Answer:
3x^2+8x-3
Step-by-step explanation:
(x+3)(3x-1)
x*3x + x*(-1) + 3x*3 + 3*(-1)
x*3x = 3x^2
x*(-1) + 3x*3 = -x+9x = 8x
3*(-1) = -3
= 3x^2+8x-3
The expression for the trinomial ( x + 3 ) ( 3x - 1 ) is [tex]3x^{2} + 8x - 3[/tex]
Given data:
The expression is represented as (x + 3)(3x - 1)
To express the product of (x + 3)(3x - 1) as a trinomial, perform the multiplication.
Using the FOIL method (First, Outer, Inner, Last), we multiply the terms:
(x + 3)(3x - 1) = x(3x) + x(-1) + 3(3x) + 3(-1)
Simplifying further:
[tex]3x^{2} - x + 9x - 3[/tex]
Combining like terms:
[tex]3x^{2} + 8x - 3[/tex]
Hence, the expression (x + 3)(3x - 1) can be expressed as the trinomial [tex]3x^{2} + 8x - 3[/tex] .
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Which expression is equivalent to (m-4/m+4)/(m+2)?
A) m-4/(m+4)(m+2)
B) (m+4)(m+2)/m-4
C) (m-4)(m+2)/m+4
D) m+4/(m-4)(m+2)
Hello,
[tex] \frac{ \frac{m - 4}{m + 4} }{m + 2} = \frac{ \frac{m - 4}{m + 4} }{ \frac{m + 2}{1} } = \frac{m - 4}{m + 4} \times \frac{1}{m + 2} = \frac{m - 4}{(m + 4)(m + 2)} [/tex]
If a polynomial f(x) is divided by (x+a) and leaves the reminder is a and b are constants then
f(-a) is the remainder when f(x) is divided by (x+a). This can be obtained by remainder theorem for polynomials.
What is the required remainder?Given that f(x) is divided by (x+a) and leaves a reminder
Using the remainder theorem for polynomials we get,
f(x) = (x+a)·g(x) + r, where g(x) is the quotient and r is the remainder.
Put x = -a, then
f(-a) = (-a+a)·g(-a) + r
f(-a) = (0)·g(x) + r
f(-a) = r
f(-a) is the remainder.
Hence f(-a) is the remainder when f(x) is divided by (x+a).
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You are in a hot air balloon looking down at two ponds. Pond A which is in front of your balloon, is at an angle of depression that is your birth month (October), in degrees. (October = 10 degrees). Pond B, which is behind the balloon, is at angle of depression that is your Birth Day in degrees. (October 8 = 8 degrees). The balloon is 875 m in the air.
a. Draw and label a diagram
b. Find the distance from the hot air balloon to both Pond A and B
c. FInd the distance between the two ponds.
a. See attachment for the labelled diagram
b. Using the sine ratio, distance from the hot air to pond A is 5,038.9 m, while distance from the hot air to pond B is 6,287.1 m.
c. Distance between the two ponds is 1,263.5 m.
What is the Sine and Tangent Ratios?Sine ratio, is: sin ∅ = opposite side/hypotenuse length
Tangent ratio is: tan ∅ = opposite side/adjacent length.
a. The diagram with the appropriate labels is shown in the image attached below.
b. Use the sine ratio to find the distance from the hot air to pond A (CA) and to pond B (CB):
CA = hypotenuse
∅ = 10°
Opposite = 875 m
sin 10 = 875/CA
CA = 875/sin 10
CA ≈ 5,038.9 m (distance from the hot air to pond A)
CB = hypotenuse
∅ = 8°
Opposite = 875 m
sin 8 = 875/CB
CB = 875/sin 8
CB ≈ 6,287.1 m (distance from the hot air to pond B)
c. Distance between the two ponds, BA = BD + DA.
Apply the tangent ratio to find BD and DA
tan 8 = 875/BD
BD = 875/tan 8
BD = 6,225.9 m
tan 10 = 875/DA
DA = 875/tan 10
DA = 4,962.4 m
Distance between the two ponds = 6,225.9 - 4,962.4 = 1,263.5 m.
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PLEASE HELP BRAINLIST ANSWER PLEASE
Answer:
y = -3x + 4
Step-by-step explanation:
y = mx + c
y = gradient(x intercept) + y intercept
y = -3x + 4
Find the value of x.
