Find the area of the surface.
The part of the sphere
x2 + y2 + z2 = 81
that lies above the plane
z = 4.
Answers
Parameterize this surface by
r (u, v) = 9 cos(u) sin(v) i + 9 sin(u) sin(v) j + 9 cos(v) k
with 0 ≤ u ≤ 2π and 0 ≤ v ≤ arccos(4/9)
See the attached sketch to see how we arrive at the upper limit for v.
Take the normal vector to the surface to be
n = ∂r/∂u × ∂r/∂v
n = -81 cos(u) sin²(v) i - 81 sin(u) sin²(v) j - 81 sin(v) cos(v) k
which has magnitude
||n|| = 81 sin(v)
Then the surface area is
[tex]\displaystyle \iint_S \|\mathbf n\| \,\mathrm du\,\mathrm dv = 81 \int_0^{2\pi} \int_0^{\arccos(4/9)} \sin(v)\,\mathrm dv\,\mathrm du \\\\ = 162\pi \int_0^{\arccos(4/9)} \sin(v)\,\mathrm dv \\\\ = -162\pi (\cos\left(\arccos\left(\frac49\right) - \cos(0)\right) \\\\ = 162\pi \left(1 - \dfrac49\right) \\\\ = \boxed{90\pi}[/tex]
If you're not familiar with surface integrals, you can instead use what's sometimes called the projection method. Let
z = f(x, y) = √(81 - x ² - y ²)
(where we take the positive square root because we're looking at a part of the top half of the sphere)
Projecting the surface down onto the (x, y)-plane, we see that it casts a "shadow" of a disk with radius √(9² - 4²) = √65. (Use the Pythagorean theorem to solve for the missing side of the triangle in the sketch.)
Then the surface S considered above is hovering over the set in the (x, y)-plane,
D = {(x, y) : x ² + y ² ≤ 65}
The area is then
[tex]\displaystyle \iint_D \sqrt{1 + \left(\dfrac{\partial f}{\partial x}\right)^2 + \left(\dfrac{\partial f}{\partial y}\right)^2} \,\mathrm dx\,\mathrm dy[/tex]
It's easier to compute this integral in polar coordinates, so we take
x = r cos(t )
y = r sin(t )
dx dy = r dr dt
and the region D is given by the set
{(r, t ) : 0 ≤ r ≤ √65 and 0 ≤ t ≤ 2π}
Then the integral would be
[tex]\displaystyle \iint_D \sqrt{1 + \left(\dfrac{\partial f}{\partial x}\right)^2 + \left(\dfrac{\partial f}{\partial y}\right)^2} \,\mathrm dx\,\mathrm dy = \iint_D \sqrt{1 + \frac{x^2+y^2}{81-x^2-y^2}}\,\mathrm dx\,\mathrm dy \\\\ = \iint_D \sqrt{\frac{81}{81-x^2-y^2}}\,\mathrm dx\,\mathrm dy \\\\ = \int_0^{2\pi} \int_0^{\sqrt{65}} r \sqrt{\frac{81}{81-r^2}}\,\mathrm dr\,\mathrm dt[/tex]
Substitute s = 81 - r ² and ds = -2r dr :
[tex]\displaystyle \int_0^{2\pi} \int_0^{\sqrt{65}} r \sqrt{\frac{81}{81-r^2}}\,\mathrm dr\,\mathrm dt = 2\pi \int_0^{\sqrt{65}} r \sqrt{\frac{81}{81-r^2}}\,\mathrm dr \\\\ = -9\pi \int_{81}^{16} \frac{\mathrm ds}{\sqrt s} \\\\ = 9\pi \int_{16}^{81} s^{-1/2}\,\mathrm ds \\\\ = 18\pi \left(\sqrt{81} - \sqrt{16}\right) \\\\ = \boxed{90\pi}[/tex]
Related Questions
If p(x)=x3-3x2,then find the value of p(1)
Answers
Answer:
p=[tex]x^{2}[/tex]-3x
Step-by-step explanation:
Divide each term by x
Simplify the left side
Cancel the common factor of x
Cancel the common factor
Divide p by 1
Simplify the right side
Simplify each term
Cancel the common factor of x^3 and x
Factor x out of x^3
Cancel the common factors
Raise x to the power of 1
Factor x out of x^1
Cancel the common factor
Rewrite the expression
Divide x^2 by 1
Cancel the common factor of x^2 and x
Factor x out of −3x^2
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Raise x to the power of 1
Factor x out of x^1
Cancel the common factor
Rewrite the expression
Divide −3x by 1
p=x^2−3x
Can someone please help me?
