The average squared distance between the points is their variance
The average squared distance is 8/3
How to determine the average squared distance?The given parameters are:
R={(x,y): 0<=x<=2, 0<=y<=2}
The point = (2,2).
f(x,y) = (x-2)² + (y-2)²
The squared distance is calculated as:
D² = f(x,y) = (x-2)² + (y-2)²
Where the area (A) is:
A = xy
Substitute the maximum values of x and y in A = xy
A = 2 * 2
A = 4
The minimum values of x and y are 0.
So, the limits for the integrals are 0 to 2
The integral becomes
[tex]D\² = \int \int (x-2)\² + (y-2)\² dx dy[/tex]
Expand
[tex]D\² = \int \int x\² -4x + 4 + (y-2)\² dx dy[/tex]
Evaluate the inner integral with respect to x from 0 to 2
[tex]D\² = \int \frac 13x^3 -2x^2 + 4x + x(y-2)\² |\limits^2_0 dy[/tex]
Expand
[tex]D\² = \int [\frac 13(2)^3 -2(2)^2 + 4(2) + (2)(y-2)\²] dy[/tex]
Simplify
[tex]D\² = \int \frac 83 + (2)(y-2)\² dy[/tex]
Expand
[tex]D\² = \int \frac 83 + 2y^2-8y + 8 \ dy[/tex]
Evaluate the integral with respect to y from 0 to 2
[tex]D\² = [\frac 83y + \frac 23y^3- 4y^2 + 8y ]|\limits^2_0[/tex]
Expand
[tex]D\² = \frac 83*2 + \frac 23*2^3- 4*2^2 + 8*2[/tex]
D² = 32/3
The average squared distance (AD²) is calculated as:
AD² = D²/A
So, we have:
AD² = 32/3 [tex]\div[/tex] 4
Evaluate the quotient
AD² = 32/12
Simplify
AD² = 8/3
Hence, the average squared distance is 8/3
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The average squared distance between the points of R = {(x,y): 0<=x<= h, 0<=y<=k } is 8/3.
How do determine the average squared distance?
From the given parameters R = {(x,y): 0<=x<= h, 0<=y<=k }
The squared distance from point (h, k) is calculated as
D² = f(x,y) = (x-2)² + (y-2)²
Where the area (A) will be
A = xy
Substitute the maximum values of x and y
A = 2 × 2 = 4
The minimum values of x and y are 0.
So, The integral becomes
[tex]D^{2} = \int \int(x-2)^2 +(y-2)^2 dx dy\\\\D^{2} = \int \int\limits^2_0 (x)^2 -4x + 4+(y-2)^2 dx dy\\\\D^{2} = \int1/3(x)^3-2x^{2} +4x + 4+x(y-2)^2 dy\\\\D^{2} = \int8/3+2y^{2} -8y +8 dy[/tex]
[tex]D^2 = {8/3y +2/3y^2 -4y^2+8y[/tex]
By expanding we get,
D^2 = 32/3
The average squared distance (AD²) is calculated as:
AD² = D²/A
AD² = 32/3 4
AD² = 32/12
AD² = 8/3
Hence, the average squared distance is 8/3
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Which two expressions are equivalent to 8(6x + 9)
A. 8(6x) + 8(9)
B. 8(6x) + 9
C. 14x + 17
D. 48x + 17
E. 48x + 72
Explanation:
The distributive property allows us to go from the original expression to choice A. We multiply the outer term 8 by each term inside the parenthesis.
In general, the distributive property is a(b+c) = ab+ac. Example: 2(3+4) = 2(3)+2(4) = 6+8 = 14.
After simplifying the expression in choice A, it leads directly to choice E. This is because 8(6x) = 48x and 8(9) = 72.
Side note: A common mistake students do is to multiply the outer term by only one of the terms inside the parenthesis, while forgetting about the other term inside. Choice B shows this mistake. It's possible your teacher put choice B in there as a trap/trick answer.
A trapezoidal prism undergoes a dilation. The surface area of the pre-image is 35 m². The surface area of the image is 315 m². The height of the pre-image is 5. 8 m. What is the height of the image? Enter your answer as a decimal in the box. M.
The original object is called pre-image, and its dilated version is called image. The height of the image for the considered case is 17.4 m
How does dilation affect length, area, and volume of an object?Suppose a figure (pre image) is dilated (dilated image) by scale factor of k.
