Answer:
π/6 [(29)^³/₂ − 1]
81.247
Step-by-step explanation:
Surface area is:
S = ∫∫ √(fₓ² + fᵧ² + 1) dA
The partial derivatives are:
fₓ = -2x
fᵧ = -2y
Substituting:
S = ∫∫ √(4x² + 4y² + 1) dA
To make this easier, we can convert to polar coordinates.
S = ∫∫ √(4r² + 1) dA
S = ∫∫ √(4r² + 1) r dr dθ
S = ⅛ ∫∫ 8r √(4r² + 1) dr dθ
The limits for θ are 0 to 2π. The minimum of r is 0. The maximum of r is:
-6 = 1 − x² − y²
-6 = 1 − r²
r² = 7
r = √7
Integrating the first integral:
S = ⅛ ∫ ⅔ (4r² + 1)^³/₂ |₀ᴿ dθ
S = ⅛ ∫ [⅔ (4(7) + 1)^³/₂ − ⅔ (4(0) + 1)^³/₂] dθ
S = ⅛ ∫ [⅔ (29)^³/₂ − ⅔] dθ
S = ¹/₁₂ [(29)^³/₂ − 1] ∫ dθ
Integrating the second integral:
S = ¹/₁₂ [(29)^³/₂ − 1] θ |₀²ᵖⁱ
S = ¹/₁₂ [(29)^³/₂ − 1] (2π − 0)
S = π/6 [(29)^³/₂ − 1]
S ≈ 81.247
The area of a shape is the amount of space it covers.
The surface area is approximately 81.3 unit squares
The given parameters are:
[tex]\mathbf{z = 1 - x^2 - y^2}[/tex]
[tex]\mathbf{z = -6}[/tex]
Calculate the partial derivatives of [tex]\mathbf{z = 1 - x^2 - y^2}[/tex]
[tex]\mathbf{f_x = -2x}[/tex]
[tex]\mathbf{f_y = -2y}[/tex]
So, the surface area is:
[tex]\mathbf{A = \int\limits^a_b\int\limits^a_b {\sqrt{(f_x^2 + f_y^2 +1 )}} \, dA }[/tex]
So, we have:
[tex]\mathbf{A = \int\limits^a_b\int\limits^a_b {\sqrt{((-2x)^2 + (-2y)^2 +1 )}} \, dA }[/tex]
[tex]\mathbf{A = \int\limits^a_b\int\limits^a_b {\sqrt{(4x^2 + 4y^2 +1 )}} \, dA }[/tex]
Factor out 4
[tex]\mathbf{A = \int\limits^a_b\int\limits^a_b {\sqrt{(4(x^2 + y^2) +1 )}} \, dA }[/tex]
Substitute
[tex]\mathbf{r^2 = x^2 + y^2}\\\mathbf{dA = rdr d\theta}[/tex]
So, we have:
[tex]\mathbf{A = \int\limits^a_b\int\limits^a_b {\sqrt{(4r^2 +1 )}} \, r\ dr\ d\theta }[/tex]
Multiply by 1
[tex]\mathbf{A = \int\limits^a_b\int\limits^a_b {\sqrt{(4r^2 +1 )}} \, r \times 1\ dr\ d\theta }[/tex]
Express 1 as 8/8
[tex]\mathbf{A = \int\limits^a_b\int\limits^a_b {\sqrt{(4r^2 +1 )}} \, r \times \frac{8}{8}\ dr\ d\theta }[/tex]
Rewrite as:
[tex]\mathbf{A =\frac{1}{8} \int\limits^a_b\int\limits^a_b {8r \sqrt{(4r^2 +1 )}}, \ dr\ d\theta }[/tex]
From the question,
[tex]\mathbf{z = -6}[/tex]
So, we have:
[tex]\mathbf{1 - r^2 = -6}[/tex]
Add 6 to both sides
[tex]\mathbf{7 - r^2 = 0}[/tex]
So, we have:
[tex]\mathbf{r^2 = 7}[/tex]
Integrate [tex]\mathbf{A =\frac{1}{8} \int\limits^a_b\int\limits^a_b {8r \sqrt{(4r^2 +1 )}}, \ dr\ d\theta }[/tex]
[tex]\mathbf{A =\frac{1}{8} \int\limits^a_b {\frac{8}{12} (4r^2 +1 )^{\frac 32}}, |\limits^R_0 d\theta }[/tex]
[tex]\mathbf{A =\frac{1}{8} \times \frac{8}{12} \int\limits^a_b { (4r^2 +1 )^{\frac 32}}, |\limits^R_0 d\theta }[/tex]
[tex]\mathbf{A =\frac{1}{12} \int\limits^a_b { (4r^2 +1 )^{\frac 32}}, |\limits^R_0 d\theta }[/tex]
[tex]\mathbf{A =\frac{1}{12} \int\limits^a_b { (4r^2 +1 )^{\frac 32} -(4 \times 0^2 +1 )^{\frac 32} }, d\theta }[/tex]
Substitute [tex]\mathbf{r^2 = 7}[/tex]
[tex]\mathbf{A =\frac{1}{12} \int\limits^a_b { (4\times 7 +1 )^{\frac 32} - 1^{\frac 32}}, d\theta }[/tex]
