The second side of triangle has the length is 12cm.
What is the Pythagorean theorem theory?The Pythagorean theorem, sometimes known as Pythagoras' theorem, is a basic Euclidean geometry relationship between a right triangle's three sides.Pythagoras discovered that the square of the hypotenuse is equal to the sum of the squares of the other two sides for a right angle triangle with one of the angles being 90 degrees: a2+b2=c2.A, B, and C are the three positive integers, and the Pythagorean triples are written as a2+b2 = c2. These triples are shown as (a,b,c). Here, the right-angled triangle's base, hypotenuse, and perpendicular are denoted by letters a, b, and c, respectively. The most popular and diminutive triplets include (3,4,5).Given data :
We know that,
BY p Pythagorean theorem,
a² + b² = c²
9² + b² = 15²
b² = 15² - 9²
b² = 144
b = 12
Therefore,
The second side means the hypotenuse is equal to 12cm.
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A company's 'dividend yield' is defined as the ratio between its dividend per share and share price, i.e. Dividend Yield =Dividend per SharePrice per Share. If a company has a dividend yield of 2.5%, a share price of $18 per share, how much will I receive in dividends if I own 10,000 shares? Do not include commas or the dollar sign in your answer.
So if you own 10,000 shares of the company, you would receive $4,500 in dividends.
How do we calculate the receivable dividend?A dividend is a portion of a company's profits and retained earnings that it distributes to its shareholders and owners. When a company makes a profit and accumulates retained earnings, those earnings can be reinvested in the company or distributed to shareholders as a dividend.
The amount of dividends you will receive if you own 10,000 shares can be calculated as follows:
Dividend per share = Share price * dividend yield
Dividend per share = $18 * 0.025
Dividend per share = $0.45
Total dividends received = number of shares * dividend per share
Total dividends received = 10,000 * $0.45
Total dividends received = $4,500
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nora's grandfather lived to be 87, and her father is still alive at the age of 98. nora concludes that longevity is inherited and anticipates a very long life. nora should know that . question 38 options: the lifespans of fraternal twins are just as similar as the lifespans of identical twins, indicating that longevity is probably not a heritable trait there is no reliable data on the heritability of longevity, so the cumulative effects of random events probably contributed to the long lives of her relatives the heritability of longevity is high, ranging from .75 to .95 for age at death, indicating that longevity is directly related to genetic factors rather than inheriting longevity directly, people probably inherit risk and protective factors that influence their chances of dying earlier or lat
Nora should know that rather than inheriting longevity directly, people probably inherit risk and protective factors that influence their chances of dying earlier or later. Hence, the correct answer is option C.
The given problem here requires knowledge of both biology and mathematics in order to solve the question. The mathematical component of this question is the probability of someone inheriting logetivuty or risk factors in their life.
The biological factor here is the inheritance of a trait. The traits in question are either longevity or risk factors. The probability of someone inheriting longevity is far less compared to the probability of inheriting a health condition that might be a risk factor for the person in the long run. Hence, option C is the correct answer.
The complete question that you might be looking for is given below-
Nora's grandfather lived to be 87, and her father is still alive at the age of 98. Nora concludes that longevity is inherited and anticipates a very long life. Nora should know that __________.
A) there is no reliable data on the heritability of longevity, so the cumulative effects of random events probably contributed to the long lives of her relatives
B) the lifespans of fraternal twins are just as similar as the lifespans of identical twins, indicating that longevity is probably not a heritable trait
C) rather than inheriting longevity directly, people probably inherit risk and protective factors that influence their chances of dying earlier or later
D) the heritability of longevity is high, ranging from .75 to .95 for age at death, indicating that longevity is directly related to genetic factors
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Please help! Thank you so much!