So, if a side of the figure is of length L units, and that of its similar figure is of M units, then:
L = k × M
where 'k' will be called as scale factor.
The linear things grow linearly like length, height etc.
The quantities which are squares or multiple of linear things twice grow by square of scale factor. Thus, we need to multiply or divide by k² to get each other corresponding quantity from their similar figures' quantities.
So area of first figure = k² × area of second figure
Similarly,
Volume of first figure = k³ × volume of second figure.
It is because we will need to multiply 3 linear quantities to get volume, which results in k getting multiplied 3 times, thus, cubed.
For this case, we're given that;
Surface area of pre-image = 35 m²Surface area of dilated image = 315 m²Height of pre-image = 5.8 mHeight of dilated image.Let the scale fator of dilation be k.
Then:
So area of pre-image = k² × area of dilated image
Thus, we get:
[tex]35 = k^2 \times 315\\\\k = \sqrt{\dfrac{35}{315}} = 1/3[/tex]
(took positive root as scaling is done by non-negative factors as only magnitude matters mostly)
Thus, we get:
Height of pre-image = (1/3) × Height of dilated image (say h)
[tex]5.8 = (1/3) \times h\\h = 3 \times 5.8 = 17.4 \rm \: m[/tex]
Thus, the original object is called pre-image, and its dilated version is called image. The height of the image for the considered case is 17.4 m
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What percentage of take-home pay is spent on expenses? (round to nearest whole percent)
A) 67% B) 68% C) 84% D) 85%
c) 84%
hope this helps :)
85% of take-home pay is spent on expenses option (D)85% is correct.
It is given that in the chart there are various expenses.
It is required to find the percentage of take-home pay is spent on expenses.
What is the percentage?It is defined as the ratio of two numbers expressed in the fraction of 100 parts. It is the measure to compare two data, the % sign is used to express the percentage.
We have a total take-home salary = $5388
Total expenses = $4558
In percentage it can be expressed as follows:
[tex]=\frac{4558}{5388}\times100[/tex]
[tex]=0.84595\times100\\\\= 84.595 \ \%[/tex]
or ≈ 85% (round to nearest whole)
Thus, 85% of take-home pay is spent on expenses.
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Solve this system of equations by graphing. First graph the equations, and then type the solution. y=3x–2 y= – x+6 Click to select points on the graph. The solution is , .
Answer:
(2 , 4)
Step-by-step explanation:
set 3x-2 = -x + 6
4x = 8
x = 2
3(2) - 2 = 4
Evaluate:
-5-(x+y)
2
where x=
3 and y = 6
answer
For each
x
in the expression you substitute
−
3
and for each
y
you substitute
5
. Make sure to handle the signs correctly. Remember
(
−
−
=
+
)
(
(
2
⋅
−
3
)
−
5
)
3
−
(
5
−
(
−
3
⋅
5
)
−
6
)
(
−
6
−
5
)
3
−
(
5
−
(
−
15
)
−
6
)
−
11
⋅
3
−
(
5
+
15
−
6
)
−
33
−
(
20
−
6
)
−
33
−
14
−
47
Answer linkStep-by-step explanation:
Let's see
-5-(x+y)Put x and y
-5-(3+6)-5-9-14Done!
What is the value of X
Using sines
sin alpha= a/c
sin 23° = x/13
x= sin 23° ×13
x≈ 5.08
Find the surface area of a 6*6 h cube
Answer:
Surface Area = 6×62 = 216 inches2
Step-by-step explanation:
Formula for surface area of cube is 6 into side²
EXPLANATION
Cube has all sides of equal dimensions and like that it has 6 faces.
So total surface area of cube is equal to the combined area of all the six faces.
Now each face has equal dimensions so same area
Therefore,
area of a single face =side²
Therefore, area of entire cube = 6 times the area of a single face
= 6 * side²
Find the value of c that makes the trinomial a perfect square. .
x² – 24x7c
Answer:
144
Step-by-step explanation:
The equation to find c is:
[tex](\frac{b}{2} )^{2}[/tex]
So put -24 into b:
[tex](\frac{-24}{2} )^{2}[/tex]
Which simplifies to:
144
So if you want to test this:
x^2 - 24x + 144
is
(x-12)^2
How many liters are in 950mL
Answer:
0.950 L
Step-by-step explanation:
ml/1000 = L
Please help extremely confused
Find the value of x. Round to the nearest tenth. The diagram is not drawn to scale. (picture of diagram included)
Answer:
basic geometry and trig ratios
Step-by-step explanation:
sin 10 = 500/x
x = 500/sin 10
x = 2 879,4 m
please help asap have a good day and ily
Answer:
Pretty sure it's 2:30 P.M.