Answers
The derivative of:
[tex]f(x) = ( {3x}^{2} + 2x - 1) 6[/tex]
is....
[tex]6( {3x}^{2} + 2x - 1)5(6x + 2)[/tex]
Hope this helped you- have a good day bro cya)
HELP!!! Type the correct answer in the box.
The formula for the volume, V, of a cone having the radius, r, and the height, h, is shown below.
[tex]V=\frac{1}{3} \pi r^{2} h[/tex]
Write the formula to calculate the height, h.
Answers
Answer:
cone height formulas are, h = 3V/πr 2 and h = √l2 - r2, where V = Volume of the cone, r = Radius of the cone, and l = Slant height of the cone.
Estimate and then solve 6,699 ÷ 72 = ___.
92 r 4
93 r 3
93 r 4
94 r 3
Answers
Answer:
It would be 93 remainder 3/the second one.
Charlie drove 495 miles in 9 hours. How long would it take to drive 715 miles?
Answers
Answer:
13 hours
Step-by-step explanation:
if he drives 495 miles in 9 hours, he drives 55 miles each hour. 55x13=715
Answer:
13 hours
Step-by-step explanation:
Charlie drove 495 miles in
= 9 hours
∴ Rate
= miles/hours
= 495/9
= 55 miles per hour
∴ No. of hours to travel 715 miles
= miles/miles per hour
= 715/55 hours
= 13 hours
Thus, 13 hours would it take to drive 715 miles.
I need help with Part B does anybody know
Answers
Answer:
Ako pilipino
Step-by-step explanation:
I'm on american hehe pilipino talaga ako
janet obtained 32% for her statistics test, which was out of a total of 300 marks. how much should she score in the computer science test that is out of 200 marks in total if she is to get proportionally weighted average of 46%
Answers
Her score in computer to get a proportionally weighted average of 46% will be 134.
The total score for both subjects = 200 + 300 = 500
Since the proportionally weighted average is 46%, then her score will be:
= 46% × 500
= 0.46 × 500
= 230
Since she had 32% for her statistics test, which was out of a total of 300. Her score will be:
= 32% × 300
= 0.32 × 300
= 96
Therefore, her score in computer will be:
= 230 - 96
= 134
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A square has a side lenght 9cm
Answers
Answer:
9cm
Step-by-step explanation:
A "square" always has an equal amount of length and height. If the length is 9cm then the height is also 9cm. a square is like a cube. All of its sides are equal.
-kiniwih426
Find the value of x. NO LINKS AT ALL!!
Answers
Answer:
x= -7
The whole thing is just an awkward mirror but the lower triangle has an angle that is 1 degree smaller so the whole lower section is going to be bigger or smaller and the angle that mirrors the 80 degree angle is now 79 degrees which makes the other angle on that line 1 degree bigger which would be 101 so you put it in the expession to make it the equation -12x+17=101 and you solve for x which you would then get -7.
Help or no if so thanks<3
Answers
4/21 of a spool
Step-by-step explanation:
2/7 * 2/3 = 4/21
The lengths of the worms (in inches) are listed here.
Which line plot shows this data?
Answers
Answer:
The second one shows the data of the lengths of the worms.
Step-by-step explanation:
By counting the number of times a number was represented on the number line.
Like 1, four times.
1 1/2, three times.
1 3/8, three times.
Likewise to all other numbers.
Hi i need help with this
Answers
Answer:
is there an equation for the problem?
