Respuesta :
Answer:
Step-by-step explanation:
GIVEN: Suppose you want to make a cylindrical pen for your cat to play in (with open top) and you want the volume to be [tex]100[/tex] cubic feet. Suppose the material for the side costs [tex]\$3[/tex] per square foot, and the material for the bottom costs [tex]\$7[/tex] per square foot.
TO FIND: What are the dimensions of the pen that minimize the cost of building it.
SOLUTION:
Let height and radius of pen be [tex]r\text{ and }h[/tex]
Volume [tex]=\pi r^2h=100\implies h=\frac{100}{\pi r^2}[/tex]
total cost of building cylindrical pen [tex]C=3\times \text{lateral area}+7\times\text{bottom area}[/tex]
[tex]=3\times2\pi r h+7\times\pi r^2=\pi r(6h+7r)[/tex]
[tex]=\frac{600}{r}+7\pi r^2[/tex]
for minimizing cost , putting [tex]\frac{d\ C}{d\ r}=0[/tex]
[tex]\implies -\frac{600}{r^2}+44r=0 \Rightarrow r^3=\frac{600}{44}\Rightarrow r=2.39\text{ feet}[/tex]
[tex]\implies h=5.57\text{ feet}[/tex]
Hence the radius and height of cylindrical pen are [tex]2.39\text{ feet}[/tex] and [tex]5.57\text{ feet}[/tex] respectively.
