Let S be the sphere of radius 1 centered at (3, 5, 7). Find the distance from S to the plane x + y + z = 0. (HINT: Use Lagrange multipliers to find the distance from the plane to the center of the sphere.)
Answers
The distance from sphere S of radius 1 centered at (3, 5, 7) to the plane x + y + z = 0 is 1 unit.
We are given that the sphere has a radius of 1 and a center at (3,5,7).
Then, the equation of the sphere is:
(x-3)^2+(y-5)^2+(z-7)^2=1.
So, we have that:
F(x,y,z)=x+y+z+d((x-3)^2+(y-5)^2+(z-7)^2-1)
Now, partially differentiating F(x,y,z) with respect to x:
\(\mathrm{F}_{\mathrm{x}}\)=1+2d(x-3)
Now, partially differentiating F(x,y,z) with respect to y:
\(\mathrm{F}_{\mathrm{y}}\)=1+2d(y-5)
Now, partially differentiating F(x,y,z) with respect to z:
\(\mathrm{F}_{\mathrm{y}}\)=1+2d(z-7)
Putting \(\mathrm{F}_{\mathrm{x}}\)=0:
1+2d(x-3)=0
x=3-(1/2d)
Putting \(\mathrm{F}_{\mathrm{y}}\)=0:
1+2d(y-5)=0
y=5-(1/2d)
Putting \(\mathrm{F}_{\mathrm{z}}\)=0:
1+2d(z-7)=0
y=7-(1/2d)
Substituting these values in the equation of the sphere:
(x-3)^2+(y-5)^2+(z-7)^2=1
3/(4d^2)=1
d=\(\pm \frac{\sqrt{3}}{2}\).
Therefore,
x=3-1/\(\sqrt{3}\),
y=5-1/\(\sqrt{3}\),
z=7-1/\(\sqrt{3}\).
Using the distance formula, we get:
Distance = \(\sqrt{\left(3-\frac{1}{\sqrt{3}}-3\right)^2+\left(5-\frac{1}{\sqrt{3}}-5\right)^2+\left(7-\frac{1}{\sqrt{3}}-7\right)^2}\)
Distance =\(\sqrt{\frac{3}{3} }\)
Distance = 1 unit.
Therefore, the distance of the sphere S to the plane is 1 unit.
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Related Questions
A bag contains 11 numbered chips inside. Two chips are labeled 1, three are labelled 2, and six are labelled 3. Let the random variable X = "number on a chip in the bag."
What is the expected number of a chip that is randomly chosen from the bag?
Answers
Answer: 2.36
Step-by-step explanation:
Using the μ=∑[x⋅P(X=x)
U will need to do
2/11 because you have 2 labeled 1
3/11 because you have 3 labeled 2
6/11 because you have 6 labeled 3
Then you will do:
1 x 2/11 = 0.18
2 x 3/11 = 0.5454 = 0.55
3 x 6/11 = 1.63
Then add them all together to find the μ
0.18 + 0.55 + 1.63 = 2.36
Hope that helps, plz put a good rating
9⟌204 it is long division
Answers
2 2
9⟌204
18
20-18=24
ans: 22r6
Please help me I'm stuck.
Answers
\(\cfrac{89.40~~ - ~~\stackrel{ \textit{minus the tax of each} }{0.50-0.50-0.50-0.50-0.50-0.50}}{6}\implies \cfrac{86.40}{6}\implies \stackrel{ each }{14.40}\)
Complete The Proof That RST is congruent to QUT. 10 points really need help ASAP. Image is below!
Answers
The proof that ΔRST is congruent to ΔQUT is given below.
What is Congruence of Triangles?Two or more triangles are said to be congruent if and only if sides and angles of one triangle is equal to the corresponding sides and angles of another triangle.
1. It is given that ΔQRT is equilateral. So all the sides and angles are of equal measure.
2. Given ∠TQU ≅ ∠SRT
3. Given sides QU ≅ RS
4. Then since sides QT and RT are side of equilateral triangle ΔQRT, they are equal .
So sides QT ≅ RT
5. So we have, two sides and included angle of ΔRST is equal to corresponding two sides and included angle of ΔQUT.
By SAS Congruence Theorem, both the triangles ΔRST and ΔQUT are congruent to each other.
