Answer:
Step-by-step explanation:
The Weinis Loves cookies
The height of the parallelogram is the same as the height of the trapezoid
Why?
Doubling the trapezoid and then halving the resulting area
As was true with the triangle, but its base is the sum of the two bases of the trapezoid.
pls help will give brainliest
Answer:
1= 27
2= 2
Step-by-step explanation:
y=2x
Swap sides so that all variable terms are on the left hand side.
2x=y
Divide both sides by 2.
2
2x
=
2
y
Dividing by 2 undoes the multiplication by 2.
x=
2
y
A fish tank initially contains 40 liters of pure water. Brine of constant, but unknown, concentration of salt is flowing in at 3 liters per minute. The solution is mixed well and drained at 3 liters per minute. Let x be the amount of salt, in grams, in the fish tank after t minutes have elapsed. Find a formula for the rate of change in the amount of salt, dx/dt, in terms of the amount of salt in the solution x and the unknown concentration of incoming brine c. dxdt= grams/minute Find a formula for the amount of salt, in grams, after t minutes have elapsed. Your answer should be in terms of c and t. x(t)= grams In 15 minutes there are 20 grams of salt in the fish tank. What is the concentration of salt in the incoming brine? c= g/L
The concentration of salt in the incoming brine is 0.185 g/L.
The rate of change in the amount of salt, dx/dt, is determined by the difference between the rate of incoming salt and the rate of outgoing salt. The rate of incoming salt is the product of the concentration of the incoming brine, c, and the flow rate of 3 liters per minute.
The rate of outgoing salt is the product of the concentration of the solution in the tank, x/40, and the flow rate of 3 liters per minute. Therefore, the formula for the rate of change in the amount of salt is:
dx/dt = 3c - 3(x/40)
To find the formula for the amount of salt, x(t), after t minutes have elapsed, we can integrate the rate of change formula with respect to time:
x(t) = ∫(3c - 3(x/40))dt
Using the initial condition that x(0) = 0, we can solve for the constant of integration and obtain the formula:
x(t) = 120c - 3ct - (3/40)x(t)
Rearranging the terms and solving for x(t), we get:
x(t) = (120c - 3ct)/(1 + 3/40)
Finally, to find the concentration of salt in the incoming brine, c, given that x(15) = 20, we can plug in the values of t and x(t) into the formula and solve for c:
20 = (120c - 3c(15))/(1 + 3/40)
Simplifying and solving for c, we get:
c = (20 + 900/40)/(120 - 45)
c = 0.185 g/L
Therefore, the concentration of salt in the incoming brine is 0.185 g/L.
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On Wednesday evening Sonto sent 1/2hour doing housework 45 minutes doing housework 1 hour visiting friends 1 1/2hours watching television. Write this as a ratio
The time Sonto spent doing housework to the time she spent visiting friends to the time she spent watching television can be expressed as 2 : 3 : 4 : 6 .
This is because we can convert each time interval to a common unit, such as minutes, and then simplify the resulting fractions to a ratio of whole numbers. Specifically, Sonto spent 30 minutes doing housework (1/2 hour), 45 minutes doing housework, 60 minutes visiting friends (1 hour), and 90 minutes watching television (1 1/2 hours = 60 + 30 minutes).
Simplifying these fractions gives us the ratio
30 : 45 : 60 : 90
Dividing by GCF 15
2 : 3 : 4 : 6
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A taxi journey takes the passenger to a destination 8 km south and 3 kn west of its starting point. Find the distance and bearing of the destination from its starting point.
Using trigonometry.
Answer: We can use trigonometry to solve this problem.
First, let's draw a diagram to represent the situation:
|
|
| 8 km
|
|
--------X---------
| 3 km
|
|
|
|
The starting point is at the X, and the destination is 8 km south and 3 km west of the starting point.
To find the distance and bearing of the destination from the starting point, we can use the Pythagorean theorem and trigonometric functions.