Step-by-step explanation:
If her track practice started at 12:00 P.M., and and ran relays for one hour, it would already be 1:00 P.M.
Then she practiced the long jump for an hour and thirty minutes. Add another hour to that and it would be 2:00 P.M. Add thirty minutes, and it would be 2:30 P.M.
Hopefully that makes sense to you
help would be appreciated
Answer:
The answer is C 33.75 inches
Step-by-step explanation:
look at the explanation photo
AABC and AXYZ are similar. Find the missing side length.
Y
B.
6.
?
15
5
A
4
С
X
12
Z
(The triangles are not drawn to scale.)
Answer:
2
Step-by-step explanation:
As they are SIMILAR, they have a ratio!
15:5=3
So, the scale factor is 3 from triangle ABC to triangle XYZ
6/3=2
Geometry question pic Below
Arc JK = ______°
M<1 = ______°
Answer:
Step-by-step explanation:
PLEASE HELPPPP
Jason observed a series of circles and saw the following pattern. What color will the next circle be?
red, blue, green, green, red, blue, green, green, ...
- yellow
- red
- green
- blue
Answer:
Red
Step-by-step explanation:
Hope it helps :)
The base of a solid S is the region bounded by the parabola x^2=8y
Cross-sections perpendicular to the yyy-axis are equilateral triangles.
Determine the exact volume of solid SSS.
Step-by-step explanation:Updated answer is the second one
54 x 8181 x 12 x 8271 x 1301 x 812
Answer:
4.6320447e+16
I used a calculator for this LOL.
A triangle has side lengths of 6, 8, and 10. Is it a right triangle? Explain.
Answer:
yes
Step-by-step explanation:
Using the Pythagorean Theorem, considering that the sum of the squares of the lengths of the two smaller sides is equals to the square of the length of the greater side, the triangle is a right triangle.
Pythagorean Theorem:
The Pythagorean Theorem states that for a right triangle, the sum of the squares of the lengths of the two smaller sides is equals to the square of the length of the greater side.
so it is a right triangle.
The bases of a trapezoid are 11 meters and 14 meters. It’s height is 10 meters. What is the area of the trapezoid?
Answer:
120 m²
Step-by-step explanation:
A = (1/2) · (p + q) · h
A = (1/2)(14 + 10)(10)
A = (1/2)(24)(10)
A = 120
Hope this helps :)
5-(6- 9x)=1-(4-10 ) please
[tex]\qquad\qquad\huge\underline{{\sf Answer}}♨[/tex]
Let's solve ~
[tex]\qquad \sf \dashrightarrow \:5 - ( 6 - 9x) = 1 - (4 - 10)[/tex]
[tex]\qquad \sf \dashrightarrow \:5 - 6 + 9x = 1 - 4 + 10[/tex]
[tex]\qquad \sf \dashrightarrow \: 9x - 1 = 7[/tex]
[tex]\qquad \sf \dashrightarrow \:9x = 8[/tex]
[tex]\qquad \sf \dashrightarrow \:x = \dfrac{8}{9} [/tex]
A fuel tank is in the shape of a right circular cylinder. The radius of the base and height of the tank are
10
feet and
25
feet, respectively. Due to safety measures, the tank can be filled to only
80
%
of its capacity.
If
100
cubic feet of fuel are equal to
2. 78
gallons, what is the maximum number of whole gallons of fuel the tank can contain due to safety measures?
The maximum number of whole gallons of fuel the tank can contain due to safety measures is 174.584 gallons
Volume of a cylinderv = πr²hr = radius
h = height
Therefore,
r = 10 ft
h = 25 ft
The tank can only be 80% filled due to safety measures. Therefore,
v = 3.14 × 10² × 25
v = 3.14 × 100 × 25
v = 3.14 × 2500
v = 7850 ft³
80% capacity of 7850 ft³ is as follows:
80 / 100 × 7850 = 628000 / 100 = 6280 ft³
100 ft³ = 2.78 gallons
6280 ft³ = ?
cross multiply
maximum number of whole gallons = 6280 × 2.78 / 100 = 17458.4 / 100 = 174.584 gallons
Therefore, the maximum number of whole gallons of fuel the tank can contain due to safety measures is 174.584 gallons
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Find the values of the mode when median is given to be 5 and mean is 7.
