The water surface in the conical tank is rising at a rate of approximately 0.16 feet per minute when the water depth is 5 feet.
To solve for this, we can use related rates. Since the tank is being filled with water at a constant rate of 2 ft³/min , we know that the rate of change of the volume of water in the tank is constant. We can use the formula for the volume of a cone to relate the volume of water in the tank to the height of the water in the tank:
V = (1/3) * pi * r² * h
Taking the derivative of both sides with respect to time gives:
dV/dt = (1/3) * pi * r² * dh/dt
We know that dV/dt = 2 ft³/min, r = 2 ft, and h = 5 ft. Solving for dh/dt:
dh/dt = (3 / pi * r²) * dV/dt
dh/dt = (3 / 4 * pi) * 2
dh/dt = 0.16 ft/min
Therefore, the water surface is rising at a rate of approximately 0.16 feet per minute when the water depth is 5 feet.
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if the coconut from the shorter tree takes time t to reach the ground, how long (in terms of t ) will it take the other coconut to reach the ground?
2t
Question:
If the coconut from the shorter tree takes time t to reach the ground, how long (in terms of t) will it take the other coconut to reach the ground?
Solution:
We know that the time taken by the coconut to reach the ground is given by the formula:
t = √(2h/g)
Where,
t = time taken to reach the ground
h = height of the tree
g = acceleration due to gravity
If the height of the coconut from the shorter tree is h, then the time taken by it to reach the ground is t.
Now, the coconut from the taller tree has a height of 2h. Therefore, its time taken to reach the ground can be calculated as follows:
t_1 = √(2(2h)/g)
t_1 = √(4h/g)
t_1 = 2√(h/g)
Therefore, the time taken by the coconut from the taller tree to reach the ground is 2 times the time taken by the coconut from the shorter tree to reach the ground.
Hence, the answer is, the time taken by the other coconut to reach the ground is 2t.
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A primary credit card holder has a current APR of 16.75%. What is the monthly periodic interest rate, rounded to the nearest hundredth of a percent?
To calculate the monthly periodic interest rate from an annual percentage rate (APR), we need to divide the APR by 12 (the number of months in a year). We can use the following formula:
Monthly periodic interest rate = APR / 12
In this case, the APR is 16.75%, so we can plug it into the formula and simplify:
Monthly periodic interest rate = 16.75% / 12
Monthly periodic interest rate = 1.395833...%
Rounding to the nearest hundredth of a percent, we get:
Monthly periodic interest rate = 1.40%
Therefore, the monthly periodic interest rate for the primary credit card holder is 1.40%.
Justify the last two steps of the proof. Given: ABCD is a rectangle. Prove: ΔABC ΔCDA ABDC is a rectangle. ABCD is a parallelogram. AB DC and BC DA AC AC ΔABC ΔCDA Given Definition of a rectangle. Opposite sides of a parallelogram are congruent. _____________________ _____________________
Definition of a rectangle. Opposite sides of a parallelogram are congruent. Reflexive Property of congruent , SSS, option D.
Every component of the set is connected to itself, according to the reflexive feature of sets. The reflexive property of congruence is known when the relation specified on a set is congruence, and the reflexive property of equality is known when the relation defined on a set of numbers is equality. When this occurs, the relation can be referred to as a reflexive relation or as a reflexive property being satisfied on that set.
Any geometric figure compared to itself is congruent to itself so this is why:
AC ≅ CA
∠B ≅ ∠B
Since we have a parallelogram, therefore we can say:
BC ≅ DA
BA ≅ Dc
CA ≅ AC
Both triangles ABC and CDA satisfy the side to side to side congruence, since their 3 sides are congruent.
So, It's D.
Notice that the angle measure information is not included in the data above that's why we cannot say it is SAS congruence.
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Complete question:
Given: ABCD is a rectangle.
Prove: ΔABC is congruent to ΔCDA ABDC is a rectangle.
