Answer:
Distance formula: (x1,y1), (x2,y2)
d = √[(x2-x1)2+(y2-y1)2]
d = √[(5-2)2+(4-1)2]
d = √(32+32)
d = √18
d = 3√2
double this distance: The length is 6√2
it's of first one
Step-by-step explanation:
its of second one
The formula for midpoint = (x1 + x2)/2, (y1 + y2)/2. Substituting in the two x coordinates and two y coordinates from the endpoints, we get (–1 + 3)/2
When multiplying units, use the same principle that you use for multiplying fractions. In addition, ifone unit is in the numerator and the identical unit is in the denominator, they cancel each other out (to essentially equal 1). Any remaining units are used in the answer. ( centimheter) ( centimeter meter )=1 meter = meter You can also multiply several units together at once using the same principle as for fractions containing numbers. ( second centipheters )( centimeter meter )( meters kilopreter )( kilopeter megameter )= second megameter It is very important to note that if a unit appears only once in the numerator but more than once in the denominator, we can only cancel one of the unit expressions in the denominator. Think of this concept in terms of fractions. If the number 4 is in the numerator, and two 45 are in the denominator of different fractions, we only cancel out one of the 45 on the bottom, not both. (54)(43)(41)=(5∗4)(3∗1)=203=0.15 Evaluate the following unit expression. Enter the resulting units as your answer. Do not abbreviate the units and do not use parentheses. Parentheses mean multiplication. ( kilogram )( kilogram grams )( gram milligrams )= Evaluate the following unit expression. Enter the resulting units as your answer. Do not obbreviate the units and do not use parentheses. Parentheses mean multiplication. ( mole )( mole grams )( gram liter )= Evaluate the following unit expression. Enter the resulting units as your answer. Do not abbreviate the units. Enter units exactly as they anpear in the problem. Use ^ for exponents (so ft∧2forft2 ). Use " for multiplication and / for division. Do not include any spaces and not use parentheses. Parentheses mean multiplication. (mL)(cmmg)( mLcm3)=(mL)(cmmg)( mLcm∗ cm∗ cm)= Evaluate the following unit expression. Enter the resulting units as your answer. Do not abbreviate the units. Enter units exactly as they annear in the problem. Use ∧ for exponents ( so ft∧2for22). Use " for multiplication and / for division. Do not include any spaces and not use parentheses. Parentheses mean multiplication. (f2g)(mLft)(gmL)=(ft∗ftg)(mLft)(gmL)=
The resulting units are kilograms grams milligrams, mole grams liter, ft^2g mL ft gmL
To evaluate the given unit expressions, we can apply the rules of multiplying units similar to multiplying fractions.
(kilogram)(kilogram grams)(gram milligrams):
The units cancel out as follows:
(kilogram)(kilogram grams)(gram milligrams) = (kilogram)(grams)(milligrams)
Therefore, the resulting units are kilograms grams milligrams.
(mole)(mole grams)(gram liter):
The units cancel out as follows:
(mole)(mole grams)(gram liter) = (mole)(grams)(liter)
Therefore, the resulting units are mole grams liter.
(mL)(cmmg)(mLcm3):
The units cancel out as follows:
(mL)(cmmg)(mLcm3) = (mL)(cm mg)(cm^3)
Therefore, the resulting units are mL cm mg cm^3.
(f^2g)(mLft)(gmL):
The units cancel out as follows:
(f^2g)(mLft)(gmL) = (ft^2g)(mLft)(gmL)
Therefore, the resulting units are ft^2g mL ft gmL.
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To which linear equations is the coordinate a solution?
Select two options.
y = 2x + 13
y=-x-2
y = 3x - 5 2. >4 -2 2. 4
y=-x+6 -2 .
y=-2x-2
The linear equation y = -x - 2 and y = 2x + 13 form the solution (-5, 3)
Linear equation
A linear equation is in the form:
y = mx + b
where y, x are variables, m is the rate of change and b is the y intercept.
Given the coordinate (-5, 3) as a solution.
The linear equation y = -x - 2 satisfies this this equation because 3 = -(-5) -2
Also, the linear equation y = 2x + 13 satisfies this this equation because 3 = 2(-5) + 14
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Answer:
View attached graph
Step-by-step explanation:
y = 2x + 13 and y = -x - 2
Hope this helps!
