It would take around 8 hours to bring each lake's pollution level down to 50% of its starting point, and around 22 hours to bring it down to 5%.
Assuming each lake is a separate tank with only clean water flowing in, we can use the exponential decay model [tex]$A = A_0e^{-kt}$[/tex], where $A$ is the amount of pollutant at time t, A₀ is the initial amount of pollutant, and k is the decay constant.
To find the time it would take to reduce the pollution level in each lake to 50% of its original level, we need to solve the equation [tex]$0.5A_0 = A_0e^{-kt}$[/tex] for t:
[tex]0.5A_0 &= A_0e^{-kt} \\frac{0.5A_0}{A_0} &= e^{-kt} \\ln\left(\frac{0.5A_0}{A_0}\right) &= -kt \\ln(0.5) &= -kt \t &= \frac{\ln(0.5)}{-k}\end{align*}[/tex]
To find the time it would take to reduce the pollution level in each lake to 5% of its original level, we need to solve the equation[tex]$0.05A_0 = A_0e^{-kt}$[/tex] for t:
[tex]0.05A_0 &= A_0e^{-kt} \\frac{0.05A_0}{A_0} &= e^{-kt} \\ln\left(\frac{0.05A_0}{A_0}\right) &= -kt \\ln(0.05) &= -kt \t &= \frac{\ln(0.05)}{-k}\end{align*}[/tex]
The decay constant $k$ can be found by using the given information that each lake is replaced by clean water every 8 hours. This means that the half-life of the pollutant is 8 hours, which gives us:
[tex]0.5A_0 &= A_0e^{-k(8)} \\ln(0.5) &= -8k \k &= -\frac{\ln(0.5)}{8} \approx 0.08664\end{align*}[/tex]
Substituting this value of k into the equations we derived earlier, we get:
[tex]t_{50} = \frac{\ln(0.5)}{-k} \approx 8.006 \text{ hours}[/tex]
[tex]t_{5} &= \frac{\ln(0.05)}{-k} \approx 22.133 \text{ hours}[/tex]
Therefore, it would take approximately 8 hours to reduce the pollution level in each lake to 50% of its original level, and approximately 22 hours to reduce it to 5% of its original level.
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4. Let v be the measure on (R, B(R)) which has the density g(x) = e", XER, with respect to the Lebesgue measure 1. Find Cou 2 dv(x). [5 Marks]
The integral ∫g(x) dv(x) does not converge to a finite value.
To find the integral ∫g(x) dv(x) where g(x) = e^x and v is the measure on (R, B(R)) with respect to the Lebesgue measure:
1. Identify the given density function, g(x) = e^x.
2. Note that we need to find the integral of g(x) with respect to v(x), i.e., ∫g(x) dv(x).
3. Since v is a measure with density g(x) with respect to the Lebesgue measure, we can rewrite the integral with respect to the Lebesgue measure, i.e., ∫g(x) dλ(x), where λ is the Lebesgue measure.
4. Now, we can evaluate the integral ∫e^x dλ(x) on the real line (R).
However, since e^x is not bounded on the real line, this integral will diverge. Therefore, the integral ∫g(x) dv(x) does not converge to a finite value.
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Ten percent of an airline’s current customers qualify for an executive traveler’s club membership.
A) Find the probability that between 2 and 5 (inclusive) out of 20 randomly selected customers qualify for the membership.
B) Find the expected number and the standard deviation of the number who qualify in a randomly selected sample of 50 customers
The probability between 2 and 5 is P(2 ≤ X ≤ 5) = 0.285 + 0.296 + 0.179 + 0.066 = 0.826. We can expect around 5 customers out of 50 to qualify for the membership.
The standard deviation of the number of customers who qualify for the membership in a randomly selected sample of 50 customers is 1.5. This tells us that the distribution of X is relatively narrow and tightly clustered around the expected value of 5.
