Answer:
Brand x: 80
Brand y: 45
Step-by-step explanation:
x + y = 125
20x + 15y = 2275
-20x - 20y = -2500
(+) 20x + 15y = 2275
--------------------------------
-5y = -225
y = 45
x + y = 125
x + 45 = 125
x = 80
Answer:
Brand x: 80
Brand y: 45
What is meant by a biased sample?
A biased sample refers to a sample that is not representative of the population it is intended to represent. In a biased sample, certain characteristics or groups within the population are either overrepresented or underrepresented, leading to a distortion or skew in the data.
Bias can occur in various ways during the sampling process. Here are a few examples:
1. Selection Bias: When the method used to select the sample systematically favors or excludes certain individuals or groups from being included. This can lead to an overrepresentation or underrepresentation of specific characteristics in the sample.
2. Nonresponse Bias: When a portion of the selected sample does not participate or respond to the survey or study, resulting in a biased representation of the population.
3. Volunteer Bias: When individuals self-select to participate in a study or survey, which can introduce bias as those who volunteer may have different characteristics or motivations compared to the general population.
4. Measurement Bias: When the measurement instrument or procedure used to collect data systematically produces errors or inaccuracies that favor or exclude certain groups or characteristics.
Biased samples can lead to misleading or inaccurate conclusions about the population of interest since the sample does not accurately reflect the diversity and characteristics of the entire population. It is essential to strive for representative and unbiased samples to make valid inferences and generalizations about the population.
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Solve for x.
-3(x+5)=-9
Enter your answer in the box. X=__
Answer:
x=-2
Step-by-step explanation:
Divide both sides by the numeric factor on the left side, then solve.
1/5 x 11 simplified if can
Answer:
2.2
Step-by-step explanation:
Answer:
11/5
Step-by-step explanation:
A cheese shop has some bulk cheese in blocks measuring 30 cm x 20 cm x 8 cm. How much paper is needed to cover the block of cheese
Answer:
232cm
Step-by-step explanation:
A cheese usually has the shape of a cuboid.
Characteristics of a Cuboid
A cuboid is a convex polyhedronIt has 12 edgesIt has 8 facesIts base shape is a rectangleIt has 6 facesThe amount of paper that would be needed to cover the block of cheese can be determined by calculating the perimeter of the block of cheese which is shaped as a cuboid
Perimeter of a cuboid = 4 x (length + breadth + height)
4 x (30 + 20 + 8) = 232cm
A trapezoid has angles 54 degrees, 54 degrees, x, and x. Look at the trapezoid shown. What is the measure of angle x? The measure of each angle x is 54° 90° 126° 252°
Answer: the answer is C 126°
which fractions are the least common denominator of 84
A. 3/4, 1/7 and 1/6
B. 2/3, 1/6 and 1/42
C. 2/27, 1/7 and 1/3
D. 2/3, 1/6 and 1/7
Answer:
C
Step-by-step explanation:
HURRY PLEASEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE
Answer:
Option 3 Or number 5
Step-by-step explanation:
Answer:
8
Step-by-step explanation:
Eight people are at least 169.5 cm tall because the frequency of people that height is five. The frequency of people taller than 169.5 (because it said 'at least') is three. 5 + 3 = 8
The National Teacher Association survey asked primary school teachers about the size of their classes. Nineteen percent responded that their class size was larger than 30. Suppose 760 teachers are randomly selected, find the probability that more than 22% of them say their class sizes are larger than 30.
The probability for more than 22% of the given data say their class sizes are larger than 30 is equal to 0.0864, or 8.64%.
To find the probability that more than 22% of the randomly selected teachers say their class sizes are larger than 30,
Use the binomial distribution.
Let us denote the probability of a teacher saying their class size is larger than 30 as p.
19% of the teachers responded with a class size larger than 30, we can estimate p as 0.19.
Now, calculate the probability using the binomial distribution.
find the probability of having more than 22% of the 760 teachers .
which is equivalent to more than 0.22 × 760 = 167 teachers saying their class sizes are larger than 30.
P(X > 167) = 1 - P(X ≤167)
Using the binomial distribution formula,
P(X ≤167) = [tex]\sum_{i=0}^{167}[/tex] [C(760, i) × [tex]p^i[/tex] × [tex](1-p)^{(760-i)[/tex]]
where C(n, r) represents the combination 'n choose r' the number of ways to choose r items from a set of n.
Using a statistical calculator, the probability P(X ≤ 167) is determined to be approximately 0.9136.