Can you solve this graph problem?
a) what is the difference in temperatures between 7 am to 5 pm
b) which part of the day was the temperature rising the fastest a, b, c, d, or e?
Answers:
a) The difference in temperature is 25 degrees Fahrenheit. b) The temperature was rising fastest during interval 'a'=========================================================
Explanations:
a) We subtract the y coordinates of the left and right endpoints. We get 65-40 = 25b) The fastest rising temperature will visually correspond to the steepest part of the curve. That would be region 'a'. After this interval, region b is still increasing, but not as much as the previous interval (since it's not as steep here). Then it gets less steep for region c, and d sort of flattens out as well. So during the morning hours it appears the temperature is going up the fastest.what's the surface area of a cylinder with radius 3 feet and height 4 feet?
Answer: The answer is 226.2 feet.
Step-by-step explanation: The formula to find the total surface area of a cylinder is 2\(\pi\) x radius squared x height. If you use the formula you will get 226.19 feet as your answer and I rounded that to 226.2 feet.
UR MOM
what is the slope of each side of the triangle? the sides are A(-2,4) B(-1,1) C(2,3)
Answer:
The slopes of three sides of triangle are as follows:
AB = -3
BC = 2/3
AC = -1/4
Step-by-step explanation:
The slope is denoted by m and is calculated using the formula
\(m = \frac{y_2-y_1}{x_2-x_1}\)
The given vertices are:
A(-2,4) B(-1,1) C(2,3)
The sides will be:
AB, BC, AC
Let m1 be the slope of AB
Let m2 be the slope of BC
Let m3 be the slope of AC
Now
\(Slope\ of\ AB = m_1 = \frac{1-4}{-1+2} = \frac{-3}{1} = -3\\Slope\ of\ BC = m_2 = \frac{3-1}{2+1} = \frac{2}{3}\\Slope\ of\ AC = m_2 = \frac{3-4}{2+2} = \frac{-1}{4} = -\frac{1}{4}\)
Hence,
The slopes of three sides of triangle are as follows:
AB = -3
BC = 2/3
AC = -1/4
i need to know how to solve this question please
ANSWER
w = 6.71 or -6.71
EXPLANATION
Determine whether the lines CD and EF in the image are parallel for the given angle measures
The lines CD and EF in the image are not parallel for the given angle measures
How to determine if the lines are parallel for the given angle measures?Question #9
The measures of the angles are given as:
BPG = 139
GPC = 95
BQF = 110
As a general rule, the co-interior angles between a parallel, when added have a sum of 180 because they are supplementary angles.
This means that, at least 2 of the given angles must add up to 180.
So, we have:
139 + 95 = 234
139 + 110 = 249
95 + 110 = 205
None of the angles add up to 180.
Hence, the lines CD and EF in the image are not parallel for the given angle measures
Question #10
The measures of the angles are given as:
BPD = 35
APG = 115
EQA = 35
As a general rule, the co-interior angles between a parallel, when added have a sum of 180 because they are supplementary angles.
This means that, at least 2 of the given angles must add up to 180.
So, we have:
35 + 115 = 150
35 + 115 = 150
35 + 35 = 70
None of the angles add up to 180.
Hence, the lines CD and EF in the image are not parallel for the given angle measures
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Find the number of ways of arranging the letters of the word MOSHOESHOE, if the letter M must always begin a word
Answer:
7560.
Step-by-step explanation:
There are 9 letters after the M consisting of 3 O's, 2 H's, 2 H's and 2 E's.
So the number of ways is 9! / (3!*2!*2!*2!)
= 362880 / (6*2*2*2)
= 362880 / 48
= 7560.
The number of ways the word MOSHOESHOE can be rearranged, if the letter M must always begin a word is 75,600 ways
Given:
MOSHOESHOE
n = total number of letters = 10
a = number of times O appears = 3
b = number of times S appears = 2
c = number of times E appears = 2
d = number of times H appears = 2
number of ways the word can be rearranged if M must always begin a word
= n! / (a! b! c! d!)
= 10! / (3!, 2!, 2!, 2!)
= (10 × 9 × 8 × 7 × 6 × 5 × 4× 3 × 2 × 1) / (3 × 2 × 1, 2 × 1, 2 × 1, 2 × 1)
= 3,628,800 / (6 × 2 × 2 × 2)
= 3,628,800 / 48
= 75,600 ways
Therefore, the number of ways the word can be rearranged if M must always begin a word is 75,600 ways
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3. Find \( y^{\prime} \) for the following implicit function \( y^{2}-x^{2} y=-2 \)
The derivative \(\( y' \)\) of the implicit function \(\( y^2 - xy = -2 \)\) is 0, indicating a constant slope with no change in relation to \(\( x \)\).
To find \(\( y' \)\)for the implicit function \(\( y^2 - xy = -2 \)\), we can differentiate both sides of the equation with respect to \(\( x \)\) using the chain rule. Let's go step by step:
Differentiating \(\( y^2 \)\) with respect to \(\( x \)\) using the chain rule:
\(\[\frac{d}{dx}(y^2) = 2y \cdot \frac{dy}{dx}\]\)
Differentiating \(\( xy \)\) with respect to \(\( x \)\) using the product rule:
\(\[\frac{d}{dx}(xy) = x \cdot \frac{dy}{dx} + y \cdot \frac{dx}{dx} = x \cdot \frac{dy}{dx} + y\]\)
Differentiating the constant term (-2) with respect to \(\( x \)\) gives us zero since it's a constant.
So, the differentiation of the entire equation is:
\(\[2y \cdot \frac{dy}{dx} - (x \cdot \frac{dy}{dx} + y) = 0\]\)
Now, let's rearrange the terms:
\(\[(2y - y) \cdot \frac{dy}{dx} - x \cdot \frac{dy}{dx} = 0\]\)
Simplifying further:
\(\[y \cdot \frac{dy}{dx}\) \(- x \cdot \frac{dy}{dx} = 0\]\)
Factoring out:
\(\[(\frac{dy}{dx})(y - x) = 0 \]\)
Finally, solving:
\(\[\frac{dy}{dx} = \frac{0}{y - x} = 0\]\)
Therefore, the derivative \(\( y' \)\) of the given implicit function is 0.
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30 POINTS an object with a mass of 5kg accelerates 5m/s(2) when an unknown force is applied to it. what is the amount of force ?! ( use unit of measure ). 30 POINTS
Answer:
25 N
Step-by-step explanation:
please help me for this
At the beginning of unit 10, information was introduced about the significance of e. which of the following statements is not true regarding ?
a. The number e is equal to about 2.718
b. The number e is called the "natural" exponential because it arises naturally in math and science
c. The number e is considered a special irrational number in mathematics
d. The number e is another way to express the number π
Answer:
d. The number e is another way to express the number π
Step-by-step explanation:
You want to know the false statement among those offered.
a. 2.718The first few digits of the irrational number e are 2.718281828459045...
(true)
b. NaturalLeonard Euler identified e as the value of 1 compounded continuously at an annual rate of 100%. More than 100 years earlier, John Napier computed and published tables of the logarithms of trig functions. The base was related to e, but he didn't call it that (or even know its value).
(true)
c. SpecialThe value e is sufficiently "special" that most scientific calculators have a button for it. It shows up in many formulas, especially those related to growth, decay, and logarithms.
(true)
d. PiSome expressions involving both e and π can make it look like there might be a relation.
In complex numbers, Euler's identity e^(iπ)+1 = 0 involves both irrational numbers. However, there is no known algebraic relationship between π and e.
(false)
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chuck bought 19 movie tickets for $174.50. The adult tickets cost $11.50 each and child tickets cost $7.50 each. How many child tickets did he purchase?
4.5. Let N be a nonnegative integer-valued random variable. For nonnegative values aj.J > = I. show that Then show that and
We have shown that P(N < aJ) ≤ 1 - J for nonnegative values aj.N is a nonnegative integer-valued random variable
To prove the given inequality, let's start by defining the indicator random variable Ij, which takes the value 1 if aj ≤ N and 0 otherwise.
We have:
Ij = {1 if aj ≤ N; 0 if aj > N}
Now, we can express the expectation E(Ij) in terms of the probabilities P(aj ≤ N):
E(Ij) = 1 * P(aj ≤ N) + 0 * P(aj > N)
= P(aj ≤ N)
Since N is a nonnegative integer-valued random variable, its probability distribution can be written as:
P(N = n) = P(N ≤ n) - P(N ≤ n-1)
Using this notation, we can rewrite the expectation E(Ij) as:
E(Ij) = P(aj ≤ N) = P(N ≥ aj) = 1 - P(N < aj)
Now, let's consider the sum of the expectations over all values of j:
∑ E(Ij) = ∑ (1 - P(N < aj))
Expanding the sum, we have:
∑ E(Ij) = ∑ 1 - ∑ P(N < aj)
Since ∑ 1 = J (the total number of values of j) and ∑ P(N < aj) = P(N < aJ), we can write:
∑ E(Ij) = J - P(N < aJ)
Now, let's look at the expectation E(∑ Ij):
E(∑ Ij) = E(I1 + I2 + ... + IJ)
By linearity of expectation, we have:
E(∑ Ij) = E(I1) + E(I2) + ... + E(IJ)
Since the indicator random variables Ij are identically distributed, their expectations are equal, and we can write:
E(∑ Ij) = J * E(I1)
From the earlier derivation, we know that E(Ij) = P(aj ≤ N). Therefore:
E(∑ Ij) = J * P(a1 ≤ N) = J * P(N ≥ a1) = J * (1 - P(N < a1))
Combining the expressions for E(∑ Ij) and ∑ E(Ij), we have:
J - P(N < aJ) = J * (1 - P(N < a1))
Rearranging the terms, we get:
P(N < aJ) = 1 - J * (1 - P(N < a1))
Since 1 - P(N < a1) ≤ 1, we can conclude that:
P(N < aJ) ≤ 1 - J
Therefore, we have shown that P(N < aJ) ≤ 1 - J for nonnegative values aj.
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Help me find the slope please guys
Im struggling
Answer:
negative 1
Step-by-step explanation:
\( \frac{y2 - y1}{x2 - x1} = m\)
the 2 points: (-2,0), (-3,1)
(1-0)/(-3+2)
=-1
m=-1
brainliest please
At a coffee shop, the manager recorded the number of customers who visited the store at the end of each hour. The graph shows the recordings for a 24-hour period. The function describing this graph is a transformation of the parent sine function, y=sin(x)
Which value is closest to the amplitude of the transformed function?
O 83 customers
O 27 customers
O 54 customers
O 30 customers
Answer:
Step-by-step explanation:
The amplitude of a sine function is equal to one-half the distance between the maximum and minimum values of the function.
In this case, the maximum value is approximately 84 customers, and the minimum value is approximately 30 customers
Therefore, the value that is closest to the amplitude of the transformed function is 27 customers.
A SALESMAN BOUGHT A COMPUTER FROM A MANUFACTURER. THE SALESMAN THEN SOLD THE COMPUTER FOR $15,600 MAKING A LOSS OF 25%. WHAT AMOUNT DID THE SALESMAN PAY THE MANUFACTURER FOR THE COMPUTER?
Answer:
20800 dollars
Step-by-step explanation:
15600*(1/0.75)
=20800
the original price was 20800 dollars.
weirdly expensive computer though
The generic metal A forms an insoluble salt AB(s) and a complex AC5(aq). The equilibrium concentrations in a solution of AC5 were found to be [A] = 0. 100 M, [C] = 0. 0360 M, and [AC5] = 0. 100 M. Determine the formation constant, Kf, of AC5. The solubility of AB(s) in a 1. 000-M solution of C(aq) is found to be 0. 131 M. What is the Ksp of AB?
A theater charges $5.25 per ticket. The theater has already sold 40 tickets. Write and solve an inequality that represents how many more tickets the theater needs to sell to earn at least $400.
NO LINKS PLEASE :]
Need help ASAP! Ty
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゚ Black was not An Impostor. 。 .
' 1 Impostor remains 。
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Answer:
Its 22 squared feet
Step-by-step explanation:
you multiply everything and then divide by 1/2 because it's a triangle, so you would use the equation for that.
Answer:
the first one
Step-by-step explanation:
on a circle of radius 2, center (0, 0), find the x and y coordinates at angle 270 degrees (or 3π/2 in radian measure).
At an angle of 270 degrees (or 3π/2 radians) on a circle with a radius of 2 and center at (0, 0), the x-coordinate is 0 and the y-coordinate is -2.
To find the x and y coordinates at an angle of 270 degrees (or 3π/2 in radian measure) on a circle of radius 2 with center (0, 0), we can use the trigonometric definitions of sine and cosine.
The x-coordinate (x-value) represents the horizontal position on the circle, while the y-coordinate (y-value) represents the vertical position.
For a point on the unit circle (circle with radius 1) at a given angle θ, the x-coordinate is given by cos(θ) and the y-coordinate is given by sin(θ).
In this case, the circle has a radius of 2, so we need to multiply the cosine and sine values by 2 to get the x and y coordinates, respectively.
Using the angle 270 degrees (or 3π/2 in radian measure):
x-coordinate = 2 * cos(3π/2)
y-coordinate = 2 * sin(3π/2)
Evaluating these expressions:
x-coordinate = 2 * cos(3π/2) = 2 * 0 = 0
y-coordinate = 2 * sin(3π/2) = 2 * (-1) = -2
Therefore, at an angle of 270 degrees (or 3π/2 radians) on the circle of radius 2 with center (0, 0), the x-coordinate is 0 and the y-coordinate is -2.
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4.80 for 16 ounces of cereal how much did i spend per ounce?
Answer:
0.3 per ounce
4.80 / 16 = 0.3
0.3 × 16 = 4.80
Mark recently took a road trip across the country. The number of miles he drove each day was normally distributed with a mean of 450. If he drove 431.8 miles on the last day with a z-score of -0.7, what is the standard deviation?
Answer:
The (population) standard deviation is 26 miles or \( \\ \sigma = 26\) miles.
Step-by-step explanation:
We can solve this question using the concept of z-score or standardized value, which is given by the formula:
\( \\ z = \frac{x - \mu}{\sigma}\) [1]
Where
\( \\ z\) is the z-score.
\( \\ x\) is the raw score.
\( \\ \mu\) is the population's mean.
\( \\ \sigma\) is the population standard deviation.
Analyzing the question, we have the following data to solve this question:
The random variable number of miles driven by day is normally distributed.The population's mean is \( \\ \mu = 450\) miles.The raw score, that is, the value we want to standardize, is \( \\ x = 431.8\) miles.The z-score is \( \\ z = -0.7\). It tells us that the raw value (or raw score) is below the population mean because it is negative. It also tells us that this value is 0.7 standard deviations units (below) from \( \\ \mu\).Therefore, using all this information, we can determine the (population) standard deviation using formula [1].
Then, substituting each value in this formula:
\( \\ z = \frac{x - \mu}{\sigma}\)
Solving it for \( \\ \sigma\)
Multiplying each side of the formula by \( \\ \sigma\)
\( \\ \sigma*z = (x - \mu) * \frac{\sigma}{\sigma}\)
\( \\ \sigma*z = (x - \mu) * 1\)
\( \\ \sigma*z = x - \mu\)
Multiplying each side of the formula by \( \\ \frac{1}{z}\)
\( \\ \frac{1}{z}*\sigma*z = \frac{1}{z}*(x - \mu)\)
\( \\ \frac{z}{z}*\sigma = \frac{x - \mu}{z}\)
\( \\ 1*\sigma = \frac{x - \mu}{z}\)
\( \\ \sigma = \frac{x - \mu}{z}\)
Then, this formula, solved for \( \\ \sigma\), will permit us to find the value for the population standard deviation asked in the question.
\( \\ \sigma = \frac{431.8 - 450}{-0.7}\)
\( \\ \sigma = \frac{-18.2}{-0.7}\)
\( \\ \sigma = 26\)
Thus, the (population) standard deviation is 26 miles or \( \\ \sigma = 26\) miles.
For a criminal trial, 8 active and 4 alternate jurors are selected. Two of the alternate jurors are male and two are female. During the trial, two of the active jurors are dismissed. The judge decides to randomly select two replacement jurors from the 4 available alternates. What is the probability that both jurors selected are female? 1/12 1/6 1/2 1/4
The probability that both jurors selected are female is 1/6. To calculate the probability that both jurors selected are female,.