A) To find the probability that between 2 and 5 (inclusive) out of 20 randomly selected customers qualify for the membership, we can use the binomial distribution formula: P(2 ≤ X ≤ 5) = P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
where X is the number of customers who qualify for the membership. We can calculate each probability using the binomial distribution formula:
P(X = k) =
[tex]n choose k) * p^k * (1 - p)^(n - k)[/tex]
where n is the sample size, k is the number of successes, and p is the probability of success. In this case, n = 20, k = 2, 3, 4, 5, and p = 0.1. Plugging these values into the formula, we get: P(X = 2) =
[tex](20 choose 2) * 0.1^2 * 0.9^18 = 0.285[/tex]
P(X = 3) =
[tex] (20 choose 3) * 0.1^3 * 0.9^17 = 0.296[/tex]
P(X = 4) =
[tex] (20 choose 4) * 0.1^4 * 0.9^16 = 0.179[/tex]
P(X = 5) =
[tex](20 choose 5) * 0.1^5 * 0.9^15 = 0.066[/tex]
B) To find the expected number and standard deviation of the number who qualify in a randomly selected sample of 50 customers, we can use the binomial distribution again. The expected value of X is given by: E(X) =
[tex]n * p[/tex]
where n = 50 and p = 0.1. Plugging these values in, we get: E(X) =
[tex]50 * 0.1[/tex]
= 5 The standard deviation of X is given by: SD(X) =
[tex] \sqrt{} (n \times p \times (1 - p))[/tex]
Plugging in n = 50 and p = 0.1, we get: SD(X) = sqrt(50 * 0.1 * 0.9) = 1.5
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Given the data below, what is the upper extreme?
4, 4, 1, 3, 8, 9, 15, 13, 4, 1
1
9
15
14
The upper extreme of the given data set is 15.
Now, the upper extreme of the data set, we need to find the highest value in the set.
The given data set is;
⇒ 4, 4, 1, 3, 8, 9, 15, 13, 4, 1
Thus, find the upper extreme, we need to sort the data set in ascending order:
⇒ 1, 1, 3, 4, 4, 4, 8, 9, 13, 15
Thus, The highest value in the data set is 15, which is the upper extreme.
Therefore, the upper extreme of the given data set is 15.
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if y=8 when x=4 and z=2 what is y when x=9 and z=10
The requried, for a given proportional relationship when x = 9 and z = 10, y is equal to 0.72.
If y varies directly with x and inversely with the square of z, we can write the following proportion:
y ∝ x / z²
To solve for k, we can use the initial condition:
y = k (x / z²)
When x = 4 and z = 2, y = 8. Substituting these values into the equation, we get:
8 = k (4 / 2²)
k = 8
So, the equation for the variation is:
y = 8 (x / z²)
To find y when x = 9 and z = 10, we substitute these values into the equation:
y = 8 (9 / 10²)
y = 0.72
Therefore, when x = 9 and z = 10, y is equal to 0.72.
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Assume that it costs a manufacturer approximately C(x) = 1,152,000 + 340x + 0. 0005x² dollars to manufacture x gaming systems in an hour. How many gaming systems should be manufactured each hour to minimize average cost?. Gaming systems per hour What is the resulting average cost of a gaming system?. $
If fewer than the optimal number are manufactured per hour, will the marginal cost be larger, smaller, or equal to the average cost at that lower production level? a The marginal cost will be larger than average cost. B The marginal cost will be smaller than average cost. C The marginal cost will be equal to average cost
The resulting average cost of a gaming system is approximately $678.58.
To find the number of gaming systems that should be manufactured each hour to minimize average cost, we need to find the minimum point of the average cost function. The average cost function is given by:
A(x) = C(x)/x
where C(x) is the cost function.
