of the following.
12/49=0.24
A real - life representation of 12 / 49 = 0. 24 is the cost of a single donut in a 49 pack of donuts that cost $ 12.00
What real - life representation shows the equation ?The equation is 12 / 49 = 0. 24. A real - life representation of this could have to do with unit cost, and total cost.
The 12 could be the total cost of a package of donuts that have 49 donuts inside.
The unit cost of each of these donuts would therefore be:
= Total cost of donuts / Number of donuts in package
= 12 / 49
= $ 0.24 per donut
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The lengths of the sides of triangle XYZ are written in terms of the variable m, where m ≥ 6.
Triangle X Y Z is shown. The length of side X Y is m + 8, the length of side Y Z is 2 m + 3, the length of side Z X is m minus 3.
Which is correct regarding the angles of the triangle?
mAngleX < mAngleZ < mAngleY
mAngleY < mAngleZ < mAngleX
mAngleY < mAngleX < mAngleZ
mAngleZ < mAngleY < mAngleX
The correct arrangement regarding the angles of the triangle is;
m∠Y < m∠Z < m∠X
What are the missing angles of the triangle?In triangle XYZ, the lengths of sides are;
XY = m + 8
YZ = 2m + 3
ZX = m - 3
If m ≥ 6, then;
m + 8 ≥ 14
2m + 3 ≥ 15
m - 3 ≥ 3
Thus;
m - 3 ≤ m + 8 ≤ 2m + 3
The greatest angle is opposite to the greatest side, the smallest angle is opposite to the smallest side, so;
the greatest side is YZ = 2m + 3- the greatest angle is ∠X
the smallest side is ZX = m - 3 - the smallest angle is ∠Y
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From a 15-inch by 15-inch piece of metal, squares are cut out of the four corners so that the sides can be folded up to make a box. Let x represent the length of the sides of the squares, in inches, that are to be cut out. Express the volume of the box as a function of x.
By applying the formula of cuboid volume, the volume of the box can be expressed as 225x - 60x² + 4x³.
Cuboid is a 3-D shape formed by three pairs of rectangles (6 sides), 12 edges, and 8 vertices.
The volume of a cuboid can be calculated as follows:
Volume = length * width * height
Initially, we have a 15 x 15-inch piece of metal, then we cut out the four corners so that the sides can be folded up to make a box.
Now we have a shape as the attached image, where x represents the length of the sides of the squares that are to be cut out.
Thus, after the cutting, the length of the new side is 15 - 2x.
This value also represents the length and width of the box, whilst the height is x.
Now we can calculate the volume of the box:
Volume = length * width * height
= (15 - 2x) * (15 - 2x) * x
= (225 - 60x + 4x²) * x
= 225x - 60x² + 4x³
Thus, the volume of the box can be expressed as 225x - 60x² + 4x³.
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lian's truck has a 25 gallon tank and uses 0.05 gallon of gas for every mile driven.when creating a table for the function y=-0.05x + 25 ,lian argues that he can only use positive rational numbers for the input .is lian correct? justify your answer
Since it is possible to drive a fraction of a mile, so it will be appropriate to use fractions; however, it is not possible to drive a negative number of miles, so only positive rational numbers are appropriate.
How to interpret Linear equations?A rational number is said to be positive if its numerator and denominator are both positive.
Now, we are told that lian's truck uses 0.05 gallon of gas for every mile driven. Thus;
Rate of consumption of gas = 0.05 gallons per mile
The function for number of gallons is;
y = -0.05x + 25
where;
x is number of miles
y is number of gallons
We can use a fraction for the number of miles because it is possible but then the number of miles must be positive rational numbers because number of miles cannot be negative.
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What is the remainder of the quantity 2 x squared plus 7 x minus 39 end quantity divided by the quantity x minus 7 end quantity? Please show all work
The remainder of the division is (x-3)(x+13)/x-7
How to determine the reminderFrom the information given, we have that;
2 x squared plus 7 x minus 39 end quantity quantity x minus 7 end quantityThis is represented as;
2x² + 7x - 39/x - 7
Now, factorize the numerator
Find the pair factors of -78 that sum up to 7, then substitute the value, we have;
2x² + 13x - 6x - 39/x- 7
(2x² + 13x) - (6x-39)/x-7
Factor the common terms
x(x + 13) - 3(x + 13)/x-7
Then, we get;
(x-3)(x+13)/x-7
Hence, the value is (x-3)(x+13)/x-7
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Find the exact value of sin(x + y), given sin(x) = Negative one-half, cos(y) = Negative one-third, x is an angle in quadrant III, and y is an angle in quadrant II.
StartFraction 2 minus StartRoot 5 EndRoot Over 6 EndFraction
StartFraction 1 minus 2 StartRoot 6 EndRoot Over 6 EndFraction
StartFraction 6 minus StartRoot 5 EndRoot Over 6 EndFraction
StartFraction StartRoot 3 EndRoot minus StartRoot 5 EndRoot Over 6 EndFraction
The exact value of sin(x + y) is (1 - 2√6) /6. The 2nd option is the answer.
How to find the exact value of sin(x + y)?We have sin(x) = -1/2 and cos(y) = -1/3
x is an angle in quadrant III, and y is an angle in quadrant II
For x:
Using trigonometric ratio
sin(x) = -1/2 (sin = opposite/hypotenuse)
adjacent = √(2²-(-1)²) = √3
cos(x) = (-√3)/2 (cos = adjacent/hypotenuse)
Note: adjacent in quadrant III is negative
For y:
Using trigonometric ratio
cos(y) = -1/3 (cos = adjacent/hypotenuse)
opposite = √(3²-(-1)²) = √8 = 2√2
sin(y) = (2√2)/3 (sin = opposite/hypotenuse)
Using trigonometric identity:
sin (x + y) = sin (x)·cos(y) + cos (x)·sin (y)
sin (x + y) = -1/2·(-1/3) + [(-√3)/2]·[(2√2)/3]
sin (x + y) = 1/6 - (√6)/3
sin (x + y) = (1 - 2√6) /6
Thus, the exact value of sin(x + y) is (1 - 2√6) /6
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Given that 5 x : 9 = 8 : 3 Calculate the value of x . Give your answer in its simplest form.