Ask Question
15 August, 11:24

An open-top rectangular tank with a square base and a volume of 32 ft3 is to be built. What dimensions minimize the amount of material required to build this tank? Show that your result is a minimum.

+5
Answers (1)
  1. 15 August, 12:59
    0
    x = 8 ft

    h = 1/2 ft

    Step-by-step explanation:

    Let x be side of the base then area of the base is x²

    Let h be the height of the tank

    Tank volume is 32 ft³ and is 32 = x²*h then h = 32 / x²

    Area of base + lateral area = total area (A)

    A = x² + 4*x*h ⇒ A = x² + 4*x * (32/x²) A = x² + 128/x

    A (x) = x² + 128/x (1)

    Taking derivatives on both sides of the equation

    A' (x) = 2x - 128/x² A' (x) = 0 2x - 128/x² = 0

    (2x² - 128) / x² = 0

    2x² - 128 = 0

    x² = √64

    x = 8 ft

    The result is minimum since replacing in equation (1) x = 8 we get

    A (x) > 0

    And

    h = 32/x²

    h = 1/2 ft
Know the Answer?
Not Sure About the Answer?
Find an answer to your question ✅ “An open-top rectangular tank with a square base and a volume of 32 ft3 is to be built. What dimensions minimize the amount of material ...” in 📘 Mathematics if you're in doubt about the correctness of the answers or there's no answer, then try to use the smart search and find answers to the similar questions.
Search for Other Answers