Find the point on the graph of y=x^2+1 that’s closest to the point 8, 1.5. Hint: Remember
the distance formula.
Answers
Answer:
The point on the graph that is closest to the point (8, 1.5) is:
[tex]\left(\sqrt[3]{4}, 2 \sqrt[3]{2}+1\right) \approx \left(1.587,3.520)[/tex]
Step-by-step explanation:
To find the point on the graph of y = x² + 1 that is closest to the point (8, 1.5), we need to find the point on the parabola that is at the shortest distance from (8, 1.5). We can use the distance formula to do this.
[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Distance Formula}\\\\$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$\\\\\\where:\\ \phantom{ww}$\bullet$ $d$ is the distance between two points. \\\phantom{ww}$\bullet$ $(x_1,y_1)$ and $(x_2,y_2)$ are the two points.\\\end{minipage}}[/tex]
Any point (x, y) on the parabola y = x² + 1 can be defined as (x, x²+1).
Therefore:
(x₁, y₁) = (8, 1.5)(x₂, y₂) = (x, x²+1)Substitute these points into the distance formula to create an equation for the distance between any point on the parabola and (8, 1.5):
[tex]d = \sqrt{(x - 8)^2 + (x^2+1 - 1.5)^2}[/tex]
Simplifying this expression for d², we get:
[tex]d = \sqrt{(x - 8)^2 + (x^2-0.5)^2}[/tex]
[tex]d^2 = (x - 8)^2 + (x^2-0.5)^2[/tex]
[tex]d^2 = x^2-16x+64 + x^4-x^2+0.25[/tex]
[tex]d^2=x^4-16x+64.25[/tex]
To find the x-coordinate that will minimize this distance, take the derivative of the expression with respect to x, set it equal to zero and solve for x:
[tex]\implies 2d \dfrac{\text{d}d}{\text{d}{x}}=4x^3-16[/tex]
[tex]\implies \dfrac{\text{d}d}{\text{d}{x}}=\dfrac{4x^3-16}{2d}[/tex]
Set it equal to zero and solve for x:
[tex]\implies \dfrac{4x^3-16}{2d}=0[/tex]
[tex]\implies 4x^3-16=0[/tex]
[tex]\implies 4x^3=16[/tex]
[tex]\implies x^3=4[/tex]
[tex]\implies x=\sqrt[3]{4}[/tex]
Finally, to find the y-coordinate of the point on the graph that is closest to the point (8, 1.5), substitute the found value of x into the equation of the parabola:
[tex]\implies y=\left(\sqrt[3]{4}\right)^2+1[/tex]
[tex]\implies y=\sqrt[3]{4^2}+1[/tex]
[tex]\implies y=\sqrt[3]{16}+1[/tex]
[tex]\implies y=\sqrt[3]{2^3 \cdot 2}+1[/tex]
[tex]\implies y=\sqrt[3]{2^3} \sqrt[3]{2}+1[/tex]
[tex]\implies y=2 \sqrt[3]{2}+1[/tex]
Therefore, the point on the graph that is closest to the point (8, 1.5) is:
[tex]\left(\sqrt[3]{4}, 2 \sqrt[3]{2}+1\right) \approx \left(1.587,3.520)[/tex]
Additional information
To find the minimum distance between the point on the graph and (8, 1.5), substitute x = ∛4 into the distance equation:
[tex]\implies d = \sqrt{(\sqrt[3]{4} - 8)^2 + ((\sqrt[3]{4})^2-0.5)^2}[/tex]
[tex]\implies d = 6.72318283...[/tex]
Related Questions
use the methods taught in this class to find the slope and y-intercept of the regression line. round your answers to 3 decimal places.
Answers
The slope and Y-intercepts of the regression lines can be found using the formulae: m = (nΣxy - ΣxΣy) / (nΣx² - (Σx)²) and b = (Σy - mΣx) / n
To find the slope and y-intercept of a regression line, you can use the formula y = mx + b, where m is the slope and b is the y-intercept. Regression Line is a method in which the dependent variable serves as the explanatory variable, and the former is utilized to infer information about the latter and create a forecast for it.