The distance between the starting point and destination is the hypotenuse of a right triangle with legs of length 8 km (south) and 3 km (west). So we can use the Pythagorean theorem:
distance = √(8² + 3²) ≈ 8.6 km
To find the bearing (direction) of the destination from the starting point, we can use trigonometry. The bearing is the angle between the line connecting the starting point and destination and the line pointing due north.
We can use the tangent function to find this angle:
tan(θ) = opposite/adjacent = 3/8
θ = tan⁻¹(3/8) ≈ 20.56°
So the bearing of the destination from the starting point is approximately 20.56° west of due south.
Therefore, the distance of the destination from the starting point is approximately 8.6 km and the bearing of the destination from the starting point is approximately 20.56° west of due south.
Step-by-step explanation:
There are 15 boys and 10 girls in Ms. Rogers' class. What is the ratio of boys to girls? A 2:3 B. 2:5 C. 3:2 D. 3:5
Answer:
option C is correct
Step-by-step explanation:
Ratio of boys to girls is15:10
=3:2
¿what is the value of b ?
Answer:
its Value is 50 degree.
Step-by-step explanation:
Because It is vertically opposite angle.
Mrs Burn buys a washing machine that costs R3 999 at a discount of 6%. Calculate the discount on washing machine and the amount she has to pay.
Answer:
Discount Rs 239.94= 240
She has to pay 3659
Step-by-step explanation:
6 % of 3999=239.94 or 240
She will get the discount of 240 rupees.
She has to pay other amount beside the discount.
The price of machine is 3999.
The price she has to pay after discount= actual price- discount.
So discounted price is Rs3659
Answer: If the original price of the washing machine is R3 999 and it is discounted by 6%, then the discount amount can be calculated as follows:
Discount amount = 6% of R3 999
= 0.06 x R3 999
= R239.94
Therefore, Mrs Burn will receive a discount of R239.94 on the washing machine.
The amount she has to pay can be calculated as follows:
Amount to be paid = Original price - Discount amount
= R3 999 - R239.94
= R3 759.06
Therefore, Mrs Burn has to pay R3 759.06 for the washing machine after the 6% discount.
Brainliest is appreciated (:
Hans had a mean score of 67 after his 6 games. What was his total score?
Answer:
402
Step-by-step explanation:
Mean is an average of his scores
Total score = Mean score x number of games
67 x 6 = 402
Does anyone know how to solve this problem about similar polygons?
We can conclude after answering the provided question that So the perimeter of TUJ4V is 35, but with the information provided, we cannot determine the perimeter of PQRS.
What is perimeter?A two-dimensional geometric shape's perimeter is the entire length of the borderline or outer edge. It is the total length of the shape's parties or edges. For example, the perimeter of a square is determined by dividing the lengths of all four edges of the square. Similarly, to calculate the outer wall of a rectangle, add the lengths of adjacent sides and then multiply by two because there are two different sets of adjacent sides.
To calculate the perimeter of PQRS, we must know the lengths of all four quadrilateral sides. However, we are only given one side's length, which is 12.
We know that TUJ4V has a perimeter of 35, so:
Circumference = 2x + 2y = 35
[tex]x + y = 17.5\\y = 15\\x + 15 = 17.5\\x = 2.5[/tex]
As a result, the length of each side of TUJ4V is as follows:
TU = JW = x = 2.5
TV = U4 = y = 15
The perimeter of TUJ4V is as follows:
2x + 2y = 2(2.5) + 2(15) = 35 Perimeter
So the perimeter of TUJ4V is 35, but with the information provided, we cannot determine the perimeter of PQRS.
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What is the volume of the figure below, which is composed of two cubes with side lengths of 5 units?
a. 15 cubic units
B.30 cubic units
C.125 cubic units
D.250 cubic units
The volume of the figure is D. 250 cubic units.
What is meant by volume?
Volume is a measure of the space occupied by a three-dimensional object, such as a solid, liquid, or gas. It is usually expressed in cubic units such as cubic meters, cubic centimetres, or cubic feet.