On solving the provided question, we can say that from the graphs the equation we have [tex]f(x) = y^2 + 5y -9[/tex] => f(1/3) = -29/9
What is graphs?Mathematicians use graphs to logically convey facts or values using visual representations or charts. A graph point will typically reflect a relationship between two or more things. Nodes, or vertices, and edges make form a graph, a non-linear data structure. Glue together the nodes, often referred to as vertices. This graph has vertices V=1, 2, 3, 5, and edges E=1, 2, 1, 3, 2, 4, and (2.5), (3.5). (4.5). Statistical charts (bar charts, pie charts, line charts, etc.) graphical representations of exponential growth. a logarithmic graph in the shape of a triangle
here,
from the graphs the equation we have
[tex]f(x) = y^2 + 5y -9\\f(1/3) = 1/9 + 5/3 - 9\\f(1/3) = 1 + 15 - 45 / 9\\f(1/3) = -29/9[/tex]
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2 Jai paddles 8 miles on a kayak
each day for 4 days. On the fifth
day, he paddles some more miles.
In 5 days, he paddles 40 miles.
How many miles does he paddle
on the kayak on the fifth day?
Jai paddles 8 miles each day for 4 days, a total of 8*4 = 32 miles.
What is an example of a unitary method?
A single or distinct unit is referred to by the word unitary. Therefore, the goal of this strategy is to establish values in reference to a single unit. The unitary technique, for instance, can be used to calculate how many kilometers a car will go on one litre of gas if it travels 44 km on two litres of fuel.
We know that in 5 days he paddles 40 miles in total.
So on the fifth day he paddles 40-32 = 8 miles.
Therefore, Jai paddles 8 miles on the kayak on the fifth day.
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Which function is best represented by this graph?
Responses
f(x)=x2+1
f(x)=x2−1
f(x)=−x2+1
f(x)=−x2+x−1
The function that best represented by this graph is f(x) = −x² + 1.
The correct option is C.
What is a quadratic function?A polynomial function with one or more variables, where the largest exponent of the variable is two, is referred to as a quadratic function. In other terms, a "polynomial function of degree 2" is a quadratic function.
A graph of a function is given in the image.
The parent function of the graph is g(x) = x².
And the graph is negative of the parent function.
And shifted upwards to 1 point.
That means, the function is f(x) = −x² + 1.
Therefore, the function is f(x) = −x² + 1.
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. If 16+1=17, does (16+1)-1=17-2?
Answer: False.
Step-by-step explanation:
We will compute the different expressions to see if the equations are true.
16 + 1 = 17
17 = 17 ✓
(16 + 1) -1 = 17 - 2
(17) -1 = 15
16 ≠ 15 ✗
The second equation is incorrect. We can tell right away because 2 is being subtracted from 2, but nothing is being taken away from the 16 (1 - 1 = 0).
7
The product of (a - b)(a - b) is a perfect square trinomial. (1 point)
Sometimes
Always
Never
Find the product of (x + 5)². (1 point)
Ox²-10x+25
Ox²-25
O x²+25
Ox²+10x+25
Product of (a - b)(a - b) is never a perfect square trinomial, the product of (x + 5)² is x² + 10x + 25.
An algebraic expression is what?In mathematics, an expression that incorporates variables, constants, and algebraic operations is known as an algebraic expression (addition, subtraction, etc.). Terms comprise expressions.
The algebraic expression (a + b) (a b) should be multiplied.
(a + b) = a² - ab + ba - b² (a + b)
(a + b) = a² - b² (a b) [ - ab + ba = 0 ]
Since there are only 2 terms in the final formula, a² - b² is a binomial.
As a result, the result of (a + b) (a b) is not a trinomial with a perfect square.
product of (x + 5)²
⇒ (x+5)(x+5)
⇒ x² + 5x + 5x + 25
⇒ x² + 10x + 25
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4x-2 ≤ -22 i need a answer
Answer:
x < 5
Step-by-step explanation:
4x-2 < - 22
=4x < -22 + 2
=4x < -20
=4x/4 < -20/4
x < 5.
determine whether the relation represents y as a function of x. x^2 + y^2 = 25
Answer: The relation x^2 + y^2 = 25 does not represent y as a function of x. This is because for a given value of x, there are multiple values of y that would satisfy the equation. For example, for x = 3, y = 4 and y = -4 both satisfy the equation. In a function, each value of the independent variable (in this case, x) must correspond to only one value of the dependent variable (in this case, y).
It's an equation of a circle with a center (0,0) and a radius of 5.
Step-by-step explanation:
At what rate was an investment made that obtains $50.40 on $210 over four years?
Answer:
Rate=6
Step-by-step explanation:
First multiply the interest by 100
50.40×100=5040
then multiply the investment by the number of years
210×4=840
then divide the two answers together
5040÷840=6
You purchase a new laptop today. The value of that laptop depreciates based on the function f(t) = 4,300(0.83)t, where t is measured in years after purchase. How much is the laptop worth after five halves years, rounded to the nearest dollar?
$2,105
$1,601
$2,699
$3,215
The laptop worth after five halves years will be $1,543. Then the correct option is E.
What is an exponent?Let a be the initial value and x be the power of the exponent function and b be the increasing factor.
The exponent is given as
y = a(b)ˣ
You purchase a new laptop today. The value of that laptop depreciates based on the function is given below.
f(x) = 4,300(0.83)ˣ.
Where 'x' is the number of years.
The laptop worth after five halves years will be given as,
[tex]\rm f(x) = \$4,300 \times (0.83)^{5.5}[/tex]
f(x) = $4,300 × 0.35886
f(x) = $1,543
The laptop worth after five halves years will be $1,543. Then the correct option is E.
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The missing options are given below.
$2,105
$1,601
$2,699
$3,215
$1,543
The worth of the laptop after 2.5 years is $2,699.
What is a function?A function is a relationship between inputs where each input is related to exactly one output.
Example:
f(x) = 2x + 1
f(1) = 2 + 1 = 3
f(2) = 2 x 2 + 1 = 4 + 1 = 5
The outputs of the functions are 3 and 5
The inputs of the function are 1 and 2.
We have,
f(t) = 4,300 [tex](0.83)^t[/tex]
Now,
For t = 5/2,
t = 2.5
f(5) = 4300 x [tex]0.83^{2.5}[/tex]
f(5) = 2699
Thus,
The laptop is worth $2,699 after 2.5 years.
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What is the answer for there’s expressions
Answer:
a) 12
b) 56
Step-by-step explanation:
In this problem we are asked to plug the given a and b values into the expressions [tex]3b \div (2a - 1) + b[/tex] and [tex]9b+a^4\div8[/tex] then evaluate them.
[tex]a = 2[/tex], [tex]b=6[/tex]
a) [tex]3b \div (2a - 1) + b[/tex]
↓ plug in the given [tex]a[/tex] and [tex]b[/tex] values
[tex]3(6) \div (2(2) - 1) + 6[/tex]
↓ simplify value inside parentheses
[tex]3(6) \div 3 + 6[/tex]
↓ execute multiplication
[tex]18 \div 3 + 6[/tex]
↓ execute division
[tex]6 + 6[/tex]
↓ execute addition
[tex]12[/tex]
b) [tex]9b+a^4\div8[/tex]
↓ plug in the given [tex]a[/tex] and [tex]b[/tex] values
[tex]9(6)+2^4\div8[/tex]
↓ simplify exponent
[tex]9(6)+16\div8[/tex]
↓ execute multiplication
[tex]54+16\div8[/tex]
↓ execute division
[tex]54+2[/tex]
↓ execute addition
[tex]56[/tex]
The density function for the random variable, T, that denotes the life time of a bulb (in years) is given by: 6 - t f(t) = 12,0
A continuous random variable takes on an uncountably infinite number of possible values. For a discrete random variable X that takes on a finite or countably infinite number of possible values, we determined p(X=x) for all of the possible values of X and called it the probability mass function ("p.m.f").
For continuous random variables, as we shall soon see, the probability that X takes on any particular value x is 0. That is, finding p(X=x) for a continuous random variable X is not going to work. Instead, we'll need to find the probability that X falls in some interval a,b that is, we'll need to find.
We will do that using a probability density function ("p.d.f."). We'll first motivate a p.d.f. with an example, and then we'll formally define it.
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5. Sadie drew a number line diagram to find the number
of fluid ounces in 4 cups. What numbers for fluid
ounces will complete the diagram?
cups 0
fluid ounces 0
8
2
3
4
Fluid ounces are usually measured in milliliters, but if Sadie is using ounces, then the numbers she will need to complete the diagram are 32, 64, 96, and 128.
What is number?Number is a mathematical concept that represents a quantity or amount. It is used to count, measure, and label objects and phenomena in the physical and natural world. Number can be divided into two categories: cardinal numbers and ordinal numbers. Cardinal numbers are used to denote the total number of items in a set, while ordinal numbers are used to indicate the position of an item within a set. Number is also used to represent relationships between quantities, such as addition, subtraction, multiplication, and division. Number is an essential part of mathematics and is used in a variety of ways in the real world.
This is because 1 cup is equal to 8 fluid ounces, and 4 cups is therefore equal to 32 ounces, which is the same as 64 milliliters. Similarly, 2 cups is equal to 16 ounces, or 96 milliliters, and 3 cups is equal to 24 ounces, or 128 milliliters.
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2 more than n is the quotient of n and 6
Answer:
6n+2
Step-by-step explanation:
What is the image of (12,-12)(12,−12) after a dilation by a scale factor of 1/3 centered at the origin?
The image point of.tge given pair of coordinates; (12, -12) after a dilation by a scale factor of 1/3 centered at the origin is; (4, -4).
What is the image point after a dilation centered at the origin.It follows from transformations that; when a point (x, y) is dilated by a scale factor of 1 / 3 centered at the origin, the resulting image point is; ( kx , ky ).
On this note, since the transformation in discuss is such that; the scale factor is 1/3 centered at the origin; the resulting image point is;
( 1/3 • 12 , 1/3 • -12 )
= ( 4, -4 ).
Ultimately, the image point which results from the dilation is; ( 4, -4 ).
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An ice chest contains 6 cans of apple juice, 9 cans of grape juice, 8 cans of orange juice, and 4 cans of pineapple juice. Suppose that you reach into the container and randomly select three cans in succession. Find the probability of selecting no grape juice.
The probability of randomly selecting a can that is not of grape is:
P = 0.67
How to find the probability?We know that on the ice chest there are:
6 cans of apple juice.9 cans of grape juice8 cans of orange juice.4 cans of pineapple juice.For a total of 6 + 9 + 8 + 4 = 27
The probability of selecting a can that is not of grape juice is equal to the quotient between the numbers of cans that are not of grape, and the total number of cans.
There are 27 cans in total and 18 of these aren't of grape, then the probability is:
P = 18/27 = 0.67
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This is calculated by multiplying values by the frequency of their occurrence, adding the total of all the products, and then dividing by the total number of occurrences.
a. mean
b. arithmetic mean
c. mode
d. weighted mean
The mean is calculated by taking the sum of all the values and then dividing by the total number of occurrences.
This is known as the arithmetic mean. The mode is the most frequently occurring value, and the weighted mean takes into account the frequency of each value when calculating the mean by multiplying each value by its frequency and then dividing by the total number of occurrences.
The mean is calculated by taking the sum of all the values and dividing by the total number of occurrences, which is known as the arithmetic mean. The mode is the most frequently occurring value, and the weighted mean takes into account the frequency of each value when calculating the mean by multiplying each value by its frequency and then dividing by the total number of occurrences.
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A stunt motorcyclist makes a jump from one ramp 20 feet off the ground to another ramp 20 feet off the ground. The jump between the ramps can be modeled by y=-1/640 x to the power of 2 + 1/4 x + 20 where x is the horizontal distance (in feet) and y is the height above the ground (in feet).A) what is the motorcycle's height r when it lands on the ramp?B) what is the distance d between the ramps?C) what is the horizontal distance h the motorcycle has traveled when it reaches its maximum height?D) what is the motorcycle's maximum height k above the ground?
A) 20 ft B) 80 ft C) 40 ft D) 24 ft. The equation y=-1/640 x2 + 1/4 x + 20 can be used to calculate the height, distance, maximum height, and horizontal distance of the stunt motorcyclist's jump.
A) The motorcycle's height when it lands on the ramp is 20 feet. This can be calculated by plugging 0 in for x in the equation y=-1/640 x2 + 1/4 x + 20. When you plug in 0 for x, you get y=-1/640 (0)2 + 1/4 (0) + 20, which simplifies to y=20. This means that when the motorcycle lands on the ramp, it is 20 feet off the ground.