Step-by-step explanation:
Answer:
11/24 pound
i think so right
he heights of one hundred trees in a certain forest are measured. It’s found that the mean height of the trees is 48 feet. Use this information to answer the following:
Would it be possible to have a standard deviation of 7.5 feet? If so, what would this tell us about the tree heights in the forest?
Would it be possible to have a standard deviation of 1? If so, what would this tell us about the tree heights in the forest?
Would it be possible to have a standard deviation of 0? If so, what would this tell us about the tree heights in the forest?
Would it be possible to have a standard deviation of -2.5? Why or why not?
Answers
Using the concept of standard deviation, it is found that:
It would be possible to have standard deviations of 7.5 feet and 1 feet, which would mean that the distribution of tree heights are normal.It would be possible to have a standard deviation of 0 feet, which would mean that all trees have the same height.It would not be possible to have a standard deviation of -2.5 feet, as it is always a positive value.The standard deviation of a data-set is given by the square root of the sum of the differences squared of each observation and the mean, divided by the one less than the number of values. It is always a positive value, thus, a standard deviation of -2.5 feet, for example, is not possible.If a data-set has a standard deviation of 0, it means that all values in the data-set are the same.In a normal distribution, 99.7% of the measures are within 3 standard deviations of the mean.
[tex]48 \pm 3(7)[/tex] and [tex]48 \pm 3(1)[/tex] are both reasonable distributions for the tree heights, thus, it is possible to have standard deviations of 7.5 and 1 feet.
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A coin is tossed 6 times, find the probability of getting 3 heads.
Answers
Answer:
3/6
Step-by-step explanation:
Answer:
0.31 if you get exactly 3 heads or 0.66 probability if you get at least 3 heads, or 50%
Step-by-step explanation:
0.31 is the probability of getting exactly 3 Heads in 6 tossed
0.66 is the probability of getting at least 3 heads
NO LINKS!!! Part 2:
[tex]y = \sqrt{x} [/tex]
Shape:
Domain:
Range:
Answers
Given:
A square toot function y = √x--------------------------------------------------------------------
Shape: half of a parabola
Domain: Non- negative numbers x∈ [0, +∞)
Range: Non- negative numbers y∈ [0, +∞)
Swimming Pool On a certain hot summer's day, 492 people used the public swimming pool. The daily prices are $1.50 for children and $2.50 for adults. The receipts for admission totaled $1060.00. How many children and how many adults swam at the public pool that day?
There were ______ children at the public pool.
Answers
Answer:
# of children = 170, # of Adults = 492 - 170 = 322
Step-by-step explanation:
Define C and A as
C = # of children
A = # of adults
Then,
A + C = 492
1.5C + 2.5A = 1060, so that
1.5C + 2.5(492 - C) = 1060
1.5C + 1230 - 2.5C = 1060
-C = 1060 - 1230
C = 170
Solve the problems. Write the complete proof in your homework and for online (only) respond to questions or statements (if any) that are parts of your proof or related to it.
Answers
Step-by-step explanation:
Well we know that ZACB=LABC because
AABC is isosceles, so base angles theorem applies.
m<ACB=70°
So mzDCE=m/DEC-35° because of base angles theorem, and because mzDCE=m/ACB-m<ECB=70°-35°
Finally, DE || CB because the alternate interior angles ECB=<DEC
Entrance to a state park costs $6 per car, plus $2 per person in the vehicle. How might you determine the entrance cost for a bus with 50 people?
Answers
Answer: 130 dollars because 26 times 5 is 130 and 5 stands for 50 people
Assuming each family had one car, 2 people would cost $10, 4 would cost $14, 10 would cost $26.
Let me know if you want me to expand further... Happy to help :)
Brainliest please
Step-by-step explanation:
Solve: x^3+3x^2+4x=0
Answers
Answer:
x^3 + 3x^2 + 4x = 0
0^3 + 3(0)^2 + 4(0) = 0
Step-by-step explanation:
x = 0
-7/6 or -11/15
which is the greatest?