To calculate the slope of a regression line, you can use the formula m = (nΣxy - ΣxΣy) / (nΣx² - (Σx)²), where n is the number of data points, x and y are the variables, and Σ represents summation.
To calculate the y-intercept of a regression line, you can use the formula b = (Σy - mΣx) / n.
Once you have calculated these values, you can substitute them into the equation y = mx + b to get your regression line.
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Correct question is:
Use the methods taught in this class to find the slope and y-intercept of the regression line. Round your answers to 3 decimal places.
The Venn diagram here shows the cardinality of each set. Use this to find the cardinality of the given set.
Answers
n(A ∩ B ∩ C^c) = 15 there are 15 elements in the intersection of sets A and B but not in set C.
What is a circle?
A circle is a two-dimensional geometric figure that consists of all the points in a plane that are equidistant from a given point called the center. The distance from the center to any point on the circle is called the radius, which is denoted by the letter "r".
To find the cardinality of n(A∩B∩C^c), we need to know the number of elements that are in the intersection of A and B but not in C.
One way to approach this is to use the principle of inclusion and exclusion. This states that:
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ C) - n(B ∩ C) + n(A ∩ B ∩ C)
We can rearrange this formula to solve for n(A ∩ B ∩ C^c):
n(A ∩ B ∩ C^c) = n(A ∩ B) - n(A ∩ B ∩ C) - n(B ∩ C) + n(A ∩ C) + n(B) + n(C) - n(A)
Plugging in the values we have been given, we get:
n(A ∩ B ∩ C^c) = 4 - 3 - 9 + 7 + 11 + 15 - 10
Simplifying this expression, we get:
n(A ∩ B ∩ C^c) = 15
Therefore, there are 15 elements in the intersection of sets A and B but not in set C.
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Keekee bought a magazine for $6 and some binders for $9 each. She spent a total of $78. How
many binders did she buy? Step by step Please!!
Answers
Answer:
10 binders ........... .......... .....
Answer:
For this one I assigned y as the missing number
We know that $6 magazine plus $9 times a specific amount would equal $78
so 6 + 9(y) = 78
6 + 9(8) = 78
9(8)=72
72+6=78
So there were 8 binders bought at $9 each and a 1 magazine bought for $6.
Step-by-step explanation:
PLS HELP 40 POINTS AND BRAINLIEST!!
Answers
Step-by-step explanation:
All of the EXTERNAL angles of a polygon sum to 360 °
95 + 44 + 42 + 78 + p = 360°
p =101°
then q + p form a straight line = 180 °
so q = 79 °
Estimate a 15% tip on a dinner bill of $58.12 by first rounding the bill amount to the nearest ten dollars.
Answers
Answer: an estimated 15% tip on a dinner bill of $58.12 would be $8.72.
Step-by-step explanation:
First, we can calculate 10% of $58.12 by moving the decimal point one place to the left, which gives us $5.81.
Then, we can calculate half of 10%, which is 5%, by dividing $5.81 by 2, giving us $2.91.
Finally, we add 10% and 5% to get the estimated tip amount:
$5.81 + $2.91 = $8.72
Therefore, an estimated 15% tip on a dinner bill of $58.12 would be $8.72.
A store purchased a stylus for $22.00 and sold it to a customer for 20% more than the purchase price. The customer was charged a 6% tax when the stylus was sold. What was the customer’s total cost for the stylus?
Answers
Answer: $27.98
Step-by-step explanation:
22.00 × .2= 4.40
22 + 4.40 = 26.40
26.40 × .06 = 1.584
26.40 + 1.584 = 27.984
Round to the nearest hundred so the total paid by the customer would be 27.98
write the equation for the relationship between x and y. x=1,2,4 y= 0,-5,-15
Answers
Answer:
y = -5x + 5
Step-by-step explanation:
y = mx + b
y = __x + __
To write the equation we need the slope (m) and the y-intercept (b).