B) The distance between the ramps is 80 feet. This can be calculated by finding the value of x when y=20. Solve the equation y=-1/640 x2 + 1/4 x + 20 for x when y=20. When you solve this equation, you get x=80. This means that the distance between the ramps is 80 feet.
C) The horizontal distance the motorcycle has traveled when it reaches its maximum height is 40 feet. This can be calculated by finding the value of x when y=24. Solve the equation y=-1/640 x2 + 1/4 x + 20 for x when y=24. When you solve this equation, you get x=40. This means that the horizontal distance the motorcycle has traveled when it reaches its maximum height is 40 feet.
D) The motorcycle's maximum height above the ground is 24 feet. This can be calculated by finding the maximum value of y in the equation y=-1/640 x2 + 1/4 x + 20. When you calculate the maximum value of y, you get y=24. This means that the motorcycle's maximum height above the ground is 24 feet.
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A certain drug is made from only two ingredients: compound A and compound B. There are 7 millimeters of compound A used for every 5 millimeters of compound B. If a chemist wants to make 900 millimeters of the drug, how many millimeters of compound A are needed?
525 millimeters of compound A are needed.
What is a mixture?
Combining two or more substances and identifying a property of the ingredients or the resulting combination is the focus of mixture problems. For instance, we could need to figure out how much water to add to a saline solution to make it more diluted, or we might need to know how much of an orange juice bottle is concentrated.
Here, we have
Given: A certain drug is made from only two ingredients: compound A and compound B. There are 7 millimeters of compound A used for every 5 millimeters of compound B. If a chemist wants to make 900 millimeters of the drug.
Given the following ratio
A : B = 7 : 5
Mixture = 900 millimeters
To calculate the amount of compound A needed, we make use of the following formula:
A = A/(A+B)×mixture
A = 7/(7+5)×900
A = 525 millimeters.
Hence, 525 millimeters of compound A are needed.
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a rock found in a gold mine weighs 12 ounces.If it contains 14% gold, how many ounces of gold does the rock contain?
Answer: 1.68
Step-by-step explanation: Okay, to find this we need to use this formula:
Amount = Part percent * whole number / 100
So, now let's plug in the numbers:
Amount = 14 * 12 / 100
= 168/100
= 1.68
Therefore the amount of gold in the rock is 1.68 oz out of the 12 ounces.
I hope this helped!
Answer: 1.68
Step-by-step explanation: So, if we want to find how much gold is in the rock, it's a good idea to set up a proportion. So, it would be like this:
[tex]\frac{14}{100} = \frac{?}{12}[/tex]
Now, we would cross-multiply the 12 and the 14, which is 168, and then divide it by 100. That can be found by moving the decimal 2 places left of 168.00. So, that would be 1.68. Therefore, there are 1.68 ounces of gold in the rock. I hope this helped!
Suppose you work for a large coffee distributor that has a secret coffee blend it sells to local stores. You mix the Tanzanian blend with the Gazebo blend, but always in the same proportion. Yesterday, you mixed 80 pounds of the Tanzanian blend with 24 pounds of the Gazebo blend. Today, there is 20 pounds of the Tanzanian coffee left in stock. How many pounds of the Gazebo coffee should you mix with it to get your secret blend?
6 pounds of Gazebo blend is required to be mixed with Tanzanian blend to get the secret blend.
What is a ratio?The quantitative relation between two amounts shows the number of times one value contains or is contained within the other. for example-"the ratio of computers to students is now 2 to 1"
Given here: A secret coffee blend requires 80 pounds of Tanzanian blend with 24 pound of Gazebo blend.
Now, A proportion is an equation in which two ratios are set equal to each other. Ratio of the required secret blend is 80:24=10:3
Amount of Tanzanian blend left in stock is 20 pounds and let x pounds of Gazebo blend be required to make the secret blend thus to preserve the proportion we must have;
10/3=20/x
x/3=20/10
x=6
Hence, 6 pounds of Gazebo blend is required to get the secret blend.
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Determine if f (x )is continuous at x equals 6. if not, select the option with the correct reasoning as to why not.
The limit of f(x), which must exist at x = 1 and be equal to k, is required for f(x) to be continuous at x = 1. (1).
What is meant by mathematical function?A mathematical function is a rule that determines the value of the dependent variable based on the values of the independent variables. A table, a formula, or a graph are just a few examples of how a function might be expressed. A function from a set X to a set Y in mathematics assigns each element of X exactly one element of Y. The set X is known as the function's domain, while the set Y is known as the function's codomain. Initially, functions were just an idealized representation of the relationship between two changing quantities.
The limit of f(x), which must exist at x = 1 and be equal to k, is required for f(x) to be continuous at x = 1. (1).
While the limit of f(x) is 1 - (1)2 = 0 as x approaches 1 from the left, it is 1+1 = 2 as x approaches 1 from the right. The limit of f(x) as x approaches 1 therefore does not exist because the one-sided limits are not the same.
As a result, at x = 1, f(x) is not continuous.
The function has a gap at x = 1, which can be seen if you graph it. Gaps and holes don't exist in continuous function graphs.
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a system of linear equations is graphed. which ordered pair is the best estimate for the solution to the system?
The best estimate for the solution to a system of linear equations can be found by using the Substitution Method.
To solve for the solution, the equations must be written in the form y = mx + b. The slope, m, and the y-intercept, b, of each equation can then be calculated. The x-coordinate of the solution can be found by setting the two equations equal to each other to create an equation in one variable. This equation can then be solved for x. The y-coordinate of the solution can be found by substituting the x-coordinate into one of the original two equations. The ordered pair that is the estimated solution can then be formed.
For example, consider the system of equations y = 3x + 2 and y = -x + 5. The slope of the first equation is m1 = 3, and the y-intercept is b1 = 2. The slope of the second equation is m2 = -1, and the y-intercept is b2 = 5. Setting the two equations equal to each other yields 3x + 2 = -x + 5. Solving for x gives x = 3. Substituting this x-coordinate into either of the original equations yields y = 3(3) + 2 = 11. The ordered pair that is the estimated solution to the system is (3, 11).
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Jimmy paid $8,200 in property taxes. His adjusted gross income is $146,000. What percentage of his adjusted gross income is he paying in property tax?
5.62% percentage of his adjusted gross income is he paying in property tax.
What is percentage?A percentage is a number or ratio that can be expressed as a fraction of 100. A percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also used. A percentage is a dimensionless number; it has no unit of measurement.
here, we have,
Jimmy paid $8,200 in property taxes.
His adjusted gross income is $146,000.
The percentage of his adjusted gross income is he paying in property tax is, (8200/146000)*100%
=5.62%
Hence, 5.62% percentage of his adjusted gross income is he paying in property tax.
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Consider the problem of predicting Y using another variable, X, so that the prediction of Y is some function of X, say g(X). Suppose that the quality of the prediction is measured by the squared prediction error made on average over all predictions, that is, by E{[Y – g(x)]?}. This exercise provides a non-calculus proof that of all possible prediction functions g, the best predic- tion is made by the conditional expectation, E(Y|X). a. E(Y|X), and let u = Y – û denote its prediction error. Show that E(u) = 0. (Hint: Use the law of iterated expectations.) b. Show that E(uX) = 0. c. Let Ỹ = g(x) denote a different prediction of Y using X, and let v = Y Ỹ denote its error. Show that E[(Y – Ỹ)?] > E[(Y - Ỹ)2]. [Hint: Let h(X) = g(x) – E(Y|X), so that v = [Y - E(Y|X)] - h(X). Derive E(v2).] a. Let û =
E(Y|X) is the conditional expectation of Y given X. We can use the law of iterated expectations to show that E(u) = 0.
This law states that E(E(X|Y)) = E(X). Therefore, E(u) = E(Y - E(Y|X)) = E(Y) - E(E(Y|X)) = E(Y) - E(Y) = 0.
b. We can use the law of iterated expectations again to show that E(uX) = 0. This law states that E(E(XY|Z)) = E(X)E(Y|Z). Therefore, E(uX) = E(YX - E(YX|X)) = E(YX) - E(E(YX|X)) = E(YX) - E(Y)E(X) = 0.
c. Let Ỹ = g(x) denote a different prediction of Y using X and let v = Y - Ỹ denote its error. We can use the equation h(X) = g(x) - E(Y|X), so that v = [Y - E(Y|X)] - h(X). Using this equation, we can derive E(v2) = E[(Y - E(Y|X))2] - 2E[(Y - E(Y|X))h(X)] + E[h(X)2]. Since E(Y - E(Y|X)) = 0, this reduces to E(v2) = E[h(X)2] < E[(Y - Ỹ)2], which shows that E[(Y - Ỹ)?] > E[(Y - Ỹ)2].
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Let v1 and v2 be two vectors in a vector space V. Show that v1 and v2 are linear dependent if and only if one of the vectors is a scalar multiple of the other.
In the case where v1 and v2 are two vectors in the vector space V and v1 is a scalar multiple of v2, let's say v1 = k * v2, where k is a scalar, we can write:
v1 = k * v2
What is linearly dependent vectors?Linearly dependent vectors are a set of vectors that can be expressed as a linear combination of the other vectors in the set. In other words, one vector in the set can be written as a scalar multiple of the other vectors in the set.
As a result, v1 can be expressed as the linear combination of v2 and k, where k is a scalar. Thus, v1 and v2 are linearly dependent on one another.
In contrast, if variables v1 and v2 are linearly dependent, v1 can be expressed as a linear combination of variables v2, and vice versa. That is, scalars k1 and k2 can exist such that:
v2 = k2 * v1 and v1 = k1 * v2.
This suggests that v1 = k1 + k2 + v1. We can divide both sides by v1 because it is not a zero vector, and the result is:
1 = k1*k2
Therefore, let's argue that v1 is a scalar multiple of v2, the other vector.
In conclusion, if and only if one of the vectors is a scalar multiple of the other, v1 and v2 are linearly dependent.
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Three equal mass satellites A, B, and C are in coplanar orbits around a planet as shown in the figure. The magnitudes of the angular momenta of the satellites as measured about the planet are LA, LB, and LC. Which of the following statements is correct? (A) LA > LB > LC (B) LC > LB > LA (C) LB > LC > LA (D) LB > LA > LC (E) The relationship between the magnitudes is different at various instants in time
(B) LC > LB > LA is the correct statement for satellites
The magnitudes of the angular momenta of the satellites as measured about the planet are LA, LB, and LC.
The angular momentum of each satellite is conserved independently so we can compare the orbits at any location. Looking at the common point between orbit A and B shows that satellite A is moving faster at that point than satellite B, showing LA > LB. A similar analysis at the common point between B and C shows LB > L A because three equal mass satellites are in coplanar orbits around a planet
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a city has a population of 50,000 in 2012. if the population of the city grows at an annual rate of 2%, the year in which the population will reach 100,000 is and the year it will reach 200,000 is .
The population will reach 100,000 in 2047 and the population will reach 200,000 in 2082.
Total population of the city = 50,000
Growth annual rate = 2%
Using the formula of compound growth -
A = P ( 1 +r)^t
Substituting the values -
P=50,000, r =2% = 0.02 and A=100,000
Thus,
1,00,000 = 50,000 ( 1 + 0.02)^t
= 1.02^t = 1,00,000/50,000
= 1.02^t = 2
Taking ln both sides, thus -
= In( 1.02^t) = In (2)
t = In( 1.02^t)/ In (2)
t = 35
The population will reach 100,000 in the year -
= 2012+35
= 2047
Similarly,
P=50,000, r =2% = 0.02 and A=2,00,000
Thus,
2,00,000 = 50,000 ( 1 + 0.02)^t
= 1.02^t = 2,00,000/50,000
= 1.02^t = 4
Taking ln both sides, thus -
= In( 1.02^t) = In (4)
t = In( 1.02^t)/ In (4)
t = 70
The population will reach 100,000 in the year -
= 2012 + 70
= 2082
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y=2r-5 r+y=-8 plot both lines
The plot of both lines is given below.
y = 2r-5,
r + y = -8.
What is the system of equations?One or many equations having the same number of unknowns that can be solved simultaneously called as simultaneous equation. And simultaneous equation is the system of equation.
Given:
The system of equations,
y = 2r-5,
r + y = -8.
And the intersection point is the solution of the equation.
The plot of the lines is given in the attached image.
Therefore, plots are given in the attached image.
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