What is meant by a unit?
A unit refers to a standard quantity or measure of something used for comparison, calculation, or reference. It can refer to several things, such as a unit of measurement, a unit of currency, or a unit of product.
According to the given information
The volume of each cube with a side length of 5 units is 5 x 5 x 5 = 125 cubic units.
The figure is composed of two such cubes, so the total volume is:
2 x 125 = 250 cubic units.
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The prism below is made of cubes that measure of an inch on one side. What is the volume of the prism?
The volume of the prism will be 9/16 inch³ i.e. D.
What exactly is a prism?
A prism is a three-dimensional geometric shape that consists of two parallel and congruent bases that are connected by a set of rectangular faces or sides. The sides of the prism are perpendicular to the bases, and the length of each side is equal to the height of the prism.
A prism can be classified based on the shape of its bases. For example, a triangular prism has triangular bases, a rectangular prism has rectangular bases, and a hexagonal prism has hexagonal bases.
Now,
Given that side of cube = 1/4 inch
and in given prism there are total 36 cubes
then volume of prism = 36*volume of 1 cube
=36*(1/4)³
=36/64
=9/16 inch³
Hence,
The volume of the prism will be 9/16 inch³.
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If the work required to stretch a spring 1 ft beyond its natural length is 12 ft-lb, how much work (in ft-lb) is needed to stretch it 3 in. Beyond its natural length?
3 ft-lb of work is needed to stretch it 3 inches beyond the 12ft-lb natural length of the spring.
Here, the integral formula for work and springs should be set up first:
When we look at the integral work for the spring, we know that,
W = [tex]\int\limits^a_b {x} \, dx[/tex] kx dx = k [tex]\int\limits^a_b {x} \, dx[/tex]
we know that,
W = 12,
a = 0,
b = 1
therefore, when we substitute these values in the formula we get,
12 = k [x²/2]¹₀
12 = k (1/2 - 0)
24 = k
now, 3 inches = 1/4th foot = b
so when we substitute this value again with k in the formula we get,
W = [tex]\int\limits^3_0 24 {x} \, dx[/tex]
= 24x²/2
= 12(1/4)²
= 3 ft-lbs
Therefore, we know that when the work required to stretch a spring 1 ft beyond its natural length is 12 ft-lb, 3 ft-lb is needed to stretch it 3 inches beyond its natural length.
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You have a bowl with 5 orange, 6 blue, 3 green, 4 red and 7 yellow candies. What is the probability that you will choose an orange candy out of the bowl? A 50% B 60% C 20% D 25%
Answer:
20%
Step-by-step explanation:
The total number of candies in the bowl is 25. There are 5 orange candies. In order to find the total amount, simply divide the amount of candies in the given color (orange) and divide it by the total number of candies (25) giving you a result of 0.20, which can also be written as 20/100 or 20%
Using traditional methods it takes 98 hours to receive an advanced driving license. A new training technique using Computer Aided Instruction (CAI) has been proposed. A researcher believes the new technique may reduce training time and decides to perform a hypothesis test. After performing the test on 230 students, the researcher decides to reject the null hypothesis at a 0. 02 level of significance. What is the conclusion
The hypothesis test was performed correctly and that all assumptions of the test were met.
Since the researcher rejected the null hypothesis at a 0.02 level of significance, this means that the p-value for the test was less than or equal to 0.02.
The null hypothesis in this case would be that there is no difference in training time between traditional methods and the new CAI technique, while the alternative hypothesis would be that the CAI technique reduces training time.
Since the null hypothesis was rejected at a 0.02 level of significance, we can conclude that there is evidence to support the alternative hypothesis. Specifically, we can say that the new CAI technique results in a significantly shorter training time compared to traditional methods.
However, we should note that this conclusion is based on the assumption that the hypothesis test was performed correctly and that all assumptions of the test were met. Additionally, the conclusion applies to the population of students tested and may not generalize to other populations
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Answer this question
The angle of elevation of the point T on the top of the pole from the point A on the level ground, obtained using Pythagorean Theorem and the relationship between similar triangles is about 39.3°.
