At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Ask your questions and receive accurate answers from professionals with extensive experience in various fields on our platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.

Question Help
Reasoning A cone with radius 6 and height 9 has its radius quadrupled. How many times greater is the
volume of the larger cone than the smaller cone?l


Sagot :

Let's see

For initial cone

  • r=6
  • h=9

Volume

  • 1/3πr²h
  • 1/3π(6)²(9)
  • 3(36)π
  • 108π units³

For new cone

  • r=4(6)=24
  • h=9

Volume

  • 1/3π(24)²(9)
  • 3(576)π
  • 1728π

So

  • 1728π/108π
  • 16times

Answer:

16 times

Step-by-step explanation:

[tex]\textsf{Volume of a cone}=\sf \dfrac{1}{3} \pi r^2 h \quad\textsf{(where r is the radius and h is the height)}[/tex]

If only the radius is changed, the change in volume will be proportionate to the multiplicative factor squared.

[tex]\sf \implies Volume =\dfrac{1}{3} \pi (ar)^2h=\dfrac{1}{3} \pi (a^2)r^2h[/tex]

Therefore, if the cone is quadrupled (multiplied by 4), the volume of the larger cone will be 4² times greater than the volume of the smaller cone, so 16 times greater than the smaller cone.

Proof

Given:

  • radius = 6
  • height = 9

Substituting the given values into the formula:

[tex]\sf \implies Volume =\dfrac{1}{3} \pi (6)^2(9)=108 \pi \: \:cubic\:units[/tex]

If the radius is quadrupled:

  • radius = 6 × 4 = 24
  • height = 9

Substituting the new given values into the formula:

[tex]\sf \implies Volume =\dfrac{1}{3} \pi (24)^2(9)=1728 \pi \: \:cubic\:units[/tex]

To find the number of times greater the volume of the large cone is than the volume of the smaller cone, divide their volumes:

[tex]\sf \implies \dfrac{V_{large}}{V_{small}}=\dfrac{1728\pi}{108\pi}=16[/tex]

So the volume of the larger cone is 16 times greater than the volume of the smaller cone.