10
Q
12
x = [?]
Answer:
10
Step-by-step explanation:
It's just symmetry. You see that the geometrical construction above has been rotated, so its properties will remain the same.
22 A circle passes through the points
P(3, 0) and Q(0, 5). Its centre lies on
the line y = x + 2.
(i) Find the equation of the perpendicular bisector of PQ.
(ii) Hence show that the coordinates of the centre of the circle are (-1, 1).
(iii) Find the equation of the circle.
A second circle with equation
2x² + y² + ax + by - 14 = 0 has the
same centre as the first circle.
(iv) Write down the value of a and of b.
(v) Show that the second circle lies
inside the first circle.
The equation of the first circle is (x + 1)^2 + (y - 1)^2 = r^2 and the equation of the second circle is (x + 1)² + (y - 1)² = 16
The equation of the perpendicular bisectorThe points are given as:
P(3, 0) and Q(0, 5)
The midpoint of PQ is
Midpoint = 0.5(3 + 0, 0 + 5)
Midpoint = (1.5, 2.5)
Calculate the slope of PQ
m = (y2 - y1)/(x2 - x1)
m = (5 - 0)/(0 - 3)
m = -5/3
A line perpendicular to PQ would have a slope (n) of
n = -1/m
This gives
n = -1/(-5/3)
n = 0.6
The equation is then calculated as:
y = n(x - x1) + y1
Where
(x1, y1) = (1.5, 2.5)
So, we have:
y = 0.6(x - 1.5) + 2.5
y = 0.6x - 0.9 + 2.5
Evaluate the sum
y = 0.6x + 1.6
Hence, the equation of the perpendicular bisector of PQ is y = 0.6x + 1.6
The center of the circleWe have:
y = x + 2
Substitute y = x + 2 in y = 0.6x + 1.6
x + 2 = 0.6x + 1.6
Evaluate the like terms
0.4x = -0.4
Divide
x = -1
Substitute x = -1 in y = x + 2
y = -1 + 2
y = 1
Hence, the center of the circle is (-1, 1)
The circle equationWe have:
Center, (a, b) = (-1, 1)
Point, (x, y) = (0, 5) and (3, 0)
A circle equation is represented as:
(x - a)^2 + (y - b)^2 = r^2
Where r represents the radius.
Substitute (a, b) = (-1, 1) in (x - a)^2 + (y - b)^2 = r^2
(x + 1)^2 + (y - 1)^2 = r^2
Substitute (x, y) = (0, 5) in (x + 1)^2 + (y - 1)^2 = r^2
(0 + 1)^2 + (5 - 1)^2 = r^2
This gives
r^2 = 17
Substitute r^2 = 17 in (x + 1)^2 + (y - 1)^2 = r^2
(x + 1)^2 + (y - 1)^2 = r^2
Hence, the circle equation is (x + 1)^2 + (y - 1)^2 = r^2
The value of a and bThe equation of the second circle is
2x² + y² + ax + by - 14 = 0
Rewrite as:
2x² + ax + y² + by = 14
For x and y, we use the following assumptions
2x² + ax = 0 and y² + by = 0
Divide through by 2
x² + 0.5ax = 0 and y² + by = 0
Take the coefficients of x and y
k = 0.5a k = b
Divide by 2
k/2 = 0.25a k/2 = 0.5b
Square both sides
(k/2)² = 0.0625a² (k/2)² = 0.25b²
Add the above to both sides of the equations
x² + 0.5ax +0.0625a² = 0.0625a² and y² + by + 0.25b² = 0.25b²
Express as perfect squares
(x + 0.25a)² = 0.0625a² and (y + 0.5b)² = 0.25b²
Add both equations
(x + 0.25a)² + (y + 0.5b)² = 0.0625a² + 0.25b²
So, we have:
2x² + ax + y² + by = 14 becomes
(x + 0.25a)² + (y + 0.5b)² = 0.0625a² + 0.25b²+ 14
Comparing the above equation and (x + 1)^2 + (y - 1)^2 = r^2, we have:
0.25a = 1 and 0.5b = -1
Solve for a
a = 4 and b = -2
This means that the value of a is 4 and b is -2
Show that the second circle is in the firstWe have:
a = 4 and b = -2
Substitute these values in (x + 0.25a)² + (y + 0.5b)² = 0.0625a² + 0.25b²+ 14
This gives
(x + 0.25*4)² + (y - 0.5*2)² = 0.0625*4² + 0.25*(-2)²+ 14
(x + 1)² + (y - 1)² = 16
The equation of the first circle is
(x + 1)² + (y - 1)² = 17
The radii of the first and the second circles are
R = √17
r = √16
√17 is greater than √16
Since they have the same center, and the radius of the first circle exceeds the radius of the second circle, then the second circle lies inside the first circle.
