The calculated t-statistic is:t = (-2.5 - 0) / 5.303 = -0.471Since |-0.471| < 1.980, we fail to reject the null hypothesis.
a. The coefficients for testing the contrast between the average growth at 50% control and the average growth at 0% and 100% control can be calculated as follows: c = [0, 1, 0, -1/2, 0, -1/2]
The coefficients correspond to the contrast c = μ50% - (μ0% + μ100%)/2, where μi represents the population mean for the i-th level of vegetation control. The contrast can also be written as c = [0, 1, 0, -1/2, 0, -1/2] * [μ0%, μ50%, μ100%, (μ0% + μ100%)/2, (μ0% + μ100%)/2, μ50%], where * denotes the dot product.
b. To perform the test, we can use a t-test for the contrast c. The test statistic is given by:t = (ĉ - c0) / SE(ĉ), where ĉ is the sample estimate of the contrast, c0 is the null hypothesis value (in this case, c0 = 0), and SE(ĉ) is the standard error of the contrast estimate.
The sample estimate of the contrast can be calculated as:ĉ = y50% - (y0% + y100%)/2, where yi is the sample mean for the i-th level of vegetation control. Plugging in the values, we get:ĉ = 75 - (50 + 120)/2 = -2.5.
The standard error of the contrast estimate can be calculated as:SE(ĉ) = sqrt{[(s^2/n50%) + (s^2/n0%) + (s^2/n100%)] * [1/2 + 1/(2n50%) + 1/(2n0%) + 1/(2*n100%)]}, where s is the pooled standard deviation, n50%, n0%, and n100% are the sample sizes for the 50%, 0%, and 100% control groups, respectively.
Plugging in the values, we get:SE(ĉ) = sqrt{[(30^2/40) + (30^2/40) + (30^2/40)] * [1/2 + 1/(240) + 1/(240) + 1/(2*40)]} = 5.303.
The degrees of freedom for the t-test are df = n - k, where n is the total sample size and k is the number of groups (in this case, k = 3). Plugging in the values, we get df = 117. Using a significance level of 0.05 and consulting a t-distribution table with 117 degrees of freedom, we find that the critical value for a two-tailed test is ±1.980.
The calculated t-statistic is:t = (-2.5 - 0) / 5.303 = -0.471Since |-0.471| < 1.980, we fail to reject the null hypothesis. There is not enough evidence to support the claim that the average growth at 50% control is less than the average of 0% and 100% levels.
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In how many ways can 345 be written as the sum of an increasing sequence of two or more consecutive positive integers
There are 7 ways for 345 to be written as the sum of an increasing sequence of two or more consecutive positive integers.
We need to find consecutive numbers (an arithmetic sequence that increases by 1) that sum to 345. This calls for the sum of an arithmetic sequence given that the first term is k, the last term is g, and with n elements, which is:
n(k+g)/2.
There are 7 ways can 345 be written as the sum of an increasing sequence of two or more consecutive positive integers.
We look for sequences of n consecutive numbers starting at k and ending at k + n − 1. We can now substitute g with k+n- 1. Now we substitute our new value of g into n(k + g)/2 to get the sum i.e.
n(k+k+n-1)/2 = 345
This simplifies to n(2k+n-1)/2 = 345.
This gives a nice equation. We multiply out the 2 to get that n. (2k+n-1): =690. This leaves us with 2 integers that multiply to 690 which leads us to think of factors of 690.
We know the factors of 690 are:
1, 2, 3, 5, 6, 10, 15, 23, 30, 46, 69, 115, 138, 230, 345, 690
So, through inspection (checking), we see that only 2, 3, 5, 6, 10, 15, and 23 work. This gives us the answer to 7 ways.
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Julie wanted to match Lisa's obstacle course record of 68.2 seconds. She has already spent thirty-nine and one fourth seconds on wall climbing and 12.84 seconds on the ropes. How much time did she have left to match the record? 12.24 seconds 16.11 seconds 26.41 seconds 28.95 seconds
The time she has left to match the record is 16.11 seconds. The correct answer would be an option (B).