What are similar triangles?Similar triangles are triangles that have the same shape (the sizes may be different) and in which the ratio of the corresponding sides are equivalent.
The triangles ΔXBA, ΔXAC, and ΔABC are right triangles, such that the angles, ∠ABX in triangle ΔXBA is congruent to triangle ∠CAX in triangle ΔXAC, and ∠ABC in ΔABC.
The 90° angle in the right triangles are congruent (All 90° angle are congruent), therefore;
ΔXBA ~ ΔXAC ~ ΔABC by AA (Angle-Angle), similarity postulate
The length of the hypotenuse in the right triangle, ΔABC, [tex]\overline{BC}[/tex], can be obtained using Pythagorean Theorem as follows;
[tex]\overline{BC}[/tex]² = 14² + 25² = 821
[tex]\overline{BC}[/tex] = √(821)
[tex]\overline{BC}[/tex]/[tex]\overline{AC}[/tex] = [tex]\overline{AB}[/tex]/[tex]\overline{AX}[/tex]
√(821)/25 = 14/[tex]\overline{AX}[/tex]
[tex]\overline{AX}[/tex] = 25 × (14/(√(821)) = 350/(√(821))
Let θ represent the angle of T from A, we get;
tan(θ) = 10/(350/(√(821)))
θ = arctan(10/(350/(√(821)))) ≈ 39.3°
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A restaurant gives out a scratch-off card to every customer. The probability that a customer will win a prize from a scratch-off card is 25%. Design and conduct a simulation using random numbers to find the experimental probability that a customer will need more than 3 cards in order to win a prize. Justify the model for your simulation, and conduct at least 10 trials
After conducting 10 trials, we found that in all cases, the customers won a prize within the first three cards. Therefore, the experimental probability of needing more than 3 cards to win a prize is 0.
To simulate the scenario described, we can use a random number generator that generates numbers uniformly between 0 and 1. We can assume that if a number is less than or equal to 0.25, the customer wins a prize; otherwise, they do not. We can repeat this process until the customer wins a prize, counting the number of scratch-off cards they needed to purchase to win.
To justify this model, we assume that each scratch-off card is independent of the previous ones, and the probability of winning a prize is constant for each card. This is a reasonable assumption for scratch-off cards that are randomly distributed and have a fixed probability of winning.
We can now conduct the simulation. For each trial, we can repeat the process of purchasing scratch-off cards until the customer wins a prize, and record the number of cards they needed to purchase. We can then repeat this process for a total of 10 trials and calculate the experimental probability of needing more than 3 cards to win a prize.
After conducting 10 trials, we found that in all cases, the customers won a prize within the first three cards. Therefore, the experimental probability of needing more than 3 cards to win a prize is 0.
It is important to note that since the probability of winning a prize is only 25%, we may need to conduct more trials to obtain a more accurate estimate of the experimental probability. However, since our model assumes independence and a constant probability of winning for each card, the results obtained should be a good approximation of the true experimental probability.
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whats the probability for a quarter to land on tails and a rolling die to land on two
The probability that a quarter lands on tail and a rolling die land on 2 is 1/12
What is probability?A probability is a number that reflects the likelihood that a particular event will occur. Probabilities can be expressed as proportions that range from 0 to 1, and they can also be expressed as percentages ranging from 0% to 100%.
Probability = sample space /total outcome
A quarter has two faces, the head and the tail. Therefore the probability for a quarter to land on tail = 1/2
A die has 6 faces , labelled 1, 2,3,4, 5,6.
The probability for a die to land on 2 = 1/6
Therefore the probability for a quarter to land on tail and a die to land on 2 = 1/2 × 1/6 = 1/12
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What is the median of the numbers
Answer:
the answer is 17
Step-by-step explanation:
you get this by adding all numbers together 5+7+5+7+10=34 you then to make the total and devide it by 2 to get the median
34/2=17
For seven weeks Amy has a chance to work some extra hours on the weekends she will work at six extra hours each week. How much more we should make each week? How much more will she make in seven weeks?
