Respuesta :
Answer:
The answer is h = 2r cm
Explanation:
To construct the container, a rectangle-shaped piece is cut to make the cylindrical body of said container, and two circular-shaped pieces for its lids. We establish A as the total area of the container, At will be the area of the lids, Ac will be the area of the body, r will be the radius of the cylindrical body and h will be its height.
The total area A of the cylindrical container will be given by the following equation:
A = 2 π r^2 + 2 π r h
To solve the equation we will use the information from the exercise: the volume of the container is 1 liter. This means that:
Volume = π r^2 h = 1000
Clearing the height h, we have:
h = 1000/(π r^2)
Substituting this expression in the equation for area A:
A = 2 π r^2 + 2 π r h = 2 π r^2 + 2 π r (1000 / (π r^2)) = 2 π r^2 + 2000/r
A(r) = 2 π r^2 + 2000/r, condition r> 0
Optimizing this equation:
A’(r) = 4 π r - 2000/r^2 = 4 (π r^3 - 500)/r^2
If A’(r) = 0
Solving for r:
r = 3√(500/π) cm
Substituting in the equation for height:
h = 1000/(π r^2),
h = 2 3√ (500/π) = 2 r cm