This implies,
The probability of having more than 22% of the randomly selected teachers say their class sizes are larger than 30 is,
P(X > 167)
= 1 - P(X ≤ 167)
≈ 1 - 0.9136
≈ 0.0864
Therefore, the probability for the given condition is approximately 0.0864, or 8.64%.
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a) 3= x + 7
b) -7a-49
Answer: b im pretty sure hope this helps :)
Step-by-step explanation:
Answer:
A: x = -4
B: a = -7 (I'm assuming the '-' in front of 49 was supposed to be an '=')
Step-by-step explanation:
A: You need to group the like terms together, and isolate the variable, which in this case is x.
Take the 7 over to the other side, so the equation looks a little bit like this.
3-7 = x.
3-7 is -4, so we've gotten our answer.
x = -4.
B: -7a = 49.
When two negative numbers are multiplied together, the product becomes positive.
49 is 7x7.
-7 times -7 would be 49.
So, in this situation, a would be -7.
a = -7.
Nine bearings made by a certain process have a mean diameter of 0.404 cm and a standard deviation of 0.003 cm. What can we say about the maximum error if we use x = 0.404 cm as an estimate of the mean diameter of bearings made by that process:
a) With a confidence of 95%
b) With a confidence of 99%
To determine the maximum error when using [tex]$x = 0.404$[/tex] cm as an estimate of the mean diameter of bearings made by the process, we can calculate the margin of error for different confidence levels.
a) With a confidence of 95%:
For a 95% confidence level, we can use the standard normal distribution and the formula for the margin of error:
[tex]\[\text{{Margin of error (E)}} = z \times \left(\frac{{\sigma}}{{\sqrt{n}}}\right)\][/tex]
where [tex]$z$[/tex] is the critical value corresponding to the desired confidence level, [tex]$\sigma$[/tex] is the standard deviation, and [tex]$n$[/tex] is the sample size.
Since we only have the population standard deviation [tex]($\sigma$)[/tex] and not the sample size [tex]($n$)[/tex] , we cannot calculate the margin of error without additional information. Please provide the sample size [tex]($n$)[/tex]to compute the maximum error with a 95% confidence level.
b) With a confidence of 99%:
Similarly, for a 99% confidence level, we can use the standard normal distribution and the formula for the margin of error:
[tex]\[\text{{Margin of error (E)}} = z \times \left(\frac{{\sigma}}{{\sqrt{n}}}\right)\][/tex]
Using a 99% confidence level, the critical value [tex]($z$)[/tex] is 2.576 (obtained from the standard normal distribution table).
Therefore, the maximum error with a 99% confidence level can be calculated as:
[tex]\[\text{{Margin of error (E)}} = 2.576 \times \left(\frac{{0.003}}{{\sqrt{n}}}\right)\][/tex]
Again, we need the sample size [tex]($n$)[/tex] to compute the maximum error with a 99% confidence level. Please provide the sample size to proceed with the calculation.
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A rubber ball is dropped from a height of 26 feet, and on each bounce it rebounds up 62% of its previous height. Step 2 of 2: Assuming the ball bounces indefinitely, find the total vertical distance traveled. Round your answer to two decimal places.
The total vertical distance traveled by the rubber ball, assuming it bounces indefinitely, is approximately 85.71 feet.
To find the total vertical distance traveled, we need to sum up the heights achieved by the ball during each bounce. The ball initially drops from a height of 26 feet, so we start with this value. On each bounce, the ball rebounds up 62% of its previous height. This means that after the first bounce, the ball reaches a height of 26 feet * 0.62 = 16.12 feet.
For subsequent bounces, we continue to multiply the previous height by 0.62 to find the new height. Therefore, after the second bounce, the height becomes 16.12 feet * 0.62 = 9.99 feet.
We can see that the heights achieved during each bounce form a geometric sequence with a common ratio of 0.62. The sum of an infinite geometric sequence can be calculated using the formula,
Sum = a / (1 - r), first term is a and 'r' is the common ratio is r.
In this case, 'a' is the initial height of 26 feet and 'r' is 0.62. Plugging these values into the formula, we get,
Sum = 26 / (1 - 0.62) = 26 / 0.38 ≈ 68.42 feet.
Therefore, adding all the distances,
Distance = 68.42 + 9.99 + 16.12
Distance = 85.71 feet, total vertical distance traveled by the rubber ball, rounded to two decimal places, is approximately 85.71 feet.
