Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 8
Answers
The volume of a cone is given
V= 1/3(πr²h)
We will use some Calculus to find the maximum volume of the cone, but first, we need to get V in one term only, either in term of 'r' or in term of 'h'
Referring to the diagram, we can start with the right-angle triangle that has two short sides; the radius of the cone and the (h-r) and the hypotenuse 'R' which is the radius of the sphere.
Forming an equation based on Pythagoras theorem
r² = R² - (h-R)² ⇒ We know that R = 8
r² = 8² - (h - 8)²
r² = 64 - (h² - 16h + 64)
r² = 64 - h² + 16h - 64
r² = 16h - h²
Substitute r² = 16h - h² into the formula for V, we have
V = 1/3 (π) (16h - h²) h
V = 1/3 (π) (16h² - h³)
V = (16/3)πh² - (1/3)πh³
-------------------------------------------------------------------------------------------------------------
Now we have V in term of 'h', we will use Calculus to find the value of 'h' that will give the maximum value of V
First, we will differentiate V, then set dV/dh = 0
V = (16/3)πh² - (1/3)πh³
dV/dh = (32/3)πh - πh²
Setting dV/dh = 0
0 = (32/3)πh - πh²
0 = πh [(32/3) - h]
πh = 0 and (32/3) - h = 0
h = 0 and h = 32/3
---------------------------------------------------------------------------------------------------------------
The value of h that we will choose is h = 32/3 since h = 0 is unlikely
Substituting h = 32/3 back into V, we have
V = 1/3 (π) [16 (32/3)² - (32/3)³]
V = 1/3 π × 606.814
V = 635.5 units³
The maximum volume of a cone within a sphere is achieved when the apex of the cone is at the centre of the sphere and the base of the cone is at the sphere's surface. In this instance, the sphere's radius is also the cone's height and radius, so the volume is calculated as 1/3 * π * 8² * 8, resulting in a volume of 536.48.
Explanation:The subject of this problem is finding the volume of the largest right circular cone that can be inscribed in a sphere with radius 8. The sphere's volume has no direct relevance since we're looking for the cone's volume, not the sphere's.
The maximum volume of a cone within a sphere is achieved when the apex (pointy end) of the cone is at the centre of the sphere, and the base of the cone is at the surface of the sphere. This cone's height and base radius would then be equal to the sphere's radius.
The formula for the volume of a cone is V = 1/3 * π * r² * h . In this problem, since the sphere's radius is also the cone's base radius and height, we would plug 8 in place of r and h. So the answer would be V = 1/3 * π * 8² * 8 which gives us V = 536.48 cubic units .
Learn more about Volume of a Cone here:https://brainly.com/question/29767724
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Related Questions
Choose the system of equations which matches the following graph:
A. 5x - y = 7
2x - y = 0
B. 5x - y = 7
2x + y = 0
C. 5x + y = -7
2x + y = 0
D. 5x + y = -7
2x - y = 0
Answers
Answer:
Option B : [tex]5x - y = 7[/tex]
[tex]2x + y = 0[/tex]
Step-by-step explanation:
Intersecting point on the graph = (1,-2)
Substitute this point in the given options .
A. [tex]5x - y = 7[/tex]
[tex]2x - y = 0[/tex]
Substitute (1,-2)
[tex]5(1) - (-2) = 7[/tex] and [tex]2(1) -(-2) = 0[/tex]
[tex]7 = 7[/tex] and [tex]4 \neq 0[/tex]
So, this system of equations does not matches the graph.
B. [tex]5x - y = 7[/tex]
[tex]2x + y = 0[/tex]
Substitute (1,-2)
[tex]5(1) - (-2) = 7[/tex] and [tex]2(1)+(-2) = 0[/tex]
[tex]7 = 7[/tex] and [tex]0= 0[/tex]
So, this system of equations matches the graph.
C.[tex]5x + y = -7[/tex]
[tex]2x + y = 0[/tex]
Substitute (1,-2)
[tex]5(1) - (-2) = -7[/tex] and [tex]2(1)+(-2) = 0[/tex]
[tex]7\neq -7[/tex] and [tex]0= 0[/tex]
So, this system of equations does not matches the graph.
D. [tex]5x + y = -7[/tex]
[tex]2x - y = 0[/tex]
Substitute (1,-2)
[tex]5(1) - (-2) = -7[/tex] and [tex]2(1) -(-2) = 0[/tex]
[tex]7\neq -7[/tex] and [tex]4 \neq 0[/tex]
So, this system of equations does not matches the graph.
