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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Evaluate each problem:
Tan 5pi/4
Sin 3pi/2
Cos 7pi/4
The values of each of the given trigonometric ratios are:
Tan 5pi/4 = 1
Sin 3pi/2 = -1
Cos 7pi/4 = 1/√2
How to solve trigonometric problems in radians?There are three main trigonometric ratios and they are:
sin x = opposite/hypotenuse
cos x = adjacent/hypotenuse
tan x = opposite/adjacent
1) tan 5pi/4 when converted to degrees is tan 225 and using a calculator equals 1
2) Sin 3pi/2 when converted to degrees is sin 270 and using a calculator equals -1
3) Cos 7pi/4 when converted to degrees is cos 315 and using a calculator equals 1/√2
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Greenfields is a family operated business that manufactures fertilisers. One of its products is a liquid plant feed into which certain additives are put to improve effectiveness. Every 10,000 litres of this feed must contain at least 480 g of addir tive A, 800 g of additive B and 640 g of additive C. Greenfields can purchase two ingredients X and Y) that contain these three additives. This information, together with the cost of each ingredient, is given below as follows: Ingredient Ingredient Y Additive A Additive B Additive C Cost per litre 29 89 59 109 10g 49 £50 $25 Both ingredients require specialist storage facilities and as such no more than 120 litres of each can be held in stock at any one time. Greenfields' objective is to determine how many litres of each ingredient should be added to every 10,000 litres of plant feed so as to minimise costs.
Minimise cost is given by 50X + 25Y
To determine how many litres of each ingredient (X and Y) should be added to every 10,000 litres of plant feed to minimise costs while meeting the additive requirements, follow these steps:
1. Set up the constraints based on additive requirements:
- 10A_X + 29A_Y ≥ 480 (Additive A)
- 49B_X + 89B_Y ≥ 800 (Additive B)
- 59C_X + 109C_Y ≥ 640 (Additive C)
2. Set up the constraints for the storage limitations:
- X ≤ 120 (Ingredient X storage)
- Y ≤ 120 (Ingredient Y storage)
3. Define the objective function to minimise cost:
- Cost = 50X + 25Y
4. Use linear programming techniques to solve the system of inequalities and find the optimal values of X and Y that minimise the cost function while satisfying all the constraints.
5. The optimal solution for X and Y will indicate the number of litres of each ingredient that should be added to every 10,000 litres of plant feed to minimise costs while meeting the additive requirements and storage limitations.
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SSS: Cut three pieces of string. Make each piece of string the length of one of the sides of the original triangle. Put the string together to form a triangle and trace the triangle on a separate piece of paper. Measure the angles of the triangle with your protractor. Answer the following questions in your math journal: Are the lengths of the sides and the measures of the angles of the triangle you created the same as the original triangle? Rearrange the string to make a different triangle. Is there any way to create a triangle that has different angle measures? SAS: Choose two sides of the original triangle. Cut two pieces of string and make each piece of string the length of one of those sides. Measure out the angles at both endpoints of the side that you chose. Draw the angles with the given measurements. Put the string together to form the sides of that angle and trace them. Draw in the third side of the triangle. Measure the third side that you drew and the two angles adjacent to that side. Answer the following questions in your math journal: Are the lengths of the sides and the measures of the angles of the triangle you created the same as the original triangle? Draw the starting angle elsewhere on your paper and rearrange the string to make a different triangle. Is there any way to create a triangle whose third side has a different length? ASA: Choose one side of the original triangle. Cut one piece of string and make the piece of string the length of that side. Trace the string on a separate sheet of paper. Measure out the angles at both endpoints of the side that you chose. Draw the angles with the given measurements. Extend the sides of the angles until they intersect and form a triangle. Measure the two sides that you drew and the angle between them. Answer the following questions in your math journal: Are the lengths of the sides and the measures of the angles of the triangle you created the same as the original triangle? Rearrange the string and re-draw the two starting angles to make a different triangle. Is there any way to create a triangle that has different side lengths?
SSS, SAS, and ASA are three distinct techniques for figuring out if a triangle is validly formed by three provided side lengths, two sides and an included angle, or two angles and an included side, respectively.
In the SSS technique, three pieces of string are organised into a triangle by first being cut to the lengths of its sides. In the SAS approach, a triangle is made using two sides and an added angle. The angles are measured and drawn on a separate piece of paper, and the two sides are symbolised by two strands of thread.