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if 121 ml of a 1.0 m glucose solution is diluted to 550.0 ml , what is the molarity of the diluted solution?
The molarity of the diluted solution is approximately 0.220 M.
The concentration of a solute in a solution is measured by its molarity. The amount of solute that dissolves in one liter (L) of solution is the number of moles. One of the most used units of concentration is t, represented by the symbol M. Number of moles of solute contained in 1 liter of solution is how it is defined.
To calculate the molarity of a solution, you need to use the formula:
M₁V₁ = M₂V₂
Substituting these values into the formula:
(1.0 M)(121 ml) = M₂(550.0 ml)
Rearranging the equation to solve for M₂:
M₂ = (1.0 M)(121 ml) / (550.0 ml)
M₂ = 121 / 550 ≈ 0.220 M
Therefore, the molarity of the diluted solution is approximately 0.220 M.
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There is 20 million m3 of water in a lake at the beginning of a month. Rainfall in this month is a random variable with an average of 1 million mº and a standard deviation of 0.5 million mº. The monthly water flow entering the lake is also a random variable, with an average of 8 million mº and a standard deviation of 2 million mº. Average monthly evaporation is 3 million m3 and standard deviation is 1 million mº. 10 million m’ of water will be drawn from the lake this month. a Calculate the mean and standard deviation of the water volume in the lake at the end of the month. b Assuming that all random variables in the problem are normally distributed, calculate the probability that the end-of-month volume will remain greater than 18 million m3.
The probability that the end-of-month volume will remain greater than 18 million m³ is approximately 0.1922.
a)The mean water volume in the lake at the end of the month can be calculated using the formula given below:
Mean water volume = Starting water volume + Total rainfall + Total flow - Total evaporation - Water drawn from the lake
Given:
Starting water volume = 20 million m³
Total rainfall = random variable with mean = 1 million m³ and standard deviation = 0.5 million m³
Total flow = random variable with mean = 8 million m³ and standard deviation = 2 million m³
Total evaporation = 3 million m³
Water drawn from the lake = 10 million m³
Now, let's calculate the mean water volume at the end of the month.
Mean water volume = Starting water volume + Total rainfall + Total flow - Total evaporation - Water drawn from the lake= 20 + 1 + 8 - 3 - 10= 16 million m³
Therefore, the mean water volume at the end of the month is 16 million m³.
The standard deviation of the water volume in the lake at the end of the month can be calculated using the formula given below:
σ = √{σr² + σf² + σe²}
σr = standard deviation of rainfall = 0.5 million m³
σf = standard deviation of flow = 2 million m³
σe = standard deviation of evaporation = 1 million m³σ = √{σr² + σf² + σe²}σ = √{0.5² + 2² + 1²}= √{5.25}≈ 2.29 million m³
Therefore, the standard deviation of the water volume in the lake at the end of the month is approximately 2.29 million m³.b)Given that all the random variables in the problem are normally distributed, we can find the probability that the end-of-month volume will remain greater than 18 million m³ using the z-score formula.
z = (x - μ) / σ
Where,
z = z-scorex = 18 μ = 16σ = 2.29
Now, let's calculate the z-score.
z = (x - μ) / σ= (18 - 16) / 2.29= 0.87
Using the z-table, we can find that the probability of z being less than 0.87 is 0.8078.
Therefore, the probability of the end-of-month volume being greater than 18 million m³ is:
1 - 0.8078 = 0.1922 (rounded to 4 decimal places)
Hence, the probability that the end-of-month volume will remain greater than 18 million m³ is approximately 0.1922.
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Find the volume of this square pyramid
Answer:
216
Step-by-step explanation:
Answer:
72yd
Step-by-step explanation:
Hope that helps
hsobsnsjns
Write two expressions that are equivalent to 7 x 10 to the -4
[tex]\huge\text{Hey there!}[/tex]
[tex]\mathsf{7\times 10^{-4}}[/tex]
[tex]\rightarrow\mathsf{7\times\dfrac{1}{10\times10\times10\times10}}[/tex]
[tex]\mathsf{= \dfrac{7}{10\times10\times10\times10}}[/tex]
[tex]\mathsf{10\times10\times10\times10=100\times100=10,000=\bf 10^4}[/tex]
[tex]\mathsf{=\dfrac{7}{10^4}}[/tex]
[tex]\mathsf{=\dfrac{7}{10,000}}[/tex]
[tex]\boxed{\boxed{\large\textsf{Possible ANSWERS: }\mathsf{\bf {\dfrac{7}{10^4}\ or \ \dfrac{7}{10,000}}}}}\huge\checkmark[/tex]
[tex]\text{Good luck on your assignment and enjoy your day!}[/tex]
~[tex]\frak{Amphitrite1040:)}[/tex]
Convert 456,300,000 to scientific notation.