Answer:
B. 5x - y = 7
2x + y = 0
Step-by-step explanation:
took test
How many different nine player batting orders can be chosen from a baseball team of 16?
Answers
[tex]_nP_r = \frac{n!}{(n - r)!}[/tex]
If I plug in n = 16 and r = 9
[tex]\frac{16!}{(16-9)!} = \frac{16!}{7!} = 4151347200[/tex]
So there are 4151347200 different nine player batting orders that can be chosen from a baseball team of 16 :)
A caters fee is based on the number of meals she provides. How much is the price per meal if the total fee is $1,088 for 64 meals?
Answers
Which equivalent expression will be generated by applying the Distributive Property and combining like terms in the expression 11 + 4(x + 2y + 4)?
Answers
11 + 4x + 8y + 16 =
4x + 8y + 27
5b-2=13 what does b equal
Answers
5b=15 divide both sides by 5
b=3
5b-2=13
add 2 to both sides
5b=15
divide both sides by 5
b = 15/5 = 3
b=3
The following table shows the monthly charges for subscribing to the local newspaper.
Number of Months
1
2
3
4
5
Total Cost ($)
15.25
30.50
45.75
61.00
76.25
Identify the independent and dependent variables in the newspaper subscription table.
a.
independent -- Total Cost
dependent -- Number of Months
c.
independent -- Total Cost
dependent -- Month of the Year
b.
independent -- Cost per Paper
dependent -- Number of Months
d.
independent -- Number of Months
dependent -- Total Cost
Answers
dependent -- Total Cost
The dependent Variable depends on the independent variable so, as the number of months change, the total cost changes.
Vincent is giving cakes to some children at a carnival. He has 2 strawberry cakes, 6 pineapple cakes, and 7 chocolate cakes. If Vincent selects a cake randomly without looking, what is the probability that he will give a pineapple cake to the first child and then a strawberry cake to the second child? (5 points)
6 over 15 multiplied by 2 over 14 is equal to 12 over 210
6 over 15 plus 2 over 14 is equal to 114 over 210
6 over 15 multiplied by 2 over 15 is equal to 12 over 225
6 over 15 plus 2 over 15 is equal to 8 over 15
Answers
2+6+7 = 15 total cakes
6 pineapple so probability is 6/15
then there would be 14 cakes
2 strawberry for a probability of 2/14
6/15 x 2/14 = 12/210
the first answer is the right one
Answer:
[tex]\frac{12}{210}[/tex]
Step-by-step explanation:
Strawberry cakes = 2
Pineapple cakes = 6
Chocolate cakes = 7
Total cakes = 15
Now we are given that Vincent selects a cake randomly without looking, and he will give a pineapple cake to the first child
So, probability of getting a pine apple cake in first draw = [tex]\frac{\text{No. of pineapple cakes}}{\text{Total no. of cakes}}[/tex]
= [tex]\frac{6}{15}[/tex]
Now total no. of remaining cakes = 15-1 =14
Now we are given that Vincent selects a cake randomly without looking, and he will give a strawberry cake to the second child
So, So, probability of getting a strawberry cake in second draw= [tex]\frac{2}{14}[/tex]
So, the probability that he will give a pineapple cake to the first child and then a strawberry cake to the second child = [tex]\frac{6}{15} \times\frac{2}{14} [/tex]
= [tex]\frac{12}{210}[/tex]
Hence Option A is true.
The probability that he will give a pineapple cake to the first child and then a strawberry cake to the second child is [tex]\frac{12}{210}[/tex]
NEED HELP!!! 80 POINTS! (sike)
At the airport, ten people are waiting on standby. Two seats become available, one first class and one coach. How many ways are there to fill the two seats?
A. 10
B. 20
C. 45
D. 90
Answers
1st seat there are 10 people so 10 different ways to fill it
2nd set there are 9 people left so 9 different ways to fill it
10 x 9 = 90
there are 90 ways to fill the seats
Answer:
d. 90
Step-by-step explanation:
Find the outlier of the set of data: 24, 37, 33, 31, 28, 25, 33, 12
Answers
The outlier is therefore 12, because it is not in the general range of values, which is on average around 29.
Answer:
The outlier of the set is 12.
Step-by-step explanation:
Consider the provided set of data:
4, 37, 33, 31, 28, 25, 33, 12
The outlier is a kind of data which is abnormal distance from the other points.