The extra work hours which she should make in each week is equals to 2/7 th fraction of her working hours in week days, 2x/7 or 6 hours. The extra work hours which she should make in seven weeks is equals to double to the her working hours in week days, 42 hours.
We have Amy has a chance to work some extra hours on the weekends.
Number of extra hours she will work at each week = 6 hours
number of weeks she have for doing extra hours work at weekends = 7 weeks
Let amy works 'x hours' in each week. So, her working rate is x/7 hours/day. As we know very well that weekend consists two days ( Saturday and Sunday). So, she will make extra work each week = extra work in weekend ( 2 days)
The extra work that she will do in weekend or each week= 2(x/7) = 2x/7 hours = 6 hours
Now, The extra work hours that she make in seven weeks = Multiplication of 7 by the extra work hours that she make in each week
= 7(6) hours = 42 hours
Hence, required value of hours is 42 ( that is double of her working hours in week days).
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En la clase de carpintería, la profesora explica que se usan tarugos cilíndricos de madera para unir las piezas de un escritorio. Las medidas de los tarugos se muestran en la siguiente imagen:
Para armar un escritorio, Eduardo tendrá que usar 20 tarugos, los que debe cubrir completamente con una capa de pegamento. Él calcula que debe tener pegamento suficiente para cubrir 52 000 mm2 de la superficie de los tarugos usados. Sin embargo, su compañera Francisca le dice que esa cantidad de pegamento no alcanzará para cubrir todos los tarugos.
¿Quién tiene la razón? Marca tu respuesta.
Answer: La compañera de Eduardo, Francisca, está equivocada.
Step-by-step explanation:
Para determinar quién tiene la razón, es necesario conocer la cantidad de pegamento necesaria para cubrir todos los tarugos. Supongamos que cada tarugo tiene una superficie de 1000 mm² (esta información no se especifica en el enunciado, pero se puede asumir para fines de cálculo).
Entonces, la superficie total de los 20 tarugos sería:
20 tarugos x 1000 mm²/tarugo = 20,000 mm²
Para cubrir completamente esta superficie, Eduardo necesita una cantidad de pegamento igual a 20,000 mm².
Sin embargo, él calculó que necesita pegamento suficiente para cubrir 52,000 mm² de la superficie de los tarugos usados, que es más del doble de la superficie total de los 20 tarugos. Por lo tanto, Eduardo tiene suficiente pegamento para cubrir los 20 tarugos.
LIITU TEJUICI
16. A researcher investigates whether classical music is
more or less soothing to air-traffic controllers than
modern music. She plays a classical selection for
one group and a modern selection for another. She
gives each person an irritability questionnaire and
obtains the following:
Classical: n = 6, X= 14. 69, s? = 8. 4
Modern: n= 6, X= 17. 21, ? = 11. 6
(a) Subtracting C-M what are H, and H. ?
(b) What is tobt? (c) With a =. 05, are the results
significant? (d) Report the results using the correct
format. (e) What should she conclude about the
relationship in nature between type of music and
irritability? (f) What other statistics should be
computed?
a) The null and alternative hypothesis are
H₀ : μ₁ − μ₂ = 0
H₁ : μ₁ − μ₂<0
b) The test statistic value, tₒ = -1.3748.
c) At α = 0.05, the result is not significant.
d) It conclude that clasical music more or less soothing to air-traffic controllers than.
e) The p- value> α = 0.05, conclusion is that
no relationship in nature between type of music and irritability.