ABCD is a parallelogram. AB is congruent to DC and BC is congruent to DA AC is congruent to AC ΔABC is congruent to ΔCDA Given Definition of a rectangle. Opposite sides of a parallelogram are congruent. _____________________ _____________________
A. Symmetric Property of congruent; SAS
B. Reflexive Property of congruent to; SAS
C. Symmetric Property of congruent; SSS
D. Reflexive Property of congruent; SSS
RS=???
Not sure in what the answer might be.
Answer:
18
Step-by-step explanation:
What measurement is equal to 4 quarts 1 quarts = 2 pints or 1 ping = 2 cups
Answer:
8 pints=4 quarts,16 cups is 8pints
Step-by-step explanation:
It is multiplication, if you write it down then think about how the numbers connect its easier to understand.
a. Is there a value of x, for -3≤x≤2, such that g(x)= 0
b. Find the absolute minimum value of g and the absolute maximum value of g on the interval -7≤x≤9. Justify your answer.
For x = -2, g(-2) = 2(-2)^3 - 5(-2)^2 + 4(-2) - 1 = 0, so there is a value of x such that g(x) = 0 for -3 ≤ x ≤ 2.
The absolute minimum value of g on the interval -7 ≤ x ≤ 9 is -765, and the absolute maximum value of g on the interval is 1720.
How to Solve the Problem?a. To determine if there is a value of x such that g(x) = 0 for -3 ≤ x ≤ 2, we can plug in each value of x in the interval into the equation and see if we get 0.
g(x) = 2x^3 - 5x^2 + 4x - 1
For x = -3, g(-3) = 2(-3)^3 - 5(-3)^2 + 4(-3) - 1 = -55, which is not 0.
For x = -2, g(-2) = 2(-2)^3 - 5(-2)^2 + 4(-2) - 1 = 0, so there is a value of x such that g(x) = 0 for -3 ≤ x ≤ 2.
b. To find the absolute minimum and maximum values of g on the interval -7 ≤ x ≤ 9, we can use the Extreme Value Theorem, which states that a continuous function on a closed interval will have both an absolute minimum and maximum value on that interval.
To find these values, we can take the derivative of g(x) and set it equal to 0 to find critical points, and then evaluate g(x) at those critical points as well as at the endpoints of the interval.
g(x) = 2x^3 - 5x^2 + 4x - 1
g'(x) = 6x^2 - 10x + 4 = 2(3x-2)(x-1)
Setting g'(x) = 0, we get critical points x = 2/3 and x = 1.
g(-7) = -765, g(2/3) = -23/27, g(1) = 0, and g(9) = 1720.
Therefore, the absolute minimum value of g on the interval -7 ≤ x ≤ 9 is -765, and the absolute maximum value of g on the interval is 1720.
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Layla makes necklaces to sell online. she spends $46 on supplies at the first store and then $30 at the second store. she press to make a necklaces and sell each necklace at a markup of 75%. how much will she charge for each necklace?
a)12.55
b)1663
c)28.45
d)133.04
Answer:
Step-by-step explanation:
First, we need to determine the cost of making one necklace.
The total cost of supplies is $46 + $30 = $76.
If she makes one necklace, that cost is divided by one, so the cost of making one necklace is $76.
To find the selling price, we need to add a markup of 75%.
Markup = 75% x Cost = 0.75 x $76 = $57
The selling price will be the cost plus the markup:
Selling price = Cost + Markup = $76 + $57 = $133.
Therefore, Layla should charge $133 for each necklace.
The answer is d) 133.04, although it should be rounded to the nearest cent, so the final answer would be $133.
what is the probability that the random variable will assume a value between 45 and 55 (to 4 decimals)?
Answer:
Step-by-step explanation:
When answering questions on Brainly, it is important to be factually accurate, professional, and friendly. One should be concise and not provide extraneous amounts of detail, while also ensuring to answer the entire question asked. It is important to use the terms and concepts relevant to the subject and question at hand in order to provide a clear and accurate answer. Furthermore, it is important to refrain from using any offensive or derogatory language or behavior, as Brainly values respect and positivity in its community. Finally, it is important to cite any sources used to support one's answer, and to avoid plagiarism by providing original and unique responses.
