Mathematics: Look into the sky and how many stars do u see

A) Radius: 3.44 cm.Height: 6.88 cm.B) Radius: 2.73 cm.Height: 10.92 cm.Step-by-step explanation:We have to solve a optimization problem with constraints. The surface area has to be minimized, restrained to a fixed volumen.a) We can express the volume of the soda can as:This is the constraint.The function we want to minimize is the surface, and it can be expressed as:To solve this, we can express h in function of r: And replace it in the surface equationTo optimize the function, we derive and equal...

Look into the sky and how many stars do u see

Ответ:

sduquesnay23ocesdcom

13.05.2022

Mathematics

A) Radius: 3.44 cm.

Height: 6.88 cm.

B) Radius: 2.73 cm.

Height: 10.92 cm.

Step-by-step explanation:

We have to solve a optimization problem with constraints. The surface area has to be minimized, restrained to a fixed volumen.

a) We can express the volume of the soda can as:

$V=\pi r^2h=256$

This is the constraint.

The function we want to minimize is the surface, and it can be expressed as:

$S=2\pi rh+2\pi r^2$

To solve this, we can express h in function of r:

$V=\pi r^2h=256\\\\h=\frac{256}{\pi r^2}$

And replace it in the surface equation

$S=2\pi rh+2\pi r^2=2\pi r(\frac{256}{\pi r^2})+2\pi r^2=\frac{512}{r} +2\pi r^2$

To optimize the function, we derive and equal to zero

$\frac{dS}{dr}=512*(-1)*r^{-2}+4\pi r=0\\\\\frac{-512}{r^2}+4\pi r=0\\\\r^3=\frac{512}{4\pi} \\\\r=\sqrt[3]{\frac{512}{4\pi} } =\sqrt[3]{40.74 }=3.44$