To find the minimum point of A(x), we can differentiate it with respect to x and set it equal to zero:
A'(x) = [C'(x)x - C(x)]/[tex]x^2[/tex] = 0
Solving for x, we get:
C'(x)x - C(x) = 0
340 + 0.001x = C(x)/x
Substituting the cost function C(x) = 1,152,000 + 340x + 0.0005x^2, we get:
340 + 0.001x = (1,152,000 + 340x + 0.0005[tex]x^2[/tex])/x
Multiplying both sides by x, we get:
340x + [tex]x^2[/tex]/2000 = 1,152,000/x
Multiplying both sides by 2000x, we get:
340[tex]x^2[/tex] + [tex]x^3[/tex] = 2,304,000
Dividing both sides by [tex]x^2[/tex], we get:
[tex]x^2[/tex] + 340x - 2,304,000/[tex]x^2[/tex] = 0
Let y =[tex]x^2,[/tex] then the equation becomes:
[tex]y^2[/tex] + 340y - 2,304,000 = 0
Solving for y using the quadratic formula, we get:
y = (-340 ± √([tex]340^2[/tex] + 4*2,304,000))/2
y ≈ 3,177.56 or y ≈ -6,517.56
Since y =[tex]x^2[/tex], we take the positive root:
[tex]x^2[/tex] ≈ 3,177.56
x ≈ 56.37
Therefore, the optimal number of gaming systems that should be manufactured each hour to minimize average cost is approximately 56.37.
To find the resulting average cost of a gaming system, we plug this value into the average cost function:
A(56.37) = C(56.37)/56.37 ≈ $678.58
Therefore, the resulting average cost of a gaming system is approximately $678.58.
If fewer than the optimal number are manufactured per hour, the marginal cost will be larger than the average cost at that lower production level. This is because the marginal cost is the derivative of the cost function with respect to x, and the cost function is a quadratic function that increases with x. At lower production levels, the marginal cost will be higher than the average cost because the cost function is increasing at an increasing rate.
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At a local carnival, kid's tickets cost $10 apiece and adult tickets cost $20 apiece. These are the only two types of tickets sold. At the recent show, 29 total tickets were sold for a total revenue of $430. How many children’s tickets were sold?
There are 15 children’s tickets were sold.
Given that;
At a local carnival, kid's tickets cost $10 apiece and adult tickets cost $20 apiece.
And, These are the only two types of tickets sold. At the recent show, 29 total tickets were sold for a total revenue of $430.
Let number of children’s tickets = x
And, Number of adult tickets = y
Hence, We can formulate;
⇒ x + y = 29 .. (i)
And, 10x + 20y = 430
⇒ x + 2y = 43
⇒ x = 43 - 2y
Plug above value in (i);
⇒ x + y = 29
⇒ 43 - 2y + y = 29
⇒ 43 - 29 = y
⇒ y = 14
From (i);
⇒ x + y = 29
⇒ x + 14 = 29
⇒ x = 29 - 14
⇒ x = 15
Thus, There are 15 children’s tickets were sold.
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Find the value of the following expression. 26 25 25 24+24-23-23-22+22-21-21-20+ +20 19 19 18+18 17-17 16 16 15 15 · 14
The value of the expression is 295.
We can simplify the expression by grouping the terms that have the same value:
26 + (25 + 25) + (24 + 24) - (23 + 23) - (22 + 22) - (21 + 21) - (20 + 20) + (19 + 19) + (18 + 18) + 17 - (16 + 16) + (15 + 15) + (14)
= 26 + 50 + 48 - 46 - 44 - 42 - 40 + 38 + 36 + 17 - 32 + 30 + 14
= 295
The given expression involves a series of numbers where some of them are added and some of them are subtracted. To simplify this expression, we need to group the terms that have the same value. We can see that the expression has pairs of numbers that add up to the same value, such as (25 + 25), (24 + 24), and so on. We can combine these pairs and simplify the expression further.
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The fraction of the time she worked was 7/9
Answer:
what
Step-by-step explanation:
3. Let C = { v, w, x,y,z }.
a).What is the cardinality of C? What is the
cardinality of P(C)?
b) Draw a tree showing all possible strings of letters
of length 5 or less starting with the letter z. What
is the cardinality of the set M = {all strings of
length 5 or less with letters from C}?
c) Sketch a tree showing all possible strings (of any
length). What is the cardinality of the set K= {all
strings using letters from C}?
a) there are 32 possible subsets of C.
b)The cardinality of set M is the sum of these numbers, which is 781.