Hence completed the proof using SAS Congruence theorem.
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escribe el número que corresponda a la escritura polinomica 8x10³+4x10²+0x10¹+9x10⁰
Answers
Answer:
8409 es 8×10³ es 8 4×10² es 4 0×10¹ es 4 9×10⁰ es 9
How many grams of HCl will make a solution with 100g of water saturated at 60 degrees C?
(please be quick!!!!)
110g HCl
55g of HCl
27.5g of HCl
Answers
The amount of HCL needed to make a solution with 100g of water saturated at 60 degrees C is 55 g.
option B.
What is the solubility of the HCL?
The solubility of HCl in water increases with temperature. At 60°C, the solubility of HCl in water is approximately 45 g/100g of water. This means that at 60°C, 100g of water can dissolve up to 45g of HCl.
Therefore, to make a saturated solution of HCl at 60°C using 100g of water, you would need to add 45g of HCl.
100 g - 45 g = 55 g
So, the answer is B. 55g of HCl is more than what is needed to saturate the solution, and 27.5g of HCl would not be enough to fully saturate the solution.
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2x-3y+z=0
3x+2y=35
4y-2z=14
Which of the following is a solution to the given system?
A. (2, 3, 5)
B. (3, 2, 0)
C. (1, 16, 0)
D. (7, 7, 7)
Answers
Based on the evaluations, only Option D: (7, 7, 7) satisfies all three equations and is a solution to the given system.
To determine which of the given options is a solution to the system of equations, we can substitute the values into the equations and check if they satisfy all three equations simultaneously. Let's evaluate the options one by one:
Option A: (2, 3, 5)
Checking the equations:
2(2) - 3(3) + 5 = 4 - 9 + 5 = 0 (satisfies the first equation)
3(2) + 2(3) = 6 + 6 = 12 (does not satisfy the second equation)
4(3) - 2(5) = 12 - 10 = 2 (does not satisfy the third equation)
Option B: (3, 2, 0)
Checking the equations:
2(3) - 3(2) + 0 = 6 - 6 + 0 = 0 (satisfies the first equation)
3(3) + 2(2) = 9 + 4 = 13 (does not satisfy the second equation)
4(2) - 2(0) = 8 - 0 = 8 (does not satisfy the third equation)
Option C: (1, 16, 0)
Checking the equations:
2(1) - 3(16) + 0 = 2 - 48 + 0 = -46 (does not satisfy the first equation)
3(1) + 2(16) = 3 + 32 = 35 (satisfies the second equation)
4(16) - 2(0) = 64 - 0 = 64 (does not satisfy the third equation)
Option D: (7, 7, 7)
Checking the equations:
2(7) - 3(7) + 7 = 14 - 21 + 7 = 0 (satisfies the first equation)
3(7) + 2(7) = 21 + 14 = 35 (satisfies the second equation)
4(7) - 2(7) = 28 - 14 = 14 (satisfies the third equation)
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The half-life of Radium-226 is 1590 years. If a sample contains 400 mg, how many mg will remain after 1000 years? At the beginning of an experiment, a scientist has 368 grams of radioactive goo. After 195 minutes, her sample has decayed to 11.5 grams. What is the half-life of the goo in minutes? Find a formula for G(t), the amount of goo remaining at time t. G(t) = How many grams of goo will remain after 11 minutes?
Answers
Approximately 251.98 mg of Radium-226 will remain after 1000 years. the half-life of the goo is 39 minutes. Approximately 311.89 grams of goo will remain after 11 minutes.
To determine the amount of a radioactive substance remaining after a certain time, we can use the formula:
A(t) = A₀ * (1/2)^(t/h),
where A(t) is the amount remaining at time t, A₀ is the initial amount, t is the elapsed time, and h is the half-life of the substance.
Let's solve the problems using this formula:
The half-life of Radium-226 is given as 1590 years, and the initial amount is 400 mg. We can calculate the amount remaining after 1000 years:
A(t) = 400 * (1/2)^(1000/1590)
= 400 * (1/2)^(10/16)
= 400 * (1/2)^(5/8)
≈ 400 * 0.629961
≈ 251.98 mg.
