Ask Question
23 March, 17:54

The base and sides of a container is made of wood panels. The container does not have a lid. The base and sides are rectangular. The width of the container is x c m. The length is fourth times the width. The volume of the container is 400 c m 3. Determine the minimum surface area that this container will have.

+3
Answers (1)
  1. 23 March, 19:03
    0
    A (mim) = 300 cm²

    Step-by-step explanation:

    Area of the base is:

    A = 4*x*x ⇒ A = 4*x²

    Let call h the height of the container, then area lateral are:

    A₁ = 2*4*x*h A₁ = 8*x*h and

    A₂ = 2 * x * h A₂ = 2*x*h

    From the volume of the container we have:

    400 = Area of the base*h

    400 = 4*x²*h ⇒ h = 100 / x²

    Now Total area of the container is:

    A = 4*x² + 8*x*h + 2*x*h ⇒ A = 4*x² + 10*x*h

    As h = 100/x²

    A (x) = 4*x² + 10*x * 100/x²

    A (x) = 4*x² + 1000/x

    Taking derivatives on both sides of the equation we get:

    A' (x) = 8*x - 1000/x²

    A' (x) = 0 ⇒ 8*x - 1000/x² = 0

    8*x³ = 1000 = 0 ⇒ x³ = 1000 / 8 ⇒ x = ∛ (1000) / 8

    x = 5 cm

    Now minimum area is:

    Area of the base

    4*5*5 = 100 cm²

    A₁ = 8*x*h h = 100 / x² h = 100 / 25 h = 4 cm

    A₁ = 8*5*4

    A₁ = 160 cm²

    A₂ = 2*5*4

    A₂ = 40 cm²

    A (mim) = 300 cm²
Know the Answer?
Not Sure About the Answer?
Find an answer to your question ✅ “The base and sides of a container is made of wood panels. The container does not have a lid. The base and sides are rectangular. The width ...” 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