What are Arithmetic operations?Arithmetic operations can also be specified by subtracting, dividing, and multiplying built-in functions.
To match Lisa's obstacle course record of 68.2 seconds, Julie needs to complete the course in 68.2 seconds.
We know that she has already spent 39.25 seconds on wall climbing and 12.84 seconds on the ropes.
So, the time she has left to match the record is:
68.2 - (39.25 + 12.84) = 68.2 - 52.09 = 16.11 seconds
In the above step, use the arithmetic operations to solve.
Hence, the answer is 16.11 seconds.
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pls help with 1-6 pls !!
1. Write an equation in slope-intercept form of the line given the slope=" 1/4 pass through
(-2,1).
2. Write an equation in slope-intercept form of the line that passes through (0,4) and (5,19)
3. Write a linear function f with the given values.
a. g(0) = 2, f (2) = -3
b. g (-4) = -5, g (3) = - 1.
4. The corresponding data for electricity generated by hydropower are 248 million megawatt hours in 2007 and 277 million megawatt hours in 2012. Write a linear model that represents the number of megawatt hours generated by hydropower as a function of the number of years since 2007.
5. Write an equation in point-slope form of the line that passes through (-2, -5) and (4, -1).
6. In a city, if a customer is charged $12 for using 1,000 gallons and $16 for using 1,800
gallons.
a. Write a linear model that represents the monthly cost for water as a function of the number of gallons you will used.
b. Find the cost of 2,000 gallons.
Answer:
See below, and note the interpretation of the question.
Step-by-step explanation:
See the attached image for a graph of all the lines and points.
1. Write an equation in slope-intercept form of the line given the slope=" 1/4 pass through (-2,1).
1. Look for an equation of the form y=mx+b, where m is the slope and b is the y-intercept (the value of y when x is zero).
We are given the slope, so use that for m: y = (1/4)x + b.
We need a value of b that forces the line through point (-2,1). This can be done by entering the point into the above equation and solving for b:
y = (1/4)x + b
1 = (1/4)*(-2) + b [for (-2,1)]
1 = -0.5+ b
b = 1.5
The equation is y = (1/4)x + 1.5
2. Write an equation in slope-intercept form of the line that passes through (0,4) and (5,19)
1. Look for an equation of the form y=mx+b
2. Determine the slope by calculating the Rise and Run of the two given points:
Going from (0,4) to (5,19):
Rise = 19-4 = 15
Run = 5 - 0 = 5
Rise/Run, or slope, m = (15/5) or 3
The equation with the slope of 5:
y = 5x+b
3. To find b, enter either of the 2 points into the equation and solve for b:
y = 5x+b
4 = 5*(0)+b for (0,4)
b = 4
The equation is y=5x+4
3. Write a linear function f with the given values.
Are the expressions written correctly? I see f and g. I'll assume the forst two are meant to be f, and the last two a new function, g(x).:
a. f(0) = 2, f(2) = -3
b. g(-4) = -5, g(3) = - 1.
These results supply us with 2 points for each function [line]:
f: (0,2) and (2,-3)
g: (-4,-5) and (3,-1)
We can use these two points to find the slope for lines f and g:
f: (0,2) and (2,-3)
Rise = -3 - 2 = -5
Run = 2 - 0 = 2
Slope = (-5/2) or -2.5
The equation for f is y = (-2.5)x + b
Enter one of the 2 points to find b:
y = (-2.5)x + b
y = (-2.5)*(0) + b for (0,2)
2 = b
The equation for f is f(x) = -2.5x+2
Do the same for g:
(-4,-5) and (3,-1)
Rise = (-1 - (-5)) = 4
Run = (3 - (-4)) = 7
Slope = 4/7 or 0.571
y = 0.571x + b
Substituting point (3,-1) and solving for b:
-1 = 0.571*(3) + b
b = -2.71
The equation for g is g(x) = 0.571c-2.71
See the attached image.
can someone help me solve this?
The answer is B: 1/16
7) What is the (x, y) solution for the following s [y=-4x+2 [y=6x-8
solution for the two equation is x = 1, and y = - 2
What is solution?
It is a mathematical term that acts as the substitute of variables present in one or more equations. It is the value of unknown numbers that makes an equation true.