C) there are an infinite number of possible strings, the cardinality of set K, which contains all possible strings using letters from C, is also infinite.
a) The cardinality of set C is 5, as there are 5 distinct elements in the set. The cardinality of the power set of C, denoted as P(C), is 2^5 = 32, as there are 32 possible subsets of C.
b) A tree showing all possible strings of letters of length 5 or less starting with the letter z would look like:
z
├── v
│ ├── v
│ ├── w
│ ├── x
│ └── y
├── w
│ ├── v
│ ├── w
│ ├── x
│ └── y
├── x
│ ├── v
│ ├── w
│ ├── x
│ └── y
├── y
│ ├── v
│ ├── w
│ ├── x
│ └── y
└── z
├── v
├── w
├── x
└── y
The cardinality of set M, which contains all possible strings of length 5 or less with letters from C, is equal to the sum of the cardinalities of all sets of strings of each length. Thus,
Set of strings Number of strings
Length 1 1
Length 2 5
Length 3 5^2 = 25
Length 4 5^3 = 125
Length 5 5^4 = 625
The cardinality of set M is the sum of these numbers, which is 1 + 5 + 25 + 125 + 625 = 781.
c) A tree showing all possible strings of any length would have an infinite number of branches. Each node in the tree would represent a different string, and the branches emanating from each node would represent the next letter that could be added to the string. Since there are an infinite number of possible strings, the cardinality of set K, which contains all possible strings using letters from C, is also infinite.
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The total surface area of the
prism is
A. 180 cm
B. 244 cm
C. 200 cm
D. 190 cm
The surface area of the prism is 200 cm².
What is the total surface area of the prism?The total surface area of the prism is calculated by applying the formula for total surface area of prism.
S.A = bh + (s₁ + s₂ + s₃)L
where;
b is the base of the triangleh is the height of the triangles₁ is the first triangular faces₂ is the second triangular faces₃ is the third triangular faceL is the length of the prismThe surface area of the prism is calculated as;
S.A = 8 cm (15 cm) + (8 cm + 15 cm + 17 cm) x 2cm
S.A = 120 cm² + 80 cm²
S.A = 200 cm²
Thus, the surface area of the prism is calculated using the formula for surface of right prism.
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Find the area of the rectangle.
5.5 in
20.45 in
Answer:112.475
Step-by-step explanation:
What is the rule for the transformation formed by the translation 8 units right and 5 units down followed by a 180 degree rotation
The rule for the transformation formed by the translation 8 units right and 5 units down followed by a 180 degree rotation is (x, y) changes to (-x - 8, 5 - y).
Consider a point (x, y).
When this point is translated such that it is translated 8 units right and 5 units down, then the point becomes,
(x, y) changes to (x + 8, y - 5).
This point is rotated 180 degrees.
When a point (x, y) is rotated 180 degrees, then the point becomes (-x, -y).
So, (x + 8, y - 5) changes to (-x - 8, -y + 5) = (-x - 8, 5 - y).
Hence the rule for the given transformation is (-x - 8, 5 - y).
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what would be the difference in predicted price of two wines that both have a rating of 90, but one is produced in california, and one is produced in oregon? make sure to use your rounded coefficients from the estimated regression equation to calculate this. round your final answer to 2 decimal places. the model predicts that the california wine would be more expensive than the oregon wine.
The model predicts that California wine would be more expensive than Oregon wine by $28.00.
To calculate the difference in predicted price between the two wines, we need to use the estimated regression equation and substitute the values for the variables. Let's say our estimated regression equation is:
Price = 50 + 2.5(Rating) + 10(California) - 8(Oregon)
Both wines have a rating of 90, so we can substitute that value in:
Price of California wine = 50 + 2.5(90) + 10(1) - 8(0) = 295
Price of Oregon wine = 50 + 2.5(90) + 10(0) - 8(1) = 267
Therefore, the predicted price of California wine is $295 and the predicted price of Oregon wine is $267. The difference between the two is $28.00.
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A market research firm calls a simple random sample of customers to determine whether they are satisfied with their current internet service provider. Out of 500 people surveyed, 389 say they are satisfied. If we are going to create a confidence interval for the percent of customers in the population who are satisfied, we will need a box model. Fill in the blank: The number of tickets in the box labeled 1 is a quantity that is _______.
a. fixed and known
b. fixed and estimated
c. random and known
d. random and estimated
d. random and estimated. The number of tickets in the box labeled 1 represents the number of customers in the population who are satisfied with their internet service provider.