[tex]\mathbf{A =\frac{1}{12} \int\limits^a_b {29^{\frac 32} - 1}, d\theta }[/tex]
Rewrite as:
[tex]\mathbf{A =\frac{1}{12} (29^{\frac 32} - 1), \int\limits^a_b d\theta }[/tex]
Integrate
[tex]\mathbf{A =\frac{1}{12} (29^{\frac 32} - 1), \times \theta |\limits^{2\pi}_0}[/tex]
So, we have:
[tex]\mathbf{A =\frac{1}{12} (29^{\frac 32} - 1), \times (2\pi - 0) }[/tex]
[tex]\mathbf{A =\frac{1}{12} (29^{\frac 32} - 1), \times 2\pi}[/tex]
Rewrite as:
[tex]\mathbf{A =\frac{2\pi}{12} (29^{\frac 32} - 1)}[/tex]
[tex]\mathbf{A =\frac{\pi}{6} (29^{\frac 32} - 1)}[/tex]
[tex]\mathbf{A =81.3}[/tex]
Hence, the surface area is approximately 81.3 unit squares
Read more about surface areas at:
https://brainly.com/question/3621496
What is the distance, d, between the points (3,5/2) and (5/3,1)?
Enter your answer in the box. Enter your answer in simplest radical form.
The distance between the two points is of:
[tex]D = \frac{\sqrt{145}}{6}[/tex]
------------------------------
Distance between two points:
Two points, [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex]. The distance between them is given by:[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
------------------------------
The points are [tex](3, \frac{5}{2})[/tex] and [tex](\frac{5}{3},1)[/tex].Thus, the distance between them is given by:[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
[tex]D = \sqrt{(\frac{5}{3} - 3)^2+(1 - \frac{5}{2})^2}[/tex]
[tex]D = \sqrt{(\frac{5}{3} - \frac{9}{3})^2 + (\frac{2}{2} - \frac{5}{2})^2}[/tex]
[tex]D = \sqrt{(-\frac{4}{3})^2 + (-\frac{3}{2})^2}[/tex]
[tex]D = \sqrt{\frac{16}{9} + \frac{9}{4}}[/tex]
[tex]D = \sqrt{\frac{64 + 81}{36}}[/tex]
[tex]D = \sqrt{\frac{145}{36}}[/tex]
[tex]D = \frac{\sqrt{145}}{6}[/tex]
A similar problem is given at https://brainly.com/question/14849255
Answer:
The answer is \/145/6 (The answer above)
Step-by-step explanation:
I took the Test.
what is the slope???
hello please help will give brainliest!
Jasmine was asked to find the product of 14.8 and 3.65. Her work is shown below. Estimate: 15 × 4 = 60 Actual: 14.8 × 3.65 = 540.2 Which statements about Jasmine’s work are true? Check all that apply. Jasmine did not make a good estimate. Jasmine did not insert her decimal point correctly when estimating. Jasmine did not insert her decimal point correctly after multiplying the actual amount. Jasmine did not make a mistake. Jasmine’s estimate and actual product don’t match closely enough.
Answer:
0 Jasmine did not insert her decimal point correctly after multiplying the actual amount
0 Jasmine’s estimate and actual product don’t match closely enough.
Answer:
oh that answer was correct but why did you put 0
Step-by-step explanation:
....
6x4x8x5=
that is the
drag numbers to write an equation for the line in slope intercept form number is may be used once more than once or not at all
Answer:
y = -4x + 5
Step-by-step explanation:
The equation for the line, in slope-intercept form will take the following equation format which is y = mx + b.