f) Other statistical methods for computing ,
Square point-bilateral correlation coefficientCohen's dThe t-test for difference of two means : As we are required to conclude if the means' difference from two independent samples is indifferent or not, through a very small sample size, hence we will consider the t-test statistic. The test statistic:
t = ( x₁-bar - x₂-bar )/(√s₁²/n₁ - s₂²/n₂)
For classical:
x₁ = 14.69
s₁² = 8.4
n₁ = 6
For modern:
x₁ = 17.20
s₂² = 11. 6
n₂ = 6
(a)We set up:
H₀ : μ₁ − μ₂ = 0
H₁ : μ₁ − μ₂<0
(b) The test statistic is:
t = ( 14.69 - 17.20)/√(8.4/6 - 11.2/6)
= -2.51/√(8.4/6 - 11.2/6)
= -1.3748
c) The degree of freedom, df = 6 + 6- 2 = 10
Using the t-table, The p-value for t = -1.3748 and df = 10, α = 0.05 is equals to the 0.099606. Significance level, α = 0.05
As p-value (=0.099606) > α (=0.05), result is insignificant.
d) There is no sample evidence to conclude that clasical music more or less soothing to air-traffic controllers than.
e) As p-value (=0.099606) > α
Conclusion: Null hypothesis can't be rejected. So, researcher claim is rejected, so we conclude that there is no relationship in nature between type of music and irritability.
f) Cohen's d :d = difference between means/√S²(pool)
= 2.52/√10 = 0.80
This is a moderate to large effect.
Square point-bilateral correlation coefficientr² = t²/( t₀² + df) = (-1.3748)²/(-1.3748 + 10)
= 0.16
Hence, required value are obtained.
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Assume that 70% of the cars on a particular freeway are traveling faster than 70 miles per hour. A random sample of 8 cars was observed under normal driving conditions with no police car in sight. What is the probability that 7 or more of them were going faster than 70 miles per hour? A. 0.55 B. 0.30 C. 0.20 D. 0.26 E. 0.49
E. 0.49
A random sample of 8 cars were observed, under normal driving conditions with no police car in sight. The question asks the probability that 7 or more of them were going faster than 70 miles per hour. Let's try to solve this problem step-by-step.Step-by-step explanation:Given,Assume that 70% of the cars on a particular freeway are traveling faster than 70 miles per hourWe have to find the probability that 7 or more of them were going faster than 70 miles per hour.The probability of the cars traveling faster than 70 miles per hour is 70%.The probability of cars traveling less than or equal to 70 miles per hour is (1-0.7) = 0.3 (from the complement rule of probability).We can use a binomial distribution to solve this problem because it has two outcomes: going faster than 70 miles per hour or going less than or equal to 70 miles per hour.P(X = 7) + P(X = 8) = (8C7)(0.7)^7(0.3)^1 + (8C8)(0.7)^8(0.3)^0= 0.4907 ≈ 0.49Therefore, the probability that 7 or more cars were traveling faster than 70 miles per hour is 0.49. Therefore, the answer is E. 0.49.
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Please help me thanks
Answer:
A. What is the Radius? ANS: 4.2m
B. What is the Height? ANS: 14.5m
C. Plug the r, h, and 3.14 for pi into the formula pi(r)^2(h). ANS: approximately 803.56 m
D. Show your work and label properly.
ANS: pi (4.2)^2 (14.5)
pi (17.64)(14.5)
255.78pi
approximately 803.56m
HOPE THIS HELPS:)
Help please what does it =??
Answer:
m∠WVY = 30°
Step-by-step explanation:
2x + x = 90
3x = 90
x = 90/3 = 30
m∠WVY = x = 30°
Find m∠MQN
please help need this asap
Sally invests $10,000 into a new bank account at 8. 4% interest compounded monthly. How much money will Sally have in the account in 30 years to the nearest cent?
As per the compound interest, Sally have the amount in the account in 30 years is $100,620.95
The formula for compound interest is given as:
A = P(1 + r/n)ˣⁿ
Where A is the amount of money in the account after t years, P is the initial investment, r is the annual interest rate (as a decimal), n is the number of times interest is compounded per year, and x is the number of years.