This quantity is not fixed or known, as we are using a sample to estimate the proportion of the population who are satisfied. The tickets in the box are randomly selected from the population, and the number in the box is estimated based on the proportion of satisfied customers in the sample. Therefore, the quantity is both random and estimated. we can calculate the sample proportion and construct a confidence interval to estimate the true proportion of satisfied customers in the population.
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Our friend purchased a medium pizza for $10. 31 with a 30% off coupon. What is the price of a medium pizza without a coupon?
Therefore, the original purchased price of the medium pizza without a coupon is $10.31.
A coupon is a ticket or document that may be used in marketing to obtain a financial discount or refund when making a purchase of a good. Customers receive a discount on their initial purchase thanks to the First Order Coupon. The first order coupon sales rule may be configured by admin in the admin area.
It aids in improving conversion rates. Frequently, yearly percentages are used to describe coupon payments. For instance, a bond with a $1,000 face value and an annual payment of $30 is said to have a 3% coupon. If the friend purchased a medium pizza for $10.31 with a 30% off coupon, then the price of the pizza after the discount is:
= 10.31 - 0.30(10.31)
= 10.31 - 3.09
= $7.22
So the price of the medium pizza without a coupon is $7.22 / (1 - 0.30) = $10.31.
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Could the number of cars owned be related to whether an individual has children? In a local town, a simple random sample of 200 residents was selected. Data was collected on each individual on how many cars they own and whether they have children. The data was then presented in the frequency table:
Number of Vehicles Do you have children Total
No Yes
Zero: 24 50 74
One: 27 25 52
Two or more: 57 17 74
Total: 108 92 200
Part A: What proportion of residents in the study have children and own at least one car? Also, what proportion of residents in the study do not have children and own at least one car? (2 points)
Part B: Explain the association between the number of cars and whether they have children for the 200 residents. Use the data presented in the table and proportion calculations to justify your answer. (4 points)
Part C: Perform a chi-square test for the hypotheses.
H0: The number of cars owned by residents of a local town and whether they have children have no association.
Ha: The number of cars owned by residents of a local town and whether they have children have an association.
What can you conclude based on the p-value?
The probability of number of 1-2 Children in car and 3 plus children in car is 0.203.
We have,
The possibility of the result of any random event is known as probability. This phrase refers to determining the likelihood that any given occurrence will occur.
The probability of P(1-2 children| car). P (3 plus children| car) is given by:
P = 63/88 × 25/88
P=0.203
The probability of P(Bus| 1-2 children). P (Bus | 3 plus children) is given by:
P = 38/101 × 49/74
P=0.249
The probability of P(Car |1-2 Children) is given by:
P= 63/101
P=0.624
The probability of P(3 plus children | Bus)is given by:
P=49/87
P=0.563
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complete question:
Drag the tiles to the boxes to form correct pairs. Not all tiles will be used.
The table shows the mode of transportation to school for families with a specific number of children.
Mode of Transportation
Car
Number of
Children
0.284
1-2
63
38
3+
25
49
Total
88
87
A family from the survey is selected at random. Match the probability to each event.
0.662
Bus
0.203
0.249
101
74
175
0.624
P (3+ Children Bus)
Total
P(1-2 Children Car) - P (3+ Children Car)
Reset
P (Car 1-2 Children)
0.563
P (Bus 1-2 Children) - P (Bus 3+ Children)
▸
18. Determine the equation of the line through the points (2,8) and (-4,5). Express the line in slope-interceptorm.
The equation of the line through the points (2, 8) and (-4, 5) in slope-intercept form is y = (1/2)x + 7.
To determine the equation of the line through the points (2, 8) and (-4, 5) and express it in slope-intercept form, follow these steps:
1. Calculate the slope (m) of the line using the formula: m = (y2 - y1) / (x2 - x1)
In our case, (x1, y1) = (2, 8) and (x2, y2) = (-4, 5).
m = (5 - 8) / (-4 - 2) = (-3) / (-6) = 1/2
2. Use the slope-intercept form equation, y = mx + b, and plug in the slope (m) and one of the points (x, y) to solve for the y-intercept (b).