Slope:
The is the change in y over the change in x. As the y is decreeing by 5, the x is increasing by 1. This gives us the slope -5/1 = -5
y-intercept:
The y intercept is the point (0,b). It is when x equals zero. If we start at 1 and go back to 0. That mean that the y increase by 5, so the intercept is at the point (0,5) which makes the y-intercept 5.
y = -5x + 5
Helping in the name of Jesus.
Find the surface area of the triangular pyramid. The side lengths of the base are equal.
Answers
The surface area of the triangular pyramid is the sum of the lateral surface area and the base area.
Lateral Surface Area: The lateral surface area of a triangular pyramid is equal to the product of the slant height and the perimeter of the base.
Slant height = √ ( (length of side)2 + (height)2 )
Perimeter of the base = 3 x length of side
Therefore, the lateral surface area = √ ( (length of side)2 + (height)2 ) x 3 x length of side
Base Area: The base area of the triangular pyramid is equal to one-half of the product of the three side lengths.
Therefore, the base area = 1/2 x (length of side) x (length of side) x (length of side)
Surface Area: The total surface area of the triangular pyramid = lateral surface area + base area
Therefore, the surface area = √ ( (length of side)2 + (height)2 ) x 3 x length of side + 1/2 x (length of side) x (length of side) x (length of side)
I need help I really don’t get this
Answers
The result of multiplying [tex]5,230[/tex] by [tex]25[/tex] is a quotient of [tex]209.2[/tex] rupees.
What in arithmetic is a quotient?In this example, the number being divided (15) is known as the reward, and the number being divided by (3 in this instance) is known as the divisor. The consequence of divide is the quotient.
What is the ratio between two numbers?When we divided one number by another, the result is a quotient. As an instance, when we divide 6 with 3, we obtain the answer 2, which corresponds to the division.This quotient may be either an integer or a decimal. For describing the presence or intensity of a quality in someone or something, the quotient is utilized.
25)5230(209.2
-50
--------------
230
-225
-----------------
50
-50
------------
0
-----------
So the quotient is [tex]209.2[/tex]
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PLEASE HELP AND SHOW THE WORK
Answers
An equation of the line that goes through the point (-1, -3) and (3, 5) is y = 2x - 1.
An equation of the line in slope-intercept form that is perpendicular to the equation for obstacle 1 is y = -x/2 + 3.
How to determine an equation of this line?In Mathematics, the point-slope form of a straight line can be calculated by using the following mathematical expression:
y - y₁ = m(x - x₁) or [tex]y - y_1 = \frac{(y_2- y_1)}{(x_2 - x_1)}(x - x_1)[/tex]
Where:
m represent the slope.x and y represent the points.At data point (-1, -3), a linear equation in slope-intercept form for this line can be calculated by using the point-slope form as follows:
[tex]y - y_1 = \frac{(y_2- y_1)}{(x_2 - x_1)}(x - x_1)\\\\y - (-3) = \frac{(5- (-3))}{(3-(-1))}(x -(-1))\\\\y +3 = \frac{(5+3)}{(3+1)}(x +1)[/tex]
y + 3 = 2(x + 1)
y = 2x + 2 - 3
y = 2x - 1
In Mathematics, a condition that must be met for two lines to be perpendicular is given by:
m₁ × m₂ = -1
2 × m₂ = -1
m₂ = -1/2.
At point (-4, 5), an equation of the line can be calculated by using the point-slope form as follows:
y - y₁ = m(x - x₁)
y - 5 = -1/2(x + 4)
y = -x/2 - 2 + 5
y = -x/2 + 3
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5x+6/1/3=3x/0.4 HELP RSM.
Answers
Answer:
x = 0.8
Step-by-step explanation:
5x+6/1/3=3x/0.4
5x + 2 = 3x/0.4
2x + 0.8 = 3x
x = 0.8
Which equation defines y as a linear function of x?
Select all that apply.
□ A. y = 3x² +7
□ B. x - 4y = 5
□ C. 5x² + y = 3
OD. 2(x+3y) = 6
O E. y = 5x - 3x + 4
Answers
Among the given equations, the equation E. y = 5x - 3x + 4 is the only equation that is in the form y = mx + b.
What is linear equation ?