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What is the slope of the line containing (-3, 1) and (1, -2)?
[tex]\textbf{Heya !}[/tex]
use the slope-formula:-
[tex]\sf{\cfrac{y2-y1}{x2-x1}}[/tex]
put-in the values
[tex]\sf{\cfrac{-2-1}{1-(-3)}}[/tex]
[tex]\sf{\cfrac{-3}{1+3}}[/tex]
[tex]\sf{\cfrac{-3}{4}[/tex]
[tex]\sf{-\cfrac{3}{4}}[/tex]
`hope it's helpful to u ~
Given the right triangle below, for the Pythagorean theorem, in which a 2 + b 2 = c 2, which side would represent the side "c." Hint: look at this triangle and the names of the angles.
Answer:
line segment AC (hypotenuse)
Step-by-step explanation:
The Pythagorean Theorem states that the sum of squares of the base/height of the triangle will equal the hypotenuse squared which is what c is equal to. So the c would represent the hypotenuse which is the side from the points A to C in the diagram.
The horizontal viewing angle is the angle subtended by a straight line from
each side of the screen to the seating position.
THX
THX Ltd., is a company founded in 1983 by George Lucas that develops
audio/visual reproduction standards for movie theaters. According to THX,
the viewing angle in a theater should be no less than 26 degrees and the best
viewing angle seems to be around 45-50 degrees and towards the center.
Suppose seat G11 has a horizontal viewing angle of 45°. This would be considered the best seat in the theater.
3. What is the measure of the arc the screen subtends?
The measure of the arc is given as π/2. See the explanation below.
What is an arc?An "arc" is a curve that connects two points in mathematics.
It can also be depicted as a section of a circle. It is essentially a portion of a circle's circumference. An arc is a kind of curve.
What is the calculation for the above solution?Note that the viewing angle is 45°.
Thus, the center angle is:
45 X 2 = 90°
Measure of the arc therefore is:
= (π/180°) x 90
= π/2
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There are 6 fifth grade classrooms that share 7 packs of paper. How much paper should each classroom get?
Answer:1.166 reams of paper.
Step-by-step explanation:
Just use simple division and divide 7/6.
Each classroom will get [tex]\dfrac{7}{6}[/tex] paper when there are 6 fifth grade classrooms that share 7 packs of paper.
A fraction is defined as a part of a whole. The upper part of the fraction is called the numerator, and the lower part is called the denominator.
Given that:
Number of fifth grade classrooms = 6
Number of packs of paper = 7
The number of paper each classroom get in fractions can be obtained by dividing the number of packs of paper by the number of fifth grade classrooms.
Each classroom get = [tex]\dfrac{Number \ of \ packs \ of \ paper}{ \ Number \ of \ fifth \ grade \ classrooms}[/tex]
Each classroom get = [tex]\dfrac{7}{6}[/tex]
Each classroom will get [tex]\dfrac{7}{6}[/tex] paper.
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Please help quickly I need the answer now please you’ll be brainleist
Answer:
B
Step-by-step explanation:
See the attached image.
Parabola a can be represented using the equation (x 3)2 = y, while line b can be represented using the equation y = mx 9. isabel claims one solution to the system of two equations must always be the vertex of parabola a. which best describes the reasonableness of her claim?
Isabel is incorrect because the point of intersection between line B and parabola A that can be determined is the y-intercept of the equations, not the vertex. This can be obtained using finding y intercept of both parabolas and vertex of parabola A.