Let's use the point (2, 8).
8 = (1/2) * 2 + b
8 = 1 + b
b = 7
3. Now, plug the slope (m) and y-intercept (b) back into the slope-intercept form equation.
y = mx + b
y = (1/2)x + 7
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please help asap!!!!
Answer:
Step-by-step explanation:
1, 3 and 4
Someone help me please! question is in the attachment
Answer: 0.3%
Step-by-step explanation:
Factor 44+38. Write your answer in the form a(b+c) where a is the GCF of 44 and 38
44 + 38 can be written in the form a(b + c) as:
44 + 38 = 2(22 + 19) = 2(41)
To solve this problemWe may use the distributive property to factor 44 + 38 by first determining their greatest common factor (GCF), which is 2, and then writing the result as follows:
44 + 38 = 2(22) + 2(19)
By removing the second from the equation, we may further reduce it: 44 + 38 = 2(22 + 19).
Therefore, 44 + 38 can be written in the form a(b + c) as:
44 + 38 = 2(22 + 19) = 2(41)
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Solve the following:
(If you answer for the points I will be reporting you)
(2x3 + 4x3 - ) - (-7x2 + x -5)
(-6y2 + 2y - 2) - (y2 - 3y +10)
(5x2 -4x +11) + (-12x2 +4x -1)
(10x2 -5x +3) - (8x2 + 6x + 4)
Answer:
Bellow
Step-by-step explanation:
(2x³ + 4x³ - ) - (-7x² + x -5)
= 6x³ + 7x² - x + 5
(-6y² + 2y - 2) - (y² - 3y +10)
= -6y² + 2y - 2 - y² + 3y - 10
= -7y² + 5y - 12
(5x² -4x +11) + (-12x² +4x -1)
= -7x² + 0x + 10
= -7x² + 10
(10x² -5x +3) - (8x² + 6x + 4)
= 10x² - 5x + 3 - 8x² - 6x - 4
= 2x² - 11x - 1
I hope this helps!
The expressions are s
6x³ + 7x² - x + 5
-7y² + 5y - 12
-7x² - 8x + 10
2x² - 11x -1
What are algebraic expressions?Algebraic expressions are defined as expressions that are composed of coefficients, variables, constants, terms and factors.
These algebraic expressions are also made up of some arithmetic operations. These operations are;
BracketParenthesesMultiplicationSubtractionAdditionDivisionFrom the information given, we have that;
1. (2x3 + 4x3 - ) - (-7x2 + x -5)
expand the bracket
6x³ + 7x² - x + 5
2. (-6y2 + 2y - 2) - (y2 - 3y +10)
expand the bracket
-6y² + 2y -2 - y² + 3y - 10
collect the like terms
-7y² + 5y - 12
3. (5x2 -4x +11) + (-12x2 +4x -1)
expand the bracket
5x² - 4x + 11 - 12x² - 4x - 1
-7x² - 8x + 10
4. (10x2 -5x +3) - (8x2 + 6x + 4)
expand the bracket
2x² - 11x -1
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6. Caleb wants to buy a skateboard that costs $73.56. If sales tax is 7%, how much would his total purchase be?
Step-by-step explanation:
Total cost will be
$ 73.56 + 7% of 73.56
$ 73.56 + .07 * $73.56
(1.07) ( 73.56) = $ 78 . 71
During Hari Raya Aidilfitri, there is a promotion in ketupat sales. The original price of each ketupat (rice dumpling) is RM2.00. With a discount of less than 20% from the selling price, the total sales of that day is RM85.00. Do you know how many ketupat are sold on that day?
Answer:
53.125 or 53 dumplings.
Step-by-step explanation:
20 percent of 2.00 is 0.40 so 2.00 minus 0.40 is equal to 1.60. Since 85 dumpling were sold we divide 85 with 1.6 to get 53.125
While on vacation in Hawaii, Ellie and her friends go to a coconut farm to harvest fresh
coconuts. Ellie uses a pole pruner to release a bunch of coconuts from a canopy 24 meters
above the ground. Then, Ellie's friends catch the coconuts with a net situated 1.5 meters
above the ground.
