the probability that a top-selling red and voss tire wears out before 80,000 miles is about 2.56%.
The student question asks about the probability that a top-selling red and voss tire, rated for 80,000 miles, wears out before reaching 80,000 miles. The tire's lifespan follows a normal distribution with a mean of 94,000 miles and a standard deviation of 7200 miles.
To find the probability, we need to calculate the z-score first. The z-score is a measure of how many standard deviations away from the mean a particular value is. We can use the following formula to calculate the z-score:
z = (X - μ) / σ
where X is the value (in this case, 80,000 miles), μ is the mean (94,000 miles), and σ is the standard deviation (7200 miles).
Calculate the z-score:
z = (80,000 - 94,000) / 7200
z = -14,000 / 7200
z ≈ -1.944
The z-score is approximately -1.944, which means the tire wearing out at 80,000 miles is about 1.944 standard deviations below the mean.
Find the probability:
Now, we can use the z-score to find the probability. We can look up the z-score in a standard normal distribution table or use a calculator with a built-in function for this purpose.
Using a standard normal distribution table or calculator, we find that the probability corresponding to a z-score of -1.944 is approximately 0.0256 or 2.56%.
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Mackenzie measured a line to be 5.3 inches long. If the actual length of the line is 5.4 inches, then what was the percent error of the measurement, to the nearest tenth of a percent?
The percent error of the measurement is 1.9%.
What is the percent error?
The difference between an exact value and an approximation to it is the approximation error in a data value. Either an absolute error or a relative error might be used to describe this error. Measurement mistakes or computation machine precision can generate an approximation error.
Here, we have
Given: Mackenzie measured a line to be 5.3 inches long. If the actual length of the line is 5.4 inches.
We have to find the percent error of the measurement.
If the actual length of the line is 5.4 inches
So, the difference in length of the line = 5.4-5.3
= 0.1
Now, the percent error = Difference in length of line/Actual length × 100
= 0.1/5.3 × 100
= 0.01887 × 100
= 1.9 %
Hence, the percent error of the measurement is 1.9%.
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The school auditorium seats 350
students. There are 309
students already seated. Write
and solve an inequality that
represents the additional number
of students that can be seated.
As a result, more students than or equal to 41 can be added to the current student body by inequality.
what is inequality defined as?A mathematical statement known as an inequality compares two expressions using one of the following symbols:, >,, or.
For instance:
x + 2 < 5
2y - 3 > 7
3z ≤ 9
4w + 1 ≥ 13
Now,
The disparity that indicates the extra students who can be seated is as follows:
350 - 309 ≤ x
where x is the maximum number of extra pupils who can sit in a classroom.
When we simplify this inequality, we obtain:
41 ≤ x
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when a study sample shows representativeness, what can be concluded about the study? group of answer choices the study results show evidence of sampling bias the study used strict inclusion and exclusion criteria to limit the sample size the study is most likely not generalizable because the sample was obtained using specific inclusion criteria. the study is most likely generalizable, with results that can be applied to the target population
When a study sample is representative of the target population, it allows for the results to be generalizable, meaning they can be applied to the broader population.
This is an important aspect of research, as it ensures the study's findings are relevant and useful beyond the specific sample studied.
When a study sample shows representativeness, it means that the sample accurately reflects the characteristics of the target population.
In this case, the study is most likely generalizable, with results that can be applied to the target population.
To achieve representativeness, researchers often use random sampling techniques, which help minimize the potential for sampling bias.
Sampling bias occurs when certain groups within the target population are over- or under-represented in the sample, which can lead to inaccurate conclusions.
In contrast, if a study used strict inclusion and exclusion criteria to limit the sample size or relied on specific inclusion criteria, it could result in a sample that is not representative of the target population.
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suppose that 11% of all steel shafts produced by a certain process are nonconforming but can be reworked (rather than having to be scrapped). consider a random sample of 200 shafts, and let x denote the number among these that are nonconforming and can be reworked. (round your answers to four decimal places.) (a) what is the (approximate) probability that x is at most 30? 0.9649 0.9726 (b) what is the (approximate) probability that x is less than 30? 0.9429 0.9550 (c) what is the (approximate) probability that x is between 15 and 25 (inclusive)?
The approximate probability that x is between 15 and 25 (inclusive) is 0.3026.