We need to determine the number of favorable outcomes (two female jurors selected) divided by the total number of possible outcomes.
In this scenario, there are two female alternate jurors available out of a total of four alternates. Since we need to select two jurors, we can use combinations to calculate the number of possible outcomes.
The number of possible outcomes is given by selecting 2 jurors out of 4, which can be calculated as:
C(4, 2) = 4! / (2! * (4-2)!) = 6
Therefore, there are 6 possible outcomes.
Out of these possible outcomes, we are interested in the favorable outcome where both selected jurors are female. Since there are two female alternate jurors available, we can calculate the number of favorable outcomes by selecting 2 female jurors out of 2, which is:
C(2, 2) = 2! / (2! * (2-2)!) = 1
Therefore, there is 1 favorable outcome.
Now, we can calculate the probability:
Probability = Number of favorable outcomes / Number of possible outcomes
= 1 / 6
= 1/6
Thus, the probability that both jurors selected are female is 1/6.
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for a standard normal distribution, find: p(-1.62 < z < 2.01)
The probability of the interval -1.62 < z < 2.01 in a standard normal distribution is approximately 0.9262 or 92.62%.
In a standard normal distribution, the mean is 0 and the standard deviation is 1. The z-score represents the number of standard deviations a data point is from the mean. To find the probability of a specific interval, we calculate the area under the curve between the corresponding z-values.
Given the interval -1.62 < z < 2.01, we need to find the area under the standard normal curve between these two z-values. This can be done using a standard normal distribution table or by using a statistical software or calculator.
By looking up the z-values in the table or using software, we find the corresponding probabilities: P(z < -1.62) = 0.0526 and P(z < 2.01) = 0.9788.
To find the probability of the interval -1.62 < z < 2.01, we subtract the probability of the lower bound from the probability of the upper bound: P(-1.62 < z < 2.01) = P(z < 2.01) - P(z < -1.62 = 0.9788 - 0.0526 = 0.9262.
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At a carnival you win a prize if you get a heads, you must first choose a coin. There is a fair and a biased coin, while choosing each coin is equally likely, the biased coin has a 78% of landing tails. What is the probability of choosing the biased coin if you won a prize.
Probability of choosing the biased coin if you won a prize is 0.30
Let "B" be the event of selecting biased coin and "H" be the event of getting head.
P(B) = 0.5
P(getting head when coin was biased) = 100% - 78%
= 22% = 0.22
Using conditional Probability that biased coin was selected given that you have won the prize that is getting head
we have to calculate ,
P(B | H ) = P(B∩H)/P(H)
here , P(B∩H) = P(biased coin selected and getting head) = 0.5 × 0.22
and P(H) = P(getting head)
P(getting head when coin was biased) + P(getting head when coin was unbiased) = 0.5 × 0.22 + 0.5 × 0.5
putting all together ,
P(B | H ) = P(B∩H)/P(H) = 0.5 × 0.22 / 0.5 × 0.22 + 0.5 × 0.5
cancelling 0.5 from numerator and denominator
= 0.22 / 0.5+ 0.22
= 0.22 / 0.72 = 22/72
=0.30
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please help need this urgently
Answer:
a
Step-by-step explanation:
Answer:
The answer is A. y=1/3x+4
Explanation:
Perpendicular means that the slope is flipped and the only answer that has -3 flipped is A.
Which of the following equations could represent the result of a vertical change to the graph of y = -2x – 1? Selectall that applyA)y = -4x - 2B)y = -2x C)x = -2y - 5D)y = 10 - 2xE)y = 2x +1F)2x + y = 4
The vertical change is represented by the equations x = - 2y - 5, y = 10 - 2x and 2x + y = 4.
The equation is given as:
y = - 2 x - 1
For vertical change, there should be a change in the y axis coordinate.
A)
y = - 4 x - 2
y = 2 (- 2 x - 1)
No vertical change
B)
y = - 2x
No vertical change
C)
x = - 2y - 5
Vertical change is there.
D)
y = 10 - 2x
y = - 2 x - 1 + 11
y - 11 = - 2 x - 1
Vertical change is there.