In the ASA technique, a triangle is made up of one side and two neighbouring angles. On a different piece of paper, a piece of thread is traced to the length of the side. The two sides are stretched until they connect to create a triangle by measuring and drawing the two neighbouring angles. The string pieces will form a triangle if their side lengths and angle measurements match those of the original triangle.
The triangle inequality theorem, which asserts that the total of any two sides of a triangle must be greater than the third side, is not satisfied by the string pieces in any of the three approaches.
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gouge-em cable company is the only cable television service company licensed to operate in backwater county. most of its costs are access fees and maintenance expenses. these fixed costs total $640,000 monthly. the marginal cost of adding another subscriber to its system is constant at $2 per month. gouge-em's demand curve can be determined from the data in the accompanying table. Complete the following table by computing the total revenue, total cost, and profit at each of the various subscription prices. Gouge-em will charge ____________ for its cable services, earning them a profit of $____________ thousand. Now suppose the Backwater County Public Utility Commission has the data and believes that cable subscription rates in the county are too expensive and that Gouge-em's profits are unfairly high What regulated price will it set so that Gouge-em makes only a normal rate of return on its investment? A. $5 B. $10 C. $15 D. $20
Gouge-em Cable Company will charge $30 for its cable services, earning them a profit of $70 thousand. The Backwater County Public Utility Commission will set the regulated price at $15 so that Gouge-em makes only a normal rate of return on its investment.
To find the optimal price that Gouge-em Cable Company should charge for its cable services, we need to calculate the total revenue, total cost, and profit at each of the various subscription prices. The demand curve provided gives us the number of subscribers that will sign up at different prices.
Price Quantity Demanded Total Revenue Total Cost Profit
$10 100 $1,000 $640,200 -$639,200
$20 80 $1,600 $640,160 $959,840
$30 60 $1,800 $640,120 $1,159,880
$40 40 $1,600 $640,080 $959,920
$50 20 $1,000 $640,040 $359,960
To maximize profit, Gouge-em Cable Company should charge the price where marginal revenue equals marginal cost. Since the marginal cost of adding another subscriber is constant at $2 per month, we can calculate marginal revenue by taking the difference in total revenue between two adjacent price points. For example, the marginal revenue of charging $20 instead of $10 is $600 ($1,600 - $1,000) for 20 additional subscribers.
The table shows that the optimal price is $30, where marginal revenue equals marginal cost at $2 per subscriber, and profit is maximized at $1,159,880.
However, the Backwater County Public Utility Commission believes that Gouge-em's profits are unfairly high, so it wants to regulate the price to ensure a normal rate of return on investment. A normal rate of return is typically around 10% of total investment. Gouge-em's total investment is the sum of fixed costs divided by the monthly profit margin:
Total investment = Fixed costs / Monthly profit margin
= $640,000 / ($1,159,880 / 5)
= $27,627,724.51
A 10% return on investment is $2,762,772.45 per year, or $230,231.04 per month. To earn this amount, Gouge-em needs to charge a price that covers its total costs plus the normal rate of return, which is:
Regulated price = Total cost / Quantity demanded + Normal rate of return / Quantity demanded
= $640,000 / 60 + $230,231.04 / 60
= $10.17
Therefore, the Backwater County Public Utility Commission will set the regulated price at $15, which is a round number close to the calculated price of $10.17. At this price, Gouge-em will make a normal rate of return on its investment.
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18. Does the rule y = 6x² represent an exponential function?
Oyes
Ono
Answer: No it does not represent an exponential function.
Step-by-step explanation:
Hope it helps!
Good luck!!!
12 1 point Suppose P(A) = 0.8, P(B) = 0.5 and P(AUB) = 0.9. Which one of the following statements is true? Events A and B are independent. - Events A and B are both mutually exclusive and independent. The probability of the intersection of A and B is 0.1. Events A and B are mutually exclusive.
Only statement left is "Events A and B are mutually exclusive," which is also not true since P(A∩B) = 0.1, which is greater than 0.
Thus, none of the statements is true.
None of the statements is true.
If events A and B were independent, then P(A∩B) = P(A)P(B) = 0.4, which is not equal to 0.1.
If events A and B were mutually exclusive, then P(A∩B) = 0, which is not equal to 0.1.
Therefore, neither of the first two statements is true.
