A conical paper cup is to hold a fixed volume of water. Find the ratio of height to base radius of the cone which minimizes the amount of paper needed to make the cup. Use the formula πr√r^2+h^2 for the area of the side of a cone, called the lateral area of the cone.
Answers
Given : A conical paper cup is to hold a fixed volume of water.
The formula πr√(r²+h²) for area of the side of a cone, called the lateral area of the cone.
To Find : the ratio of height to base radius of the cone which minimizes the amount of paper needed to make the cup.
Solution:
Volume of cone = V fixed/constant
V = (1/3)πr²h
=> h = 3V/πr²
Area of paper needed = πr√(r²+h²)
Substitute h = 3V/πr²
A = πr√(r²+ (3V/πr²)²)
As Area is +ve hence Maximize A² will also maximize A
Z = A² = π²r² (r²+ 9V²/π²r⁴)
=> Z = π²r⁴ + 9V²/r²
dZ/dr = 4π²r³ - 18V²/r³
dZ/dr = 0
=> 4π²r³ - 18V²/r³ = 0
=> r⁶ = 9V²/2π²
=> r³ = 3V/π√2
d²Z/dr² = 12π²r³ + 54V²/r⁴ > 0
Hence Z is minimum so Area is minimum
r³ = 3V/π√2
V = (1/3)πr²h => 3V/π = r²h
=> r³ = r²h /√2
=> r = h /√2
=> h/r = √2
ratio of height to base radius of the cone = √2
Learn More:
this problem is about maxima and minima. find the area of the ...
brainly.in/question/13355753
The area of the largest possible square inscribed in a circle of unit ...
brainly.in/question/12227090