Now coming to your question, the given random variable lies between 45 and 55. Hence the required probability is given by:P(45 < X < 55)So, the probability that the random variable will assume a value between 45 and 55 is P(45 < X < 55) to 4 decimals.
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4 Applying torque A machine fastens plastic screw-on
caps onto containers of motor oil. If the machine
applies more torque than the cap can withstand, the
cap will break. Both the torque applied and the strength
of the caps vary. The capping-machine torque T follows
a Normal distribution with mean 7 inch-pounds and
standard deviation 0.9 inch-pound. The cap strength C (the torque that would break the cap) follows a normal distribution with mean 10 inch pounds and standard deviation 1.2 inch pounds
A find the probability that a randomly selected cap had a strength greater than 11 inch pounds
The probability that a randomly selected cap had a strength greater than 11 inch-pounds is approximately 0.2033 or 20.33% (rounded to two decimal places).
We know that cap strength, C, follows a normal distribution with mean μ = 10 inch-pounds and standard deviation σ = 1.2 inch-pounds. Therefore, we can standardize the variable C using the z-score formula:
z = (x - μ) / σ
where x is the value of cap strength we are interested in. In this case, we want to find the probability that a randomly selected cap had a strength greater than 11 inch-pounds, so x = 11. Plugging in the values for μ and σ, we get:
z = (11 - 10) / 1.2 = 0.83
Next, we need to find the probability that a standard normal random variable Z is greater than 0.83. This can be looked up in a standard normal distribution table or calculated using software. Using a standard normal distribution table, we find that the probability of Z being greater than 0.83 is approximately 0.2033.
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Use the traditional method to test the given hypothesis. Assame that the population is normally distributed and that the sample has been randomly selected. Select the appropriate response.In one town, monthly incomes for men with college degrees are found to have a standard deviation of $650. Use a 0.01 significance level to test the claim that for men without college degrees in that town, incomes have a higher standard deviation. A random sample of 22 men without college degrees resulted in incomes with a standard deviation of $926. Select the correct test statistic and critical value.
Test statistic: x^2 = 25.691. Critical values: x^2= 38.932.
Test statistic: x^2= 42.620. Critical values: x^2= 43.978
Test statistic: x^2 = 21.087. Critical values: x^2= 29.806.
Test statistic: x^2= 42.620. Critical values: x^2=38.932.
The test statistic: X² = 42.620, Critical values: X²=38.932.
What are critical values?
A critical value is one that can be used to compare the test statistic to determine if the null hypothesis can be rejected or not.
Here, we have
Given: In one town, monthly incomes for men with college degrees are found to have a standard deviation of $650. A random sample of 22 men without college degrees resulted in incomes with a standard deviation of $926.
s = 926, n = 22
The null and alternative hypotheses:
H₀ : σ = 650; H₁ : σ > 650
Test statistic:
X² = (n-1)s²/σ²
X² = (22-1)926²/650²
x = 42.62
Critical values:
X² = X²a(n-1)
X² = X²(0.01)(21)
X² = 38.932.
Hence, the test statistic: X² = 42.620, Critical values: X²=38.932.
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A soap manufacturer decreases the size of its trial-size bottle of dishwashing soap to 4.5
ounces, which is 10%
less than the original size.
What is the original size of the bottle?
Answer:
Let's assume that the original size of the trial-size bottle of dishwashing soap is "x" ounces.
According to the problem, the new size of the bottle is 10% less than the original size, which means that the new size is 90% of the original size. So we can set up an equation as follows:
0.90x = 4.5
To solve for x, we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by 0.90:
x = 4.5 ÷ 0.90
x = 5
Therefore, the original size of the trial-size bottle of dishwashing soap is 5 ounces.