Is her claim correct?Parabola A, (x + 3)² = y = x² + 6x + 9
Parabola B, y = mx + 9
For parabola A, when x = 0, y = (0+3)²x=0, y = 9
y coordinate of parabola A is (0,9)
⇒The vertex coordinate,
(-b/2a , -(b²-4ac)/4a) = (-3,0)
For parabola B, when x = 0, y = 0 + 9x=0, y = 9
y coordinate of parabola B is (0,9)
Isabel claims one solution to the system of two equations must always be the vertex of parabola A which is wrong.
Hence Isabel is incorrect because the point of intersection between line B and parabola A that can be determined is the y-intercept of the equations, not the vertex.
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Disclaimer: The question was given incomplete on the portal. Here is the complete question.
Question: Parabola A can be represented using the equation (x + 3)2 = y, while line B can be represented using the equation y = mx + 9. Isabel claims one solution to the system of two equations must always be the vertex of parabola A. Which best describes the reasonableness of her claim?
The line plot (shown in the attached image) shows the prices of sunglasses at a department store.
a. Find the mean, median, and mode.
b. Which measure best describes the data? Why?
c. Which measure might be misleading in describing the average price of sunglasses? Explain your reasoning.
Please help!!
a. The mean, median and mode for this plot are 70.8, 70 and 60 respectively.
b. Mean best represents the data for this plot.
c. When there are outliers, the mean might be a deceptive center tendency measure.
Mean, Median, and Mode
Mean = Sum of prices / Total number of sunglasses
Mean = (20 + 20 + 50 + 50 + 50 + 60 + 60 + 60 + 60 + 60 + 60 + 70 + 70 + 70 + 80 + 80 + 80 + 80 + 90 + 90 + 90 + 90 + 100 + 100 + 130) / 25
Mean = 70.8
Median is equal to (n/2 + 1) when n is an odd number.
Here, the number of sunglasses is represented by n.
⇒ (25/2 + 1)th term is the median
Median = 13th term
Median = 70
Mode is the most frequent data from the plot.
⇒ Mode = 70
Mean Being the Best Central Measure
The mean is often the ideal measure of central tendency to use when your data distribution is continuous and symmetrical, such as when your data are normally distributed. So, mean here denotes the typical cost of the sunglasses.
Misleading Measure
When there are outliers, the data mean is significantly impacted. As a result, it may be inaccurate when examining the data's average price.
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How many of the list 729 positive integers are perfect squares, cubes ,or both?
Answer:
there will be 17 integers!
List :
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
256
289
They're all squares, like for example, 1^2, 2^2, 3^2, etc
Answer:
cheapt
Step-by-step explanation:
Solve for X in the diagram below
Step-by-step explanation:
the sum of all angles around a single point on one side of a line must be 180°.
because that line can be seen as the extension of the diameter of a circle, and the point would be the center of that circle. so, one side of the line represents a half-circle, which stands for 180°.
so we have
180 = 40 + 40 + (2x + 30) = 80 + 2x + 30 = 110 + 2x
70 = 2x
x = 35
11. What is the y-intercept of a line that passes through the point (5,17)
and has a slope of 4?
1. 17
2. 11
3. -3
4. 46
5. 2
[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]
y - intercept = -3[tex]\textsf{ \underline{\underline{Steps to solve the problem} }:}[/tex]
Equation of straight line in point - slope form :
[tex]\qquad❖ \: \sf \:y - y1 = m(x - x1)[/tex]
( m = slope = 4, point (5 , 17) )
[tex] \qquad◈ \: \: \sf \: y - 17 = 4(x - 5)[/tex]
[tex] \qquad◈ \: \: \sf \: y - 17= 4x - 20[/tex]
[tex] \qquad◈ \: \: \sf \: y = 4x - 20 + 17[/tex]
[tex] \qquad◈ \: \: \sf \: y = 4x - 3[/tex]
Now, it's the form of line in slope - intercept form, ( slope = coefficient of x = 4 and y - intercept = -3 )
[tex] \qquad \large \sf {Conclusion} : [/tex]
[tex] \qquad◈ \: \: \sf \: y \: \: int = - 3[/tex]
A group of 8 friends went to lunch and spent a total of $76, which included the food bill and a tip of $16. They decided to split the bill and tip evenly among themselves.
Which equations and solutions describe the situation? Select two options.
The equation StartFraction 1 over 8 EndFraction (x + 16) = StartFraction 76 over 8 EndFraction represents the situation, where x is the food bill.