Answer:
Mode = 1
Step-by-step explanation:
Relation between the Central Measures of Tendency
Mean, Median, and Mode are commonly referred to as the Central Measures of TendencyThe formula between the three is given by :⇒ Mode = 3Median - 2Mean or Mode + 2Mean = 3MedianSolving
We know that :Median = 5Mean = 7Therefore,
Mode = 3(5) - 2(7)Mode = 15 - 14Mode = 1[tex]\qquad\qquad\huge\underline{{\sf Answer}}♨[/tex]
We know the relation between mean, Median and mode. that is :
[tex]\qquad \sf \dashrightarrow \:mode = 3 \: median - 2 \: mean[/tex]
now, plug in the values ~
[tex]\qquad \sf \dashrightarrow \:mode = 3(5) - 2(7)[/tex]
[tex]\qquad \sf \dashrightarrow \:mode = 15 - 14[/tex]
[tex]\qquad \sf \dashrightarrow \:mode = 1[/tex]
Hence, value of mode is 1
A ratio of boys to girls in the classroom is 4 to 5 if there are 20 girls how many boys would there be
Answer:
16 to 20
Step-by-step explanation:
Let's do our math.
4 to 5
8 to 10
12 to 15
16 to 20
In order to get to 20, you'd have to multiply 5 by 4. So in order to get an equal ratio, you multiply 4 by 4 and get 16.
I hope this helps!
-No one
Manny has 36 coins in
his piggybank.
3 of the coins are silver dollars.
• The remaining coins are dimes (10€), quarters (25€) and nickels (5¢).
• There is an equal number of dimes, quarters and nickels.
How many nickels does Manny have?
Answer:
q=50
d=22
Step-by-step explanation:
simply 2x + x + x + 2y x 3y x y
Answer:
[tex]4x + 6y^{3} x^{2}[/tex]
Step-by-step explanation:
Simplify
A yogurt machine makes
gallon of yogurt
every minute. The total amount y that the
machine makes in x minutes is given by
the equation y
]×. Determine whether the
relationship between x and y is linear and, if
so, if it is proportional,
Step-by-step explanation:
the relationship between x and y is defined by a constant factor.
you can "go" from x to y and from y to x consistently and without any interruptions by applying that constant factor.
that makes it a linear relationship (a line on a chart, hence the name).
proportional relationships are relationships between two variables where their ratios are equivalent. another way to think about them is that, in a proportional relationship, one variable is always a constant value times the other. That constant is know as the "constant of proportionality".
so, yes, it is proportional.
and the table is then
x y
0 0
3 3×2/3 = 2
6 6×2/3 = 4
9 9×2/3 = 6
please help ill give pointssssss
Based on the formula given of 27π = πn, we can find the value of n after substitution in an equation to be 27.
What is the value of n?The value of n can be found by making n the subject of the formula as shown below;
27π = πn
n = 27π / π
Solving of n therefore gives:
n = 27π/π
n = 27
In conclusion, n is 27.
Find out more on finding terms at https://brainly.com/question/25827678.
what equation is the following graph
Answer:
y = -1/2x + 1
Step-by-step explanation:
We find slope using rise-over-run and by counting I got 1/2 but the slope is also negative. The y-intercept is 1 because the points are counting by 1/2 and the line goes through where half of 2 is on the y-axis. Hope this helps!
Step by Step solution please.
Thanks
Derivation of f(x):
[tex]f'(x) = \frac{g(x)*8x-4x^2*g'(x)}{(g(x))^2} -\frac{1}{x+2}[/tex]
Since g(3) = 6 and g'(3) = 3
[tex]f'(3) = \frac{g(3)*8*3-4(3)^2*g'(3)}{(g(3))^2} -\frac{1}{3+2} \\f'(3) = \frac{6*8*3-4*9*3}{36} -\frac{1}{5} \\f'(3) = \frac{36}{36} -\frac{1}{5} =1-\frac{1}{5} =\frac{4}{5}\\ f'(3) = 0.8[/tex]
Thus f'(3) = 0.8
Hope that helps!
A couple of identities I used in derivation:
[tex]\frac{d}{dx} (ln x) = \frac{1}{x}[/tex]
Please help me! There is a picture.
Answer:
A. is the answer because 1/2 x 20 will equal 40 gallons