Answer:
I believe it would be 4.563 x 10^8
You drop a ball vertically from a height of 1 m. It returns to a height of 0.6 m. What is the coefficient of restitution between the ball and the ground?
The coefficient of restitution between the ball and the ground is 0, indicating a completely inelastic collision.
The coefficient of restitution (e) is a measure of the elasticity or bounciness of a collision between two objects. It is defined as the ratio of the relative velocity of separation after the collision to the relative velocity of approach before the collision.
In this case, the ball is dropped vertically from a height of 1 m and returns to a height of 0.6 m. We can assume that the collision with the ground is approximately elastic, meaning that kinetic energy is conserved.
When the ball hits the ground, its initial velocity is zero, and the final velocity after the collision is also zero since it momentarily comes to rest before bouncing back up. Therefore, the relative velocity of separation is zero.
The relative velocity of the approach is the velocity just before the collision. Since the ball is dropped vertically, its velocity just before hitting the ground is given by the equation:
[tex]v = \sqrt(2gh)[/tex]
where v is the velocity, g is the acceleration due to gravity (approximately[tex]9.8 m/s^2[/tex]), and h is the initial height (1 m).
Plugging in the values:
[tex]v = \sqrt(2 * 9.8 * 1)[/tex]
[tex]= \sqrt(19.6)[/tex]
≈ 4.427 m/s
Therefore, the relative velocity of the approach is approximately 4.427 m/s.
Since the relative velocity of separation is zero, we can calculate the coefficient of restitution (e) as:
e = 0 / 4.427
= 0
Therefore, the coefficient of restitution between the ball and the ground, in this case, is 0, indicating a completely inelastic collision where the ball comes to a stop upon hitting the ground and does not bounce back.
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What is 3.72 of 0.6?
Answer:
Since I do not know the context of the question I will list answers I think it could be based on what you asked:
1. 3.72 x 0.6 = 2.232
2. 3.72 ÷ 0.6 = 6.2
3. 3.72% of 0.6 = 0.02232
The answer is probably the first one. I can't give a definite solution without knowing the exact question being asked, sorry!
Help please will give brainlist!!!
MODELING REAL LIFE The equation y=2x + 3 represents the cost y(in dollars) of mailing a package that weighs x pounds.
a. Use a graph to estimate how much costs to mail the package.
b. Use the equation to find exactly how much it costs to mail the package.
It costs $ to mail the package.
Answer:
I can't read the weight of the package due to the image quality, could you type it out please.
what is the area of a trapezoid
Answer:
28 in² = 28
Step-by-step explanation:
[tex]a = \frac{a + b}{2}h[/tex]
[tex]a = 6[/tex]
[tex]b = 8[/tex]
[tex]h = 4[/tex]
[tex]a = \frac{6 + 8}{2}4[/tex]
[tex] \frac{14}{2} 4[/tex]
[tex]7 \times 4[/tex]
[tex] = 28[/tex]
Answer = 28 _
Solve the logarithmic equation with Properties of Logs
Answer:
Simplifying
logx + -4 = 0
Reorder the terms:
-4 + glox = 0
Solving
-4 + glox = 0
Solving for variable 'g'.
Move all terms containing g to the left, all other terms to the right.
Add '4' to each side of the equation.
-4 + 4 + glox = 0 + 4
Combine like terms: -4 + 4 = 0
0 + glox = 0 + 4
glox = 0 + 4
Combine like terms: 0 + 4 = 4
glox = 4
Divide each side by 'lox'.
g = 4l-1o-1x-1
Simplifying
g = 4l-1o-1x-1
Answer:
[tex]x=40000[/tex]
Step-by-step explanation:
[tex]log(x)-log(4)=4\\log(\frac{x}{4})=4\\\frac{x}{4} = 10^4 \\x=4*10^4=40000[/tex]
The arm span and foot length were both measured (in centimeters) for each of 20 students in a biology class. The computer output displays the regression analysis.
Which of the following is the best interpretation of the coefficient of determination r2?
About 37% of the variation in arm span is accounted for by the linear relationship formed with the foot length.
About 65% of the variation in foot length is accounted for by the linear relationship formed with the arm span.
About 63% of the variation in arm span is accounted for by the linear relationship formed with the foot length.