Answers
Answer:
-11/15 is the greatest because ita in negative
Answer:
-11/15 is > -7/6
Step-by-step explanation:
Find polynomial function of degree 3. Such that f(2)=24 and x-1, x and x+2 are factors of the polynomial
Answers
Answer:
5
Step-by-step explanation:
pls I need help pls I need it
Answers
Answer:
You would need 23 inches to build it
The length of a side of an equilateral triangle is 10 cm. Find the height of the triangle
Answers
Answer:
h = 5√3
Step-by-step explanation:
h = (√3/2)a
h = 10
h = (√3/2)10
h = 5√3
which of the following is not a perfect tube 1, 9, 8,27
Answers
Answer:
9
Step-by-step explanation:
9 is not a cube number; it is 3 squared. 3³ is 27, 1³ is 1 and 2³ is 8.
9 is not a perfect cube because ∛9 is not a whole number.
Hope this helps!
Engineering An overpass intersects two
lanes of a highway. What must the value of x
be to ensure the two lanes are parallel?
Answers
Answer:
Look down :)
Step-by-step explanation:
The two equations must equal 180 degrees. So what you could do it 4x + (2x + 12) = 180
then solve. If this isnt enough information just comment :)
If no I hope this helped
For the two lanes to be parallel, the value of x must be 28 degrees,
The given information states that the same-side interior angles are 4x degrees and (2x + 12) degrees.
For parallel lines, the sum of these angles must equal to 180 degrees.
As per the concept of linear sum,
4x + (2x +12) = 180
Adding the like terms, we get:
(4x + 2x) + 12 = 180
Simplifying, we get:
6x + 12 = 180
Subtract 12 to both sides, and we get:
6x = 180 - 12
Simplifying, we get:
6x = 168
x = 28
Therefore, the value of x is 28 degrees.
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Write an equation of each parabola in vertex form vertex : (7, 4) point: (5, 16)
Answers
Answer:
y=3(x-7)^2+4
Step-by-step explanation:
A chef will make a soup with five ingredients: one of four meats, one of four vegetables, one of
four kinds of noodles, one of four kinds of broth, and one of four spices. How many different
combinations of ingredients could the chef put into the soup?
A. 20
B.256
C. 1,024
D. 3,125
Answers
A chef will make a soup with five ingredients: one of four meats, one of four vegetables, one of
four kinds of noodles, one of four kinds of broth, and one of four spices. How many different
combinations of ingredients could the chef put into the soup?
A. 20B. 256
C. 1,024
D. 3,125
Step-by-step explanation:
4 kinds of meats
4 kinds of vegetables
4 kinds of noodles
4 kinds of broth
4 kinds of spices
In a total of: 20 different combinations of ingredients.
I hope it helps!The value of different combinations of ingredients could the chef put into the soup is,
⇒ 20
What is Combination?A combination is a technique to determines the number of possible arrangements in a collection of items where the order of the selection does not matter.
Given that;
A chef will make a soup with five ingredients: one of four meats, one of four vegetables, one of four kinds of noodles, one of four kinds of broth, and one of four spices.
Hence, The value of different combinations of ingredients could the chef put into the soup is,
⇒ ⁴C₁ + ⁴C₁ + ⁴C₁ + ⁴C1
⇒ 4 + 4 + 4 + 4 + 4
⇒ 20
Thus, The value of different combinations of ingredients could the chef put into the soup is,
⇒ 20
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Josh wrote 483 out of an expected 500 words for his essay. How would this ratio be expressed in decimal form?
A 0.034
B 0.483
С 0.966
D 4.83
Answers
Answer:
b
Step-by-step explanation:
just trust me
A phone sells for $250. It is now on sale for 1/5 off the original price. April has a coupon for an extra 10% off the sale price. To the nearest dollar, how much less than the original price will April pay for the phone?
Answers
Answer:
$180
Step-by-step explanation:
Answer:
£180
Step-by-step explanation:
250 / 5 = 50
250 - 50 = 200
200 / 10 = 20
200 - 10 = 180
simplify this please
Answers
Answer:
0? I tried solving it myself then I put it into my calculator and both times I got 0