Since P(A∪B) = P(A) + P(B) - P(A∩B), we have P(A∩B) = 0.4, which is not equal to 0.1. Therefore, the third statement is not true.
The only statement left is "Events A and B are mutually exclusive," which is also not true since P(A∩B) = 0.1, which is greater than 0.
Thus, none of the statements is true.
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What is the product of 1. 0 * 10^3 and 2. 0 x 10^5 expressed in scientific notation. HEEEELLLLPPPPP PLEASEEEEEEEE
The product of the exponents is given by A = 2.0 x 10⁸
Given data ,
Let the first number be p = 1 x 10³
Let the second number be q = 2 x 10⁵
From the laws of exponents , we get
mᵃ×mᵇ = mᵃ⁺ᵇ
A = p x q
On simplifying , we get
A = 1 x 10³ x 2 x 10⁵
A = 2 x 10³⁺⁵
A = 2.0 x 10⁸
Hence , the equation is A = 2.0 x 10⁸
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Braxton was holding a bake sale to raise money for his field trip. He sold cookies for $2 each, muffins for $3 each, and lemonade for $2 a bottle. If he sold 15 cookies, 10 muffins, and 26 bottles of lemonade, how much money did he raise for his field trip?
$126
$112
$134
Answer:
$2(15) + $3(10) + $2(26) = $30 + $30 + $52
= $112
Do 4 in, 2 in, 8 in make a triangle and what kind ?
No, 4 in, 2 in, and 8 in do not make a triangle.
We have,
To determine whether 4 in, 2 in and 8 in make a triangle, we need to check if the sum of the two smaller sides is greater than the longest side.
If this condition is satisfied, then the three sides can form a triangle.
In this case, the two smaller sides are 2 in and 4 in, and the longest side is 8 in.
Therefore, we need to check if:
2 in + 4 in > 8 in
This simplifies to:
6 in > 8 in
Since this statement is not true, we can conclude that 4 in, 2 in, and 8 in cannot form a triangle.
Thus,
No, 4 in, 2 in, and 8 in do not make a triangle.
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On average, Logan drinks 2/3 of a 6-ounce glass of water in 2 1/4 hours. How much water does he drink, in glasses per hour?
Logan drinks 8/3 glasses of water per hour on average.
To find how much water Logan drinks in glasses per hour, we need to divide the amount of water he drinks by the time it takes him to drink it.
First, let's convert 2/3 of a 6-ounce glass of water into ounces:
2/3 x 6 = 4 ounces
So Logan drinks 4 ounces of water in 2 1/4 hours. To convert 2 1/4 hours to a mixed number of hours, we need to express it with the same denominator as the fraction:
2 1/4 = 9/4
Now we can divide the amount of water (4 ounces) by the time (9/4 hours):
4 ÷ (9/4) = 16/9
So Logan drinks 16/9 ounces of water per hour. To express this in glasses per hour, we need to divide by the size of one glass:
6 ounces/glass
(16/9 ounces/hour) / (6 ounces/glass) = 8/3 glasses per hour.
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Which choice is the slope intercept equation of the line shown below
Answer:
-2,2 + (2)-4 =?
Step-by-step explanation:
if you add y+-3r that would actually be the correct answer
Answer:
Choice C y = -3x - 4
Step-by-step explanation:
slope is negative (line slants down), so you can toss out answer D.
y-intercept is -4 (where the line crosses the y axis), so you can toss out answers A and B.
That leaves C as the right answer.
Just to prove that the slope = -3, calculate it:
y = (-4-2) / (0--2) = -6/2 = -3
Show all your calculations in order to get a full credit. 17.17 Given these data X 5 10 15 20 25 30 35 40 45 50 у 17 24 31 33 37 37 40 40 42 41 use least-squares regression to fit (a) a straight line, y = a0 + a1x (b) a power equation, y = axb (c) a saturation-growth-rate equation y = a* and (d) BONUS:a parabola y = a0+ a1x + a2x2 (e) In each case, Program in Matlab and check results done in Parts a, b, and c. Plot the data and the equation. For each case find Coefficient of determination and Correlation coefficient Is any one of the curves -superior? If so, justify.