12.8 x 3/4 also One more is
One-fifth the sum of one-half and one-third this one u have to write in equivalent expression
Answer:
\
Step-by-step explanation:
Given expression: 3x+9
Take 3 outside from the expression, we get,
= 3(x+3), which is called the equivalent expresion
1. Divide and simplify. **Please show all work to receive full credit 1/x+4 ÷ x-3/ x²+7x+12
After dividing and simplifying we get (x-3)/(x+3)
What is division?One of the four fundamental operations of arithmetic, or how numbers are combined to create new numbers, is division. The additional operations are addition, subtraction, and multiplication.
At a fundamental level, counting the instances in which one number is contained within another is one interpretation of the division of two natural numbers. There is no requirement that this quantity be an integer. For instance, if there are 20 apples and they are divided equally among four people, each person will get 5 apples (see picture).
The integer quotient, which is the quantity of times the second number is entirely contained in the first number, and the remainder are both produced by the division with remainder or Euclidean division of two natural numbers.
1/(x+4) divide by [tex](x-3)/(x^2 + 7x +12)[/tex]
Simplify : [tex]x^2 +7x +12[/tex]
[tex]x^2 + (3+4)x + 12[/tex]
[tex]x^2 + 3x +4x + 12[/tex]
[tex]x(x + 3) + 4(x + 3)[/tex]
[tex](x+3)(x+4)[/tex]
[tex]1/(x + 4) x (x-3)/(x+3)(x-4)[/tex]
After dividing and simplifying we get the following answer;
[tex]= (x-3)/(x+3)[/tex]
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A 2-yard piece of rope costs $22. 86. What is the price per foot?
The price per foot of the 2-yard rope is $6.36.
To find the price per foot of the 2-yard rope, we need to first convert the length to feet. Since there are 3 feet in a yard, a 2-yard rope is equal to 6 feet.
Next, we divide the total cost of the rope ($22.86) by the length of the rope in feet (6 feet) to get the price per foot:
$22.86 ÷ 6 = $3.81 per yard
To convert this to the price per foot, we divide $3.81 by 3 (since there are 3 feet in a yard):
$3.81 ÷ 3 = $1.27 per foot
Therefore, the price per foot of the 2-yard rope is $1.27 x 5 = $6.36.
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Y'all help me out here I'm literally the laziest person in the world lol
Answer:
it's b or 4.03
Step-by-step explanation:
your welcome
it is given that in a group of 3 students, the probability of 2 students not having the same birthday is 0.992. what is the probability that the 2 students have the same birthday?
Question: It is given that in a group of 3 students, the probability of 2 students not having the same birthday is 0.992. What is the probability that the 2 students have the same birthday?
The probability that 2 students do not have the same birthday when 3 students are chosen is given as 0.992.Probability of two students not having the same birthday = 0.992
We need to find the probability that 2 students have the same birthday when 3 students are chosen.
Let's represent the probability that two students have the same birthday by p.
Since the number of possible outcomes when 3 students are chosen from 365 days in a year is 365^3.
Now, we can use the complement rule to find p.
Probability of 2 students having the same birthday
p = 1 - Probability of 2 students not having the same birthday p = 1 - 0.992p = 0.008
Thus, the probability that the 2 students have the same birthday is 0.008.
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a postal employee is counting how many houses on the route have dogs. statistically, 48\% of homes have dogs, according to the postal employees research. on the route of 30 homes, the postal employee encounters 12 dogs. what is the variance?
The variance is 7.2.
We can use the binomial distribution formula to calculate the variance. The formula for the variance of a binomial distribution is
Variance = n × p × (1 - p)
where n is the number of trials, p is the probability of success on each trial, and (1 - p) is the probability of failure on each trial.
In this case, n = 30 (the number of homes on the route), p = 0.48 (the probability of a home having a dog), and (1 - p) = 0.52 (the probability of a home not having a dog).