About 63% of the variation in foot length is accounted for by the linear relationship formed with the arm span.
Answer:
The guy above me is completely wrong hahaha.
The correct answer should have a 63%.
It's probably D. The terminology is a little confusing.
The equation of the output looks like:
-7.611 + .186(x) = y
x --> Arm span
y --> Foot span
The linear relationship is formed with the armspan.
Answer: It’s D
Step-by-step explanation:
I took the test
a) 5x-12=2x-3
b) 8x+2=5x+8
c) 7x-5=4x-2
please
What's the question?
What are we supposed to do?
Evaluate the following integral over the Region R. (Answer accurate to 2 decimal places).
10(x +y))dA
R = (1, y) 16 < x² + y2 < 25, x < 0
∫ ∫R 10(x+y) dA R={(x,y)∣16≤x2+y2≤25,x≤0} Hint: The integral and Region is defined in rectangular coordinates.
The value of the integral is 15.87.
The given integral is:∫∫R 10(x+y) dAwhere R={(x,y)∣16≤x²+y²≤25,x≤0} in rectangular coordinates.In rectangular coordinates, the equation of circle is x²+y² = r², where r is the radius of the circle and the equation of the circle is given as: 16 ≤ x² + y² ≤ 25 ⇒ 4 ≤ r ≤ 5We need to evaluate the integral over the region R using rectangular coordinates and integrate first with respect to x and then with respect to y.∫∫R 10(x + y) dA = 10∫ from 4 to 5 ∫ from -√(25-y²) to -√(16-y²) (x+y) dx dy...[since x < 0]
Now, integrating ∫(x+y) dx we get ∫(x+y) dx = (x²/2 + xy)Therefore, 10∫ from 4 to 5 ∫ from -√(25-y²) to -√(16-y²) (x+y) dx dy = 10∫ from 4 to 5 ∫ from -√(25-y²) to -√(16-y²) [ (x²/2 + xy) ] dy dx= 10∫ from 4 to 5 [∫ from -√(25-y²) to -√(16-y²) (x²/2 + xy) dy] dxNow integrating with respect to y we get∫(x²/2 + xy) dy = (xy/2 + y²/2)
Putting the limits and integrating we get10∫ from 4 to 5 [∫ from -√(25-y²) to -√(16-y²) (x²/2 + xy) dy] dx = 10∫ from 4 to 5 [(∫ from -√(25-y²) to -√(16-y²) (x²/2 + xy) dy)] dx = 10∫ from 4 to 5 [(x²/2)[y]^(-√(16-x²) )_(^(-√(25-x²))] + [(xy/2)[y]^(-√(16-x²) )_(^(-√(25-x²)))] dx = 10∫ from 4 to 5 [-(x²/2)√(16-x²) + (x²/2)√(25-x²) - (x/2)(16-x²)^(3/2) + (x/2)(25-x²)^(3/2) ] dxNow integrating with respect to x, we get10∫ from 4 to 5 [-(x²/2)√(16-x²) + (x²/2)√(25-x²) - (x/2)(16-x²)^(3/2) + (x/2)(25-x²)^(3/2) ] dx = [ (10/3) [(25/3)^(3/2) - (16/3)^(3/2)] - 5√3 - (5/3)[(25/3)^(3/2) - (16/3)^(3/2) ] ]Ans: The value of the integral is 15.87.
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Find m so that x + 4 is a factor of 5x3 + 18x2 + mx + 16
The value of 'm' for which (x + 4) is a factor of the polynomial [tex]5x^3 + 18x^2 + mx + 16[/tex] is -4.
To find the value of 'm' for which the expression (x + 4) is a factor of the polynomial[tex]5x^3 + 18x^2 + mx + 16[/tex], we can apply the factor theorem. According to the factor theorem, if (x + 4) is a factor of the polynomial, then the polynomial evaluated at (-4) should be equal to zero.
Substituting (-4) into the polynomial, we get:
[tex]5(-4)^3 + 18(-4)^2 + m(-4) + 16 = 0[/tex]
-320 + 288 + (-4m) + 16 = 0
-16 + (-4m) = 0
Simplifying the equation, we have:
-4m - 16 = 0
-4m = 16
m = 16 / -4
m = -4
Therefore, the value of 'm' for which (x + 4) is a factor of the polynomial [tex]5x^3 + 18x^2 + mx + 16[/tex] is -4.