The equation StartFraction 1 over 8 EndFraction (x + 16) = 76 represents the situation, where x is the food bill.
The solution x = 60 represents the total food bill.
The solution x = 60 represents each friend’s share of the food bill and tip.
The equation 8 (x + 16) = 76 represents the situation, where x is the food bill.
Answer:
Each Person:
-In tips: $2
-Food Bill: $7.50
Step-by-step explanation:
Let's find the tip amount for each person. $16 for the whole group. Each person will give in $2 for the tips, for a total of $16 in tips.
Now that tips are done, we must subtract the amount of tips from the food bill, to see how much each person pays. $76 - $16 = $60.
Divide 60 by 8 to get.. 60/8= 7.5.
With 7.5 as an answer, we can conclude that each person paid $7.50 for the food bill.
I see you need to select solutions, may I see them to help you out? The description is pretty vague..
How many miles is it across the united states from the east coast to the west coast.
The thousand of miles across the United states from the east coast to the west coast.
In order to find the number of miles across the United states from the east coast to the west coast.
Hence, the thousand of miles across the United states from the east coast to the west coast.
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What is the area of parallelogram ABCD?
A. 12 units²
B. 30 units²
C. 24 units²
D. 55 units²
Answer:
C, 24 units^2
Step-by-step explanation:
The formula for area of a parallelogram is base times height.
The height of the parallelogram is 4 units
The base of the parallelogram is 6 units
4 times 6 is 24
Hope this helps:)
Answer:
C. 24 units²
Step-by-step explanation:
To calculate the area of a parallelogram we use the following formula:
A = B*h (B: base, h: height)
The base of the parallelogram is 6 units and the height is 4 units
4*6 = 24 units but, the area is presented in square units so the answer is 24 units²
( will give brainlyest)
A firework is launched from the ground at a speed of 160 feet per second. It’s height after t seconds is given by the polynomial -16t+160t. What is the height of the firework after 4 seconds?
Answer: 384 feet
Step-by-step explanation:
We will simplify and solve -16t² + 160t when t is equal to 4.
-16t² + 160t
-16(4)² + 160(4)
384 feet
Answer:
Step-by-step explanation:
**Note: The polynomial in question is -16t² + 160t, as clarified in the discussion under the question.**
Since there is a polynomial to find the height given a certain time, the height is a function of time.
This function would be h(t) = -16t² + 160t, so plugging in 4 for t would give the height.
[tex]h(4) = -16(4)^2+160(4)[/tex]
[tex]-16(16)+160(4)[/tex] [Squaring 4]
[tex]-256 + 640[/tex] [Multiplying]
[tex]384[/tex] [Combining both terms]
Hence, the height of the firework after 4 seconds is 384 feet.
What is the surface area of the right cone below?
A. 63π units²
B. 54π units ²
C. 99π units²
D. 126π units²
Answer:
B. 54π units ²
Step-by-step explanation:
solution:
given,
Radius (r) = 3
(l) = 15
we know that,
T.S.A. of cone = πr( r + l )
= π × 3(3 + 15)
= π × 3 × 18
= π × 54
= 54π units ²
Fill in the blank.
_______ are sample values that lie very far away from the majority of the other sample values.
Outliers are sample values that lie very far away from the majority of the other sample values.
We know that, an outlier is an observation that lies an abnormal distance from other values in a random sample from a population.
It differs significantly from other observations in a sample values.
Outlier is a value that is distant from the majority of the values in a data set.
For a scatter plot, an outlier is the point or points that are farthest from the regression line.
For example in the record of marks 25, 29, 7, 32, 85, 33, 27, 28 both 7 and 85 are outliers.
Therefore, outliers are sample values that lie very far away from the majority of the other sample values.
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What is the missing reason in the proof?
Given: ∠ABC is a right angle, ∠DBC is a straight angle
Prove: ∠ABC ≅ ∠ABD
A horizontal line has points D, B, C. A line extends vertically from point B to point A. Angle A B C is a right angle.
definition of angle bisector
segment addition property
definition of congruent angles
transitive property
Mark this and return
We can identify that the missing reason in the proof is: Definition of Congruent angles.
How to give proof of a congruent triangle?We know that an angle measure of 90° of ∠ABC in the triangle means that it is a right angle triangle.
We also see that ∠ADB is also a right angle and is equal to 90°.