The postal employee encountered 12 dogs, which means there were 18 homes without dogs. So, the actual probability of success in this case is
P(dogs) = 12/30 = 0.4
Using the formula above, we can calculate the variance
Variance = n × p × (1 - p)
= 30 × 0.4 × 0.6
= 7.2
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Bob works at Goodburger and gets a 20% discount. He wants to buy a burger that has a menu price of $4.75. What will his discount be?
Answer:
20÷100×4.75=0.95
4.75-0.95=$3.8
Answer:
i got 4.55$
Step-by-step explanation:
i just converted the percentage (20%) and then subtracted that number (0.2) from the original price (4.75$)
right cone has a base with diameter 16 units.
me volume of the cone is 320 cubic units.
What is the length of a segment drawn from the apex to the edge of the circular base?
Answer:
The length of the segment is 8 units.
We can use the formula for the volume of a cone to calculate the height of the cone:
V = (1/3)πr^2h
320 = (1/3)π(8^2)h
h = 320/(1/3)(π)(64)
h = 8 units
What did I do wrong? (#11)
Answer:
It's not "wrong", per se, it just didn't quite include the info that your teacher wanted! I personally would have said "Rotate 180 degrees about F", so I think that may be it.
Let me know if this helped by hitting the thanks button/marking brainliest! If not, please comment and I'll get back to you ASAP.
2 × 3/4 ab
Please include explanation,
will give Brainliest.
Answer:
Step-by-step explanation:
To solve the expression 2 × 3/4 ab, we can first simplify the fraction by multiplying the numerator (3) by the whole number (2) to get:
2 × 3/4 = 6/4 = 3/2
So the expression becomes:
3/2 ab
This means that we take the value of 3/2 and multiply it by the product of a and b. For example, if a = 4 and b = 5, then:
3/2 ab = 3/2 × 4 × 5 = 30
Therefore, the solution to the expression 2 × 3/4 ab is 3/2 ab or 1.5ab, which means we take the value of 3/2 and multiply it by a and b.
a. Is there a value of x, for -3≤x≤2, such that g(x)= 0
b. Find the absolute minimum value of g and the absolute maximum value of g on the interval -7≤x≤9. Justify your answer.
a) There are three values of x, namely -3, -2, and -1, for which g(x) = 0.
b) The absolute minimum value of g on the interval -7 ≤ x ≤ 9 is 20, and the absolute maximum value of g is 90
How can we solve?a. To check if there is a value of x for -3 ≤ x ≤ 2 such that g(x) = 0, we can plug in each value in the interval and see if any of them make g(x) equal to 0:
g(-3) = -3² + 5(-3) + 6 = 0
g(-2) = -2² + 5(-2) + 6 = 0
g(-1) = -1² + 5(-1) + 6 = 0
g(0) = 0² + 5(0) + 6 = 6
g(1) = 1² + 5(1) + 6 = 12
g(2) = 2² + 5(2) + 6 = 20
So, we can see that there are three values of x, namely -3, -2, and -1, for which g(x) = 0.
b. To find the absolute minimum and maximum values of g on the interval -7 ≤ x ≤ 9, we can first find the critical points of g by taking its derivative:
g'(x) = -2x + 5
Setting g'(x) = 0, we get:
-2x + 5 = 0
-2x = -5
x = 5/2
So, the only critical point of g is x = 5/2.
We can now check the values of g at the endpoints of the interval and the critical point:
g(-7) = -7² + 5(-7) + 6 = 20
g(9) = 9² + 5(9) + 6 = 90
g(5/2) = (5/2)² + 5(5/2) + 6 = 43.25
Therefore, the absolute minimum value of g on the interval -7 ≤ x ≤ 9 is 20, which occurs at x = -7, and the absolute maximum value of g is 90, which occurs at x = 9.
We can justify this by noting that g is a quadratic function with a negative leading coefficient, which means that it opens downward and has a maximum value at its vertex (which occurs at x = 5/2). Since the vertex is within the given interval, and the function is decreasing on the left and increasing on the right of the vertex, the maximum and minimum values of g must occur at the endpoints of the interval.