Or the data that lies outside the other value in the provided set.
Now consider the data.
12 is lies outside the other values because the distance is abnormal.
The number 12 is not close to any of the other numbers.
Hence, the outlier of the set is 12.
A regular pentagonal prism has a height of 14 in. and a base edge length of 10 in. Identify its lateral area and surface area.
Can anyone help me solve this?
Answers
So the lateral area = 5 * 10 * 14 = 700 (in squared)
Consider one of the bases of the prism.
As shown in the figure, the pentagon is made up of 5 isosceles triangles.
The vertex of each of these 5 triangles is 360°/5=72°.
Draw the altitude, which forms 2 right triangles with angle measures 36°-54°-90°.
Let he hypotenuse be s and the altitude be h in, as shown in the figure.
x*cos54°=5
x=5/(cos54°)=5/(0.588)=8.5
then,
h=x*sin54°=8.5*(0.8)=6.8 (in)
Thus, the area of one of the 5 congruent isosceles triangles is
[tex] \frac{1}{2}*10*6.8= 34[/tex] (in squared)
The area of the pentagonal base is 34*5 = 170 (in squared)
Thus the surface area = lateral area + 2 base area
= 700+2*170 = 700+340 = 1040 (in squared)
Answer:
lateral area: 700 in squared
surface area: 1040 in squared
Answer:
L = 700 in2 ; S = 1044.1 in2
Step-by-step explanation:
Lucky guess :)
what is 1 and 6/24 simplified as ?
Answers
A pudding recipe makes 4 1/2 cups of pudding. How many 1/3 cup servings does this equal?
Answers
4 1/2 cups = 9/2 cups
9/2 / 1/3=
9/2 * 3/1 = 27/2 = 13 1/2 servings at 1/3 cup each
Answer:
13 1/2
Step-by-step explanation:Just trust
Write this number in word form
Number:284028
Answers
hope this helps
Hopes this help and please give me a thanks
which of the following best describes a regular polygon when the sum of its interior angles is 1080 degrees A.) octagon B.) heptagon C.) hexagon D.) nonagon
Answers
1080= (n-2)180
1080/180=6
6=(n-2)
n=8
(This is the formula for finding the sum of interior angles in polygons. n represents the number of sides in the polygon)
Answer:
Option A is correct.
Step-by-step explanation:
Given: Sum of all the interior angles of a regular polygon = 1080°
To find: number of sides or name of polygon.
We know that Sum of all interior angle of a regular polygon is given by following formula,
Sum of all angles = ( n - 2 ) × 180° , where n = no of sides.
⇒ 1080° = ( n - 2 ) × 180°
n - 2 = 6
n = 6 + 2
n = 8
Name of regular polygon is Regular Octagon.
Therefore, Option A is correct.
Find the area of the parallelogram shown below
Answers
5 * 7 =
35
I WILL GIVE BRAINLIEST please answer both parts
If 365 work days are split among 12 employees evenly
PART: A How can you rewrite this as a fraction?
PART:B What does the fraction represent?
Answers
Part A: To understand this question, you need to read carefully and pick out key words that tells you what to do. All math word / puzzle problems have key words that tell you what you need to do.
For instance, the key word split tells us we need to divide. So, we know we have 365 work days that need to be split among 12 employees. So for Part A:
365 / 12
For Part B it wants to know what this represents. It represent the number of days each of the 12 employees will get if split evenly among each employee.
A box has dimensions of 19 inches long, 1.7 feet wide, and 6 inches high. What is the volume of the box? The formula for the volume is V = l ⋅ w ⋅ h.
193.8 cubic inches (in3)
2325.6 cubic inches (in3)
1.35 cubic inches (in3)
16.15 cubic inches (in3)
Answers
1 foot = 12 inches
1.7 feet = 1.7 * 12 = 20.4 in
The formula for the volume is:
V = l * w * h
and: l = 19 in, w = 20.4 in, h = 6 in.
V = 19 in * 20.4 in * 6 in = 2,325.6 in³
Answer: B ) 2,325.6 in³
How to show numbers are related to each other
Answers
Rita bought fabric and ribbon from a store the ribbon cost $18.50 rita paid the cashier $50.00 and received change of $5.25 how much did the fabric cost
Answers
2. I know this because...
- First, I subtracted the cost of the ribbon from how much she paid.