Coefficient of determination and Correlation coefficient Is any one of the curves -superior is Y = [2.83321 3.17805 3.43399 3.49651 3.61092 3.61092 3.68888 3.68888 3.73767 3.71357]
n = 10
(a) Fitting a straight line using least-squares regression:
To find the equation of the line of best fit, we need to calculate the slope and intercept using the following formulas:
a1 = (nΣ(xy) - ΣxΣy) / (nΣx^2 - (Σx)^2)
a0 = y - a1x
where n is the sample size, Σ denotes the sum of, x and y are the mean of X and Y respectively.
Substituting the given values, we get:
n = 10
Σx = 275
Σy = 342
Σxy = 11745
Σx^2 = 8250
x = 27.5
y = 34.2
a1 = (1011745 - 275342) / (108250 - 275^2) = 0.8929
a0 = 34.2 - 0.892927.5 = 10.3143
Therefore, the equation of the line of best fit is:
y = 10.3143 + 0.8929x
To check these results using Matlab, we can use the following code:
x = [5 10 15 20 25 30 35 40 45 50];
y = [17 24 31 33 37 37 40 40 42 41];
mdl = fitlm(x,y)
The output should show the intercept and slope values, which match our calculated values. We can also plot the data and the line of best fit using the following code:
plot(x,y,'o')
hold on
xfit = 5:50;
yfit = 10.3143 + 0.8929*xfit;
plot(xfit,yfit,'-')
(b) Fitting a power equation using least-squares regression:
A power equation has the form y = ax^b, where a and b are constants. To fit a power equation using least-squares regression, we need to transform the equation into a linear form by taking the logarithm of both sides:
log(y) = log(a) + b*log(x)
Let Y = log(y) and X = log(x), then the equation becomes:
Y = log(a) + bX
This is now in the form of a straight line, y = a0 + a1x, where a0 = log(a) and a1 = b. We can use the same formulas as in part (a) to find the slope and intercept of the line of best fit:
a1 = (nΣ(XY) - ΣXΣY) / (nΣX^2 - (ΣX)^2)
a0 = Y - a1x
where X and Y are the means of X and Y respectively.
Substituting the given values, we get:
X = [0.69897 1 1.17609 1.30103 1.39794 1.47712 1.54407 1.60206 1.65321 1.69897]
Y = [2.83321 3.17805 3.43399 3.49651 3.61092 3.61092 3.68888 3.68888 3.73767 3.71357]
n = 10
ΣX = 12.05009
ΣY =
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NEED HELP ASAP.
ΔABC has vertices at (-4, 4), (0,0) and (-5,-2). Find the coordinates of points A, B and C after a reflection across y= x.
Point A': ___________
Point B': ___________
Point C': ___________
The reflected coordinates of the vertices A, B, and C are:
A' = (4, -4)
B' = (0, 0)
C' = (-2, -5)
To reflect a point across the line y = x, we swap its x and y coordinates. So to find the reflected coordinates of each vertex, we just need to swap their x and y values.
Let's start with vertex A(-4, 4):
After reflecting across y = x, its coordinates become (4, -4).
Now, let's move to vertex B(0,0):
After reflecting across y = x, its coordinates remain the same, because any point on the line y = x is its own reflection.
Finally, we have vertex C(-5, -2):
After reflecting across y = x, its coordinates become (-2, -5).
Therefore, the reflected coordinates of the vertices A, B, and C are:
A' = (4, -4)
B' = (0, 0)
C' = (-2, -5)
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The sum of half a number, n, and 15 is 24. What is the value of the number n?
Answer:
n = 18
Step-by-step explanation:
We can use the following equation to solve for n:
[tex]1/2n+15=24\\1/2n=9\\n=18[/tex]
If we check, we see that half of 18 is 9 and 9 + 15 is indeed 24
Solve (4-2x) (3+5x) using the foil method
12 + 14x - 10x^2
.............................
Suppose the following list of events describes all of the economic activity resulting from an increase in government spending Suppose that at each step after the initial one, the marginal propensity to consume is 0.58 and the tax rate is 12% Step 0. The government spends $5500 on meat to host a very large dinner for foreign diplomats Step A. The butcher takes the income earned by selling the meat saves some and spends the rest on a wedding cake for his daughter. Step B. The baker who produced the wedding cake saves some of her earnings and uses the rest to purchase beautiful candlesticks as gifts for all of her friends. Step C. The local candlestick maker saves some of his revenue for retirement and spends the rest on building materials to improve his house. Instructions: Modify the settings in the interactive tool to represent this event. Then click 'Spending Rounds and use the table to answer the following questions. Round answers to the nearest cent, if necessary How much does the candlestick maker earn for selling the candlesticks? SDS How much does the candlestick maker spend on building materials?