To the nearest tenth of a second, how long does it take for the coconuts to land in the net?
Hint: Use the formula h = -4.9t² + S.
seconds
The time that it takes for the coconut to land on the net is given as follows:
t = 2.1 seconds.
How to model the situation?The quadratic function giving the height of the coconut after t seconds is defined as follows:
h(t) = -4.9t² + S.
In which S represents the height of the canopy.
Ellie uses a pole pruner to release a bunch of coconuts from a canopy 24 meters above the ground, hence the value of S is given as follows:
S = 24.
Thus the function is:
h(t) = -4.9t² + 24.
The coconuts hit the net when h(t) = 1.5, hence the time is obtained as follows:
1.5 = -4.9t² + 24
4.9t² = 22.5
t = sqrt(22.5/4.9)
t = 2.1 seconds.
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What is the median of the data set?
A. 42
B. 40
C. 41
D. 45
Answer:41.5
Step-by-step explanation:
By arranging the data from smallest to largest, like this:
40, 41, 42, 45, we can take the average of the two middle values divided by 2 to find the median. This is done with an equation like this:
(41+42)/2
Which comes out to be 41.5.
Figure pqrs is by a scale of with the center of dilation at the origin what are the coordinates of point s
The coordinates of S' is (-10, 6).
We have,
Dilation is a transformation in which the size of a figure is changed without altering its shape.
In the coordinate plane, a dilation changes the size of a figure by multiplying the distance between each point and the center of dilation by a scale factor.
The center of dilation is a fixed point in the plane about which the figure is dilated. If the scale factor is greater than 1, the figure is enlarged, and if it is less than 1, the figure is reduced. If the scale factor is negative, the figure is also reflected across the center of dilation.
From the figure,
S = (-5, 3)
Now,
Dilated with a scale factor of 2.
This means,
S' = (-5 x 2, 3 x 2) = (-10, 6)
Thus,
The coordinates of S' is (-10, 6).
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A factory manager records the number of defective light bulbs per case in a dot plot.
Describe the shape of the distribution and explain what the patterns mean in terms of the data.
The shape of the distributive is such that; it is skewed to the right. The pattern therefore means that the data is concentrated on the left and hence, the number of defective light bulbs per case is fewer in most case.
What is the shape of the distribution?It follows from the task content that the shape of the distribution is to be determined as required in the task content.
By observation, it can be inferred that more of the data is concentrated on the left and hence, the shape of the distribution can be termed; right-skewed.
This therefore implies that the pattern means; the number of defective light bulbs per case is fewer in most cases.
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Problem 1: Write a MATLAB program that solves the following system of equations:
2x + y - z = ri
- 3x – y +2z= r2
-2x + y +2z= R3 To get the solution, you need R1, R2, and R3 values. You can get these values from the file quiz2.mat. you must load the information in quiz2.mat. show your work
The system of equations using the backslash operator \, which performs Gaussian elimination with partial pivoting to obtain the solution x. Finally, we display the values of x, y, and z using the disp function.
Here's a MATLAB program that solves the given system of equations using the provided values of R1, R2, and R3 from the file quiz2.mat:
% Load the data from quiz2.mat
load('quiz2.mat');
% Define the coefficient matrix and the right-hand side vector
A = [2 1 -1; -3 -1 2; -2 1 2];
b = [R1; R2; R3];
% Solve the system of equations using the backslash operator
x = A \ b;
% Display the solution
disp(['x = ' num2str(x(1))]);
disp(['y = ' num2str(x(2))]);
disp(['z = ' num2str(x(3))]);
In this program, we first load the values of R1, R2, and R3 from the file quiz2.mat using the load function. We then define the coefficient matrix A and the right-hand side vector b using the given system of equations.
We solve the system of equations using the backslash operator \, which performs Gaussian elimination with partial pivoting to obtain the solution x. Finally, we display the values of x, y, and z using the disp function.
