The interest compounded equation to model the given situation is A(t) = 4000[tex]e^{0.05t}[/tex].
To model the given situation, the interest compounded can be written as the given equation of:
A(t) = A0e^(rt)
Where:
A(t) is the amount of money after t years
A0 is the initial amount of money, in this case,
r is the interest rate expressed as a decimal
t is the time in years that the money is invested.
From the case, we know that:
Initial deposit = $4,000
Interest rate = 5%
Time period = t (undetermined)
So, substituting the values given in the problem, the equation to model the given situation is:
A(t) = A0e^(rt)
A(t) = 4000e^(0.05t)
Therefore, A(t) = 4000e^(0.05t) is the required equation that models the given situation.
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Julie is backpacking to the Blue Ridge Mountains. She starts from M, travels a 3 3/8 mi to N, travels 2 1/4 mi to P, then walks 4 11/24 mi back towards M. If M, N, and P lie on a straight path, how far is Julie from the starting point M?
Julie is trekking to the Blue Ridge Mountains, therefore she is 10 1/12 miles from the beginning point M.
what is distance ?Distance in mathematics is a numerical representation of the actual area between two points. The shortest distance between those two points is how long the path is. The Pythagorean theorem, which asserts that in a right triangle, the sum of the squares of the lengths of the two legs (the sides perpendicular to one another) is equal to the square of the length of the hypotenuse, is used to determine the distance between two locations in a two-dimensional plane (the longest side, opposite the right angle).
given
We must determine Julie's total distance traveled before we can determine how far she is from the beginning location M. By combining the distances between each location, we may determine this:
3 3/8 miles from M to N
2 1/4 miles from N to P
P returning to M: 4 11/24 miles
We must identify a common denominator in order to aggregate these distances. Because it can be divided by 8, 4, and 3, we may pick 24 as the common denominator.
From M to N, the distance is 3 3/8 miles (or 27/8 kilometers). From N to P, it is 2 1/4 kilometers (or 9 kilometers).
We may now multiply the distances:
27/8 + 9/4 + 107/24 = (81 + 54 + 107) / 24
= 242/24
= 10 1/12
Julie is trekking to the Blue Ridge Mountains, therefore she is 10 1/12 miles from the beginning point M.
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Is 4.284 an irrational number?
Answer: "An irrational number is a number that cannot be expressed as a ratio between two integers and is not an imaginary number.
Since 4.284 is not the square root of a negative number, it is not imaginary
Since 4.284 is a rational number from above, 4.284 is not an irrational number"
Step-by-step explanation: /\ I looked at a calculator for irrational numbers. Should be right, considering how it's well explained. Just search for "irrational number calculator". Dont rely on that. But it's there if
you need it! :)
No
It is a rational number since it is a fraction/decimal that isn't non-terminating, not is it pi or
A Nigerian visiting India changed N70200 to rupees at the rate of 3 naira to 35 rupees. He spent 224 000 rupees and invested the remaining amount in the State Bank of India at 41.5% simple interest per annum. At the end of 8 months, he transferred the capital and interest to his account in the Modern Bank of Nigeria at the rate of 21 rupees to 2 naira. What was the amount, in naira, credited to his account, to the nearest naira?
According to the solving this to the nearest naira, the amount credited to his account is 580,163 Nigerian naira.
Describing percentage:A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to calculate a percentage of a number, we should divide it by its entirety and then multiply it by 100. The proportion, therefore, refers to a component per hundred. Per 100 is what the word percent means.
According to the given information:The Nigerian visitor changed N70200 to rupees at a rate of 3 nairas to 35 rupees. Therefore,
70200 Nigerian naira = 70200 * 35 / 3 = 819500 Indian rupees
He spent 224,000 rupees, so the amount he invested at 41.5% per annum was:
819500 - 224000 = 595500 rupees
The simple interest he earned after 8 months at a rate of 41.5% per annum is:
595500 * (41.5/100) * (8/12) = 129702.5 rupees
So, the total amount he had after 8 months was:
595500 + 129702.5 = 725202.5 rupees
He then transferred this amount to his account in the Modern Bank of Nigeria at a rate of 21 rupees to 2 naira. Therefore,
725202.5 rupees = (725202.5 / 21) * 2 = 580162.5 Nigerian naira
this to the nearest naira, the amount credited to his account is 580,163 Nigerian naira.
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Complete the equation that has $(3,4)$ as a solution.
y= _x--2
Answer: Your welcome!
Step-by-step explanation:
y= 3x-2; This is a linear equation with a slope of 3 and a y-intercept of -2, which has the solution (3,4).
Thanks! #BO
(20P) Help please and thankyou it’s due soon
A lorry travels 320km and uses 40 litres of petrol, work out the average rate of petrol usage. Amswer in km. Litre
If a lorry travels 320km and uses 40 litres of petrol, the average rate of petrol usage for the lorry is 8 km per liter.
To find the average rate of petrol usage for the lorry, we need to divide the total distance traveled by the amount of petrol used. This will give us the number of kilometers traveled per liter of petrol.
In this case, the lorry traveled 320 km and used 40 liters of petrol, so we can calculate the average rate of petrol usage as follows:
Average rate of petrol usage = Total distance traveled / Amount of petrol used
= 320 km / 40 litres
= 8 km/litre
This means that for every liter of petrol used, the lorry can travel an average of 8 kilometers. This metric can be useful in comparing the fuel efficiency of different vehicles or in calculating the cost of a particular journey based on the price of petrol per liter.
In summary, calculating the average rate of petrol usage involves dividing the distance traveled by the amount of petrol used, resulting in a unit of km per liter.
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B) The formula of connecting mass and weight is
W=m x acceleration due to gravity. What is the value of acceleration due to gravtiy on Earth
The acceleration due to gravity on Earth is approximately 9.81 meters per second squared (m/s^2)
Acceleration due to gravity is the acceleration experienced by an object when it is dropped or falls freely in a gravitational field. On Earth, the value of acceleration due to gravity is approximately 9.81 meters per second squared. This means that if an object is dropped from a certain height, its velocity will increase by 9.81 m/s^2 for every second it falls.
The formula connecting mass and weight, W = m x g, shows that weight is directly proportional to the acceleration due to gravity. This means that as the value of acceleration due to gravity changes, the weight of an object will also change accordingly. The value of acceleration due to gravity is an important factor in various fields, including physics, engineering, and astronomy.
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A machine takes 2.8 hours to make 9 parts. At that rate, how many parts can the machine make in 28.0 hours?
Answer:
The machine can make 9 parts in 2.8 hours.
To find the rate of production, we can divide the number of parts by the time: 9 parts / 2.8 hours = 3.214 parts per hour.
Now that we know the machine's rate of production, we can use it to answer the question:
In 28.0 hours, the machine will produce: 3.214 parts per hour x 28.0 hours = 89.9 parts.
Therefore, the machine can make 89.9 parts in 28.0 hours at the given rate. We can round this to 90 parts.
Step-by-step explanation:
Tristan has $1. 40 worth of nickels and dimesm he has twice as many nickels as dimes
Tristan has 14 nickels and 7 dimes worth of $1. 40.
This is because 1 nickel is worth 5 cents and 1 dime is worth 10 cents.
Let's use the following variables to represent the number of nickels and dimes Tristan has:
n = number of nickels
d = number of dimes
We know that Tristan has $1.40 worth of nickels and dimes. Each nickel is worth $0.05 and each dime is worth $0.10, so we can write an equation based on their values:
0.05n + 0.1d = 1.4
We also know that Tristan has twice as many nickels as dimes:
n = 2d
We can substitute n = 2d into the first equation and solve for d:
0.05(2d) + 0.1d = 1.4
0.1d + 0.1d = 1.4
0.2d = 1.4
d = 7
So Tristan has 7 dimes. Using n = 2d, we can find the number of nickels:
n = 2(7) = 14
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PLEASE HELP MARKING BRAINLEIST JUST ANSWER ASAP
Answer:
The perimeter is the sum of all the sides of the rectangle. So, adding up all the given sides, we get:
Perimeter = 2u + u + 10 + u + 10 + 8u
Simplifying the expression by combining like terms, we get:
Perimeter = 12u + 20
Therefore, the simplified answer for the perimeter is 12u + 20.
i need help asappppppppppppppppppppppppppppppppppppppppppppppppppppp
Answer: A ♀️
Step-by-step explanation:
25 cm 7 cm 15 cm what is the area of triangle
The area of the triangle with side lengths of 25 cm, 7 cm, and 15 cm is approximately 209.27 cm².
To calculate the area of a triangle with side lengths of 25 cm, 7 cm, and 15 cm, we can use Heron's formula, which is a formula for finding the area of a triangle when only the side lengths are known:
Area = √(s(s-a)(s-b)(s-c))
where s is the semi-perimeter of the triangle, and a, b, and c are the lengths of its sides. The semi-perimeter is half the sum of the three sides:
s = (a + b + c) / 2
Substituting the given values, we get:
s = (25 + 7 + 15) / 2 = 23.5
Now we can use Heron's formula to calculate the area:
Area = √(23.5(23.5-25)(23.5-7)(23.5-15))
= √(23.5 * (-1.5) * 16.5 * 8.5)
= √(43,822.5)
≈ 209.27 cm²
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solve for b??
5b-3 > 9b+4
Answer: b<-7/4
Step-by-step explanation:
Let's collect numbers with same variables to the same side:
-3-4>9b-5b
-7>4b
b<-7/4
Finnd the equation of straight line passing through the point (3a,0) and (0,3b) also shwo that the line passese throught the poinits (a , 2bb)
The point (a,2b) is on the line, and the equation of the line passing through (3a,0) and (0,3b) is y = (-a/b)x + 3b
We can use the two-point form of the equation of a straight line to find the equation passing through the points (3a,0) and (0,3b).
The two-point form of a straight line is given by,
y - y1 = m(x - x1)
where (x1, y1) and (x, y) are two points on the line, and m is the slope of the line.
Let's take the point (3a,0) as (x1, y1) and the point (0,3b) as (x, y). Then, we have:
y - 3b = [(0 - 3a)/(3b - 0)](x - 0)
Simplifying this, we get:
y - 3b = (-a/b)x
y = (-a/b)x + 3b
This is the equation of the line passing through (3a,0) and (0,3b).
Now, to show that this line passes through the point (a,2b), we substitute x = a and y = 2b into the equation of line:
2b = (-a/b)a + 3b
2b = -a + 3b
a = b
Therefore, the point (a,2b) is on the line
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SHOW STEPS! PLS HELP
Answer:
chocolate = $1.25
soft drink = $1.85
Step-by-step explanation:
Given price of chocolate is c and price of soft drink is s
Olivia: 5c + 2s = 9.95
Taylor: 6c + 6s = 18.60
6c + 6s = 18.60
divided by 6, we have
c + s = 3.1
=> s = 3.1 - c
Substitute s = 3.1 - c into
5c + 2s = 9.95
5c + 2(3.1 - c) = 9.95
5c + 6.2 - 2c = 9.95
3c = 9.95 - 6.2
3c = 3.75
c = 3.75/3 = 1.25
s = 3.1 - c = 3.1 - 1.25 = 1.85
Find [fog](x) and [gof](x), if they exist. State the domain and range for each.
5.f(x) = -3x
g(x) = x +8
6. f(x) = 2x²-x + 1
g(x) = 4x + 3
Geometry: solve this problems, it’s urgent
1. triangle = 180
7x7 = 49
180-49 = 131
2. 10x10 = 100 but since its a pyramid its degrees is 180
180-100=80
What is the greatest number of tens that can live in the tens house?
The greatest number of tens that can live in the tens house is 9.
The greatest number of tens that can live in the tens house depends on how many digits there are in the number.
Let's start with a two-digit number, like 99. In this number, the leftmost digit represents the number of tens, so we want to find the largest digit that can be in that place.
Since we are limited to using only the digits 0-9, the largest digit that can be in the tens place is 9.
Now, let's consider a three-digit number, like 456. Again, we want to find the largest digit that can be in the hundreds place.
Since we are limited to using only the digits 0-9, the largest digit that can be in the hundreds place is also 9. So, we can have up to 9 groups of ten (or 90) in the hundreds place.
In the tens place, we can again have up to 9 groups of ten, and in the units place, we can have any digit from 0-9.
Therefore, the greatest number of tens that can live in the tens house in a three-digit number is 9.
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The following figure is made of 3 triangles and 1 rectangle.
H
248
B
2
Figure
Triangle A
Triangle B
Rectangle C
Triangle D
Whole figure
C2
2
A
D
Find the area of each part of the figure and the whole figure.
2
4
6
Area (square units)
The area of each part of the figure and then the whole figure, can be found to be :
Triangle A - Triangle B - 2 Rectangle C - 4 Triangle D - Whole figure - How to find the area ?The area of Triangle A would be :
= 1 / 2 x Base x height
= 1 / 2 x ( 2 + 2 + 6 ) x 4
= 1 / 2 x 10 x 4
= 20 units ²
The area of Triangle D is :
= 1 / 2 x base x height
= 1 / 2 x 6 x 2
= 6 units ²
The area of the whole figure would then be:
= 20 + 2 + 4 + 6
= 32 units ²
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Angela can shovel the snow from her driveway in 2 hours. When Franklin joins her, the driveway can be finished in just 54 minutes. How long would it take Franklin to shovel the driveway alone?
By speed formula, it would take Franklin 2 hours and 30 minutes (or 150 minutes) to shovel the driveway alone.
What is speed?
Speed is defined as the distance travelled by an object in a given amount of time. Speed is a scalar quantity, meaning that it has magnitude but no direction.
Mathematically, speed is calculated as follows:
speed = distance / time
where "distance" is the distance travelled by the object, and "time" is the time it takes for the object to travel that distance.
Let's assume that Angela's shoveling rate is "a" and Franklin's shoveling rate is "f" (measured in driveways per hour). We can use the formula:
time = distance / rate
where "distance" is the length of the driveway (which we can assume to be 1 driveway) and "rate" is the shoveling rate (in driveways per hour).
According to the problem, Angela can shovel the driveway in 2 hours, so her shoveling rate is:
a = 1/2
When Franklin joins her, they can finish the driveway in 54 minutes, or 9/10 of an hour. Therefore, their combined shoveling rate is:
(a + f) = 1 / (9/10) = 10/9
We can now set up a system of equations to solve for "f".
First, we know that Angela and Franklin can finish the driveway in 9/10 of an hour:
1/2 + f = 1 / (9/10)
Multiplying both sides by 10/9, we get:
5/9 + (10/9)f = 1
Simplifying, we get:
(10/9)f = 4/9
f = (4/9) * (9/10)
f = 4/10
f = 2/5
Therefore, Franklin's shoveling rate is 2/5 of a driveway per hour. To find how long it would take him to shovel the driveway alone, we can use the formula:
time = distance/rate
time = 1 / (2/5)
time = 5/2
time = 2 1/2 hours
Therefore, it would take Franklin 2 hours and 30 minutes (or 150 minutes) to shovel the driveway alone.
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the density of apple juice is 1.04 grams per cm³
the density of fruit syrup is 1.6 grams per cm³
the density of sparkling water is 0.99 grams per cm³
35 cm³ of apple juice are mixed with 25 cm³ of fruit syrup and 270 cm³ of sparkling water to make a drink with a volume of 330 cm³
work out the density of the drink
Therefore, the density of the drink is approximately 1.04 grams per cm³.
What is volume?Volume is a measure of the amount of space occupied by a three-dimensional object or substance. It is the amount of space inside an object or container and is usually measured in cubic units such as cubic meters (m³), cubic centimeters (cm³), or cubic feet (ft³).
Given by the question.
To calculate the density of the drink, we need to first calculate the total mass of the drink, which is the sum of the masses of apple juice, fruit syrup, and sparkling water.
Mass of apple juice = volume of apple juice x density of apple juice = 35 cm³ x 1.04 g/cm³ = 36.4 g
Mass of fruit syrup = volume of fruit syrup x density of fruit syrup = 25 cm³ x 1.6 g/cm³ = 40 g
Mass of sparkling water = volume of sparkling water x density of sparkling water = 270 cm³ x 0.99 g/cm³ = 267.3 g
Total mass of the drink = mass of apple juice + mass of fruit syrup + mass of sparkling water
= 36.4 g + 40 g + 267.3 g
= 343.7 g
Now we can calculate the density of the drink by dividing the total mass by the volume of the drink:
Density of the drink = total mass of the drink / volume of the drink = 343.7 g / 330 cm³ = 1.04 g/cm³ (approx.)
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help asap assignment closes soon!
Answer:
a = 12.56637061
Step-by-step explanation:
a= 4 · π · r²
a= 4 · π · 1²
a= 4π
a= 12.56637061
a= 12.57
HELPPPPP
In 2005, a sample of a radioactive substance had a mass of 600 milligrams. Since then, the sample has decayed by 4.8% each year.
Lett be the number of years since 2005. Let y be the mass of the substance in milligrams.
Write an exponential function showing the relationship between y and t.
The exponential function that relates the mass of the substance to the number of years since 2005 is: [tex]y = 600 \times e^(-0.048t)[/tex] . [Where t is a number of years since 2005, and y is mass of the substance in milligrams at that time.]
What is an exponential function?An exponential function is a mathematical function of the form[tex]f(x) = a^x,[/tex] representing a rapid increase or decrease in value as x increases or decreases.
It represents a rapid growth or decay in value as the independent variable changes, and is used to model many natural phenomena such as population growth, compound interest, and radioactive decay.
The radioactive decay of the substance follows an exponential decay model. The formula for exponential decay is:
[tex]y = a * e^(-rt)[/tex]
Where:
y - amount of substance at time t.
a - initial amount of the substance.
r - decay rate per unit of time.
t - time elapsed since the start of the decay.
In this case, we know that the initial mass of the substance in 2005 was 600 milligrams. We also know that the substance decays by [tex]4.8[/tex] % each year, which means that the decay rate per year is [tex]0.048[/tex] (4.8% expressed as a decimal).
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the number of netflix subscribers in latin america has increased a lot in recent years the number of paid subscribers from 2018-2020 was:
Answer: 37.5 million
Step-by-step explanation:
I looked it up
you are playing super mario bros together with 2 of your friends. you got to level 4, where you encounter your nemesis bowser. bowser is very strong, and he is defeated only 41% of the times. each of you will play level 4 one time. (a) (2 points) let x be the total number of times that bowser is defeated. what is the distribution of x? (b) (3 points) what is the probability that only 1 of you defeats bowser? (c) (2 points) you want to understand how likely it is to correctly predict the number of times bowser is defeated. what is the variance of x? (d) (1 point) what is the probability that you beat bowser - regardless of whether your friends beat him or not? suppose that, after your friends are gone, you decide to play level 4 until you beat bowser. let y be the number of times you play level 4. (e) (3 points) what is the distribution of y? (f) (3 points) what is the probability that you play less than 3 times? (g) (3 points) what is the expected number of times that you play?
(a) The distribution of x is a binomial distribution with n=3 and p=0.41, where n is the number of trials (each of you playing level 4 one time) and p is the probability of success (defeating Bowser).
(b) The probability that only 1 of you defeats Bowser is given by the binomial probability formula:
P(x=1) = (3 choose 1)(0.41)^1(0.59)^2 = 0.411
(c) The variance of x is given by the formula:
Var(x) = np(1-p) = 3(0.41)(0.59) = 0.726
(d) The probability that you beat Bowser, regardless of whether your friends beat him or not, is simply the probability of success in one trial, which is 0.41.
(e) The distribution of y is a geometric distribution with p=0.41, where p is the probability of success (defeating Bowser).
(f) The probability that you play less than 3 times is given by the sum of the probabilities of playing 1 or 2 times:
P(y<3) = P(y=1) + P(y=2) = (0.41)^1(0.59)^0 + (0.41)^1(0.59)^1 = 0.651
(g) The expected number of times that you play is given by the formula:
E(y) = 1/p = 1/0.41 = 2.439
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A number is chosen from 1 to 20. Find the probability that the number chosen is a odd prime number
The probability of choosing an odd prime number from 1 to 20 is 0.35
The probability is the ratio of the number of favorable outcomes to the total number of outcomes
The odd prime numbers between 1 and 20 are 3, 5, 7, 11, 13, 17, and 19. There are 7 odd prime numbers in this range.
The total number of possible choices is 20 (since there are 20 numbers in the range 1 to 20).
Therefore, the probability of choosing an odd prime number is:
number of odd prime numbers / total number of possible choices
= 7 / 20
= 0.35
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Relative to the origin O, the position vectors of two points A and B are a and b respectively. b is a unit vector and the magnitude of a is twice that of b. The angle between a and b is 60°. Show that [a×[ob + (1-o)a] =√k, where k is a constant to be determined.
Answer:
|a × [ob + (1 - o)a]| = √(7 - 8o(a · o) - 8(a · o)^2)
where k = 7 - 8o(a · o) - 8(a · o)^2.
Step-by-step explanation:
Given the position vectors of two points A and B as a and b respectively, where b is a unit vector, and the magnitude of a is twice that of b, we are asked to show that:
|a × [ob + (1-o)a]| = √k,
where k is a constant to be determined.
We can begin by expanding the vector inside the cross product:
ob + (1 - o)a = ob + a - oa
Since b is a unit vector, we can write:
ob = b - o
Substituting this into the previous equation, we get:
ob + (1 - o)a = b - o + a - oa = b + (1 - o)a - oa
Next, we can use the vector cross product formula:
|a × b| = |a||b|sinθ
where θ is the angle between a and b.
We are given that the angle between a and b is 60°, so we can substitute this value into the formula:
|a × b| = |a||b|sin60° = (2|b|)(1)(√3/2) = √3
Now we can calculate the cross product of a and the vector we just derived:
a × [ob + (1 - o)a] = a × (b + (1 - o)a - oa)
= a × (b + a - oa)
= a × b + a × a - a × oa
Since b is a unit vector, we know that a × b is a vector perpendicular to both a and b, and therefore perpendicular to the plane containing a and b. The vector a × a is 0 since the cross product of a vector with itself is 0. Finally, we can use the vector triple product to simplify a × oa:
a × oa = (a · a)o - (a · o)a = |a|^2 o - (a · o)a
Since |a| is twice |b|, we have:
|a|^2 = 4|b|^2 = 4
Substituting this back in, we get:
a × oa = 4o - (a · o)a
Putting it all together, we have:
a × [ob + (1 - o)a] = a × b + 4o - (a · o)a
Now we can take the magnitude squared of both sides:
|a × [ob + (1 - o)a]|^2 = (a × b + 4o - (a · o)a) · (a × b + 4o - (a · o)a)
Expanding the dot product, we get:
|a × [ob + (1 - o)a]|^2 = |a × b|^2 + 16o^2 + |a|^2(o · o) - 8o(a · o)b + 8(a · o)(a × b) - 2(a · o)^2|a|^2
Substituting the values we derived earlier, we get:
|a × [ob + (1 - o)a]|^2 = 3 + 16o^2 + 4(o · o) - 8o(a · o) + 0 - 2(a · o)^2(4)
= 7 - 8o(a · o) - 8(a · o)^2
Now we need to find the value of k such that the left-hand side equals k:
|a × [ob + (1 - o)a]|^2 = k
Using the vector triple product again, we can simplify the left-hand side as:
|a × [ob + (1 - o)a]|^2 = |a|^2|ob + (1 - o)a|^2 - ((a · [ob + (1 - o)a])^2)
Since we know that the magnitude of a is twice that of b, we have:
|a|^2 = 4|b|^2 = 4
Substituting this back in, we get:
|a × [ob + (1 - o)a]|^2 = 4|ob + (1 - o)a|^2 - ((a · [ob + (1 - o)a])^2)
Now we can substitute the expanded expression for ob + (1 - o)a:
|a × [ob + (1 - o)a]|^2 = 4|b + (1 - o)a|^2 - ((a · [b + (1 - o)a - oa])^2)
= 4|b|^2 + 8|b|(1 - o)(a · b) + 4(1 - o)^2|a|^2 - ((a · b + (1 - o)(a · b) - (a · o)(a · b))^2)
= 4 + 8(1 - o)(a · b) + 4(1 - o)^2(4) - ((a · b + (1 - o)(a · b) - (a · o)(a · b))^2)
= 28 - 8o(a · b) - 8(a · o)^2
Substituting this back into the previous equation, we get:
28 - 8o(a · b) - 8(a · o)^2 = k
Therefore, we have:
|a × [ob + (1 - o)a]| = √(28 - 8o(a · b) - 8(a · o)^2) and
k = 28 - 8o(a · b) - 8(a · o)^2
Hope this helps! Sorry if it's wrong! If you need more help, ask me! :]
98 kilometers in 7 hours = how many kilometers per hour
Answer:
[tex]\huge\boxed{\sf 14 \ km}[/tex]
Step-by-step explanation:
Given that,7 hours = 98 km
Divide both sides by 7
7/7 hour = 98/7 km
1 hour = 14 km[tex]\rule[225]{225}{2}[/tex]
solve this problem for me
The discounted price of the camera is $270 and the price of the camera after the 40% increase is $378.
When the store offered a 40% discount on the original price of $450, the discounted price of the camera can be calculated as follows:
Discounted price = Original price - Discount
Discounted price = $450 - 40% x $450
Discounted price = $450 - $180
Discounted price = $270
Therefore, the discounted price of the camera is $270.
After the sale, the discounted price of the camera was increased by 40%. We can calculate the new price of the camera as follows:
New price = Discounted price + 40% x Discounted price
New price = $270 + 40% x $270
New price = $270 + $108
New price = $378
Therefore, the price of the camera after the 40% increase is $378.
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The probability that each item coming off a production line is defective is p and the probability that it is non-defective is q, 0 < p < 1, p + q = 1. At the beginning of a day’s production, a quality control officer repeatedly inspects items each coming off a production line until he inspects n items. Let X be the number of defective items he finds. (a) Write down without proof the probability that X = k, indicating the possible values of k. Hence, by considering the expansion of 〖(p+q)〗^n+〖(q-p)〗^n. Show that the probability that X is even is 1/2[1+〖(1-2p)〗^n] .(Note that 0 is an even number) (b) Find the expected value of X, and write down without proof the variance of X. (c) If E(X) = 0.0125 and Var(X) = 0.9875, find to 4 decimal places the probability that X is odd.
From the given information provided, the expected value of X = np, the variance of X = npq and the probability that X is odd is 0.4824 rounded to four decimal places.
(a) The probability of finding k defective items out of n can be calculated using the binomial distribution:
P(X=k) = C(n,k) ×[tex]p^k[/tex] × [tex]q^(n-k)[/tex], where C(n,k) is the binomial coefficient.
The possible values of K: 0, 1, 2, ..., n.
To show that the probability that X is even is 1/2[1+(1-2p)ⁿ], we use the binomial theorem to expand (p+q)ⁿ+(q-p)ⁿ as:
(p+q)ⁿ + (q-p)ⁿ = ∑[k=0,n]C(n,k) × [tex]p^k[/tex] × [tex]q^(n-k)[/tex] + ∑[k=0,n]C(n,k) × [tex](-1)^k[/tex] × [tex]p^k[/tex]× [tex]q^(n-k)[/tex]
The first sum corresponds to the probability of finding an even number of defective items, while the second sum corresponds to the probability of finding an odd number of defective items. Therefore,
P(X is even) = (1/2)[(p+q)ⁿ + (q-p)ⁿ]
= (1/2)[(p+q)ⁿ - (p-q)ⁿ] (since q-p = 1-2p)
= (1/2)[(1)ⁿ + (1-2p)ⁿ] (since p+q = 1)
Thus, the probability that X is even is 1/2[1+(1-2p)ⁿ].
(b) The expected value of X is:
E(X) = ∑[k=0,n]k × P(X=k)
= ∑[k=0,n]k × C(n,k) × [tex]p^k[/tex]× [tex]q^(n-k)[/tex]
Using the identity ∑[k=0,n]k × C(n,k) ×[tex]p^k[/tex] × [tex]q^(n-k)[/tex] = np, the expected value simplifies to:
E(X) = np
The variance of X is given by:
Var(X) = E(X²) - [E(X)]²
= ∑[k=0,n]k² × P(X=k) - (np)²
= ∑[k=0,n]k² × C(n,k) × [tex]p^k[/tex] × [tex]q^(n-k)[/tex]- n²p²
Using the identity ∑[k=0,n]k² × C(n,k) ×[tex]p^k[/tex] × [tex]q^(n-k)[/tex] = n(n-1)p² + npq, the variance simplifies to:
Var(X) = n(n-1)p² + npq - n²p²
= npq
(c) Using the formula for the expected value and variance of X, we can write:
0.0125 = E(X) = np
0.9875 = Var(X) = npq
Solving for p and q, we obtain:
p = 0.01 and q = 0.99
Therefore, the probability that X is odd can be calculated using the formula for the probability that X is even derived in part (a):
P(X is odd) = 1 - P(X is even)
= 1 - 1/2[1+(1-2p)ⁿ]
= 1 - 1/2[1+(1-2*0.01)ⁿ]
= 1/2[1-(0.98)ⁿ]
Substituting n = 1/0.98 ln(0.0125/0.01) = 24.68
P(X is odd) = 0.482
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