To solve this problem, we will use the binomial distribution, since we are dealing with a binary outcome (nonconforming vs. conforming) and a fixed sample size. Let p = 0.11 be the probability of a nonconforming shaft and q = 1 - p = 0.89 be the probability of a conforming shaft. Then, the probability mass function of x is given by:
P(X = x) = (200 choose x) * p^x * q^(200-x)
(a) To find the probability that x is at most 30, we can compute:
P(X ≤ 30) = Σ P(X = x) for x from 0 to 30
However, this is quite tiresome to tally by hand, so we can instead use a normal approximation to the binomial distribution. Specifically, if np and nq are both at least 10, then we can approximate the binomial distribution with a normal distribution with mean μ = np and standard deviation σ = sqrt(npq). In this case, we have np = 22 and nq = 178, both of which are at least 10.
Thus, we can approximate X with a normal distribution:
[tex]X ~ N(mu, σ^2) = N(22, 3.8004)[/tex]
Then, we can compute the probability that X is at most 30 by standardizing and using the standard normal distribution:
[tex]P(X ≤ 30) ≈ P(Z ≤ (30 - mu) / σ) = P(Z ≤ (30 - 22) / 1.9488) = P(Z ≤ 4.1142) = 0.9997[/tex]
(b) To find the probability that x is less than 30, we can use the same normal approximation and compute:
[tex]P(X < 30) ≈ P(Z < (30 - mu) / σ) = P(Z < (30 - 22) / 1.9488) = P(Z < 4.1142) = 0.9997[/tex]
(c) To find the probability that x is between 15 and 25 (inclusive), we can either use the binomial distribution directly or use the normal approximation as before:
P(15 ≤ X ≤ 25) = Σ P(X = x) for x from 15 to 25
However, we can use the normal approximation instead. Using the same approach as before, we get:
[tex]P(15 ≤ X ≤ 25) ≈ P((15 - μ) / σ ≤ Z ≤ (25 - μ) / σ) = P(-3.0806 ≤ Z ≤ -0.5145) = P(Z ≤ -0.5145) - P(Z ≤ -3.0806) = 0.3035 - 0.0009 = 0.3026[/tex]
Therefore, the approximate probability that x is between 15 and 25 (inclusive) is 0.3026.
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The box plot represents the number of tickets sold for a school dance.
A horizontal line labeled Number of Tickets sold that starts at 8, with tick marks every one unit up to 30. The graph is titled Tickets Sold for A Dance. The box extends from 17 to 21 on the number line. A line in the box is at 19. The lines outside the box end at 10 and 27.
Which of the following is the appropriate measure of variability for the data, and what is its value?
The IQR is the best measure of variability, and it equals 17.
The range is the best measure of variability, and it equals 4.
The IQR is the best measure of variability, and it equals 4.
The range is the best measure of variability, and it equals 17
The IQR is the best measure of variability for this data set, and its value is 4.
What is median?
Median is a measure of central tendency that represents the middle value in a dataset when the values are arranged in order of magnitude.
The correct answer is:
The IQR is the best measure of variability, and it equals 4.
The box plot shows the median (represented by the line in the box), the interquartile range (IQR) (represented by the length of the box), and the range (represented by the lines outside the box).
The IQR is the range between the first quartile (Q1) and the third quartile (Q3) of the data set, which contains the middle 50% of the data. In this case, the box extends from 17 to 21, indicating that Q1 is 17 and Q3 is 21. Therefore, the IQR is the difference between Q3 and Q1, which is 21-17=4.
The range is the difference between the maximum and minimum values in the data set, which is 27-10=17. However, the range can be affected by outliers and is not as robust as the IQR.
Therefore, the IQR is the best measure of variability for this data set, and its value is 4.
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Simon and Luke shared a pizza. Simon ate of the pizza. Luke ate of the pizza. Which model is shaded to show the fraction of the pizza that both boys ate?
A 1/4
B 2/8
C 7/12
D 7/16
The model shaded to show the fraction of the pizza that both boys ate is option C, 7/12.
To find the fraction of the pizza that both boys ate, we need to find the intersection of the two fractions that represent the amount each boy ate.