What is the (x, y) solution for the following equation bellow?
equation given, y = - 4x + 2
y = 6x - 8
if we use addition method then y = - 4x +2
y = 6x -8
- - +
0 = - 10x + 10
10 x = 10
x = 1
putting the value of x in one of the above equations we get the y value.
y = 6×1 -8
y = -2
the solution is the x = 1 and y = - 2
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how to solve simplify rational expressions
An
elementary school teacher with 25 students plans to have each of them make a poster about two different states. The teacher first numbers the states (in alphabetical order, from
01 to 50), then uses a random number table to decide which states each kid gets. The random digits are 33874 64946 29977 66233.
a) Which two state numbers does the first student get?
b) Which two state numbers go to the second student?
Therefore , the solution of the given problem of combination comes out to be a) 45 and 10 b) 17 and 10
Define combination.Combinations are mathematical procedures that count the number of possible configurations from a set of items, where the selection direction is immaterial. You can select any arrangement of the parts in any combination. It's possible to mix together combinations and permutations.
Here,
Given :
Total students = 25
a) The first state is represented by the first two-digit number in the random digits, which is 45.
The following random two-digit number, which does not represent a state, is 92.
The following two-digit number in the random sequence is 10, which stands for the first student's second state.
b) The second student's initial state is represented by the following two-digit number (17) from the random digits.
Ten symbolizes the second state for the second state and is the first two-digit number of the random digits.
Therefore , the solution of the given problem of combination comes out to be a) 45 and 10 b) 17 and 10
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WILL GIVE BRAINLIEST AND 30 POINTS IF SOMEONE HELPS
The value of the variables x and y of the given transversal lines are;
x = 4/3 and y = 5/3
How to solve Algebra Expressions?We see from the transverse lines that;
The horizontal line shows the line has been divided into two equal parts by the vertical transversal lines. Thus;
5x + 8 = 21x
Subtract 5x from both sides to get;
21x - 5x = 8
6x = 8
x = 8/6
x = 4/3
Similarly, the upper division will be equal and as such;
20y - 2 = 17y + 3
20y - 17y = 3 + 2
3y = 5
y = 5/3
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Please I’m gone faillll helpppooop
On solving the question of the polynomial The quotient = 3x+8 and remainder are and 2 respectively.
What are Polynomials ?An expression that consists of variables, constants, and exponents that is combined using mathematical operations like addition, subtraction, multiplication, and division is referred to as a polynomial.
A polynomial is an equation made up of coefficients and indeterminates that uses only the addition, subtraction, multiplication, and powers of positive-integer variables.
Polynomial long division is an extended variant of the well-known mathematical operation known as long division. It is an algorithm for dividing a polynomial by another polynomial of the same or lower degree. Due to the fact that it breaks down a more difficult division issue into simpler ones, it may be completed quickly by hand.
Dividing 3x^3 - 2X^2 + 2x - 1 by x-1
= 3x+8
The quotient = 3x+8 and remainder are and 2 respectively.
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help me out please and thank you!
The images of points A(x, y) = (- 2, 8), B(x, y) = (2, 4) and C(x, y) = (- 2, 6) are A'(x, y) = (4, - 1), B'(x, y) = (2, 0.5) and C'(x, y) = (4, 0), respectively.
How to find the image of a given set of points by rigid transformations
In this problem we find three points: A(x, y) = (- 2, 8), B(x, y) = (2, 4) and C(x, y) = (- 2, 6), whose images must be determined by means of rigid transformations, that is, transformations that modify a points without any alteration on Euclidean distances.
The transformation rule of this question is a combination of reflection about x-axis, dilation, horizontal and vertical translations, all of them rigid transformations. Now we determine the image of each point:
Point A
A'(x, y) = (- 0.5 · (- 2) + 3, - 0.5 · 8 + 3)
A'(x, y) = (1 + 3, - 4 + 3)
A'(x, y) = (4, - 1)
Point B
B'(x, y) = (- 0.5 · 2 + 3, - 0.5 · 5 + 3)
B'(x, y) = (- 1 + 3, - 2.5 + 3)
B'(x, y) = (2, 0.5)
Point C
C'(x, y) = (- 0.5 · (- 2) + 3, - 0.5 · 6 + 3)
C'(x, y) = (1 + 3, - 3 + 3)
C'(x, y) = (4, 0)
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2. What is the new balance for the credit statement shown?