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The equation y. 210(1.082) depicts the amount of money in a savings account after e years.
What does the y -intercept represent in the equation?
O the growth factor of the account
O the initial amount of the account
O the maximum amount of the account
O the minimum amount of the account
Answer:
In the equation y = 210(1.082)^e, the y-intercept is the value of y when e = 0. When e = 0, the exponent (1.082)^e is equal to 1, so the equation becomes y = 210(1), which simplifies to y = 210.
Therefore, the y-intercept represents the initial amount of the savings account, which is $210. This means that when the account is first opened (e = 0), there is $210 in the account before any interest is earned.
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The price of a 6-minute phone call is 1. 80. What is the price of a 18-minute phone call?
Answer:
5.40
Step-by-step explanation:
let X represent the price of a 18 minute phone call
X=18 ×1.80
6
= 5.40
For what values of x and y is PQRS a parallelogram?
P x+9 Q x+7 R 2x-1 S y
The values for the variables x and y in the parallelogram are equal to 10 and 17 respectively.
How to evaluate for the variables x and y in the parallelogramThe parallelogram OPSR have two pairs of parallel sides hence OP = RS and PS = OR.
We shall evaluate for variables x and y as follows:
2x - 1 = x + 9
2x - x = 9 + 1 {collect like terms}
x = 10
y = x + 7
y = 10 + 7 {substituting 10 for x}
y = 17
Therefore, the values for the variable x and y in the parallelogram are equal to 10 and 17 respectively.
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quantitative research methods focus on the collection of . group of answer choices statistical data social life as people experience it independent variables dependent variables
The gathering of statistical data is the main emphasis of quantitative research methodologies. set of potential solutions data from statistics on how people view social life independent elements dependent factors
What are quantitative research methods?Quantitative research is the approach of collecting and analyzing numerical data in order to identify patterns and trends. Quantitative methods are primarily focused on using mathematical and statistical analyses to measure and evaluate the study's data. In quantitative research, the data collection process is systematic, objective, and statistical.
Because quantitative research involves empirical evidence, it is commonly utilized in the natural and social sciences, as well as in the fields of business, finance, and economics. This approach is used to examine theories, hypotheses, and predictions in order to determine if they are supported by data or if they need to be modified.
What is the aim of quantitative research?
The goal of quantitative research is to test and validate hypotheses and theories through rigorous statistical analyses. It frequently involves the creation of surveys or questionnaires that are then given to a representative sample of the population in order to gather data. The collected data is then statistically analyzed to identify trends, correlations, and other important patterns.
There are several types of quantitative research techniques, including surveys, experiments, and observations. Each technique can be used in different ways and for different purposes, and they each have their own strengths and limitations.
Quantitative research methods focus on the collection of statistical data.
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This histogram shows the number of people who volunteered for community service and the number of hours they worked.
How many volunteers worked no more than 10 hours?
A 30 volunteers
B 35 volunteers
C 55 volunteers
D 60 volunteers
A’(10, 5) is the image of A after a translation along the vector 〈−6, 0〉. What are the coordinates of A?
To perform the opposite translation, we add the opposite of the translation vector to the image point A': the coordinates of point A are (16, 5).
what is a vector?
In mathematics, a vector is an object that represents a quantity having both magnitude (or length) and direction. Vectors can be represented geometrically as arrows, where the length of the arrow represents the magnitude of the vector and the direction of the arrow represents the direction of the vector.
To find the coordinates of point A, we need to perform the opposite translation of moving along the vector 〈−6, 0〉 from the image point A'(10, 5). This is because a translation is a rigid motion that preserves the distance between points, so the distance between A and A' is the same as the distance between their respective translations.
To perform the opposite translation, we add the opposite of the translation vector to the image point A':
A = A' - 〈-6, 0〉 = (10, 5) - (-6, 0) = (16, 5)
Therefore, the coordinates of point A are (16, 5).