We can represent each boy's portion of the pizza using the following models:
Simon = 3/12
Luke = 4/12
When these fractions are added, we have
Total = 3/12 + 4/12
Evaluate the sum
Total = 7/12
Hence, the model is 7/12
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Very confused on this question, unsure of the answer
Answer:
a = 9 , b = 5 , c = 25
Step-by-step explanation:
using the Cosine rule in the triangle
a² = b² + c² - 2bc cosA
where a is the side opposite ∠ A and b, c the sides adjacent to ∠ A
here
a = y , b = x + 5 , c = 3x and A = Θ , then
y² = (x + 5)² + (3x)² - [ 2(x + 5) × 3x × cosΘ ]
y² = x² + 10x + 25 + 9x² - [ 6x(x + 5) × [tex]\frac{1}{6}[/tex] ]
y² = 10x² + 10x + 25 - x(x + 5)
y² = 10x² + 10x + 25 - x² - 5x
y² = 9x² + 5x + 25 ← in the form y² = ax² + bx + c
with a = 9 , b = 5 , c = 25
suppose that 5% of teachers at a university attended a conference. if 4000 teachers are enrolled at the university, about how many teachers attended the conference?
Answer: 200 teachers attended the conference
Step-by-step explanation:
To answer this question, we will find 5% of 4,000. A percent divided by 100 becomes a decimal, and "of" means multiplication in mathematics.
5% / 100 = 0.05
4,000 * 0.05 = 200 teachers attended the conference
The answer is 200
If 5% of teachers at a university attended a conference and 4000 teachers are enrolled at the university, about 200 teachers attended the conference. What is the problem asking for? The problem is asking us to calculate how many teachers attended a conference given that 5% of the total number of teachers at a university attended the conference and that the university has 4000 teachers .What is the formula for percentage? The formula for percentage is given by:(Part/Whole)*100Let T be the total number of teachers at the university, and let x be the number of teachers that attended the conference. We know that 5% of teachers at a university attended a conference, therefore:(5/100)*T = x,. Given that there are 4000 teachers in the university, we can substitute this value into the equation: (5/100)*4000 = x. Simplifying the equation, we get: x = 0.05 * 4000x = 200.
Therefore, about 200 teachers attended the conference.
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44% of c is 72. find the value of c
Answer: 163.6363...
Step-by-step explanation:
44% of c is 72, so then [tex]0.44*c=72\\[/tex].
Divide both sides by 0.44 to solve for c, and you get c=163.6363...
Tip: if you are ever trying to figure out what a number is a percentage of, like in this problem, just divide the number by the percent in decimal form.
))
Which operation should be completed first to find the value of the expression
below?
6-3
12+ 30 (6-3) + 1
12+ 30
306
3+1
Don
Answer:
(6-3)
Step-by-step explanation:
In how many years will the population of a colony be 92,610 from 80,000 at the population growth rate of 5% per annum? If the growth rate is 2% less than before, what would be the difference in population for the same time? Find it.
Answer:
To find the number of years it will take for the population of the colony to grow from 80,000 to 92,610 with a growth rate of 5% per annum, we can use the formula for compound interest:
Final Population = Initial Population * (1 + Growth Rate) ^ Number of Years
92,610 = 80,000 * (1 + 0.05) ^ t
Now, we can solve for t:
(92,610 / 80,000) = (1.05) ^ t
1.157625 = (1.05) ^ t
To find t, we can use logarithms:
t = log(1.157625) / log(1.05)
t ≈ 2.967
So, it will take approximately 2.967 years for the population to grow from 80,000 to 92,610 at a 5% growth rate.
Now, let's consider a 2% lower growth rate (5% - 2% = 3%). We can use the same formula to find the final population after the same time (2.967 years):
Final Population = Initial Population * (1 + Growth Rate) ^ Number of Years
Final Population = 80,000 * (1 + 0.03) ^ 2.967
Final Population ≈ 80,000 * 1.09364
Final Population ≈ 87,491.2
To find the difference in population for the same time, we can subtract the population with the lower growth rate from the population with the higher growth rate:
Difference in population = 92,610 - 87,491.2
Difference in population ≈ 5,118.8
So, the difference in population for the same time with a 2% lower growth rate would be approximately 5,118.8.
Solve for z in the proportion.
60
90
Z =
22
Z + 17
Submit
The equation is not a proportion, and the value of z in the equation is 17
Calculating the value of z in the equationGiven that
2Z = Z + 17
To solve for z in the proportion:
2z = z + 17
We can start by simplifying the equation by subtracting z from both sides:
2z - z = 17
Simplifying further, we get:
z = 17
Therefore, the solution to the proportion is z = 17.
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please help with this question?!!
Answer: SHawn
Step-by-step explanation:
PLEASE HELP!! 50 POINTS!! if you just take the points without actually helping you will be forever cursed within my mind and never forgiven.
Christopher borrows 7,500$ to build a garage. He agrees to pay 475$ a month for 24 months but pays off the loan after 18 months.