A linear function is defined as a function that has a constant rate of change, meaning that as x increases by a certain amount, y also increases by a certain amount. Therefore, the equation that defines y as a linear function of x is of the form y = mx + b, where m is the slope and b is the y-intercept.
Among the given equations, the equation E. y = 5x - 3x + 4 is the only equation that is in the form y = mx + b. By combining like terms, we can simplify it to y = 2x + 4. This equation has a constant rate of change of 2, which means that for every 1 unit increase in x, y will increase by 2 units.
The other equations are not linear functions because they either have an x² term, which causes a non-constant rate of change, or they have multiple variables, which means that changes in x will not produce a consistent change in y.
Therefore, the answer is E. y = 5x - 3x + 4.
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The coordinates of three vertices of a parallelogram are (3,-2), (5,2) and (0,2). What are the coordinates of the fourth vertex?
Answers
The coordinate of fourth point of a parallelogram is (2,0)
We need to remember that the diagonals of a parallelogram intersect each other at a halfway point and the midpoint of each diagonal is the same.
A parallelogram is a geometric object with sides that are parallel to one another in two dimensions. It is a form of polygon with four sides (sometimes known as a quadrilateral) in which each parallel pair of sides have the same length. A parallelogram's adjacent angles add up to 180 degrees.
The midpoint formula
[tex]M=\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}[/tex]
AC=BD
We can find the coordinates of the fourth vertex (x,y) through this procedure:
For x
[tex]\frac{5-2}{2}=\frac{0+x}{2}\\\\x=2\\\\for\ y \\\frac{2-2}{2}=\frac{y+2}{2}\\\\y=0[/tex]
Hence the coordinate of fourth point is (2,0)
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The table represents some points on the graph of a linear function.
Which equation represents the same relationship?
y= -1.5x+3
y = 3x - 1.5
y= 1.5x -3
y= -3x+1.5
Answers
An equation that represents the same relationship include the following: D. y = -3x + 1.5.
How to determine an equation of this line?In Mathematics, the point-slope form of a straight line can be calculated by using the following mathematical expression:
y - y₁ = m(x - x₁) or [tex]y - y_1 = \frac{(y_2- y_1)}{(x_2 - x_1)}(x - x_1)[/tex]
Where:
m represent the slope.x and y represent the points.At data point (-3.5, 10.5), a linear equation in slope-intercept form for this line can be calculated by using the point-slope form as follows:
[tex]y - y_1 = \frac{(y_2- y_1)}{(x_2 - x_1)}(x - x_1)\\\\y - 10.5 = \frac{(3- 10.5)}{(-0.5 +3)}(x +3)[/tex]
y - 10.5 = -3(x + 3)
y = -3x - 9 + 10.5.
y = -3x + 1.5
In this context, we can reasonably infer and logically deduce that a linear function of the line that represents this table in slope-intercept form is y = -3x + 1.5.
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Graph the system of linear equations.
4x + 3y = 24
-2x + 6y = 18
Use the Line tool to graph the lines.
Answers
A line can be defined by two points, this is how standard form equations are graphed. The points are the x-intercept, (C/B, 0), and the y-intercept, (0, C/A).
A line can also be defined by a point and the slope, this is how slope-intercept equations are graphed. The -intercept, b, is the point (0,b), and the slope, m, gives the rise and the run to calculate another point on the graph.
You are buying 30 acres of farm land at $12000 per acre. What is total cost?
Answers
360000 acres
Step-by-step explanation:
12000 x 30= 360000
10. Audry spends $600 per month on rent. She makes $25,000 per year. To the nearest percent, what percent of her income does she spend on rent? F 28% G 21% H 20% J 29%
Answers
Answer:
J 29%
Step-by-step explanation:
Monthly rent: $600.
There are 12 months in 1 year, so the total rent she pays in 1 year is
12 × $600 = $7200
We need to find what percent $7200 is of $25,000.
percent = part/whole × 100%
percent = 7200/25000 × 100%
percent = 28.8%
Rounded to the nearest percent, the answer is 29%.