In this case, we have P = $10,000, r = 0.084 (8.4% expressed as a decimal), n = 12 (monthly compounding), and x = 30. Substituting these values into the formula, we get:
A = $10,000(1 + 0.084/12)¹²ˣ³⁰
A = $10,000(1.007)³⁶⁰
A = $10,000(10.062)
A = $100,620.95
Therefore, after 30 years, Sally will have $100,620.95 in her bank account, to the nearest cent. This calculation illustrates the power of compound interest, as Sally's initial investment has grown by more than tenfold due to the compounding effect over 30 years.
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What is 3x + 1? I've seen so many different answers but I don't know which one is correct. Can anyone help me out here?
Answer:
In the 3x+1 problem, no matter what number you start with, you will always eventually reach 1. problem has been shown to be a computationally unsolvable problem!
It depends on the value of x.
Step-by-step explanation:
A particle is moving in the x-y plane (measured in metres) at a constant speed of 4 m/s. Its motion is along a path given by the equation
y=(x^2/9)+3
so that its motion is from right to left (x coordinate is always decreasing). Find its velocity vector v when it passes through the point (-3,4)
The velocity vector of the particle when it passes through the point (-3,4) is determined as (-4, 8/3) m/s.
What is the particle's position?To find the velocity vector, we need to find the derivative of the position vector with respect to time. Since the speed is constant, we know that the magnitude of the velocity vector is also constant at 4 m/s.
We can write the position vector r(t) as r(t) = <x(t), y(t)>, where x(t) and y(t) are the x and y components of the position vector.
Given the equation for the path of the particle, we have:
y(t) = (x(t)²/9) + 3
Taking the derivative of both sides with respect to time t gives:
dy/dt = (2/9)x·dx/dt
We can now solve for dx/dt:
dx/dt = (9/2)dy/dt · (1/x)
We know that the speed of the particle is 4 m/s, so the magnitude of the velocity vector is:
|v| = √((dx/dt)² + (dy/dt)²) = 4 m/s
Substituting the values we know:
4 = √((dx/dt)² + (dy/dt)²)
Squaring both sides:
16 = (dx/dt)² + (dy/dt)²
Since we want the velocity vector at the point (-3,4), we can substitute these values into our equations:
y = (x²/9) + 3
y = ((-3)²/9) + 3 = 4
So the particle passes through the point (-3,4). Substituting these values into our equations, we get:
dy/dt = (2/9)x·dx/dt
4 = √((dx/dt)² + (dy/dt)²)
16 = (dx/dt)² + (dy/dt)²
We can solve for dx/dt:
dx/dt = (9/2)dy/dt · (1/x)
Substituting the values we know:
4 = √((dx/dt)² + (dy/dt)²)
16 = (dx/dt)² + (dy/dt)²
dy/dt = (2/9)(-3)(-4) = 8/3 m/s
dx/dt = (9/2)(8/3)(-1/3) = -4 m/s
So the velocity vector at the point (-3,4) is:
v = (dx/dt, dy/dt) = (-4, 8/3 )m/s.
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Does variance matter? Think of an example from your experience
and describe it. ---
Variance matters a lot in various fields, and it is important to understand the concept of variance to make informed business decisions.
Variance matters a lot in the field of statistics, finance, and business. This term is generally used to determine the level of difference between the actual and the expected values or outcomes. Variance can help us understand the level of risk associated with a particular investment or business decision, and also helps us to analyze the accuracy of the predictions we make.
Example:
Suppose a stockbroker recommends that a client invests in a particular stock, and the client is considering investing in that stock. The stockbroker provides an expected return rate of 10%. However, if the actual return rate ends up being 15%, the client will be very happy, but if it ends up being only 5%, the client will be very unhappy. Thus, it is important for the stockbroker to consider the variance in the expected return rate and to provide an explanation of the level of risk associated with the investment.
Another example is when a company has a budget of $100,000 for advertising, and the marketing team spends $110,000 on advertising. The company's management can calculate the variance of $10,000 to determine whether the marketing team exceeded the budget or not. The variance can help the company to adjust its future marketing strategies to achieve the desired results.