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Find (3x + 2x2 + 3 sin (x)) and evaluate it at x = 1. a. dx² 17.6829 b. 19.4755 20.5544 c. -15.3589 d. None
Approximate value is 7.5245.
To find the value of the expression (3x + 2x² + 3 sin(x)) and evaluate it at x = 1 using trigonometry, follow these steps:
Step 1: Substitute x = 1 into the expression:
(3(1) + 2(1)² + 3 sin(1))
Step 2: Simplify the expression:
(3 + 2 + 3 sin(1))
Step 3: Evaluate sin(1) (Note that x=1 is in radians):
sin(1) ≈ 0.8415
Step 4: Substitute the value of sin(1) back into the expression:
(3 + 2 + 3(0.8415))
Step 5: Calculate the final value:
3 + 2 + 3(0.8415) ≈ 5 + 2.5245 = 7.5245
So, the value of the expression (3x + 2x² + 3 sin(x)) evaluated at x = 1 is approximately 7.5245. The given options do not include this value, so the correct answer is d. None.
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Determine Q(Q), where Q is the cubic defined by the polynomial: (1) F(X,Y,Z) = X3 + 2Y3 – 423 € Q[X,Y,Z). (2) F(X,Y,Z) = (Y + Z)3 - 2X3 € Q[X,Y,Z). 9 Hint: For (1), study the divisibility by powers of 2 of an eventual solution, once assumed to be given by integral coordinates. For (2), note that Q is not geomet- rically irreducible and study the Galois action on the irreducible components. F(X, Y, Z) = X3 + 2Y3 – 423 € Q[X, Y, Z] F(X, Y, Z) = (Y + 2)3 – 2X3 E Q[X, Y, Z].
The Q(Q) = {(a,b,c,√2a+b+c) | a,b,c ∈ Q} ∪ {(-a,b,c,-√2a+b+c) | a,b,c ∈ Q}, where Q is the cubic.
To determine Q(Q), we need to find the set of solutions to the cubic equations defined by the polynomials F(X,Y,Z) in Q[X,Y,Z].
For F(X,Y,Z) = X3 + 2Y3 – 423 € Q[X,Y,Z], we can use the fact that any integer cube is congruent to either 0, 1, or -1 modulo 9. Thus, if we assume that there exists a solution with integral coordinates, we must have X and Y both congruent to 3 modulo 9 (since 423 is congruent to 6 modulo 9). However, this leads to a contradiction when we consider the parity of Z (odd), so there are no solutions with integral coordinates. Therefore, Q(Q) = {}.
For F(X,Y,Z) = (Y + Z)3 - 2X3 € Q[X,Y,Z], we note that Q is not geometrically irreducible since the polynomial (Y+Z)3 - 2X3 can be factored as (Y+Z-√2X)(Y+Z+√2X)(Y+Z) in Q(√2X)[Y,Z]. Thus, we need to study the Galois action on the irreducible components.
The Galois group of Q(√2X)/Q is generated by the automorphism σ(√2X) = -√2X, which fixes Q and interchanges the two roots of the irreducible polynomial Y+Z-√2X. Therefore, there are two irreducible components of Q(Q), given by Y+Z-√2X = 0 and Y+Z+√2X = 0.
To find the solutions on each component, we substitute either Y+Z-√2X or Y+Z+√2X into the original equation F(X,Y,Z) = (Y + Z)3 - 2X3 € Q[X,Y,Z] and solve for X. We obtain:
- For Y+Z-√2X = 0, we have X = (Y+Z)√2/∛2. Thus, we can express the solutions as (X,Y,Z) = (a,b,c,√2a+b+c) where a, b, and c are arbitrary rational numbers.
- For Y+Z+√2X = 0, we have X = -(Y+Z)√2/∛2. Thus, the solutions can be expressed as (X,Y,Z) = (-a,b,c,-√2a+b+c) where a, b, and c are arbitrary rational numbers.
Therefore, Q(Q) = {(a,b,c,√2a+b+c) | a,b,c ∈ Q} ∪ {(-a,b,c,-√2a+b+c) | a,b,c ∈ Q}.
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