Where,
b = y-intercept = 5 (this is where the line intercepts the y-axis, and also, x = 0 here)
m = [tex] \frac{y_2 - y_1}{x_2 - x_1} [/tex]
Use any two points on the line. Using (0, 5) and (1, 1), calculate m as follows:
m = [tex] \frac{1 - 5}{1 - 0} = \frac{-4}{1} = -4 [/tex]
Substitute m = -4, and b = 5 into y = mx + b.
y = -4x + 5
Answer:
Y=-4x+5
Step-by-step explanation:
what happens when you celebrate halloween in spring matrix multipication
Answer:
his is the day that immediately proceeds the last and final spring sabbat, ... The fact that Beltane and Samhain happen in unison twice a year lends ... You know all those Halloween recipes you've been saving on Pinterest for years now? ... Halloween and Samhain product brands as you can all year long.
Step-by-step explanation:
Can someone please help
Answer:
What is your question?
Step-by-step explanation:
do you like ainme\dislike
Answer:
I like anime, do u?
Step-by-step explanation:
Answer:
Yes I love it
Step-by-step explanation:
Write the answer to the following calculations in standard form.
10 to the power of 8 – 7 • 10 to the power of 6
what is 40% of 80 please hellppppppppp
Answer:
32
Step-by-step explanation:
please help me asap
Answer:
b, b, c
hope this helps, i can explain if needed!
Answer:
1) The second option
2) The second option
3) The third option
I did this one rather quickly so I'm hoping it's correct but pls tell me if it isn't so I can correct my mistakes
Which of the ratio below is equivalent to 4:1? Select all that apply. A) 12:3 B) 24:6 C) 8:4 D)16:4 E) 10:6
Answer:
that would be a
Step-by-step explanation:
just took the test
Answer:
A, B, D
Step-by-step explanation:
4.05 x 10^-2 Standard Form
Answer:
0.0405
Step-by-step explanation:
10^-2 is the same as 1/100, so if you multiply 4.05 x 1/100, you get 0.0405.
Another way to think about it is by looking at the exponent -2. You need to move the decimal point 2 spaces to the left -- and add zeroes to any spaces that don't already have numbers.
So if you start with 4.05, and move the decimal one space to the left, you get 0.405, and then you move it one more space, and you get 0.0405.
What is the product?
8
-1
-4.
-5
9
-32
4
O
20
-36
32
-4
-20
AnswerC
Step-by-step explanation:
John and Amelia have a baby daughter Sonia John is 23 kg heavier than Amelia who is four times as heavy as Sonia their combined weight is 122. how heavy is each person
x+4x+4x+23
9x+23=122
122-23=99
9\99=11
J=67
A=44
S=11
11+44+67=122 kg
The weight of John, Amelia and Sonia are 67 kg, 44 kg and 11 kg respectively.
What is an Equation?An equation is the statement of two expressions located on two sides connected with an equal to sign. The two sides of an equation is usually called as left hand side and right hand side.
Given that
Let j , a and s be the weights of John, Amelia and Sonia respectively.
John is 23 kg heavier than Amelia.
j = a + 23
Amelia is four times as heavy as Sonia.
a = 4s ⇒ s = a/4
Their combined weight is 122.
j + a + s = 122
Substituting the above expressions of j and s in terms of a in the equation j + a + s = 122, we get,
(a + 23) + a + (a/4) = 122
2a + a/4 + 23 = 122
9a/4 = 122 - 23
2.25a = 99
a = 44
j = 44 + 23 = 67
s = 44/4 = 11
Hence the weight of each person is 67 kg, 44 kg and 11 kg.
To learn more about Equations Solving, click :
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#SPJ2
What is 2.068 rounded to the nearest 0.1
Answer:
2.1
Step-by-step explanation:
Answer:
2.1
Step-by-step explanation:
Mai's new bedroom has a walk-in closet with a floor that measures 20 square
feet. Her closet ceiling is 9 feet from the floor.
What is the volume of her closet?
Answer:
The volume of he bedroom is about 180 feet :)
A 33 pound bag of grass seed holds 2 pounds less than 5 smaller bags of seed. Find the weight of the smaller bag of seed.