By substituting -4 for 'm' in the given polynomial, we obtain:
[tex]5x^3 + 18x^2 - 4x + 16[/tex]
When this polynomial is divided by (x + 4), the remainder will be zero, confirming that (x + 4) is indeed a factor.
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factorise:x^2-y^2-x-y
9514 1404 393
Answer:
(x -y -1)(x +y)
Step-by-step explanation:
The expression can be factored by grouping.
x^2 -y^2 -x -y = (x^2 -y^2) -(x +y)
= (x -y)(x +y) -1(x +y)
= (x -y -1)(x +y)
_____
It is useful to know that a difference of squares is factored as ...
a^2 -b^2 = (a -b)(a +b)
2. (10 Points) Show that g(x) = (3) * has a unique fixed on [0,1].
The function g(x) = (3)ˣ has a unique fixed on [0,1]
Showing that the function has a unique fixed on [0,1].From the question, we have the following parameters that can be used in our computation:
g(x) = (3)ˣ
The above function is an exponential function with the following features
Initial value = 1
Rate = 3
using the above as a guide, we have the following:
x = 0 in [0, 1]
So, we have
g(0) = (3)⁰
Evaluate
g(0) = 1
See that g(0) = 1 i.e. [0. 1]
Hence, the function has a unique fixed on [0,1]
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Mrs. Smith washed 2 5 of her laundry. Her son washed 1 3 of it. Who washed most of the laundry? How much of the laundry still needs to be washed?
Answer:
a) The person who washed the most of the laundry is Mrs Smith
b) 4/15 of the laundry is left to wash
Step-by-step explanation:
Mrs. Smith washed 2/5 of her laundry. Her son washed 1/3 of it.
a) Who washed most of the laundry?
We convert the fraction of laundry each person washed to decimal
Mrs Smith = 2/5 = 0.4
Her son = 1/3 = 0.333
Therefore, the person who washed the most of the laundry is Mrs Smith
b) How much of the laundry still needs to be washed?
Let us total laundry = 1
=1 - ( 2/5 + 1/3)
Lowest Common Denominator is 15
=1- (3 × 2 + 5 × 1/15)
= 1 - (6 +5/15)
=1 - 11/15
= 4/15
PLEASE HELP!! I only have 5 mins
Two particles, Alpha and Beta, race from the y-axis to the vertical line x = 6*pi. For t >= 0, Alpha's position is given by the parametric equations xalpha = 3t - 4sin(t) and yalpha = 3 - 3cos(t) while Beta's position is given by xbeta = 3t - 4sin(t) and ybeta = 3 - 4sin(t). Which sentence best describes the race and its outcome?
(A) Beta starts out in the wrong direction and loses.
(B) Alpha takes a shorter path and wins.
(C) Alpha moves slower and loses.
(D) Beta moves faster but loses.
(E) Alpha and Beta tie
The outcome of the race between Alpha and Beta, as described by their parametric equations, is that Beta moves faster but loses. Although Beta has a higher speed, Alpha consistently maintains a higher vertical position, leading to Alpha winning the race.
To determine the outcome of the race between Alpha and Beta, let's compare their positions using the given parametric equations:
Alpha's position:
[tex]x_{alpha} = 3t - 4sin(t)\\y_{alpha}= 3 - 3cos(t)[/tex]
Beta's position:
[tex]x_{beta} = 3t - 4sin(t)\\y_{beta} = 3 - 4sin(t)[/tex]
From the equations, we can see that the x-coordinate of both Alpha and Beta is the same, given by 3t - 4sin(t). Therefore, their horizontal positions are identical throughout the race.
To determine the vertical positions, we compare their y-coordinates. Alpha's y-coordinate is given by 3 - 3cos(t), while Beta's y-coordinate is given by 3 - 4sin(t).
Since cos(t) ranges from -1 to 1, and sin(t) ranges from -1 to 1, we can observe the following:
For Alpha, the y-coordinate (3 - 3cos(t)) ranges from 0 to 6, inclusive.
For Beta, the y-coordinate (3 - 4sin(t)) ranges from 2 to 4, inclusive.
Based on the range of their y-coordinates, we can conclude that Beta remains at a higher position throughout the race. Therefore, the correct answer is:
(D) Beta moves faster but loses.
Despite Beta moving faster, it loses the race because Alpha consistently maintains a higher vertical position.
Therefore, the outcome of the race between Alpha and Beta, as described by their parametric equations, is that Beta moves faster but loses. Although Beta has a higher speed, Alpha consistently maintains a higher vertical position, leading to Alpha winning the race.
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