Part A: Determine the amount of unearned interest.
Part B: Determine the amount needed to repay the loan using the Rule of 78.
Part C: Show your work to support your answers to Part A and Part B.
To solve the problem, we can use the formula for the future value of an annuity:
FV = PMT * ( (1 + r)^n - 1 ) / r
where:
- PMT is the periodic payment
- r is the periodic interest rate
- n is the number of periods
In this case,
- PMT = $475 (the monthly payment)
- r = the monthly interest rate
- n = 24 (the number of months in the agreement)
We first need to calculate the monthly interest rate. We can find this by dividing the yearly interest rate by 12 months:
r = 18% / 12 months = 0.015
Now we can calculate the future value (FV) of the annuity using the above formula:
FV = $475 * ( (1 + 0.015)^24 - 1 ) / 0.015 = $13,237.19
This is the amount that Christopher would owe after 24 months if he made monthly payments of $475.
However, Christopher pays off the loan after 18 months. To find out how much he owes at this point, we can calculate the future value after 18 months:
FV = $475 * ( (1 + 0.015)^18 - 1 ) / 0.015 = $10,198.66
So, after 18 months, Christopher owes approximately $10,198.66.
help with this problem
Answer:
PQ = 165.60m
Step-by-step explanation:
Since triangles are similar
PQ/70 + 210 = 124.2/210
PQ = 124.2 (70 + 210)/210 = 165.60m
dustin has 3 cups of buttermilk.does he have enough to make four batches of muffins.explain(he needs 3/4 cup butter milk to make 1 batch)
Answer:
Step-by-step explanation:
The math used shows how much cups it would take to make 4 batches, 3 cups. Therefor yes, he does have exactly enough to make 4 batches of muffins.
Can 2x divided by 3y be simplified
Answer:
No, you cannot simplify it.
Step-by-step explanation:
Because 2 and 3 are one of the lowest numbers that cannot be simplified.
The standard form of a circle is (x-5)^2+(y-5)^2=16. Convert the standard form into general form.
Therefore, the general form of the circle is:
[tex]x^2 + y^2 - 10x-10y+34=0[/tex].
What is circle?A circle is a geometric shape that is defined as the set of all points in a plane that are at a fixed distance (called the radius) from a given point (called the center). The distance from the center to any point on the circle is always the same. Circles are often represented using the symbol "O" or "⚪" and are important in mathematics, geometry, and physics. They have many properties, such as circumference (the distance around the circle), area (the space inside the circle), and diameter (the distance across the circle through the center). Circles are used in various fields, including architecture, engineering, and art.
To convert the standard form of a circle to the general form, we expand and simplify the equation as follows:
[tex](x - 5)^2 + (y - 5)^2 = 16[/tex]
[tex]x^2 - 10x + 25 + y^2 - 10y + 25 = 16[/tex] (using the identity [tex](a - b)^2 = a^2 - 2ab + b^2)[/tex]
[tex]x^2 + y^2 - 10x-10y+34=0[/tex]
Therefore, the general form of the circle is:
[tex]x^2 + y^2 - 10x-10y+34=0[/tex]
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Are these right? If not, can you tell the right answers so I can change them please?
Answer:
O=0
T=-5
c = 0
R = -2
6 = 8
2 = 8
11=-3
Step-by-step explanation: Sorry for the late answer It took me a bit to solve all the problems but I hope this helps you!!
6x^2-12x-18 factor the polynomial
Answer:
6x² - 12x - 18
the factors are -18 and 6
6x² - 18x + 6x - 18
6x(x - 3) + 6 (x - 3)
(6x + 6) or (x - 3)
the factors are:
(6x + 6) or (x - 3)
Naomi's bedroom is 6 meters long and 4 meters wide. Starting at the far left corner, Naomi walks down the length, across the width, and then diagonally back to the far left corner. How far does Naomi walk? If necessary, round to the nearest tenth.
Naomi walks a total of 6 + 4 + 7.2 ≈ 17.2 meters.
How to calculate?
Naomi walks the length of the room, which is 6 meters, then across the width, which is 4 meters.
Using the Pythagorean theorem, the length of the diagonal can be found:
diagonal = √(6²2 + 4²2)
diagonal = √(36 + 16)
diagonal = √52
diagonal ≈ 7.2 meters
Therefore, Naomi walks a total of 6 + 4 + 7.2 ≈ 17.2 meters.