A 33 pound bag of grass seed has = 2 pounds less than 5 smaller bags of seed
= 33 pounds = 2 pounds less than 5 smaller bags
Let the weight of smaller bag of seed be ' x ' . Then ,
= 33 = 5x - 2
= 5x -2 = 33
= 5x = 33 + 2 ( transposing -2 from LHS to RHS changes -2 into +2 )
= 5x = 35
= x = 35 ÷ 5 ( transposing ×5 from LHS to RHS changes ×5 to ÷5 )
= x = 7 = weight of smaller bag of seed
Then , the weight of smaller bag of seed will be 7 pounds . Let us check it by putting 7 in the place of x :
= ( 5 × 7 ) - 2 = 33
= 35 - 2 = 33
LHS = RHS
Therefore , the weight of of smaller bag of seed is 7 pounds .
If x<0 and y>0, determine the sign of the real number.
Answer:
both negative because y is negative and make both answers negative
Answers:
6) B. Negative7) A. Positive==============================================
Explanation:
x < 0 means x is negative
y > 0 tells us y is positive
Dividing a positive by a negative, or vice versa, means the result is negative. It's similar to how positive times negative = negative. So that is why the answer to problem 6 is "negative".
---------------
In problem 7, we square x to get some positive result.
Consider something like x = -7. Squaring this leads to
x^2 = (-7)^2 = (-7)*(-7) = 49
The two negatives cancel out when we square.
Since x^2 is positive, and y is positive, this means x^2*y is positive as well.
he average 30- to 39-year old man is 69.5 inches tall, with a standard deviation of 2.7 inches, while the average 30- to 39-year old woman is 64.2 inches tall, with a standard deviation of 3.2 inches. Who is relatively taller based on their comparison to their gender, LeBron James at 81 inches or Candace Parker at 76 inches
Answer:
2. LeBron is relatively taller because he has a larger z-score.
Step-by-step explanation:
Given that ,
mean =[tex]\mu[/tex] = 69.5
standard deviation [tex]=\sigma[/tex] = 2.7
x = 81
Now use the formula of z-score
[tex]z = x - \frac{\mu}{\sigma}[/tex]
z =[tex]81 - \frac{69.5}{2.7}[/tex]
z = 4.26 LeBron
mean = [tex]\mu[/tex] = 64.2
standard deviation =[tex]\sigma[/tex] = 3.2
x = 76
Now use the formula of z-score
[tex]z = x - \frac{\mu}{\sigma}[/tex]
[tex]z = 76 - \frac{64.2}{3.2}[/tex]
z = 3.69 Candace
As it can be seen that the lebron is taller as he has more z-score as compared to the Candace
The solution for x^2+2x+8<0_ is
Answer:
Step-by-step explanation:
x
2
−2x−8=0
x
2
−4x+2x−8=0
x(x−4)+2(x−4)=0
(x+2)(x−4)=0
⟹x=−2,4
Which of these is another way to write the function f(x)=5x?
a. f(x)=1/4(625x)
b. f(x)=(625x)1/125
c. f(x)=(625x)1/4
d. f(x)=1/125(625x)
Answer:
d
Step-by-step explanation:
when f(x)=5......It can also be
f(x)=1/125(625x)
every whole number over one...1(625x)=625x
also 125(1)=125 therefore (625/125)x=5x
what number is rational?
Answer:
0.33333333333333
Step-by-step explanation:
-4x-6=-3x-10
solve for x
Answer:
-4x-6 = 3x-10
-4x-3x= -10+6
-7x= -4
x= -4/-7
x= 4/7
What is the reciprocal of 3
Answer:
1/3
Step-by-step explanation:
3/1 backwards is 1/3
that's why its reciprocal is 1/3
hope this helps:)
y=12x^2-9x+4 does this have one two or no solution
I will give you branilest!
Answer:
10/3
Step-by-step explanation:
I did quiz lol and I got it right
I need help with rounding
Answer:
ok so say the number is 23 you would look at the 3 and ask yourself is the 3 greater or smaller than 5 if its 4 and under you keep the number the same and just add a zero which would make it 20 and if its 5 and over you would add 1 to the first number and put a 0 at the end making it 30. HOPE THIS HELPS!
One inlet pipe can fill an empty pool in 6 hours, and a drain can empty the pool in 8 hours. How long will it take the pipe to fill the pool if the drain is left open? Express your answer as a reduced fraction.
Answer:
24 hours
Step-by-step explanation:
Let x represent the time required to fill both pipes if they are left open
1/6 + (-1/8)= 1/x
1/6 - 1/8= 1/x
Divide through by 48
48(1/6) -48(1/8) = 48(1/x)
8 - 6 = 48 x
2= 48/x
2x= 48
x= 48/2
x= 24
Hence it will take 24 hours to fill the pipe if the drain is left open