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aleksley [76]
2 years ago
10

A light bulb is designed by revolving the graph of:

Mathematics
1 answer:
nadya68 [22]2 years ago
5 0

Answer:

\displaystyle 0.251327 \ in. \ of \ glass

General Formulas and Concepts:

<u>Pre-Algebra</u>

Order of Operations: BPEMDAS

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

<u>Algebra I</u>

  • Terms/Coefficients
  • Expand by FOIL (First Outside Inside Last)
  • Factoring

<u>Calculus</u>

Differentiation

Derivative Notation

Basic Power Rule:

  • f(x) = cxⁿ
  • f’(x) = c·nxⁿ⁻¹

Integration

  • Integration Property: \displaystyle \int\limits^a_b {cf(x)} \, dx = c \int\limits^a_b {f(x)} \, dx
  • Fundamental Theorem of Calculus: \displaystyle \int\limits^a_b {f(x)} \, dx = F(b) - F(a)
  • Area between Two Curves
  • Volumes of Revolution
  • Arc Length Formula: \displaystyle AL = \int\limits^a_b {\sqrt{1+ [f'(x)]^2}} \, dx
  • Surface Area Formula: \displaystyle SA = 2\pi \int\limits^a_b {f(x) \sqrt{1+ [f'(x)]^2}} \, dx

Step-by-step explanation:

<u>Step 1: Define</u>

\displaystyle y = \frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}\\Interval: [0, \frac{1}{3}]

<u>Step 2: Differentiate</u>

  1. Basic Power Rule:                    \displaystyle y' = \frac{1}{2} \cdot \frac{1}{3}x^{\frac{1}{2} - 1} - \frac{3}{2} \cdot x^{\frac{3}{2} - 1}
  2. [Derivative] Simplify:                \displaystyle y' = \frac{1}{6}x^{\frac{-1}{2}} - \frac{3}{2}x^{\frac{1}{2}}
  3. [Derivative] Simplify:                \displaystyle y' = \frac{1}{6\sqrt{x}} - \frac{3\sqrt{x}}{2}}

<u>Step 3: Integrate Pt. 1</u>

  1. Substitute [Surface Area]:                                                                             \displaystyle SA = 2\pi \int\limits^{\frac{1}{3}}_0 {(\frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}) \sqrt{1+ [\frac{1}{6\sqrt{x}} - \frac{3\sqrt{x}}{2}}]^2}} \, dx
  2. [Integral - √Radical] Expand/Add:                                                               \displaystyle SA = 2\pi \int\limits^{\frac{1}{3}}_0 {(\frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}) \sqrt{\frac{81x^2+18x+1}{36x}} \, dx
  3. [Integral - √Radical] Factor:                                                                         \displaystyle SA = 2\pi \int\limits^{\frac{1}{3}}_0 {(\frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}) \sqrt{\frac{(9x + 1)^2}{36x}} \, dx
  4. [Integral - Simplify]:                                                                                       \displaystyle SA = 2\pi \int\limits^{\frac{1}{3}}_0 {-\frac{|9x + 1|(3x - 1)}{18}} \, dx
  5. [Integral] Integration Property:                                                                     \displaystyle SA = \frac{- \pi}{9} \int\limits^{\frac{1}{3}}_0 {|9x + 1|(3x - 1)} \, dx

<u>Step 4: Integrate Pt. 2</u>

  1. [Integral] Define:                                                                                             \displaystyle \int {|9x + 1|(3x - 1)} \, dx
  2. [Integral] Assumption of Positive/Correction Factors:                                 \displaystyle \frac{9x + 1}{|9x + 1|} \int {(9x + 1)(3x - 1)} \, dx
  3. [Integral] Expand - FOIL:                                                                                 \displaystyle \frac{9x + 1}{|9x + 1|} \int {27x^2 - 6x - 1} \, dx
  4. [Integral] Integrate - Basic Power Rule:                                                         \displaystyle \frac{9x + 1}{|9x + 1|} (9x^3 - 3x^2 - x)
  5. [Expression] Multiply:                                                                                      \displaystyle \frac{(9x + 1)(9x^3 - 3x^2 - x)}{|9x + 1|}

<u>Step 5: Integrate Pt. 3</u>

  1. [Integral] Substitute/Integral - FTC:                                                              \displaystyle SA = \frac{- \pi}{9} (\frac{(9x + 1)(9x^3 - 3x^2 - x)}{|9x + 1|})|\limits_{0}^{\frac{1}{3}}
  2. [Integrate] Evaluate FTC:                                                                                \displaystyle SA = \frac{- \pi}{9} (\frac{-1}{3})
  3. [Expression] Multiply:                                                                                     \displaystyle SA = \frac{\pi}{27} \ ft^2

<em>It is in ft² because it is given that our axis are in ft.</em>

<u>Step 6: Find Amount of Glass</u>

<em>Convert ft² to in² and multiply by 0.015 in (given) to find amount of glass.</em>

  1. Convert ft² to in²:                    \displaystyle \frac{\pi}{27} \ ft^2 \ \div 144 \ in^2/ft^2 = \frac{16 \pi}{3} \ in^2
  2. Multiply:                                   \displaystyle \frac{16 \pi}{3} \ in^2 \cdot 0.015 \ in = 0.251327 \ in. \ of \ glass

And we have our final answer! Hope this helped on your Calc BC journey!

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