The box plots show a random sample of wait times for two rides at a water park
Answers
The median wait time for Speed Slide is 2 minutes longer than the median wait time for Wave Machine and the IQR for both rides is 6 minutes.
Define the term box plot?A box plot, also known as a box and whisker plot, is a graphical representation of a data set that shows the distribution of the data along a number line.
In the box plots show a random sample of wait times for two rides at a water park is shown in figure.
If we compare the wait times in the box plots,
then for, Speed Slide: Median = 11
IQR= Q1 - Q3 (Calculation formula of IQR)
= 12 - 6
IQR = 6 minutes
for wave slide: median: 9
IQR= Q1 - Q3 (Calculation formula of IQR)
= 11 - 9
IQR = 2 minutes
The median wait time for Speed Slide is 2 minutes longer than the median wait time for Wave Machine and the IQR for both rides is 6 minutes.
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6.) The sum of two numbers is 45.
The larger number y is 6 less than twice the smaller number
a.
Write a system of linear equations.
What is the smaller number?
Answers
Answer:
x= 13 =smaller number
Step-by-step explanation:
let x=smaller number
y=2x-6=larger number
x+y=45
x+2x-6=45 (equation)
3x=39
x=13
Answer:
[tex]x = 13[/tex] ..... smaller number
and
[tex]y = 32[/tex] ....... larger number
Step-by-step explanation:
Greetings!!!
Let the two numbers be X and Y
[tex]x + y = 45........ \:equation \: 1[/tex]
The larger number is y and is 6 less than twice the smaller number. which means:-
[tex](y - 6) = 2x........... \: equation \: 2[/tex]
so, now solve for y. from equation 2
[tex]y - 6 = 2x \\ y = 2x + 6[/tex]
Substitute equation 1 into equation 2
[tex]x + y = 45 \\ x + (2x + 6) = 45 \\ 3x + 6 = 45 \\ 3x = 45 - 6 \\ 3x = 39....divide \: both \: sides \: by \: 3 \\ x = 13[/tex]
To solve for y substitute x into the first equation
[tex](13) = + y = 45 \\ y = 45 - 13 \\ y = 32[/tex]
Finally, to be more sure make a cross check
[tex]y - 6 = 2x \\ 32 - 6 = 2(13) \\ 26 = 26[/tex]
If you have any questions tag it on comments
Hope it helps!!!
3.
h
b=32 inches
h = 22 inches
Find the area of the triangle above.
O242 sq. inches
O125 sq. inches
O 300 sq. inches
O 352 sq. inches
Answers
There are 352 square inches in the triangle's surface area. Hence, O 352 sq. inches is 's correct response.
The correct answer is 352 sq. inches
Which fundamental principle of a triangle is this?The triangle has the following characteristics: There are 180 angles in a triangle, thus they all sum up to 180. The lengths of the two sides of a triangle are greater than the length of the third side. Similar to this, the third side of a triangle has a shorter length than the angle between its 2 sides.
A = (1/2)bh if b is the quadrilateral base and h is its height.In this instance, the triangle's height is 22 inches, while its base is 32 inches. These values are substituted into the calculation to produce the following result: A = (1/2)(32)(22) A Equals 352 square inches
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A type of wood has a density of 250 kg/m3. How many kilograms is 75,000 cm3 of the wood? Give your answer as a decimal.
Answers
V=75000cm3=0,075m3
Kg=?
D=m*v
m=d/v
m=250/0.075=3333,33kg
which is correct answer?
a (a value)
b
c
d
Answers
Indeed, the intermediate value theorem demonstrates that the equation [tex]x + sin(x) = -1[/tex] must have at least one solution on the interval [-/2, /4]. Thus option B is correct.
What is the intermediate valve theorem?The intermediate value theorem states that if a continuous function exhibits values with opposite signs at two places, there must be at least one location in the interval where it equals zero (or crosses the x-axis).
Note that x + sin(x) is a continuous function to see why. When x + sin(x) is evaluated at the interval's left endpoint, x = -/2, the result is [tex]-/2 + sin(-/2) = -/2 - 1[/tex] , which is less than -1.