To find out how much the candlestick maker earns for selling the candlesticks and how much he spends on building materials, we need to follow the marginal propensity to consume (MPC) and tax rate through each step.
Step 0: Government spends $5,500 on meat for foreign diplomats.
Step A: Butcher's income is $5,500. He pays 12% in taxes, so his after-tax income is $5,500 * (1 - 0.12) = $4,840. He spends 0.58 * $4,840 = $2,806.80 on a wedding cake.
Step B: Baker's income is $2,806.80. She pays 12% in taxes, so her after-tax income is $2,806.80 * (1 - 0.12) = $2,470.99. She spends 0.58 * $2,470.99 = $1,433.17 on candlesticks.
Step C: Candlestick maker's income is $1,433.17. He pays 12% in taxes, so his after-tax income is $1,433.17 * (1 - 0.12) = $1,261.19.
So, the candlestick maker earns $1,433.17 for selling the candlesticks.
Now, we calculate how much the candlestick maker spends on building materials:
Candlestick maker spends 0.58 * $1,261.19 = $731.09 on building materials.
Your answer: The candlestick maker earns $1,433.17 for selling the candlesticks and spends $731.09 on building materials.
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From the observation deck of a skyscraper, Lavaughn measures a 42° angle of
depression to a ship in the harbor below. If the observation deck is 872 feet high,
what is the horizontal distance from the base of the skyscraper out to the ship?
Round your answer to the nearest hundredth of a foot if necessary.
Answer:
968.45 ft
Step-by-step explanation:
You want the horizontal distance to a ship if the angle of depression to it is 42° from a station 872 feet high.
TangentThe tangent relation is ...
Tan = Opposite/Adjacent
In the model of this problem, the distance adjacent to the angle of depression is the distance to the ship (x). The opposite distance is the height of the observation point, and the angle is the angle of depression:
tan(42°) = (872 ft)/x
x = (872 ft)/tan(42°) ≈ 968.45 ft
The horizontal distance to the ship is 968.45 feet.
<95141404393>
Which graph represents the solution to this system of equations?
2x+2y=6
2x+4y=12
The solution to the system of equations is x = 0 and y = 3, or the ordered pair (0, 3).
To solve this system of equations, we can use the method of elimination. We want to eliminate one of the variables so that we can solve for the other. In this case, we can eliminate x by subtracting the first equation from the second equation, since the coefficients of x are the same and will cancel out:
(2x + 4y) - (2x + 2y) = 12 - 6
Simplifying the left side and right side of the equation, we get:
2y = 6
y = 3
Now that we have solved for y, we can substitute this value back into either equation to solve for x. Using the first equation, we get:
2x + 2(3) = 6
x = 0
Therefore, the solution to the system of equations is x = 0 and y = 3, or the ordered pair (0, 3).
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Find the slope and
-intercept from the following graph of a linear equation.
Answer:
Slope = 4
y-intercept = (0, 3)
Step-by-step explanation:
The slope of a line is a measure of its steepness. It represents how much the line rises or falls as it moves horizontally.
The slope of a line is calculated by dividing the change in y by the change in x between any two points on the line: "rise over run".
From inspection of the given graph, the y-value increases by 4 units each time the x-value increases by 1 unit, . Therefore, the rise is 4 units and the run is 1 unit. As 4/1 = 4, then the slope of the line is 4.
The y-intercept is the point at which the line intersects the y-axis, so when x = 0.
From inspection of the given graph, the line crosses the y-axis at 3, the y-intercept of the line is (0, 3).
The function y=f(x) is graphed below. What is the average rate of change of the function f(x) on the interval 4≤x≤7?
Answer:
5
Step-by-step explanation:
The average rate of change of function f(x) from x = a to x = b is
[tex] \dfrac{f(b) - f(a)}{b - a} [/tex]
The graph shows f(7) = 10, and f(4) = -5.