· Equation: $50.00 - $18.50 = $31.50
- Second, I subtracted the change received by the reduced cost ($31.50)
· Equation: $31.50 - $5.25 = $26.25
3. So, the final amount of money Rita spent on fabric is $26.25.
______________________________________________________________
I am 99.9% sure that this is the correct answer. If for some reason my answer is incorrect, I am truly sorry, and please forgive me. Thank You!
Number 8 please it will help a lot
Answers
what is the value of x and y
Answers
First, square root x^2 + y^2 = 25, then you get x + y = 5
Elimination. Multiply x + y = 5 by -1, then you get -x - y = -5.
2x + y = 10
-x - y = -5
---------------
x = 5
An amusement park sells child and adult tickets at a ratio of 8 to 1. On Saturday they sold 147 more child tickets than adult tickets. How many tickets did the amusement park sell on Saturday?
Answers
c=a+147
8a=a+147
7a=147
a=147/7
a=21 Adult Tickets c=8a=8 (21)= 168 Children Tickets Total Tickets = 21+168=189 Tickets is your answer
Six less then two thirds of a number is negative ten what is the number
Answers
2/3X-6 = -10
2/3x = -4
x = -4 / 2/3 = -6
X = -6
check:
2/3(-6) - 6 = -4 -6 = -10 TRUE
explain how complex fractions can be used to solve problems involving ratios
Answers
Use the Distributive Property to simplify each expression:
6(a + 10)
(5 + w)5
14(4f - 8)
2.1(5w -15)
0(3.7x - 4.21)
-(-m + n + 1)
Answers
6a + 60
5w + 25
56 f - 122
10.5 w - 31.5
0
m - n -1
Which number in the monomial 215x18y3z21 needs to be changed to make it a perfect cube?
3
18
21
215
Answers
xyz = (x)(y)(z)
if x, y and z are perfect cubes, then xyz would also be a perfect cube
Also,
(x^n)^m = x^(mn)
if it is a perfect cube, then it can be factored into the expression (x^n)^3, wherein n is a whole number.
All should be a perfect cube, given
215x^18(y^3)(z^21)
By writing the terms separately, we would see which one is to be changed.
215 = 5*43, we see this is not a perfect cube, we could changed it to 6^3 or 216
x^18=(x^6)^3
y^3=y^3
z^21=(z^7)^3
The other terms are perfect cubes except for the first one. So the correct answer would be the last option, 215 should be changed.
Answer:
D; 215
Step-by-step explanation:
EDGE
What is the measure of angle 4?
Answers
7x + 23x = 180
30x = 180
x = 6
m<4 = 7x = 7(6) = 42
answer
B. 42 (Second choice)
Help!! Image attached, will put most brainliest
Answers
Point Slope Form: y - y₁ = m(x - x₁)
Info Given:
Coordinates - (1,4)
Slope: -2
(1,4)
x₁ y₁
Plug slope and point into formula: y - 4 = -2(x -1)
Slope = -2
Passes through (1, 4)
We simply need to plug it into the point-slope equation :)
y - y₁ = m(x - x₁)
* m = slope
y - 4 = -2(x - 1)
D) y - 4 = -2(x - 1)
~Hope I helped!~
Jack wants to fill a rectangular box with sand. The length of the sand box is 3 feet, width is 6 inches, and height is 2.4 inches. Each bag of sand contains 0.15 cubic foot of sand. How many bags of sand will Jack need to fill the box completely? [1 foot = 12 inches] numerical answers expected
Answers
3 feet = 36 inches
36 x 6 x 2.4 = 518.4 cubic inches
1 cubic foot = 1728 cubic inches
518.4/1728 = 0.3 cubic feet
0.3/0.15 = 2
he will need 2 bags
The table shows the number of students in a school who like dodgeball and/or ping pong:
Like Dodgeball
Do Not Like Dodgeball
Total
Like Ping Pong
26
18
44
Do Not Like Ping Pong
24
32
56
Total
50
50
100
What is the relative frequency, by row, of students who like both dodgeball and ping pong?
0.22
0.40
0.48
0.59
PLEASE HELP !!!
Answers
I've included the table so it's easier to visualize.
Divide the amount of students who like ping pong and dodge ball (26, it's highlighted in the image) by the total amount of students that like ping pong (44).
26/44 = 0.59
The answer is 0.59
Hope this helps! :)
Answer:
0.59
Step-by-step explanation: