The velocity v in terms of t is: v = [tex]20t^{ \frac{3}{2} } - 12t^{\frac{1}{2} } + 923[/tex]
The position s in terms of t is: s = [tex]40t^{ \frac{5}{2} } /5 - 24t^{\frac{2}{3} } /3 + 923t - 949[/tex]
Calculating Velocity and Position:To find the velocity and position functions given the acceleration and initial conditions.
Use the fact that acceleration is the second derivative of position with respect to time and that velocity is the first derivative of position with respect to time, to perform the integrations.
We also used the initial conditions given for velocity and position at a specific time to solve for the constants of integration.
Here we have
Partice moves on a coordinate line with acceleration d²s/dx² = 30√t- 6/√t subject to the conditions that ds/dt = 911 and s = 14 when t = 1.
To find the velocity, integrate the acceleration with respect to t once:
=> d²s/dx² = 30√t- 6/√t
=> ds/dt = ∫(d²s/dx²)dt
= ∫(30√t- 6/√t)dt
= [tex]20t^{ \frac{3}{2} } - 12t^{\frac{1}{2} } + C[/tex]
Using the initial condition ds/dt = 911 when t = 1, we can solve for C:
=> ds/dt = [tex]20t^{ \frac{3}{2} } - 12t^{\frac{1}{2} } + C[/tex]
=> 911 = [tex]20(1)^{ \frac{3}{2} } - 12(1)^{\frac{1}{2} } + C[/tex]
=> C = 923
Therefore, the velocity v in terms of t is:
=> v = [tex]20t^{ \frac{3}{2} } - 12t^{\frac{1}{2} } + 923[/tex]
To find the position, we can integrate the velocity with respect to t once:
=> ds/dt = [tex]20t^{ \frac{3}{2} } - 12t^{\frac{1}{2} } + 923[/tex]
=> s = ∫(ds/dt)dt = [tex]\int\limits } \, 20t^{ \frac{3}{2} } - 12t^{\frac{1}{2} } + 923[/tex]
=> s = [tex]40t^{ \frac{5}{2} } /5 - 24t^{\frac{2}{3} } /3 + 923t + C'[/tex]
Using the initial condition s = 14 when t = 1, we can solve for C':
=> 14 = [tex]40 (1)^{ \frac{5}{2} } /5 - 24(1)^{\frac{2}{3} } /3 + 923t + C'[/tex]
=> C' = -949
Therefore,
The velocity v in terms of t is: v = [tex]20t^{ \frac{3}{2} } - 12t^{\frac{1}{2} } + 923[/tex]
The position s in terms of t is: s = [tex]40t^{ \frac{5}{2} } /5 - 24t^{\frac{2}{3} } /3 + 923t - 949[/tex]
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If x and y vary directly, and X = 3 when y = 15, what is the value of x when y = 25?
Answer:
Step-by-step explanation:
Which is the area of the rectangle?
A rectangle of length 150 and width 93. Inside the rectangle, there is one segment from one opposite angle of base to the base. The length of that segment is 155.
The area of the rectangle is 13, 950 square unit.
We have,
length = 150
width= 93
So, Area of rectangle
= length x width
= 150 x 93
= 13950 square unit.
Thus, the required Area is 13, 950 square unit.
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A pole 12 feet tall is used to support a guy wire for a tower, which runs from the tower to a metal stake in the ground. After placing the pole, Jamal measures the distance from the pole to the stake and from the pole to the tower, as shown in the diagram below. Find the length of the guy wire, to the nearest foot.
Answer:
The given question is on trigonometry which requires the application of required function so as to determine the value known. So that the length of the guy wire is 67.0 feet.
Trigonometry is an aspect of mathematics that requires the application of some functions to determine the value of an unknown quantity.
Let the length of the guy wire be represented by l, and the angle that the guy wire makes with the stake be θ. So that applying the appropriate trigonometric function to determine the value of θ, we have:
Tan θ =
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
adjacent
opposite
=
11
4
4
11
Tan θ = 2.75
θ =
�
�
�
−
1
Tan
−1
2.75
= 70.0169
θ =
7
0
�
70
o
Considering triangle formed by the tower and the stake to determine the value of l, we have;
Cos θ =
�
�
�
�
�
�
�
�
ℎ
�
�
�
�
�
�
�
�
�
hypotenuse
adjacent
Cos
7
0
�
70
o
=
23
�
l
23
l =
23
�
�
�
7
0
�
Cos70
o
23
=
23
0.3420
0.3420
23
l = 67.2515
l = 67 feet
The length of the guy wire is 67 feet.
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The probability assigned to each experimental outcome must be
a. one
b. between zero and one
c. smaller than zero
d. any value larger than zero
The probability assigned to each experimental outcome must be: b. between zero and one
The probability assigned to each experimental outcome must be between zero and one. This is because probability is a measure of how likely an event is to occur, and it cannot be negative or greater than 100%. A probability of zero means that the event will not occur, while a probability of one means that the event is certain to occur. Probabilities between zero and one indicate the likelihood of an event occurring, with higher probabilities indicating greater likelihood. It is important for probabilities to add up to one across all possible outcomes, as this ensures that all possible events are accounted for and that the total probability is normalized. Probability theory is used in many fields, including statistics, finance, and engineering, and is essential for making informed decisions based on uncertain events. By assigning probabilities to different outcomes, we can calculate expected values and make predictions about future events, helping us to better understand the world around us.
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In a class of 30 students there are 18 who have passed Mathematics, 16 who have passed English and 6 who have not passed either of them. We randomly select a student from that class:
1.a) What is the probability that he has advanced in English and Mathematics?
2.b) Knowing that he has passed Mathematics, what is the probability that he has passed English?
3.c) Are the events "Pass Mathematics" and "Pass English" independent?
We can conclude that the events "Pass Mathematics" and "Pass English" are dependent
a) The probability that a student has advanced in both Mathematics and English can be calculated using the formula:
P(Math and Eng) = P(Math) + P(Eng) - P(Math or Eng)
where P(Math) is the probability of passing Mathematics, P(Eng) is the probability of passing English, and P(Math or Eng) is the probability of passing at least one of them.
From the given information, we have:
P(Math) = 18/30 = 0.6
P(Eng) = 16/30 = 0.5333
P(Math or Eng) = 1 - P(not passing either) = 1 - 6/30 = 0.8
Substituting these values into the formula, we get:
P(Math and Eng) = 0.6 + 0.5333 - 0.8 = 0.3333
Therefore, the probability that a student has advanced in both Mathematics and English is 0.3333 or approximately 33.33%.
b) If we know that a student has passed Mathematics, we can use conditional probability to calculate the probability that they have passed English:
P(Eng | Math) = P(Eng and Math) / P(Math)
We already calculated P(Eng and Math) in part (a) as 0.3333. To find P(Math), we can use the information given in the problem:
P(Math) = 18/30 = 0.6
Substituting these values into the formula, we get:
P(Eng | Math) = 0.3333 / 0.6 = 0.5556
Therefore, the probability that a student has passed English given that they have passed Mathematics is 0.5556 or approximately 55.56%.
c) To determine whether the events "Pass Mathematics" and "Pass English" are independent, we can compare their joint probability (the probability of passing both) to the product of their individual probabilities:
P(Math and Eng) = 0.3333
P(Math) = 0.6
P(Eng) = 0.5333
If the events are independent, then we should have:
P(Math and Eng) = P(Math) x P(Eng)
Substituting in the values we calculated, we get:
0.3333 ≠ 0.3198
Since the joint probability is not equal to the product of the individual probabilities, we can conclude that the events "Pass Mathematics" and "Pass English" are dependent.
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Which table shows a linear function
The table that shows a linear function include the following: B. table B.
What is a linear function?In Mathematics, a linear function is a type of function whose equation is graphically represented by a straight line on the cartesian coordinate.
This ultimately implies that, a linear function has the same (constant) slope and it is typically used for uniquely mapping an input variable to an output variable, which both increases simultaneously.
In this context, we have:
Slope = (0 - 2)/(-3 + 5) = (2 - 0)/(-3 + 1) = -1
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a) Find the five-number summary for the data. Show your work. (5 points: 1 point
for each number)
27,29,36,43,79
b) Use your results from Part a to display the data on a box plot. (2 points)
25 30 35 40 45 50 55 60 65 70 75 80 85
The five-number summary for the data are:
Minimum (Min) = 27.First quartile (Q₁) = 28.Median (Med) = 36.Third quartile (Q₃) = 61.Maximum (Max) = 79.The data is shown on a box plot below.
The interquartile range (IQR) of the data is 33.
How to determine the five-number summary for the data?In order to determine the statistical measures or the five-number summary for the data, we would arrange the data set in an ascending order:
27, 29, 36,43,79
From the data set above, we can logically deduce that the minimum (Min) is equal to 27.
For the first quartile (Q₁), we have:
Q₁ = [(n + 1)/4]th term
Q₁ = (5 + 1)/4
Q₁ = 1.5th term
Q₁ = 1st term + 0.5(2nd term - 1st term)
Q₁ = 27 + 0.5(29 - 27)
Q₁ = 27 + 0.5(2)
Q₁ = 27 + 1
Q₁ = 28.
From the data set above, we can logically deduce that the median (Med) is equal to 11.
For the third quartile (Q₃), we have:
Q₃ = [3(n + 1)/4]th term
Q₃ = 3 × 1.5
Q₃ = 4.5th term
Q₃ = 4th term + 0.5(5th term - 4th term)
Q₃ = 43 + 0.5(79 - 43)
Q₃ = 43 + 0.5(36)
Q₃ = 61
Mathematically, interquartile range (IQR) of a data set is typically calculated as the difference between the first quartile (Q₁) and third quartile (Q₃):
Interquartile range (IQR) of data set = Q₃ - Q₁
Interquartile range (IQR) of data set = 61 - 28
Interquartile range (IQR) of data set = 33.
In conclusion, a box and whisker plot for the given data set is shown in the image attached below.
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which equation is true when n = 5? A) 2n = 7 B) n + 3 =8 C) 9 -n = 14 D) n/15 = 3
The answer is B.
In order to get the answer, you replace the letter n to 5 and workout the problems to see which one is true.
B is the answer because n + 3 = 8
5 + 3 = 8
Answer:
B
Step-by-step explanation:
n+3 =
5+3=
Nayeli has a points card for a movie theater.
• She receives 40 rewards points just for signing up.
• She earns 14.5 points for each visit to the movie theater.
• She needs at least 185 points for a free movie ticket.
Use the drop-down menu below to write an inequality representing v, the number of
visits she needs to make in order to get a free movie ticket.
An inequality representing v, the number of visits Nayeli needs to make to get a free movie ticket is 40 + 14.5v ≥ 185.
What is inequality?Inequality describes a mathematical statement that states that two or more algebraic expressions are unequal.
Inequalities are represented as:
Greater than (>)Less than (<)Greater than or equal to (≥)Less than or equal to (≤)Not equal to (≠).The rewards points Nayeli has just for signing up = 40
The points earned per visit to the movie theater = 14.5
The total number of points required for a free movie ticket ≥ 185
Let the number of visits Nayeli needs to make too get a free movie ticket = v
Inequality:40 + 1.45v ≥ 185
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PLS HELP ME ASAP PLS RN
I'm not sure if this is correct but 9560 (rectangle surface area) + 900 (triangle surface area) =10460...
DON'T TRUST ME ON THIS ONE CHECK IT YOURSELF BUT IM PRAYING FOR YOU
Answer: 21120
Step-by-step explanation:
Area for the rectangluar prism
A(Rect)= LA+2B
where LA,Lateral area = Ph P,Perimeter of base= 30+120+30+120 = 300
h, height =20
LA=300(20)=6000
B,area of base=(30)(120)=3600
A(rect)=LA+2B=6000+2(3600)
=13200
Area for triangular prism, turn on side so triangle is base(columned)
A(triangle) = LA+2B
LA= Ph Perimeter of triangle = 17+17+30=64 h=120
LA=(64)(120)=7680
B, the base is the triangle=1/2 bh where b =30 h=8
B=1/2 (30)(8)
=120
A(triangle)=7680+2(120) =7920
Add the 2 areas
A(total)=13200+7920=21120
on a coordinate plane, kite h i j k with diagonals is shown. point h is at (negative 3, 1), point i is at (negative 3, 4), point j is at (0, 4), and point k is at (2, negative 1).
On a coordinate plane, kite HIJK with diagonals is shown. Point H is located at (-3, 1), point I is at (-3, 4), point J is at (0, 4), and point K is at (2, -1). The diagonals of this kite connect points H and J, and points I and K, creating an intersecting pattern within the shape.
Based on the information provided, we know that a kite shape has been loaded onto a coordinate plane with points h, i, j, and k located at specific coordinates. The coordinates for point h are (-3, 1), for point i are (-3, 4), for point j are (0, 4), and for point k are (2, -1). Additionally, we know that the kite has diagonals, but we are not given any information about their lengths or intersection points.
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Answer:
B is the answer
Step-by-step explanation:
in (x-2)+in(x+1)=2
x = -2.6047
x = 4.2312
x = 3.652
x = 3.6047
Answer:
Step-by-step explanation:
The number in your question is expressed in scientific notation, which is typically used to express numbers that are either too large or too small. The number in scientific notation is expressed as a power of 10. A positive exponent means the # is large whereas if the exponent is negative, then the # is small.
3.652 x 10-4 --> negative exponent, therefore, # is small.
All you need to do is to convert this number to standard notation by moving the decimal 4 places to the left.
3.652 x 10-4 = 0.0003652
Find the value of sin N rounded to the nearest hundredth, if necessary
The value of the identity sin N = 3/5
How to determine the valueTo determine the value of the identity, we need to know the different trigonometric identities. These identities are;
cosinesinetangentcotangentsecantcosecantFrom the information given, we have that;
The angle of the triangle is N
The opposite side of angle N is 3
The hypotenuse side of angle N is 5
Using the sine identity, we have;
sin N = 3/5
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"I am struggling with calculating the p-value. I am using z as
the test statistic and have found that z=-4.22. Please help with
finding the p-value. Thank you.
The output voltage for an electric circuit is specified to be 130. A sample of 40 independent readings on the voltage for this circuit gave a sample mean 128.6 and standard deviation 2.1. (a) Test the hypothesis that the average output voltage is 130 against the alternative that it
is less than 130. Use a test with level.05. Report the p-value as well.
The p-value is 0.00002. (a) The average output voltage is less than 130 since our t-statistic is less than the critical value therefore, we can reject the null hypothesis. The p-value is 0.007.
To find the p-value for a z-test, you need to use a z-table or a calculator that can give you the area under the standard normal curve to the left of your test statistic.
In this case, your test statistic is z = -4.22. Using a standard normal table, the area to the left of z = -4.22 is approximately 0.00002.
Therefore, the p-value for this test is p = 0.00002.
(a) Using a one-sample t-test to test the hypothesis that the average output voltage is 130 against the alternative that it is less than 130.
With a sample size of 40 and a sample mean of 128.6, the t-statistic is calculated as:
t = (128.6 - 130) / (2.1 / sqrt(40)) = -2.67
Using a t-table with 39 degrees of freedom (df = n - 1), the critical value for a one-tailed test with a level of significance of 0.05 is -1.685.
Since our t-statistic is less than the critical value, we can reject the null hypothesis and conclude that the average output voltage is less than 130.
Using a t-distribution calculator, the one-tailed p-value for a t-statistic of -2.67 with 39 degrees of freedom is approximately 0.007.
Therefore, the p-value for this test is p = 0.007.
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Determine if the statement is true or false, a justify you answer. Assume S is nontrivial and u and v are both nonzero. If u and v are vectors, then proj_v u is a multiple of u. a.True. proj_v u is a multiple of both u and v. b.True, by the definition of Projection Onto a Vector. c.False. proj_v u is a multiple of v, not u. d.False. proj_v u is not a multiple of either u or v. e.False. proj_v u is a multiple of ||u||, not u.
The statement is false.
The projection of vector u onto vector v, denoted as proj_v u, is not necessarily a multiple of vector u.
In the case of vector projection, proj_v u is a scalar multiple of vector v, not vector u. It represents the component of vector u that lies in the direction of vector v.
This projection is obtained by taking the dot product of u and v, divided by the dot product of v and itself (which is equivalent to the magnitude of v squared), and then multiplying it by vector v. The resulting projection is parallel to vector v and can be scaled by a scalar factor, but it does not necessarily align with vector u.
Therefore, option c is the correct answer. Proj_v u is a multiple of vector v, not vector u.
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y²+4y-7 evaluate the expression when y=7
Write the number in standard form 7. 1x10^4=
The number 7.1 x 10⁴ in standard form is: 71,000
In standard form, a number is expressed as a coefficient multiplied by a power of 10, where a coefficient is a number greater than or equal to 1 and less than 10, and the power of 10 represents the number of places the decimal point must be moved to obtain the number's value.
In this case, the coefficient is 7.1, which is greater than or equal to 1 and less than 10. The power of 10 is 4, which means that the decimal point must be moved 4 places to the right to obtain the value of the number. Therefore, we get 71,000.
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Show your calculation steps dearly Correct you answer to 4 decimal places and report the measurement unit when applicable. Question 1 (10 marks) A salad shop is selling fruit cups. Each fruit cup consists of two types of fruit, strawberries and blue berries. The weight of strawberries in a fruit cup is normally distributed with mean 160 grams and standard deviation 10 grams. The weight of blue berries in a fruit cup is normally distributed with mean u grams and standard deviation o grams. The weight of strawberries and blue berries are independent, and it is known that the weight of a fruit cup with average of 300 grams and standard deviation of 15 grams. (a) Find the values of u and o (b) The weights of the middle 96.6% of fruit cups are between (300 - K. 300 + K) grams. Find the value of K.
C) The weights of the middle 96.6% of fruit cups are between (L1, L2) grams. Find the values of LI and L2.
(a) The values of u is 140 g and o is 13.42 g. (b) The value of K in (300 - K. 300 + K) grams is 27.15 g. C) The weights of the middle 96.6% of fruit cups are between (L1, L2) grams. The values of LI is 272.85 g and L2 is 327.15 g.
(a) The mean weight of blueberries is:
300 g - 160 g = 140 g
The standard deviation of the weight is:
Var(X + Y) = Var(X) + Var(Y)
Adding the variances:
15^2 = 10^2 + o^2
Solving for o:
o = sqrt(15^2 - 10^2) = 13.42 g
Therefore, the values of u and o are u = 140 g and o = 13.42 g.
(b) Since the distribution is normal, we can use the standard normal distribution to find K.
The middle 96.6% of a standard normal distribution corresponds to the interval (-1.81, 1.81) (using a table or calculator). Therefore,
K = 1.81 * 15 = 27.15 g
Therefore, the weights of the middle 96.6% of fruit cups are between 300 - 27.15 = 272.85 g and 300 + 27.15 = 327.15 g.
(c) Using the standard normal distribution to find the corresponding interval on the standard normal scale:
(-1.81, 1.81)
We can then scale this interval to the distribution of the weight of fruit cups by dividing by the standard deviation and multiplying by 15 g:
L1 = 300 + (-1.81) * 15 = 272.85 g
L2 = 300 + 1.81 * 15 = 327.15 g
Therefore, the weights of the middle 96.6% of fruit cups are between 272.85 g and 327.15 g.
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Solve 3x²-14x=5 by factoring.
Answer:
(x-5)(3x+1)=0
x= 5, x= -1/3
Step-by-step explanation:
3x²-14x=5
3x²-14x-5=0
The factor that goes in are 1 and -15 which equal the sum and products.
Sum: -14
Product: -15
Therefore:
3x²+x-15x-5 = 0
Factor by grouping:
x(3x²+x) -5(-15x-5)
x(3x+1) -5(3x+1)
(x-5)(3x+1) = 0
Use Zero Product Property to solve for X
x-5 = 0 3x+1 = 0
x= 5, x= -1/3
The length of Dominic's rectangular living room is 9 meters and the distance between opposite corners is 10 meters. What is the width of Dominic's living room? If necessary, round to the nearest tenth.
Answer:
We can use the Pythagorean theorem to solve for the width of Dominic's living room. The Pythagorean theorem states that for a right triangle with legs of length a and b and hypotenuse of length c, a² + b² = c².
In this case, we can treat the length of the living room (9 meters) as one leg of the right triangle, and the distance between opposite corners (10 meters) as the hypotenuse. Let w be the width of the living room. Then the other leg of the right triangle has length w.
Applying the Pythagorean theorem, we get:
9² + w² = 10²
81 + w² = 100
w² = 19
w ≈ 4.4
Therefore, the width of Dominic's living room is approximately 4.4 meters.
Step-by-step explanation:
How do I solve for the angle and X
The measure of angle x in the given right triangle is 71.8°
Calculating the measure of an angle x in the right triangleFrom the question, we are to determine the measure of angle in the given triangle
The given triangle is a right triangle
We can determine the value of angle x by using SOH CAH TOA
sin (angle) = Opposite / Hypotenuse
cos (angle) = Adjacent / Hypotenuse
tan (angle) = Opposite / Adjacent
In the given triangle,
Adjacent = 5
Hypotenuse = 16
Thus,
cos (x) = 5/16
x = cos⁻¹ (5/16)
x = 71.7900°
x ≈ 71.8°
Hence, the measure of angle x is 71.8°
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6.2.1b: Solve for missing angles and side lengths using trigonometric
ratios.
A triangle is shown.
The values of the missing sides and angles are;
<D = 32 degrees
d = 8. 75
e = 16. 50
How to determine the valuesTo determine the value we need to note that the sum of the angles in a triangle is 180 degrees.
From the information given, we have;
<E + <D + <F = 180
substitute the values
90 + 58 + <D = 180
collect the like terms
<D = 32 degrees
Using the sine identity
sin 58 = 14/x
cross multiply the values
x = 16. 50
Using the tangent identity;
tan 58 = 14/y
cross multiply
y = 8. 75
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The ages of three men are in the ratio 3 : 4 : 5. If the difference between the ages of the oldest and the youngest is 18 years, find the sum of the ages of the three man.
Answer :
Sum of their ages = 108 years.Step-by-step explanation:
It's given that The ages of three men are in the ratio 3 : 4 : 5
Let's assume,
Age of first men = 3x Second men = 4x Third men = 5xAlso, the difference between the ages of the oldest and the youngest is 18 years.
Age of youngest men = 3x Age of oldest men = 5xDifference in their ages ,
[tex]:\implies [/tex] 5x - 3x = 18 years
[tex]:\implies [/tex] 2x = 18
[tex]:\implies [/tex] x = 18/2
[tex]:\implies [/tex] x = 9
Hence,
Age of first men = 3x
[tex]:\implies [/tex] 3 × 9
[tex]:\implies [/tex] 27 years
Age of second men = 4x
[tex]:\implies [/tex] 4 × 9
[tex]:\implies [/tex] 36 years.
Age of thrid men = 5x
[tex]:\implies [/tex] 5 × 9
[tex]:\implies [/tex] 45 years.
Now, Sum of the ages of three man
[tex]:\implies [/tex] 27 + 36 + 45
[tex]:\implies [/tex] 108 years.
Therefore, The sum of the ages of three man is 108 years.
Find the volume of the prism
Answer:
Volume formal= L × W × W
Volume formal= 6 × 8 × 9
Answer= 6×8×9= 432
Look at picture! It is all written there
Answer:
In/out
-18, -19
-17, -18
-15, -16
-7, -8
0, -1
14, 13
Step-by-step explanation:
the rule states "subtract 1" so if you look at the In, -15 for example goes to -16 for the out which proves that -15-1 is -16. so for all the In's you subtract each number by 1. to FIND a In from the out you just add 1. So just follow the rules and look at the rules carefully!
a 6. Let Xn be a bounded martingale and let T be a stopping time (NOT necessarily bounded). Prove that E[XT] = E[X] by considering the stopping times Tn= min(T, n).]
For a bounded martingale Xₙ and stopping time T (not necessarily bounded), E[XT] = E[X] is proved by considering the stopping times Tₙ= min(T, n) and using the Optional Stopping Theorem.
To prove that E[XT] = E[X], we can utilize the Optional Stopping Theorem.
First, we know that since Xₙ is a bounded martingale, it satisfies the conditions for the Optional Stopping Theorem stating that for any stopping time T, [tex]E[X_{T}] = E[X_{0}][/tex], where [tex]X_{0}[/tex] is the initial value of Xₙ.
Now, taking into consideration stopping times Tn = min(T, n). As Tn is a bounded stopping time, we utilize the Optional Stopping Theorem to get:
[tex]E[X_{Tn}] = E[X_{0}][/tex]
We can rewrite this as:
[tex]E[X_{Tn}] - E[X_{0}] = 0[/tex]
Now, if we take the limit as n→∞.As Xn is a bounded martingale, it follows that E[|Xn|] < infinity for all n. Thus, utilizing Dominated Convergence Theorem, we get:
[tex]lim_{n} E[X_{Tn}] = E[lim_{n} X_{Tn}] = E[XT][/tex]
Similarly, [tex]lim_{n} E[X_{0}] = E[X].[/tex]
Therefore, taking the limit as n→∞ in our previous equation, we get:
E[XT] - E[X] = 0
Or, equivalently:
E[XT] = E[X]
So, E[XT] = E[X] by considering the stopping times Tn= min(T, n)].
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Kim made 3 batches of this fruit punch recipe. Combine: 70 milliliters of strawberry juice 500 milliliters of pineapple juice 2 liters of apple juice How many liters of fruit punch did Kim make?
For made a fruit punch, Kim used the 3 batches of this fruit punch recipe. From unit conversion, the total quantity in litres used to fruit punch is equals to the 2.570 L.
We have Kim made 3 batches of this fruit punch recipe. It includes the combination of following,
quantity of strawberry juice = 70 mL
quantity of apple juice = 2 L
quantity of pineapple juice = 500 mL
We have to determine the number of liters of fruit punch he made. Using the unit conversion,
one liters = 1000 mililiters
=> 1 mL = 0.001 L ( conversion factor)
so, quantity of strawberry juice = 70 mL = 70× 0.001 L = 0.070 mL
quantity of pineapple juice = 500 mL = 500× 0.001 L = 0.500 L
So, total quantity of fruit punch made by Kim in liters = strawberry juice + pineapple juice + apple juice
= 0.070 L + 0.500 L + 2 L
= 2.570 L
Hence, the required value is 2.570 liters.
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Use polar coordinates to find the volume of the given solid.
Enclosed by the hyperboloid −x2 − y2 + z2 = 6 and the plane z = 3
-x2 - y2 + 9 = 6 >>> x2 + y2= 3 so r2 = 3 >>> squart 0<=r <=3
My question is that why negative square root of 3 is not included in the range???
In polar coordinates, the radial distance "r" is defined as the distance from the origin to a point in the plane. Since distance cannot be negative, we only consider the positive square root of 3 in the range for this problem. So, the correct range for "r" is 0 ≤ r ≤ √3, and negative square root of 3 is not included because it doesn't represent a valid distance in polar coordinates.
To find the volume of the given solid enclosed by the hyperboloid −x2 − y2 + z2 = 6 and the plane z = 3 using polar coordinates, we need to express the equation of the hyperboloid in terms of polar coordinates.
Substituting x = rcosθ and y = rsinθ, we get:
−r2cos2θ − r2sin2θ + z2 = 6
Simplifying, we get:
z2 = 6 - r2
Since the plane z = 3 intersects the hyperboloid, we have:
3 = √(6 - r2)
Solving for r, we get:
r = √3
Hence, the range for r is 0 ≤ r ≤ √3.
In summary, the negative square root of 3 is not included in the range of r because r represents a distance and cannot be negative. The volume of the solid can be found by integrating the function f(r,θ) = √(6 - r2) over the range 0 ≤ r ≤ √3 and 0 ≤ θ ≤ 2π using polar coordinates. The result will be in cubic units and can be obtained by evaluating the integral.
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2) A gho with a cost price of Nu 750 was sold for Nu 900. What was the percent markup?
The percent markup is 20%
The selling price of the Nu is 900
The cost price of the Nu is 750
The percent markup can be calculated as follows
= 900-750/750 × 100
= 150/750 × 100
= 0.2 × 100
= 20%
Hence the percent markup is 20%
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Q.5.f(x,y) = x + y; if both x & y are even f(x,y) = x - y; if both & y are odd f(x, y) = 2x – y2, if any one of them is odd & other is = even. Find f(2,3) - f(2,4) a)-31 b) -56 c)-11 d) -13
f(x,y) = x + y; if both x & y are even f(x,y) = x - y; if both & y are odd f(x, y) = 2x – y2, if any one of them is odd & other is = even. Then the result of f(2,3) - f(2,4) is -11, which corresponds to option c).
Determining f(2,3)
Since x = 2 (even) and y = 3 (odd)
Then, the function definition for this case is f(x,y) = 2x - y²
f(2,3) = 2(2) - 3² = 4 - 9 = -5
Determining f(2,4)
Since x = 2 (even) and y = 4 (even)
Then, the function definition for this case is f(x,y) = x + y.
f(2,4) = 2 + 4 = 6
Now, Calculating f(2,3) - f(2,4)
f(2,3) - f(2,4) = -5 - 6 = -11
Therefore, the answer is option c) -11.
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