The Pythagorean theorem is a fundamental concept in mathematics that states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
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Roxy has a box of pens. He has 11 black pens, 8 blue pens, and 6 red pens. He randomly selects a pen from the box. Describe the probability that the pen he selects is blue, using a number or a number range.
Step-by-step explanation:
The total number of pens in the box is:
Total number of pens = 11 black pens + 8 blue pens + 6 red pens
Total number of pens = 25 pens
The probability of Roxy selecting a blue pen can be expressed as the ratio of the number of blue pens to the total number of pens:
Probability of selecting a blue pen = Number of blue pens / Total number of pens
Probability of selecting a blue pen = 8 / 25
So, the probability that the pen Roxy selects is blue is 8/25 or approximately 0.32 (rounded to two decimal places).
Answer:8/25
Step-by-step explanation:
HELPPPPP DUE SOON!!!!
The distance between the points H (-9, 1), and K (-1, 8) in the coordinate plane, obtained using the distance formula is; Distance: 10.63
What is the distance formula?The distance formula is a formula that is used to find the distance between two points on the coordinate plane.
The specified points are; H = (-9, 1), and K = (-1, 8)
The distance formula can be used to calculate the distance between the points as follows;
The distance formula is; d = √((x₂ - x₁)² + (y₂ - y₁)²)
Where; (x₁, y₁) and (x₂, y₂) are the coordinates of the two points.
When, (x₁, y₁) = (-9, 1), and (x₂, y₂) = (-1, 8), we get;
d = √((-1 - (-9))² + (8 - 1)²) = √(8² + 7²) =
√(8² + 7²) = √(64 + 49) = √(113) ≈ 10.63 (rounded to the nearest hundredth)
The distance between the points is about 10.63 units
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A figure is drawn on the coordinate plane. Its points are located at D (-4, 4), E (4, 4), F (4, -2) and G (-4, -2). What is the difference between the perimeter and area of the figure.
The difference between the perimeter and area of the figure is -20 square units.
What is perimeter?In geometry, the perimeter of a shape is defined as the total length of its boundary. The perimeter of a shape is determined by adding the length of all the sides and edges enclosing the shape.
The figure is a rectangle with sides DE and FG measuring 8 units, and sides EF and GD measuring 6 units.
To find the perimeter of the figure, we can add up the lengths of all four sides:
Perimeter = DE + EF + FG + GD
Perimeter = 8 + 6 + 8 + 6
Perimeter = 28
To find the area of the figure, we can use the formula for the area of a rectangle:
Area = length * width
Area = DE * EF
Area = 8 * 6
Area = 48
Therefore, the difference between the perimeter and area of the figure is:
Perimeter - Area = 28 - 48
Perimeter - Area = -20
The difference between the perimeter and area of the figure is -20 square units.
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Draw the image of triangle△ABC under a dilation whose center is C and scale factor is 2
The original triangle ABC is down below.
PLEASE HELP
The resulting triangle would have twice the area and all sides would be twice as long as the original triangle, but the angles would remain the same.
What is triangle?A triangle is a geometric shape that consists of three straight line segments or sides that are connected at three points called vertices. Triangles are two-dimensional and are the simplest polygon that can exist in Euclidean space. Triangles can be classified according to the size and shape of their sides and angles. Triangles are used in many different areas of mathematics, science, and engineering. They are also common in everyday life, for example in construction, architecture, and design.
Here,
To dilate a triangle with a center point C and a scale factor of 2, you would first draw a ray from C through each vertex of the triangle. Then, you would measure the distance from C to each vertex of the original triangle and multiply it by the scale factor (2 in this case) to find the distance from C to each corresponding vertex of the dilated triangle. Finally, you would draw lines connecting the dilated vertices to form the dilated triangle.
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EXPANDING BRACKETS -
2 (3x + 7)
Answer:
[tex] \sf \: 6x + 14 [/tex]
Step-by-step explanation:
Now we have to,
→ Simplify the given expression.
The property we use,
→ Distributive property.
The expression is,
→ 2(3x + 7)
Let's simplify the expression,
→ 2(3x + 7)
→ 2(3x) + 2(7)
→ (2 × 3)x + 14
→ 6x + 14
Hence, the answer is 6x + 14.
Three numbers whose sum is 26 are in GP.If 5,9 and 5 are added to them respectively, then the three numbers are in AP. Find the numbers.
The three numbers are:
(-1 + √(97))/2, (-1 ± √(97))/2, (-1 ± 7√(97))/2
What is an arithmatic progression?
An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the preceding term. The fixed constant is called the common difference, denoted by d.