Therefore, approximately 251.98 mg of Radium-226 will remain after 1000 years.
In this case, the initial amount is 368 grams, and after 195 minutes, the sample has decayed to 11.5 grams. We need to find the half-life of the substance in minutes.
11.5 = 368 * (1/2)^(195/h)
To solve for h, we can divide both sides by 368:
(1/2)^(195/h) = 11.5/368
(1/2)^(195/h) ≈ 0.03125
Taking the logarithm of both sides to base 1/2:
log base 1/2 ((1/2)^(195/h)) = log base 1/2 (0.03125)
195/h = log base 1/2 (0.03125)
To solve for h, we can cross-multiply and isolate h:
1/2^(195/h) = 1/32
2^(195/h) = 32
2^(195/h) = 2^5
Since the bases are equal, the exponents must also be equal:
195/h = 5
Solving for h:
h = 195/5
h = 39
Therefore, the half-life of the goo is 39 minutes.
To find the formula for G(t), the amount of goo remaining at time t, we can use the same formula as before:
G(t) = G₀ * (1/2)^(t/h),
where G₀ is the initial amount of goo and h is the half-life of the substance.
If we want to find the amount of goo remaining after 11 minutes, we can substitute the values into the formula:
G(11) = 368 * (1/2)^(11/39)
≈ 368 * 0.847443
≈ 311.89 grams.
Therefore, approximately 311.89 grams of goo will remain after 11 minutes.
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help appreciated thanks
Answers
Answer:
No
Step-by-step explanation:
x=(b-5)/c≠(5-b)/c
supposedly
b=6
c=1
x=6-5/1=1
x=5-6/1= -1
1≠ -1
Answer:
No
Step-by-step explanation:
x = b - 5 / L equal to - (5 - b) / L
HELP ASAP PLEASE ILL MAKE BRAINLIEST
Answers
Answer:
The answer is 49 inches
Step-by-step explanation:
Divide 294 by 6 to get the answer.
Hope that helps!
Answer:
49 inches
We know that for every 6 feet in real life, there will be 1 inch in the picture.
So, that means if we divide 294 by 6, we can find how many inches wide the picture is.
294 ÷ 6 = 49
Since there are 49 sixes in 294, the picture is 49 inches wide.
¿Cual es la presión total que es perité ya un pez en su superficie si se encuentra a una profundidad de 10 m? La densidad del agua es de 1025kg/m3
Answers
So the total pressure experienced by the fish on the surface is 100450 Pa.
What is equation?An equation is a mathematical statement that shows that two expressions are equal. It consists of two parts: the left-hand side (LHS) and the right-hand side (RHS), separated by an equals sign (=). The equals sign indicates that the two expressions are equal, and the goal of solving the equation is to find the value of x that makes this statement true. Equations can be solved using various algebraic techniques, such as simplifying and rearranging the expressions, applying operations to both sides of the equation, and factoring or expanding expressions. Solving an equation involves finding the values of the variables that make the equation true.
Here,
When an object is submerged in a fluid, it experiences pressure due to the weight of the fluid above it. This pressure is called hydrostatic pressure, and it increases with depth. The formula to calculate hydrostatic pressure is:
P = ρgh
where P is the hydrostatic pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the depth of the object below the surface.
In this case, the fish is at a depth of 10m, so h = 10m. The density of water is given as 1025kg/m3, and the acceleration due to gravity is approximately 9.8m/s2. Therefore, the hydrostatic pressure experienced by the fish is:
P = (1025 kg/m3) x (9.8 m/s2) x (10 m)
= 100450 Pa
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Complete question:
"What is the total pressure experienced by a fish on the surface if it is at a depth of 10m? The density of water is 1025kg/m3"
The average weight of an adult one horned male rhinoceros found in Nepal is about 21 quintal.
Answers
The average weight of an adult one-horned male rhinoceros found in Nepal is about 21 quintals. The scientific name for the one-horned rhinoceros is Rhinoceros unicorn is.