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4. materials science a machine that produces ball bearings has initially been set so that the true average diameter of the bearings it produces is 0.500 inches. a bearing is acceptable if its diameter is within 0.004 inches of this target value. suppose, however, that the setting has been changed during the course of production, so that the bearings have normally distributed diameters with mean value 0.499 inches and standard deviation 0.002 inches. what percentage of the bearings produced will not be acceptable?
Approximately 7.30% of the bearings produced will not be acceptable.
We know that the mean value of the diameters of the ball bearings produced after the change in setting is 0.499 inches and the standard deviation is 0.002 inches. We need to find the percentage of bearings produced that will not be acceptable, i.e., the diameter of the bearing will be outside the range of (0.500 - 0.004) inches to (0.500 + 0.004) inches.
Let X be the diameter of a ball bearing produced after the change in setting. Then, X is normally distributed with mean µ = 0.499 inches and standard deviation σ = 0.002 inches.
We need to find P(X < 0.496 or X > 0.504), which is the probability that the diameter of a ball bearing will be outside the acceptable range.
Using the standard normal distribution, we can find the z-scores corresponding to the lower and upper limits of the acceptable range:
z1 = (0.496 - 0.499) / 0.002 = -1.5
z2 = (0.504 - 0.499) / 0.002 = 2.5
Using a standard normal distribution table or calculator, we can find the probabilities corresponding to these z-scores:
P(Z < -1.5) = 0.0668
P(Z > 2.5) = 0.0062
Therefore, the probability that a ball bearing produced after the change in setting will not be acceptable is:
P(X < 0.496 or X > 0.504) = P(Z < -1.5 or Z > 2.5) = P(Z < -1.5) + P(Z > 2.5) = 0.0668 + 0.0062 = 0.0730
So, the percentage of the bearings produced that will not be acceptable is:
0.0730 x 100% = 7.30%
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Which inequality does this graph show?
We found for the line passing through (-1, 0) and (0, -1.5), we can see that the correct inequality is option A is -2y + 2x ≥ 7x + y + 5.
Describe Inequality?An inequality is a mathematical statement that compares two quantities or expressions using inequality symbols such as "<" (less than), ">" (greater than), "<=" (less than or equal to), ">=" (greater than or equal to), or "!=" (not equal to).
Inequalities can involve variables or constants, and can be expressed in one variable or multiple variables. The solution to an inequality is the set of values that satisfy the inequality.
For example, the inequality 2x + 3 > 7 is true for values of x that are greater than 2, since if we substitute x = 2, we get 2(2) + 3 = 7, which is not greater than 7. On the other hand, if we substitute x = 3, we get 2(3) + 3 = 9, which is greater than 7, so the inequality is true for x > 2.
Inequalities have many applications in mathematics and other fields, such as economics, physics, and engineering. They are used to represent constraints in optimization problems, to model relationships between variables, and to describe ranges of possible values for a quantity or variable.
To determine the correct inequality, we need to find the equation of the line that passes through the points (-1, 0) and (0, -1.5).
The slope of the line can be calculated as follows:
m = (y2 - y1) / (x2 - x1)
m = (-1.5 - 0) / (0 - (-1))
m = -1.5
The y-intercept of the line can be found by substituting one of the points and the slope into the point-slope form of a line:
y - y1 = m(x - x1)
y - 0 = -1.5(x - (-1))
y + 1.5 = -1.5x - 1.5
y = -1.5x - 3
Now we can rewrite the inequality in the form y <= mx + b, where m is the slope and b is the y-intercept.
-2y + 2x >= 7x + y + 5 can be simplified to:
-3y >= 5x + 5
Dividing both sides by -3, we get:
y <= (-5/3)x - 5/3
Comparing this to the equation we found for the line passing through (-1, 0) and (0, -1.5), we can see that the correct inequality is option A:
-2y + 2x ≥ 7x + y + 5
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