1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Phantasy [73]
3 years ago
15

Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3?

Mathematics
2 answers:
QveST [7]3 years ago
8 0
The answer is 32 pie over 3
bogdanovich [222]3 years ago
6 0

First, notice that, by the Pythagorean Theorem,

x^2+y^2=3^2

meaning that:

x^2=9-y^2

Also, since the volume of a cone with radius r and height h is \frac{1}{3} \pi r^2h we know that the volume of the cone is:

\frac{1}{3} \pi x^2 (3+y) =  \frac{1}{3} \pi  (9-y^2)(3+y) =  \frac{1}{3} \pi [27+9y-3y^2-y^3]

Therefore, we want to maximize the function V(y) =  \frac{1}{3} \pi  [27+9y-3y^2-y^3] subject to the constraint 0 \leq y \leq 3.

To find the critical points, we differentiate:

V'(y)= \frac{1}{3} \pi [9-6y-3y^2] =  \pi [3-2y-y^2] =  \pi (3+y)(1-y).

Therefore, V'(y) = 0 when

                                                    \pi (3+y)(1-y)=0

meaning that y = -3 or y=1. Only y=1 is in the interval [0,3] so that’s the only critical point we need to concern ourselves with.

Now we evaluate V at the critical point and the endpoints:

V(0) =  \frac{1}{3} \pi  [27+9(0) - 3(0)^2] = 9 \pi

V(1) =  \frac{1}{3} \pi  [27+9(1)-3(1)^2-1^3] =    \frac{32 \pi }{3}

V(3) = \frac{1}{3} \pi [27+9(3) - 3(3)^2-3^2] = 0

Therefore, the volume of the largest cone that can be inscribed in a sphere of radius 3 is \frac{32 \pi }{3}


                                                   



You might be interested in
Collins spent $8 at lunch. He spent $24 on new shoes. His shoeswere how many more times expensive than his lunch
kati45 [8]

Collins’ new shoes were 3 times the amount of his lunch. You’re welcome!

5 0
3 years ago
Read 2 more answers
Write y-6=-2x in slope intercept form
Vera_Pavlovna [14]

Slope:  2

Y-Intercept:   6

2,6

8 0
3 years ago
Read 2 more answers
Divide 36x^3-96x^2+48x by 12x
sergij07 [2.7K]

Answer:

3x^2  - 8x + 4.

Step-by-step explanation:

36x^3 - 96x^2 + 48x

Dividing by 12  term by term:

= 3x^2  - 8x + 4.

5 0
3 years ago
Read 2 more answers
Heedayati drives for 180 km from Town P to Town Q. The average speed of Heedayati is 60 km/h. Heedayati took 45 minutes less for
Nadusha1986 [10]

Answer:

180 divided by 45 is equal to 4km/h

7 0
3 years ago
Do you know what1
Nikitich [7]

Answer:

42

Step-by-step explanation:

because it is the only one do t know so yeah lol

8 0
2 years ago
Other questions:
  • The definition of parallel lines requires the undefined terms line and plane, while the definition of perpendicular lines requir
    9·2 answers
  • -6x is less than or equal to 4x+2
    10·1 answer
  • How many ways can 5 different colored balls be placed into bins labeled 1 through 8 so that no bin contains more than one ball?
    14·1 answer
  • (-5, 3), (2, 1) what is the slope of the line that passes through the points
    11·1 answer
  • You roll a six sided die. What is the probability of rolling a 1 or a number greater than 4.
    8·1 answer
  • The path of pumpkin A is modeled by the equation h(x) = –0.01x2 + 0.32x + 4. The path of pumpkin B is modeled by the equation h(
    9·1 answer
  • What percent is represented by the shaded area?​
    12·2 answers
  • a woodworker wants to cut a board that is 6.432 feet long into 4 equal-length pieces. How long will each of the cut boards be?
    14·1 answer
  • Plssss i need help plsss i really need hep now asap help me pls
    8·1 answer
  • 100+300= What is it<br><br> Help plz
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!