The intermediate value theorem states that because x + sin(x) is continuous and can have values both less than and greater than -1 at either end of the range, it must also have a value of -1 somewhere in between.
Therefore, When x + sin(x) is evaluated at the interval's right endpoint, x = /4, the result is [tex]x + sin(x) = x + 2/2,[/tex] Which is greater than -1.
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How do you compute the sum of squared errors
Answers
Answer:
Relating SSE to Other Statistical Data
Variance = SSE/n, if you are calculating the variance of a full population.Variance = SSE/(n-1), if you are calculating the variance of a sample set of data.
7. Hal records the numbers of winners
of a contest in which the player
chooses a marble from a bag.
DA G
Game 1
Game 2
Game 3
Number of Number of
Players
Winners
123
52
155
63
172
65
Based on the data for all three
games, what is the experimental
probability of winning the contest?
Express the answer as a decimal.
Answers
Answer:0.40
Step-by-step explanation:
the experimental probability of winning the contest is 0.4 or 40%.
Define probabilityProbability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, where 0 denotes the impossibility of the occurrence and 1 denotes its certainty. An occurrence is more likely to occur the higher its probability.
Players: 123 + 155 + 172, or 450 total
52 plus 63 plus 65 winners make up the total of 180.
Experimental probability of winning the contest = Total number of winners / Total number of players
Experimental probability of winning the contest = 180 / 450 = 0.4
Hence, the experimental probability of winning the contest is 0.4 or 40%
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0 / 1
It is November 1 of Year 1. Sales for Dee Company for November and December of Year 1 and January of Year 2 are forecasted to be as follows:
November, 400,000; December 600,000; January, 200,000
60% of sales are credit sales; the remaining sales are cash sales. Of these credit sales, 5% are collected during the month of sale, 25% in the following month, and 65% in the second following month; 5% are never collected. Total sales for September and October of Year 1 were 100,000 and 150,000, respectively.
What is the forecasted amount of total cash collections in November of Year 1?
Answers
Answer:
Step-by-step explanation:
To find the forecasted amount of total cash collections in November of Year 1, we first need to determine the credit sales for September, October, November, and December of Year 1, and January of Year 2.
Credit sales for September and October of Year 1 have already been given as 100,000 and 150,000, respectively.
For November, credit sales are 60% of 400,000 = 240,000.
For December, credit sales are 60% of 600,000 = 360,000.
For January of Year 2, credit sales are 60% of 200,000 = 120,000.
Now, we need to calculate the amount of credit sales that will be collected in November of Year 1.
5% of credit sales for November will be collected in November itself, which is 5% of 240,000 = 12,000.
25% of credit sales for November will be collected in December, which is 25% of 240,000 = 60,000.
65% of credit sales for November will be collected in January of Year 2, which is 65% of 240,000 = 156,000.
The remaining 5% of credit sales for November, which is 5% of 240,000 = 12,000, will never be collected.
Therefore, the total cash collections in November of Year 1 will be the sum of cash sales for November, and the amount of credit sales collected in November:
Cash sales for November = 40% of 400,000 = 160,000
Credit sales collected in November = 12,000
Total cash collections in November of Year 1 = 160,000 + 12,000 = 172,000.
Hence, the forecasted amount of total cash collections in November of Year 1 is $172,000.
helps asap !!!!!
if the original price of a refrigerator is $3200 tiana buys the refrigerator at a sale for 20% off the original price she has a discount voucher that gives her a further $64 off the sale price
what percentage of the original price does tiana pay for the refrigerator ?
(can somebody show me how to do this step by step)
Answers
Answer:
she payed 78% of the origional price
Step-by-step explanation: ok so boom right 20% of 3200 is 640 so that would make 2% 64 which would be the discount voucher ( 20% = 640 take a away a 0 on both and it shows u) so u do the 20% plus the 2% and that would make 22% and 22-100 is 78%. :)
Martin has a spinner that is divided into four sections labeled A, B, C, and D. He spins the spinner twice. PLEASE ANSWER RIGHT HELP EASY THANK UU
Drag the letter pairs into the boxes to correctly complete the table and show the sample space of Martin's experiment.