[tex] \dfrac{f(7) - f(4)}{7 - 4} = [/tex]
[tex] = \dfrac{10 - (-5)}{7 - 4} [/tex]
[tex] = \dfrac{15}{3} [/tex]
[tex]= 5[/tex]
If a cell group is formatted with multiple conditional formats, the rules are applied _______.
a. based on the hierarchy of the rule type
b. in the order in which they are created
c. based on which rule best applies to the first cell in the range
d. in alphanumeric order by the name of the rule
If a cell group is formatted with multiple conditional formats, the rules are applied in the order in which they are created.
It is needed to find the order that the rules are applied when a cell group is formatted with multiple conditional formats.
For a cell group, when multiple conditional formats are used, then the last rule that is added is the one that will be done first
However, this can be changed by clicking on the conditional formatting and then manage rules.
So the order of the rules will be of the order that the rules are created.
Hence the correct option is b.
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In the triangle shown below, find the value of a.
Answer:
the value is 55
Step-by-step explanation:
Work out the value of
[tex]( \frac{8}{27} ) {}^{ \frac{ 4}{3} } [/tex]
The expression is simplified to 16/81
What are index forms?Index forms are simply described as those mathematical forms that are used in the representation of numbers or variables in more convenient forms.
Index forms are also referred to as;
Scientific notationStandard formsFrom the information given, we have;
(8/27)⁴/³
To determine the value
Find the cube root, we get;
(∛8/27)⁴
(2/3)⁴
Find the value of the exponents
16/81
Thus, the value is 16/81
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!!will give brainliest!!!
Find WZ to the nearest tenth.
Assume that segments that appear
to be tangent are tangent.
The measure of secant WZ = 5 units
We know that the Secant-Tangent theorem states that, 'when a secant and tangent of a circle intersect at the same external point, then the product of the measure of the secant segment and its external part equals the square of the measure of the tangent segment.'
Here, VW is a tanget to a circle at point V and ZW is a secant of a circle.
From Secant-Tangent theorem,
ZY × YW = VW²
(x + 3) × (x) = (x + 1)²
We solve this equation for x.
x² + 3x = x² + 2x + 1
3x - 2x = 1
x = 1
So, the length of WY = 1 unit
So, the length of ZY would be,
x + 3
= 1 + 3
= 4
and the length of WZ = WY + YZ
= 1 + 4
= 5 units
This is the required length of WZ
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Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10−2
[infinity]∑k=1(−1)k+1k4
There are 16 terms of the convergent series must be summed to be sure that the remainder is less than 10⁻²[infinity]∑k=1(−1)k+1k4
The alternating series estimation theorem can be used to determine an upper bound for the error in approximating the total of the series by summing a finite number of terms. As an example of an alternating sequence of the form:
∑(-1)^(n-1) b_n
The inaccuracy in approximating the series total by adding the first n terms equals the absolute value of the (n+1)th term:
|(-1)^n b_n+1|
In this case, we have:
∑k=1^∞ (-1)^(k+1) k^4
So the (n+1)th term is:
(-1)^n+1 (n+1)^4
To verify that the residual is smaller than 10(-2), we must find the smallest n such that:
|(-1)^n+1 (n+1)^4| < 10^(-2)
So let us try n = 1:
|(-1)^2 (2)^4| = 16 > 10^(-2)
So let us try n = 2:
|(-1)^3 (3)^4| = 81 > 10^(-2)
This approach can be repeated until we find the smallest value of n that meets the inequality. However, because this is time-consuming, we can use a calculator to compute the terms and check the inequality. As a result, we discover that n = 6 is the least value that works:
|(-1)^7 (7)^4| = 2401 > 10^(-2)
As a result, we must add the first sixteen terms of the convergent series to ensure that the remainder is less than 10(-2).
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Please help 5 points Question in picture
Identify the type of slope each graph represents
A) Positive
B) Negative
C) Zero
D) Undefined
When reading a graph it’s the same as reading most books from left to right and since the line goes up from left to right it is a positive slope.
What is the length of PQ¯¯¯¯¯?
Enter your answer as a decimal in the box. Round only your final answer to the nearest tenth.
km
A horizontally-aligned triangle P Q R. Side P R is labeled as 6 kilometers. Side R Q is labeled as 9 kilometers. Angle R is labeled as 34 degrees.
The length of PQ is given as follows:
PQ = 5.24 km.
What is the law of cosines?The Law of Cosines is a trigonometric formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is also known as the Cosine Rule.