Now, since they have exactly the same measure, these angles are congruent. Then we can say that the angles are congruent and as such:
∠ABC ≅ ∠ABD
Thus, we can identify that the missing reason is: Definition of Congruent angles.
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Calculate the total amount in an investment account if $2800 was invested at a simple interest rate of
5.5% for 18 months.
a. $3034.15
b. $3031.00
Trinh invected $2400.st.0866
c. $7340.11
d.
$5544.00
The total amount in the investment account is $3031.00
How to determine the total amount?The given parameters are
Principal, P = $2800
Rate, r = 5.5%
Time = 18 months i.e. 1.5 years
The amount is then calculated as:
A = P + PRT
This gives
A = 2800 + 2800 * 5.5% * 1.5
Evaluate
A = 3031
Hence, the total amount in the investment account is $3031.00
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Right triangle J K L is shown. Line L J has a positive slope. A dotted and horizontal line is drawn to point J to form an angle of 53 degrees.
What is the measure of the angle of elevation from point L to point J?
37°
45°
53°
137°
Considering the sum of the internal angles of the triangle, the angle of elevation from point L to point J is of 37º.
What is the sum of the internal angles of a triangle?The sum of the internal angles of a triangle is of 180º.
In this problem, the angles are:
90º, 53º, x
Hence:
90 + 53 + x = 180
x = 180 - 143
x = 37º
The angle of elevation from point L to point J is of 37º.
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Answer:
53*
Step-by-step explanation:
took on edge
How many two digit numbers have on odd digit and one evan digit?
Answer: There are only 90 two-digit numbers. It shouldn’t take long to write them out and count them
Step-by-step explanation:
Answer:
45 numbers
Step-by-step explanation:
The 10-digit numbers are 10-99. We can manually write them out to start and see if we can find a pattern.
We can start 10-19: The numbers that have one odd digit and one even digit at 10, 12, 14, 16, 18 (since 1 is odd, we can only pick the numbers that have an even unit digit)
Next, we move on to 20-29: The numbers that follow the property are 21, 23, 25, 27, 29
Both of these examples have 5 numbers that follow the property. We can notice that all of the subsequent clusters (30-39, 40-49, 50-59, 60-69, 70-79, 80-89, 90-99) will also have 5 each. The numbers that start with an odd number will have the same unit digits as the 10-19 cluster, and the numbers that start with an even number will have the same unit digits as the 20-29 cluster. Observe:
30-39: 30, 32, 34, 36, 38
40-49: 41, 43, 45, 47, 49
50-59: 50, 52, 54, 56, 58
60-69: 61, 63, 65, 67, 69
70-79: 70, 72, 74, 76, 78
80-89: 81, 83, 85, 87, 89
90-99: 90, 92, 94, 96, 98
In all, there are 9 clusters of numbers, and each one has 5 numbers that follow the property. Therefore, the total is 9*5 or 45 numbers
Write and simplify, but do not evaluate, an integral with respect to x that gives the length of the following curve on the given interval y=2cos 3x on -- An integral that gives the arc length is a d x (Type exact answers.) Enter your answer in each of the answer boxes.
The arc length of [tex]y=2\cos(3x)[/tex] on the interval [tex][a,b][/tex] is
[tex]\displaystyle \int_a^b \sqrt{1 + (y')^2} \, dx = \int_a^b \sqrt{1 + (-6\sin(3x))^2} \,dx = \boxed{\int_a^b \sqrt{1+36\sin^2(3x)} \, dx}[/tex]
what is the probability that random permutation of n numbers gets sorted after 1 pass of bubble sort
It means the probability of a random permutation of n numbers gets sorted after 1 pass of bubble sort is n!
According to the statement
we have to find the probability of random permutation of the N numbers.
So, according to the definition of A random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable.
In this we order of select the objects randomly.
So, Let There are n−1 comparisons, so 2n−1 possible sequences of actions - swap or don't swap.
To find the permutations, start with 1,2,3,4,5 and undo a sequence.
For example, if let n=5, there are 24=16 out of 5!
which is 5! =120 that end after one round of bubble sort.
So, It means the probability of a random permutation of n numbers gets sorted after 1 pass of bubble sort is n!
