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The volume of a sphere is decreasing at a constant rate of 116 cubic centimeters per second. At the instant when the volume of the sphere is 77 cubic centimeters, what is the rate of change of the radius? The volume of a sphere can be found with the equation V=\frac{4}{3}\pi r^3.V=
3
4
​
πr
3
. Round your answer to three decimal places.

Respuesta :

Space

Answer:

[tex]\frac{dr}{dt} = -1.325 \ cm/s[/tex]

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

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

Equality Properties

Calculus

Derivatives

Basic Power Rule:

  • f(x) = cxⁿ
  • f’(x) = c·nxⁿ⁻¹

Taking Derivatives with respect to time

Step-by-step explanation:

Step 1: Define

Given:

[tex]V = \frac{4}{3} \pi r^3[/tex]

[tex]\frac{dV}{dt} = -116 \ cm^3/s[/tex]

[tex]V = 77 \ cm^3[/tex]

Step 2: Solve for r

  1. Substitute:                    [tex]77 = \frac{4}{3} \pi r^3[/tex]
  2. Isolate r term:               [tex]\frac{77}{\frac{4}{3} \pi} = r^3[/tex]
  3. Isolate r:                        [tex]\sqrt[3]{\frac{77}{\frac{4}{3} \pi}} = r[/tex]
  4. Evaluate:                       [tex]2.63917 = r[/tex]
  5. Rewrite:                         [tex]r = 2.63917 \ cm[/tex]

Step 3: Differentiate

Differentiate the Volume Formula with respect to time t.

  1. Define:                                                                                                            [tex]V = \frac{4}{3} \pi r^3[/tex]
  2. Differentiate [Basic Power Rule]:                                                                   [tex]\frac{dV}{dt} = \frac{4}{3} \pi \cdot 3 \cdot r^{3-1} \cdot \frac{dr}{dt}[/tex]
  3. Simplify:                                                                                                           [tex]\frac{dV}{dt} = 4 \pi r^2 \cdot \frac{dr}{dt}[/tex]

Step 4: Find radius rate

  1. Substitute in variables:                    [tex]-116 \ cm^3/sec = 4 \pi (2.63917 \ cm)^2 \cdot \frac{dr}{dt}[/tex]
  2. Isolate dr/dt rate:                             [tex]\frac{-116 \ cm^3/s}{4 \pi (2.63917 \ cm)^2} = \frac{dr}{dt}[/tex]
  3. Evaluate:                                          [tex]-1.3253 \ cm/s = \frac{dr}{dt}[/tex]
  4. Rewrite:                                           [tex]\frac{dr}{dt} = -1.3253 \ cm/s[/tex]
  5. Round:                                             [tex]\frac{dr}{dt} = -1.325 \ cm/s[/tex]

Our radius is decreasing at a rate of -1.325 cm per second.