One-horned rhinoceroses are found in India, Nepal, Bhutan, and Indonesia. The one-horned rhinoceros is the largest animal in Nepal.
It can weigh up to 2,300 kg. Its body is covered in skin that looks like plates of armor. The one-horned rhinoceros is a herbivore, meaning it eats only plants.
They live in swamps, marshes, and forested areas near rivers, and they are excellent swimmers.
The one-horned rhinoceros has been endangered for many years due to poaching, habitat loss, and hunting.
The numbers of these animals have been increasing due to conservation efforts and strict laws in Nepal. The average weight of an adult one-horned male rhinoceros found in Nepal is about 21 quintals.
They are a protected species in Nepal, and there are several parks and conservation areas where they can be seen.
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Here is the complete question below:
The average weight of an adult one-horned male rhinoceros found in Nepal is about 21 quintal. Can you provide more information about the size, habitat, or any other relevant details regarding these rhinoceroses?
Perform the regression and write the estimated regression equation. Do the coefficient signs agree with you’re a priori expectations? (Round your answers to 4 decimal places.)Mileage and Other Characteristics of Randomly Selected Vehicles (n = 73, k = 4)Obs Vehicle CityMPG Length Width Weight ManTran 1 Acura TL 20 109.3 74.0 3968 0 2 Audi A5 22 108.3 73.0 3583 1 3 BMW 4 Series 428i 22 182.6 71.9 3470 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 71 Volkswagen Passat SE 24 191.6 72.2 3230 0 72 Volvo S60 T5 21 182.2 73.4 3528 0 73 Volvo XC90 16 189.3 76.2 4667 0 yˆy^ = − length −. width − Weight − ManTran
Answers
The estimated regression equation for the given data is ŷ = -0.0025 Length - 0.2171 Width - 0.0004 Weight - 0.6806 ManTran
In regression analysis, the estimated regression equation is obtained by fitting a model to the data. In this case, the equation is ŷ = -0.0025 Length - 0.2171 Width - 0.0004 Weight - 0.6806 ManTran, where ŷ represents the predicted value of the response variable (CityMPG) based on the given predictor variables (Length, Width, Weight, ManTran).
The coefficient signs in the estimated regression equation indicate the direction and magnitude of the relationship between each predictor variable and the response variable. Based on a priori expectations, negative signs were expected for Length, Width, and Weight, as these variables are typically associated with decreased fuel efficiency (lower CityMPG). However, the estimated coefficients for these variables have negative values, indicating that an increase in Length, Width, or Weight is associated with a decrease in CityMPG, which aligns with expectations.
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Briana Ralph is married and claims 2 allowances. Her gross weekly salary is $450. Each week she pays federal, Social Security, and Medicare taxes, $21.20
for medical insurance, and $5.00 for the credit union. What is her net pay?
Answers
Answer:
423.8
Step-by-step explanation:
you have to add the taxes and insurance together then subtract it by the gross salary and then you get your answer brainly please
Write -0.32 as a fraction in simplest form.
Answers
Answer:
-0.32
-32/100
-8/25
Step-by-step explanation:
-8/25
-0.32= -8/25
BUT FRACTIONS CANT BE NEGATIVE SO...
it would be
8/25
Put these in order starting from the smallest
6^2 3^4 2^9
5^3
Answers
Answer:
6^2, 3^4, 5^3, and 2^9
Step-by-step explanation:
6^2= 36
3^4= 81
5^3= 125
2^9= 512
A cone resting on its base, with a dashed line from the side of the circle to the center of the circle labeled r. The edge of the cone is labeled Slant height (l).
The surface area of a cone is given by the formula
S = πl + πr2. Solve the formula for l.
Answers
The equation l = (S - πr^2) / π represents the relationship between the surface area (S), radius (r), and slant height (l) of a cone. It allows us to calculate the slant height based on the given surface area and radius.
To solve the formula for the slant height (l) of a cone, we start with the given surface area formula:
S = πl + πr^2
To isolate the slant height (l), we need to get rid of the term πr^2. We can do this by subtracting πr^2 from both sides of the equation:
S - πr^2 = πl
Next, we divide both sides of the equation by π to solve for l:
(l = (S - πr^2) / π)
The final equation for the slant height (l) in terms of the surface area (S) and the radius (r) of the cone is:
l = (S - πr^2) / π
This equation allows us to calculate the slant height of a cone when the surface area and radius are known. By plugging in the values for S and r, we can find the corresponding value for l.