Learn more about probability of a random permutation here https://brainly.com/question/14615273
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Surface area=
Help me please thanks so much :))
Answer:
324[tex]\pi[/tex] units³
Step-by-step explanation:
Surface Area of Sphere = 4[tex]\pi[/tex]r²; where 'r' is the radius of the sphere.
4[tex]\pi[/tex]r² =
4[tex]\pi[/tex]9² =
4[tex]\pi[/tex]81 =
324[tex]\pi[/tex] units³
Answer:
324[tex]\pi[/tex] unit^2
Step-by-step explanation:
The formula for finding the surface area of a sphere is: [tex]A = 4\pi r^2[/tex]
--> A = Surface area of the sphere
--> r = Radius of the sphere
The image shows that the radius of the sphere is 9.
All we need to do is to replace r with 9.
--> A = 4[tex]\pi[/tex]9^2
--> A = 4[tex]\pi[/tex]81
--> A = 324[tex]\pi[/tex]
So that is your final answer! 324[tex]\pi[/tex] unit^2
--> It is unit^2 since it is an area, not a volume.
+ I'm leaving this as [tex]\pi[/tex], since the question is asking for the exact value, not the ones that are rounded up.
Use the given transformation to evaluate the integral. double integral 9xy dA R , where R is the region in the first quadrant bounded by the lines y = 2 3 x and y = 3x and the hyperbolas xy = 2 3 and xy = 3; x = u/v, y = v
It looks like the boundaries of [tex]R[/tex] are the lines [tex]y=\dfrac23x [/tex] and [tex]y=3x[/tex], as well as the hyperbolas [tex]xy=\frac23[/tex] and [tex]xy=3[/tex]. Naturally, the domain of integration is the set
[tex]R = \left\{(x,y) ~:~ \dfrac{2x}3 \le y \le 3x \text{ and } \dfrac23 \le xy \le 3 \right\}[/tex]
By substituting [tex]x=\frac uv[/tex] and [tex]y=v[/tex], so [tex]xy=u[/tex], we have
[tex]\dfrac23 \le xy \le 3 \implies \dfrac23 \le u \le 3[/tex]
and
[tex]\dfrac{2x}3 \le y \le 3x \implies \dfrac{2u}{3v} \le v \le \dfrac{3u}v \implies \dfrac{2u}3 \le v^2 \le 3u \implies \sqrt{\dfrac{2u}3} \le v \le \sqrt{3u}[/tex]
so that
[tex]R = \left\{(u,v) ~:~ \dfrac23 \le u \le 3 \text{ and } \sqrt{\dfrac{2u}3 \le v \le \sqrt{3u}\right\}[/tex]
Compute the Jacobian for this transformation and its determinant.
[tex]J = \begin{bmatrix}x_u & x_v \\ y_u & y_v\end{bmatrix} = \begin{bmatrix}\dfrac1v & -\dfrac u{v^2} \\\\ 0 & 1 \end{bmatrix} \implies \det(J) = \dfrac1v[/tex]
Then the area element under this change of variables is
[tex]dA = dx\,dy = \dfrac{du\,dv}v[/tex]
and the integral transforms to
[tex]\displaystyle \iint_R 9xy \, dA = \int_{2/3}^3 \int_{\sqrt{2u/3}}^{\sqrt{3u}} \frac{dv\,du}v[/tex]
Now compute it.
[tex]\displaystyle \iint_R 9xy \, dA = \int_{2/3}^3 \ln|v|\bigg|_{v=\sqrt{2u/3}}^{v=\sqrt{3u}} \,du \\\\ ~~~~~~~~ = \int_{2/3}^3 \ln\left(\sqrt{3u}\right) - \ln\left(\sqrt{\frac{2u}3}\right) \, du \\\\ ~~~~~~~~ = \frac12 \int_{2/3}^3 \ln(3u) - \ln\left(\frac{2u}3\right) \, du \\\\ ~~~~~~~~ = \frac12 \int_{2/3}^3 \ln\left(\frac{3u}{\frac{2u}3}\right) \, du \\\\ ~~~~~~~~ = \frac12 \ln\left(\frac92\right) \int_{2/3}^3 du \\\\ ~~~~~~~~ = \frac12 \ln\left(\frac92\right) \left(3-\frac23\right) = \boxed{\frac76 \ln\left(\frac92\right)}[/tex]