Billing
Date
Payments
& Credits
12/11/--
$70.94
Previous
Balance
$185.74
Finance
Charge
$2.71
New
Purchases
$49.80
New
Balance
So, here on solving the provided question, we can say that the New balance is as [tex]397.51[/tex]
What is balance ?A balance is the sum owing to an account in banking and accounting. The difference between the total debits and total credits posted to an account during an accounting period is referred to as a "balance" in accounting. The account will display a debit balance if the total of all debits exceeds the total of all credits.
Using the above data, first calculate the outstanding balance by using the formula below:
Previous balance-payments is as Unpaid balance
Unpaid balance is as [tex]95.85-45[/tex]
Unpaid balance is as [tex]50.85[/tex]
Finance charge is as [tex]50.85*1.5%[/tex]
Finance charge is as [tex]0.76[/tex]
New balance is as [tex]50.85+0.76+345.90[/tex]
New balance is as [tex]397.51[/tex]
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Jason's living room has 210 square feet of wall. How much wall is left to paint?
Answer:
I'm sorry, I don't have enough information to answer that question. How much of the wall has already been painted?
PLEASE HELP ME THIS IS DUE BY TOMORROW TAKE ALL MY POINTS JUST PLEASE HELP ME THE QUARTER ENDS TOMORROW SO PLEASE HELP MEEEEE
Maybe one reason people who work in the pentagon never seem to agree on anything is that there are five sides to every story
What is the value of the discriminant, b2 − 4ac, for the quadratic equation 0 = −2x2 − 3x + 8, and what does it mean about the number of real solutions the equation has?
The discriminant is −55, so the equation has 2 real solutions.
The discriminant is −55, so the equation has no real solutions.
The discriminant is 73, so the equation has 2 real solutions.
The discriminant is 73, so the equation has no real solutions.
The value of the discriminant b² - 4ac is 73.
The number of real solutions the equation has is two.
What is an equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
Example:
2x + 4 = 9 is an equation.
We have,
-2x² - 3x + 8 = 0
This is in the form of ax² + bx + c = 0
a = -2, b = -3, and c = 8
Now,
b² - 4ac
= (-3)² - 4 x (-2) x 8
= 9 + 64
= 73
This is a positive value.
This means,
The equation has two real solutions.
Thus,
The equation has two real solutions.
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What is the surface area of a triangle having a base of 6m and 8m high?
Hi !
[tex]A= \frac{b \times h}{2} = \frac{6 \times 8}{2} = 24m {}^{2} [/tex]
Have a nice day ;)
Waht number is 120% of 140
Answer:
x:b116.67
Step-by-step explanation:
120/100
140×100/120
You ride your bike for 3 days in a row. On Monday , you ride 3 and 2/3 miles. On Tuesday, you ride 1 and 1/2 times further. On Wednesday, you ride 1 and 1/4 miles less than Tuesday. Find the total number of miles.
Taking the given ride lengths for 3 consecutive days and adding them the total number of miles ridden by the rider is 14.67 miles.
How is the distance traveled on Tuesday related to Monday's distance?On Tuesday, the distance traveled is 1 and 1/2 times further than Monday's distance. This means that the distance traveled on Tuesday is 1.5 times the distance traveled on Monday.
How can we find the total distance traveled over a period of multiple days?To find the total distance traveled over a period of multiple days, we can add up the distances traveled on each day. This can be done by summing up all the individual distances traveled on each day. This can be done either in fraction or decimal. If the distances are in mixed numbers, it is good to convert them into fractions before adding them. This will give the total distance traveled over the period of multiple days.
On Tuesday, you ride 1 and 1/2 times further than on Monday, which is 3 and 2/3 miles * 1 and 1/2 = 5 and 1/2 miles.
On Wednesday, you ride 1 and 1/4 miles less than on Tuesday, which is 5 and 1/2 miles - 1 and 1/4 miles = 4 and 1/4 miles.
The total number of miles you rode on Monday, Tuesday, and Wednesday is:
3 and 2/3 miles + 5 and 1/2 miles + 4 and 1/4 miles = 13 and 5/6 miles
You can convert the mixed numbers to fraction to find the total miles.
3 and 2/3 miles = (33 + 2)/3 = 11/3 miles
5 and 1/2 miles = (52 + 1)/2 = 11/2 miles
4 and 1/4 miles = (4*4 + 1)/4 = 17/4 miles
So, the total number of miles you rode on Monday, Tuesday, and Wednesday is :
11/3 + 11/2 + 17/4 = (114 + 113 + 172)/12 = 11/3 + 33/4 = (114 + 11*3 + 34)/12 = 44/3 = 14.67 miles
You rode 14.67 miles in total for 3 days.
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Rosa likes to dance 10 hours a week to stay in shape. She tracks the amount of time
she dances each day.
The table below shows the data for the last five days.
im sorry if the picture is too blurry. but i’ll type it out too in case
1) did rosa meet her goal of dancing 10 hours a week? explain and show your work to prove whether rosa reached her goal.
2) what is the average dance time for the week? show your work to find the average.
then the numbers are on the chart.
Answer:
1) Yes
2) 2.07 hours/day
Step-by-step explanation:
1)
2.2 + 4 + 2 1/4 + 0 + 1.9 =
= 6.2 + 2.25 + 1.9
= 10.35
She danced a total of 10.35 hours, so she did meet her goal.
2)
10.35/5 = 2.07
She danced an average of 2.07 hours per day.
Nathan buys 9 items that cost b dollars each. She gives the cashier $100 dollars. Write
an expression for the change she should receive.
An expression for the change she will receive is 8b - 100 = 100
What is an expression?You should be aware that an expression in math is a sentence with a minimum of two numbers or variables and at least one math operation
The given parameters are
Nathan buys 9 items
She gives the cashier $100
The expression 8(b) -100 = 100%
Where 100 is the full amount payable
Therefore, the expression becomes 8b -100=100
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Write an equation of the passes through the given point and has the given slope? 1.(10, -4); slope - 2/5
The equation of the line that passes through the point (10, -4) and has the slope -2/5 is y = 2/5 x
How to identify the equation of the lineThe slope, m of the linear function is given as -2/5.
The equation of the line that passes through point (10, -4) is calculated using the point slope formula which is given as:
x₁ = 10
y₁ = -4
m = -2/5
substituting in to the formula
(y - y₁) = m (x - x₁)
y + 4 = -2/5 (x - 10)
y + 4 = -2/5 x + 4
y = 2/5 x + 4 - 4
y = 2/5 x
The equation of the line is given as y = 2/5 x
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A right triangle has a leg of 11 cm and a hypotenuse of 17 cm. What is the length of the other leg? round to the nearest tenth. Responses 6. 0 cm 6. 0 cm 13. 0 cm 13. 0 cm 20. 2 cm 20. 2 cm 168. 0 cm.
The length of the other leg of a right triangle, as per the Pythagoras theorem, is option B: 13.0 cm.
The Pythagorean theorem, sometimes known as Pythagoras' theorem, is a basic relationship between a right triangle's three sides in Euclidean geometry. According to this statement, the areas of the squares on the other two sides add up to the size of the square whose side is the hypotenuse. The calculation of the other leg of tbe triangle is given as follows.
According to the Pythagoras theorem:
(Hypotenuse)² = (Base)² + (Height)²
As per the question,
The height or the length of a leg of a right triangle is given equal to 11 cm and hypotenuse is given 17 cm. The other leg, or the base can be calculated as:
(Base)² = (Hypotenuse)² - (height)²
b² = (17)² - (11)²
b² = 289 - 121
b² = 168
b = √168
b = 12.96
or b = 13cm.
Thus, the base or the other leg of the right triangle is equal to 13 cm.
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Please answer the question.
The rate at which the distance is decreasing is - 30 centimeters per second. Ravi's distance from the first floor when he stepped on escalator is 1250 centimeters.
What is rate?The pace at which one quantity changes in relation to another quantity is known as the rate of change function. Simply said, the rate of change is calculated by dividing the amount of change in one thing by the equal amount of change in another.
a. From the table we can observe that as time increases, his distance from the first floor decreases.
The rate at which the distance is decreasing is given by:
Rate = (d2 - d1)/ (t2 - t1)
= (980 - 1070) / (9 - 6)
= -90 / 3
= -30
Hence, the rate at which the distance is decreasing is - 30 centimeters per second.
b. Ravi's distance from the first floor when he stepped on escalator is calculates as:
d - d1 = m(t - t1)
where, m is the rate.
Here, m = -30.
d - 1070 = (- 30) (t - 6)
d = -30t + 180 + 1070
d = -30t + 1250
The time when he stepped on escalator is 0.
d = -30(0) + 1250
d = 1250
Hence, Ravi's distance from the first floor when he stepped on escalator is 1250 centimeters.
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How do you find the perpendicular bisector of a line segment?
The perpendicular bisector can be found by solving for the equation of the line and substituting the two points.
The perpendicular bisector of a line segment can be found by using the midpoint formula. The midpoint formula is used to find the midpoint of two points (x1,y1) and (x2,y2). The formula is:
Midpoint = (x1 + x2)/2, (y1 + y2)/2
Once the midpoint is found, the slope of the perpendicular line is found by using the negative reciprocal of the original line. The formula for the slope of the perpendicular line is y = -1/m * x + b, where m is the slope of the original line.
The equation of the perpendicular bisector is then found by substituting the slope and the midpoint into the equation of a line, y - y1 = m(x - x1).
Finally, the perpendicular bisector can be found by solving for the equation of the line and substituting the two points. The resulting equation is the equation of the perpendicular bisector.
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How do you plot y =- 2x on a graph?
Answer:
below
Step-by-step explanation:
make the ordered pairsso if x=1 then y is... it's -2so if x=2 then y is... it's -4so if x=3 then y is... it's -6so now you have the ordered pairs (1,-2) and (2,-4) and (3,-6)Now plot themPlease help!
I genuinely don't know how to do this, a step-by-step would do great. Thank you.
Answer:
x = 10
Step-by-step explanation:
Because the triangles are similar, the following equations must be solved in order to find out how much bigger or smaller the triangle is.
10 * y = 15
16 * y = 24
You'll find that
y = 1.5
Then, you are able to insert y into the following equation and solve for x
(x - 4) * y = 9
(x - 4) * 1.5 = 9
1.5x - 6 = 9
1.5x = 15
x = 10
We can plug x back into the previous equation in order to check if it is correct
(x - 4) * 1.5 = 9
(10 - 4) * 1.5 = 9
6 * 1.5 = 9
9 = 9
(2,-1) y=1/4x+8 in slope-intercept form
Answer: Here is my question
Step-by-step explanation:
The local water slides have 70 employees, of which 40% are temporary. How many temporary employees are there?
How do you compare complex numbers with real numbers?
Complex numbers and real numbers cannot be directly compared as they are from different number systems.
Complex numbers have both a real and an imaginary component, whereas real numbers only have a real component. However, you can compare the magnitudes (absolute values) of complex numbers with real numbers.
The magnitude of a complex number is the distance from the origin in the complex plane, represented by the absolute value or modulus of the complex number, which is always a positive real number. By comparing the magnitudes of complex numbers, you can determine which one is larger or smaller.
For example, the magnitude of 3 + 4i is 5, and the magnitude of 2 + 3i is 3.5. Therefore, 3 + 4i has a larger magnitude than 2 + 3i.
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How do you find the distance between a point and a line vector in 3D?
Finding the distance between a point and a line vector in 3D is a bit more complicated than in 2D, but it is still doable.
What is vector?Vector is a mathematical object that has both magnitude and direction and can be represented by a line segment. It is used in many areas of mathematics and physics. In physics, vectors are used to describe forces, velocities, and accelerations. In mathematics, vectors can be used to represent points in space, transformations, and even linear equations. Vector calculus is used to analyze the behavior of vector fields, or functions of multiple variables.