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12- 1/5r = 2r + 1
R=??
Answers
Answer:
r = 5
Step-by-step explanation:
12 - 1/5r = 2r + 1
12 - 0.2r = 2r + 1
12 - 0.2r + 0.2r = 2r + 0.2r + 1
12 = 2.2r + 1
12 - 1 = 2.2r + 1 - 1
2.2r = 11
2.2r ÷ 2.2 = 11 ÷ 2.2
r = 5
Answer:
r = 5
Step-by-step explanation:
12- 1/5r = 2r + 1
12 = 11/5r + 1
5(11) = (11/5r)5
55 = 11r
r = 5
• Choose a topic from the list below: Argue why Josef Pieper conception of leisure is the best one in modernity, or instead why it might be a limited conception in comparison to another theory of leisure. • Argue why a life is better with leisure today, and why for the classical Greeks, an absence of leisure meant an absence of a happy life. • Argue why John Dewey and modern liberal thinkers did not agree with Aristotle's ideas on education or on leisure generally. • Argue how modern psychological conceptions of happiness and the classical idea of happiness in Aristotle differ. What was the "Greek Leisure Ideal" and how would it manifest today according to Sebastian De Grazia? What happened to it? • Argue why the liberal arts are so important in education and leisure, and explain its Greek origin and how that is received today. • You must choose from this list, but it can be modified slightly if you have an idea you wish to pursue. The main requirement is that you must contrast at least one ancient thinker and one modern one. • The paper must be well researched and contain a minimum of 6 sound academic sources. • Textbook or course readings may be used, but do not count in this total. DETAILS SCALCET8 1.3.039. 0/1 Submissions Used Find f o g o h. f(x) = 3x - 8, g(x) = sin(x), h(x) =x^2
Answers
To argue why the liberal arts are so important in education and leisure, one must discuss its Greek origin and how it is received today.
The term "liberal arts" comes from the Latin word "liberalis," which means free. It was used in the Middle Ages to refer to topics that should be studied by free people. Liberal arts refers to courses of study that provide a general education rather than specialized training. It encompasses a wide range of topics, including literature, philosophy, history, language, art, and science.The liberal arts curriculum is based on the idea that a broad education is necessary for individuals to become productive members of society. In ancient Greece, education was focused on developing the mind, body, and spirit.
The study of the liberal arts is necessary to create well-rounded individuals who can contribute to society in meaningful ways. While the importance of the liberal arts has been debated, it is clear that they are more important now than ever before. The study of the liberal arts is necessary to develop the skills that are required in a rapidly advancing technological world.
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What is a acute triangle and what do a acute triangle look like?
Answers
Answer:
An acute triangle is a triangle with three acute angles. An obtuse triangle is a triangle with one obtuse angle and two acute angles. Since a triangle's angles must sum to 180° in Euclidean geometry
Vhat is the area if x=6 and y=10 units?
Answers
What is the result of −58⋅−889?
Answers
Answer:
947
Step-by-step explanation:
I have had this question before lots of times
5 and 6 What are the first five terms of the recursive sequence?
Answers
Answer:
4th option and 3rd option
Step-by-step explanation:
Using the recursive rule and a₁ = 7 , then
a₂ = 3a₁ - 6 = (3 × 7) - 6 = 21 - 6 = 15
a₃ = 3a₂ - 6 = (3 × 15) - 6 = 45 - 6 = 39
a₄ = 3a₃ - 6 = (3 × 39) - 6 = 117 - 6 = 111
a₅ = 3a₄ - 6 = (3 × 111) - 6 = 333 - 6 = 327
The first 5 terms are 7, 15, 39, 111, 327
---------------------------------------------------------------------
Similarly using the recursive rule and a₁ = 9
a₂ = 3a₁ + 3 = (3 × 9) + 3 = 27 + 3 = 30
a₃ = 3a₂ + 3 = (3 × 30) + 3 = 90 + 3 = 93
a₄ = 3a₃ + 3 = (3 × 93) + 3 = 279 + 3 = 282
a₅ = 3a₄ + 3 = (3 × 282) + 3 = 846 + 3 = 849
The first 5 terms are 9, 30, 93, 282, 849
HELP ME PLEASE ILL MARK BRAINLIEST PLEASE THUS IS ALGEBRA 1 PLZZZ
Answers
Answer:
y=0x+(-3)
Step-by-step explanation:
When Anita planted a large bush in her garden, it was 5 feet tall. Now it is 40% taller. How tall is the bush now?
