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A cone has a base of [tex]5.4[/tex] and a height of [tex]8[/tex]. The diameter is [tex]6[/tex]. What is the volume? If you were to dilate the cone with a scale factor of [tex]2[/tex], what would the new radius be?

Respuesta :

Space

Answer:

(I) V = 24π ≈ 75.3982 units³

(II) V = 192π ≈ 603.186 units³

General Formulas and Concepts:

Math - Symbols

  • π - pi, the number 3.1415926535897932384626433832795

Pre-Algebra

Order of Operations: BPEMDAS

  1. Brackets
  2. Parenthesis
  3. Exponents
  4. Multiplication
  5. Division
  6. Addition
  7. Subtraction
  • Left to Right

Geometry

  • Dilations
  • Diameter: d = 2r
  • Volume of a Cone: [tex]V= \frac{1}{3} \pi r^2h[/tex]

Step-by-step explanation:

Step 1: Define

Height h = 8

Diameter d = 6

Base b = 5.4 (not needed in calculations)

Step 2: Find radius r

  1. Substitute:                    6 = 2r
  2. Isolate r:                        3 = r
  3. Rewrite:                         r = 3

Step 3: Find V of Normal Dimensions

  1. Substitute:                    [tex]V= \frac{1}{3} \pi (3)^2(8)[/tex]
  2. Exponents:                   [tex]V= \frac{1}{3} \pi (9)(8)[/tex]
  3. Multiply:                        [tex]V= 24 \pi[/tex]
  4. Evaluate:                      [tex]V \approx 75.3982[/tex]

Step 4: Find Dilations

Scale Factor: 2

  1. Define radius:                    r = 3
  2. Dilate:                                 r' = 3(2)
  3. Multiply:                              r' = 6
  1. Define height:                    h = 8
  2. Dilate:                                 h' = 8(2)
  3. Multiply:                              h' = 16

Step 5: Find V of Dilated Dimensions

  1. Substitute:                     [tex]V= \frac{1}{3} \pi (6)^2(16)[/tex]
  2. Exponents:                    [tex]V= \frac{1}{3} \pi (36)(16)[/tex]
  3. Multiply:                         [tex]V= 192 \pi[/tex]
  4. Evaluate:                       [tex]V \approx 603.186[/tex]

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