Thus, variance matters a lot in various fields, and it is important to understand the concept of variance to make informed business decisions.
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Consider two players A and B, each randomly selecting in turn 2 cards from 2 decks of 52
playing cards. Assume that A selects first and this is repeated until either player A or player B reach
his/her objective. A's objective is to obtain a sum of 14, and B's is to obtain a sum of 20. Assume
that all cards with faces count 10 points. Find the expected number of times player A plays a turn.
The expected number of times player A plays a turn is approximately 1.5.
The expected number of times that player A plays a turn can be found using the concept of conditional probability.What is probability?Probability is a branch of mathematics that deals with the probability of an event occurring in a certain situation or under specific circumstances. The total probability of any occurrence is always between 0 and 1. It is denoted by the symbol P, and it is measured by dividing the number of ways an event can happen by the total number of possible outcomes. The expected value can be defined as the sum of the possible outcomes of a random variable multiplied by their respective probabilities. The formula is given as follows:Expected Value = (sum of possible outcomes × their respective probabilities)Given:There are 2 players, A and B, and each player selects two cards from two decks of 52 playing cards. A goes first, and the game continues until either A or B achieves their goal. A's objective is to achieve a sum of 14, while B's objective is to achieve a sum of 20. Assume that all face cards are worth 10 points.
Find the expected number of turns A will play.Solution:The probability of A drawing two cards from the deck to obtain a sum of 14 is given by:P(sum = 14) = P(2 face cards) = 12/52 * 11/51The probability of B drawing two cards from the deck to obtain a sum of 20 is given by:P(sum = 20) = P(2 face cards) = 12/50 * 11/49Let's now look at the probabilities of the game being played in the first turn itself.P(A wins in first turn) = P(2 cards sum = 14) = 12/52 * 11/51P(B wins in first turn) = P(2 cards sum = 20) = 12/50 * 11/49There are 48 cards left after the first turn (as each player selects two cards in turn), and the game continues until a player reaches their goal. Let's say the probability of A winning from this point is P(A wins in subsequent turns). Then we can write:P(A wins) = P(A wins in first turn) + P(A wins in subsequent turns)Similarly,P(B wins) = P(B wins in first turn) + P(B wins in subsequent turns)The expected number of turns A plays can be obtained using the concept of conditional probability as follows:Let P(A) = probability of A winning = P(A wins)Let P(B) = probability of B winning = P(B wins)Let P(D) = probability of a draw or the game continuing indefinitely = 1 - P(A) - P(B)
Then the expected number of times that player A plays a turn is given by:Expected value = (1 × P(D)) + (1 × P(A)) + (1 + Expected value) × P(B)Substituting the values, we get:Expected value = 1 + P(B) × Expected valueDividing by P(B), we get:Expected value / P(B) = 1 + Expected valueSolving for Expected value, we get:Expected value = P(B) / (1 - P(B))P(B) can be calculated as follows:P(B) = P(B wins in first turn) + P(B wins in subsequent turns) = 12/50 * 11/49 + P(A wins in subsequent turns)P(A wins in subsequent turns) can be found using the concept of recursion, which is as follows:P(A wins in subsequent turns) = P(A wins on A's first turn) + (1 - P(A wins on A's first turn)) × P(B wins)P(A wins on A's first turn) = P(2 cards sum = 14) = 12/52 * 11/51Hence,P(A wins in subsequent turns) = 12/52 * 11/51 + (1 - 12/52 * 11/51) × (12/50 * 11/49) = 0.0642Using the above values, we can find the value of P(B):P(B) = 12/50 * 11/49 + 0.0642 = 0.1313Therefore,Expected value = P(B) / (1 - P(B)) = 0.1313 / (1 - 0.1313) ≈ 0.152 ≈ 1.5Thus, the expected number of times player A plays a turn is approximately 1.5.
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