Answers
Answer:
7 feet
Step-by-step explanation:
the new height of the bush can be represented in percentage form as 100% + 40% = 140% = 1.40
1.40 x 5 = 7 feet
Method 2
find 40% of 5 feet
0.4 x 5 = 2
Add 5 + 2 = 7 feet
0.00147 in standard form
Answers
Answer:
1.47 times 10 to the power of -3
Step-by-step explanation:
8th grade math I'm guessing lol
Help and get brainliest !!
Answers
Answer:
I think its A
Step-by-step explanation:
I eliminated C and D but am not 100% sure between a and b i think its right im not sure sorry
What is the slope of y= 3
Answers
Answer:
{36+7[6-2(10÷)+9(2)]}÷4
How much principal must be deposited to earn $400 simple interest in 2 years at a rate of 5%?
$40
$5,000
$8,000
$4,000
Answers
Answer:
$8,000 is the correct
Step-by-step explanation:
is right
Help me, please guyyyyyyusssssss.
Answers
Answer:
Step-by-step explanation:
7) Volume of the shape = Volume of triangular prism - volume of cylinder
Triangular prism:
\(\boxed{ \text{Volume of triangular prism = base area * h }}\)
b = 6 ft & height of the triangle = 5 ft
\(\sf Base \ area = \dfrac{1}{2}* 6* 5\)
= 3*5
= 15 ft²
Volume of prism = 15 * 4
= 60 ft³
Cylinder:
r = 2÷2 = 1 ft & h = 4 ft
Volume of cylinder = 3.14 * 1 * 1 * 4
= 12.56 ft³
Volume of the shape = 60 - 12.56
= 47.44 ft³
8) Volume = Volume of Pyramid - Volume of cone
\(\boxed{\text{Volume of the Pyramid = base area * h} }\)
\(\text {base area = length * width}\\\\\\\)
= 9 * 9
= 81 m²
Volume of Pyramid = 81 * 8
= 648 m³
\(\boxed{ \text{Volume of cone = \dfrac{1}{3} \pi r^{2}h}}\)\(\boxed{ \text{Volume of cone = $\dfrac{1}{3} \pi r^{2}h$}}\)
r = diameter ÷ 2 = 6÷2
r = 3 m & h = 8 m
\(\text{ Volume of cone =$\dfrac{1}{3}*3.14* 3*3*8$}\)
= 3.14 * 3 * 8
= 75.36 m³
Volume of the shape = 648 - 75.36
= 572.64 m³
9) Volume of the shape = volume of cylinder + volume of cone
Cylinder:
h = 7 in ; r = 3
\(\boxed{\text{Volume of cylinder =$ \pi r^{2}h$}}\)
\(=\dfrac{22}{7}*3*3*7\)
= 22 *3 * 3
= 198 in³
Cone:
r = 3 in & h = 4 in
\(\text{Volume of cone = $ \dfrac{1}{3}*3.14*3*3*4$}\)
= 3.14 * 3 * 4
= 37.68 in³
Volume of the shape = 198 + 37.68
= 235.68 in³
10) Volume of the shape = Volume of outer cyliner - volume of inner cylinder
Volume of outer cylinder:
R = 9 m & h = 3 m
Volume of outer cylinder = 3.14 * 9 * 9 * 3
= 763.02 m³
Volume of inner cylinder:
r = 5 m & h = 3 m
Volume of inner cylinder = 3.14 * 5 * 5 * 3
= 235.5 m³
Volume of the shape = 763.02 - 235.5
= 527.52 m³