Let the three numbers in the GP be a/r, a, and ar, where a is the middle term.
Then we have:
a/r + a + ar = 26 (sum of GP)
(a/r + 5) + (a + 9) + (ar + 5) = 3(a + 4) (sum of AP)
Simplifying the second equation, we get:
a(r - 1) = 3
Substituting this into the first equation, we get:
a/r + a + ar = 26
1/r + 1 + r = 26/a
r^2 + r - 26/a = 0
Solving for r using the quadratic formula, we get:
r = (-1 ± √(1 + 104/a))/2
Since r > 0 (because the terms are in GP), we take the positive root:
r = (-1 + √(1 + 104/a))/2
Substituting this into the equation a(r - 1) = 3, we get:
a(-1 + √(1 + 104/a))/2 - a = 3
-a + a √(1 + 104/a) - 2 = 0
a √(1 + 104/a) = -a + 2
a² (1 + 104/a) = a² - 4a + 4
104a = 4a² - 4a + 4
a² - a + 1 = 26
(a - 1/2)² + 3/4 = 26
(a - 1/2)² = 97/4
a - 1/2 = ±√(97)/2
a = (1 ± √(97))/2
Since the terms are in GP, we can use the value of a to find the other two terms:
a/r = (1 ± √(97))/(2(-1 + √(1 + 104/a))/2) = (-1 ± √(97))/2
ar = (1 ± √(97))/(2(-1 - √(1 + 104/a))/2) = (-1 ± 7√(97))/2
Therefore, the three numbers are:
(-1 + √(97))/2, (-1 ± √(97))/2, (-1 ± 7√(97))/2
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The scatter plot to the right shows the cost (y), in dollars, of orange trees based on their ages (x), in years. Based on the scatter plot, which equation represents the line of best fit for the cost of the orange trees?
A y=15.7x
B y=11.8x+29.2
C y=15.7+40.0
D y=11.8x
Answer:
The answer to your problem is, B. y=11.8x+29.2
Step-by-step explanation:
Determine the line of the best fit grpahically
Find the problem:
y = 11 x 8x + 29 x 2 B.
a classmate walks into class and states that he has an extra ticket to a festival on friday night. he asks everyone in the class to put their name on a piece of paper and put it in a basket. he plans to draw from the basket to choose the person who will attend the festival with him. if there are 22 other people in class that night, what is your chance of being chosen to attend the festival? round your answer to four decimal places, if necessary.
Your chance of being chosen to attend the festival is approximately: Probability = 1 / 23 ≈ 0.0435 or 4.35% when rounded to four decimal places.
To determine your chance of being chosen to attend the festival, you need to find the probability of your name being
drawn from the basket.
There are 22 other people in class, and each person, including you, will have their name on a piece of paper.
Total number of names in the basket = 22 (other people) + 1 (you) = 23
Since each person has an equal chance of being chosen, the probability of your name being drawn is:
Probability = (Number of favorable outcomes) / (Total number of outcomes)
Probability = 1 (your name) / 23 (total names)
So, your chance of being chosen to attend the festival is approximately:
Probability = 1 / 23 ≈ 0.0435 or 4.35% when rounded to four decimal places.
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the edges of a cube increase at a rate of 2 centimeters per second. how fast is the volume changing when the length of each cube edge is 50 centimeters ?
The required volume of the cube is increasing at a rate of 15,000 cubic centimeters per second when the length of each edge is 50 centimeters and the edges are increasing at a rate of 2 centimeters per second.
Let us consider V be the volume of the cube,
And s be the length of one edge of the cube.
Volume of a cube is equal to,
V = s^3
To find how fast the volume is changing with respect to time,
Use the chain rule of differentiation,
dV/dt = dV/ds × ds/dt
where dV/dt is the rate of change of volume with respect to time,
dV/ds is the rate of change of volume with respect to the length of one edge of the cube,
And ds/dt is the rate of change of the length of one edge of the cube with respect to time.
ds/dt = 2 cm/s,
find dV/dt when s = 50 cm.
First, find dV/ds,
dV/ds = 3s^2
Then, we can plug in s = 50 cm,
dV/ds = 3(50)^2
= 7500 cm^2
Finally, we can plug in the values we have to find dV/dt,
dV/dt = (dV/ds) × (ds/dt)
= 7500 cm^2/s × 2 cm/s
= 15,000 cm^3/s
Therefore, the volume of the cube is increasing at a rate of 15,000 cubic centimeters per second .
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