Answers
Answer:
going from left to right:
AA
BD
CB
DC
please answer question i cannot do it
Answers
The value of the height of the trapezium is 10.3 cm.
What is the area of a trapezium?The area of a trapezium is given by the formula:
Area = (1/2) x (sum of parallel sides) x (distance between the parallel sides)
The parallel sides of the trapezium are given as 18 cm and 13 cm
The area of the trapezium is given as 160 cm²
The height of the trapezium = h
The value of the height of the trapezium is calculated as follows;
160 = ¹/₂ (18 + 13) h
2 x 160 = (31)h
320 = 31h
h = 320 / 31
h = 10.3 cm
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A water cooler springs a leak and empties in 2 minutes. The graph below shows the rate at which water leaks from the cooler as a function of time.
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The amount of water that was in the cooler before it started leaking was 6 gallons.
Describe Integration?Integration is a mathematical process that involves finding the integral of a function. It is the reverse operation of differentiation, which involves finding the derivative of a function. The integral of a function is a measure of the area under the curve of the function, between two given limits of integration.
The graph shows the rate at which water leaks from the cooler as a function of time, which means that the y-axis represents the rate of leakage in gallons per minute (gal/min), and the x-axis represents the time in minutes.
Since we know that the cooler emptied in 2 minutes, we can integrate the leakage rate over the time interval [0, 2] to find the total amount of water that leaked out:
Total amount of water leaked = ∫[0,2] leakage rate(t) dt
The leakage rate is given by the graph, which consists of a straight line connecting two points: (0,6) and (2,0). We can express this line as a linear equation in slope-intercept form:
leakage rate(t) = mt + b
where m is the slope of the line and b is the y-intercept. To find the slope, we can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) = (0,6) and (x2, y2) = (2,0). Plugging in the values, we get:
m = (0 - 6) / (2 - 0) = -3
So the equation of the line is:
leakage rate(t) = -3t + 6
Now we can integrate this equation over the time interval [0, 2] to get the total amount of water leaked:
Total amount of water leaked = ∫[0,2] (-3t + 6) dt
= [-3t²/2 + 6t] from 0 to 2
= (-3(2)²/2 + 6(2)) - (-3(0)²/2 + 6(0))
= (6 - 0) - (0 - 0)
= 6 gallons
Therefore, the amount of water that was in the cooler before it started leaking was 6 gallons.
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The complete question is :
Classify each statement according to whether it describes a regression or a coefficient of determination Regression Coefficient of determination represented as R models the relationship between two variables describes how well data points fit a line values range from 0 to 1 can be positive or negative can be described by the linear equation YmX + b
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The claims under "Regression" are as follows: 1. can be described by the l
Regression is a statistical technique that connects one or more independent (explanatory) variables to a dependent variable. If there is a relationship between changes in one or more of the explanatory factors and changes in the dependent variable, it can be shown using a regression model.
Y = mx + b is how the slope-intercept form of the equation for a straight line is expressed.
In the equation y = mx + b, m denotes the slope of the line and b its intercept. The distance between the line and the x- and y-axes is denoted by the letters x and y, respectively.
The first of the following claims, "Is used to assess the fit of a model," is true.
b) By including more factors, the result may be overstated.
called the coefficient of determination.
d.) Indicates the proportion of the variation in y that the model can account for.
e.) It is equivalent to the correlation coefficient r2 in basic linear regression.
What is Regression Analysis?
A collection of statistical procedures known as regression analysis is used to estimate the correlations between a dependent variable and one or more independent variables.
SS(res) + SS(reg) = SS, R2 1 - SS(res) / SS(tot) (tot)
R2 is defined as (SS(reg)/SS(tot) = (SS(reg)/n)/(SS(tot)/n)
Where SS(tot) = (y - y(bar)), the total sum of squares (proportional to the variance of the data), 2 • The explained sum of squares, also known as the regression sum of squares, is SS(reg) = (f(i)-y(bar)). SS(res) = (y(i) - f(i))2 = e2 is the sum of squares of residuals, often known as the residual sum of squares (i)
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