E)
y = 2x + 1
y = - 1 ( - 2 x - 1)
No vertical change
F)
2x + y = 4
y = - 2 x + 4
y = - 2 x - 1 + 5
y - 5 = - 2 x - 1
Vertical change is there.
Therefore, we get that, the vertical change is represented by the equations x = - 2y - 5, y = 10 - 2x and 2x + y = 4.
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Image formation in a Pin-hole camera; Points, lines, and planes in 3D; Rotation and Stereo. A conventional pin-hole camera model is shown at the end. In this model, three points P,Q, and R, in the 3D scene are given to be: P(X1,Y1,Z1)=P(120,250,340)mm, corresponding image point: p(x,y) Q(X2,Y2,Z2)=Q(250,150,200)mm, corresponding image point: q(x,y) R(X3,Y3,Z3)=R(200,100,500)mm, and corresponding image point: r(x,y) and Focal length f=5 mm. Pixel size ps =0.010 mm. Image size =1000×1000 pixels. Image coordinate center is at the center pixel with indices (500,500). - 2+2 points) Stereo camera system (a) A second identical camera is placed with its lens center at C=(10,0,0). The coordinates axes of the two cameras are all paralle the pointing along the same directions (as in the case of a conventional parallel stereo camera system). Find the disparity (shift of its image position compared to the first camera) of the point P in the second camera. (b) The image coordinates of a point V is (x,y)=(1.0,2.0)mm in the first camera, and it is (1.5,2.0)mm in the second camera. What are the (X, Z,Z) coordinates of V in the 3D scene
In a pin-hole camera model, the three-dimensional (3D) points P, Q, and R in the scene correspond to their respective two-dimensional (2D) image points p, q, and r on the camera's image plane.
Given the coordinates of these points in the 3D scene and their corresponding image points, along with the focal length, pixel size, and image size, we can calculate various parameters. The image coordinate center is at the center pixel with indices (500,500).
For the stereo camera system, the second camera is placed parallel to the first one, with its lens center at C=(10,0,0) in the 3D scene. To find the disparity of point P in the second camera, we need to determine the difference in its image position compared to the first camera. Disparity is the horizontal shift between corresponding points in the two images. By calculating the difference in the x-coordinate of point P's image position in the two cameras, we can find the disparity.
To determine the 3D coordinates (X, Y, Z) of point V in the scene, given its image coordinates in both cameras, we can use triangulation. Triangulation involves finding the intersection point of two rays, each originating from the camera center and passing through the respective image point. By considering the known parameters of the cameras, we can compute the 3D coordinates of point V using its image coordinates in both cameras.
For the stereo camera system, the disparity of point P can be found by calculating the difference in its image position between the two cameras. To determine the 3D coordinates of point V, we can use triangulation by considering the image coordinates in both cameras along with the known camera parameters.
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The calculation of image disparity in the second camera and coordinates of a point in 3D scene involves concepts of geometry and trigonometry. The coordinates can be computed using formulas derived from rules of similar triangles.
Explanation:The given question involves the operations of a pin-hole camera and a stereo camera system. The process of imaging and finding disparities in the cameras with different lens centers is a part of computer vision in Robotics. For the first part of the question where we need to find the disparity of a certain point P, Q, and R in the second camera, the disparity can be computed using geometry and trigonometry. It entails looking at how the image's position changes when moving from one camera to another.
For the second part, where we need to find the coordinates of a point V in 3D scene. The coordinates of point V can be obtained from the disparity between two locations of point V from the first and the second camera. Using similar triangles, we can compute the coordinates as:
X = Z * (x1 - x2) / (f * pixel size)
Y = Z * (y1 - y2) / (f * pixel size)
Z = f * Base line / ((x1 - x2) * pixel size)
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To get from one term to the next in a sequence, we multiply by 2 and then
add 4.
The third term in the sequence is 48.
What is the first term in the sequence?
Answer: the first term in the sequence is 9.
Step-by-step explanation:
Let the primary term be x.
At that point the moment term is 2x + 4.
And the third term is 2(2x + 4) + 4 = 4x + 12.
Since the third term is given as 48, we will set up an condition and unravel for x:
4x + 12 = 48
4x = 36
x = 9