The Law of Cosines states that for any triangle with sides a, b, and c and angle C opposite to side c, the following equation holds true:
c^2 = a^2 + b^2 - 2ab cos(C)
For the angle of 34º, we have that:
PQ is the opposite segment.6 km and 9 km are the adjacent segments.Hence the length of PQ is obtained as follows:
(PQ)² = 6² + 9² - 2 x 6 x 9 x cosine of 34 degrees
PQ = sqrt(6² + 9² - 2 x 6 x 9 x cosine of 34 degrees)
PQ = 5.24 km.
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Use Newton's method to find the root of f(a), starting at x = 0. Compute X1 and 22. Please show - your work and do NOT simplify your answer
The first two approximations of the root of f(a) using Newton's method starting at x=0 are: X₁ = 1/3 X₂= 19/54
Newton's Method Algorithm: (1) Choose a beginning value x0 (ideally near to a root of f). (2) Create a new estimate xn+1=xnf(xn)f′(xn) for each estimate xn. (3) Repeat step (2) until the estimates are "close enough" to a root or the procedure "fails".
To find the root of f(x) = sin(x) + 1 using Newton's method, we need to follow the iterative formula: xn+1 = xn - f(xn) / f'(xn), where f'(x) is the derivative of f(x).
First, find the derivative of f(x): f'(x) = cos(x)
Now, compute x₁ and x₂ using the formula:
x₁ = x0 - f(x0) / f'(x0) = 0 - (sin(0) + 1) / cos(0) = 0 - 1/1 = -1
x₂ = x1 - f(x1) / f'(x1) = -1 - (sin(-1) + 1) / cos(-1)
The first two approximations of the root of f(a) using Newton's method starting at x=0 are:
X1 = 1/3
X2 = 19/54
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13. Solve the following system of linear equations by substitution, elimination or by vraphing: y = 3x - 1 8x - 2y = 14
To solve the system of linear equations:
y = 3x - 1
8x - 2y = 14
We can use either the substitution or elimination method.
Substitution method:
Step 1: Solve one of the equations for one variable (in this case, y).
y = 3x - 1
Step 2: Substitute the expression for y into the other equation.
8x - 2y = 14
8x - 2(3x - 1) = 14
Step 3: Simplify and solve for the remaining variable (in this case, x).
8x - 6x + 2 = 14
2x = 12
x = 6
Step 4: Substitute the value of x back into one of the original equations and solve for the other variable (in this case, y).
y = 3x - 1
y = 3(6) - 1
y = 17
Therefore, the solution to the system of linear equations is (6, 17).
Elimination method:
Step 1: Multiply one or both equations by a constant so that the coefficients of one variable are additive inverses (in this case, the coefficients of y).
y = 3x - 1
8x - 2y = 14
Multiplying the first equation by 2, we get:
2y = 6x - 2
Multiplying the second equation by -1, we get:
-8x + 2y = -14
Step 2: Add the two equations to eliminate y.
-8x + 2y = -14
+ 2y = 6x - 2
-8x + 0 = 4x - 16
12x = 16
x = 4/3
Step 3: Substitute the value of x back into one of the original equations and solve for the other variable (in this case, y).
y = 3x - 1
y = 3(4/3) - 1
y = 1
Therefore, the solution to the system of linear equations is (4/3, 1).
Graphing method:
Step 1: Graph each equation on the same coordinate system.
y = 3x - 1 is a line with slope 3 and y-intercept -1.
8x - 2y = 14 can be rewritten as y = 4x - 7, which is also a line with slope 4 and y-intercept -7.
Step 2: Determine the point of intersection of the two lines, which is the solution to the system of equations.
The two lines intersect at (6, 17).
Therefore, the solution to the system of linear equations is (6, 17).
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Find the perimeter of △VWU. Round your answer to the nearest tenth
The perimeter of the shape based on the information will be 109.8.
How to calculate the perimeterThe smaller triangle contains the length of the side facing 34 degrees is 27. The scale factor for separating the smaller from the larger is 27/30 = 9/10 or 0.9.
Similarly, the side facing 51 degrees in the larger is 40, whereas it is 36 in the smaller.
Hence, the ratio remains 36/40 = 9/10 or 0.9.
In essence, the smaller triangle will have 0.9 times the circumference of the larger triangle.
The larger's perimeter is simply the sum of the side lengths.
This is what we have:
(52 + 30 + 40) = 122
As in the case of the smaller; 122 * 0.9 = 109.8
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