The formula for finding the distance is based on the vector equation of a line.
Starting with a line given by vector $\vec{r}= \vec{a}+t\vec{v}$ , where $\vec{a}$ is the point on the line, $\vec{v}$ is a unit vector, and $t$ is a scalar parameter, and a point $\vec{b}$, the distance between the point and the line can be found by first computing the vector $\vec{p}=\vec{a}-\vec{b}$. Then, the distance is given by $d=\frac{|\vec{p}\cdot \vec{v}|}{|\vec{v}|}$.
This formula can also be used to find the closest point on the line to the point $\vec{b}$. The closest point on the line is given by $\vec{a}+t\vec{v}$, where $t=\frac{\vec{p}\cdot \vec{v}}{|\vec{v}|^2}$.
Thus, finding the distance between a point and a line in 3D involves computing the vector $\vec{p}$ and then using the formula above to compute the distance.
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To find the distance between a point and a line vector in 3D using the formula [tex]\frac{|(x_0-x_1)(x_0-x_2)|}{|x_2-x_1|}[/tex].
What is a vector?A vector is body which has both magnitude and direction. Geometrically, a vector can be imagined as a directed segment whose length corresponds to the absolute value of the vector and whose direction is indicated by an arrow. The direction of the vector is tail to head.
Suppose a 3D line is given by two points and the reference diagram attached below. where x₁ = (x₁,y₁ ,z₁) and x₂ = (x₂,y₂,z₂) are above it, giving us a vector along the line. [tex]v=[x_1+(x_2-x_1)t; y_1+(y_2-y_1)t; z_1+(z_2-z_1)t].[/tex]
The squared distance between a point on the line with parameter t and a point x₀ = (x₀, y₀, z₀) is therefore;
d² = [(x₁ - x₀) + (x₂ - x₁)t²] + [(y₁ - y₀) + (y₂ - y₁)t²] + [(z₁ - z₀) + (z₂ - z₁)t²]
To minimize the distance, set d(d²)/dt = 0 and solve for t to obtain:
t = [tex]\frac{-((x_1-x_0)(x_2-x_1))}{|x_2-x_1|^2}[/tex]
d² = (x₁ - x₀)² + (y₁ - y₀)² + (z₁ - z₀)² + 2t [(x₂ - x₁) (x₁ - x₀) + (y₂ - y₁) (y₁ - y₀) + (z₂ - z₁) (z₁ - z₀)] + t² [(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]
= [tex]|x_1-x_0|^2-2[/tex][tex]\frac{[(x_1-x_0)(x_2-x_1)]^2}{|x_2-x_1|^2} + \frac{[(x_1-x_0)(x_2-x_1)]^2}{|x_2-x_1|^2}[/tex]
= [tex]\frac{|x_1-x_0|^2|x_2-x_1|^2 - [(x_1-x_0)·(x_2-x_1)]^2 }{|x_2-x_1|^2}[/tex]
Using vector quadruple product:
(A × B)² = A²B² - (A.B)²
Where x denotes the cross product then gives:
d² = [tex]\frac{|(x_2-x_1)x(x_1-x_0)|^2)}{|x_2-x_1|^2}[/tex]
Now taking the square root results in the formula:
d = [tex]\frac{|(x_2-x_1)(x_1-x_0)|}{|x_2-x_1|}[/tex]
= [tex]\frac{|(x_0-x_1)(x_0-x_2)|}{|x_2-x_1|}[/tex]
Here, the numerator is simply twice the area of the triangle formed by points x₀, x₁, and x₂, and the denominator is the length of one of the bases of the triangle, which follows since, from the basic triangle area formula, [tex]\Delta=b \frac{d}{2}[/tex]
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please answer this question i am stuck 82 x 19 ≈80
Answer:1558
Step-by-step explanation:
Answer:1558
Step-by-step explanation:
[tex]82\\19[/tex]
put a line under it multiply the 9x2 and then 9x8 make sure to carry
put a zero under the 8 and multiply 1x2 and 1x8
now add the the numbers together
you get 1558