a conical paper cup is hold 1/4 of a liter. find height and radius of cone which minimizes the amount of paper needed to make the cup

h ≈ 7.816 cm

r ≈ 5.527 cm

Step-by-step explanation:

The volume of a cone is:

V = ⅓ π r² h

The lateral surface area of a cone is:

A = π r √ (r² + h²)

1/4 of a liter is 250 cm³.

250 = ⅓ π r² h

h = 750 / (π r²)

Square both sides of the area equation:

A² = π² r² (r² + h²)

Substitute for h:

A² = π² r² (r² + (750 / (π r²)) ²)

A² = π² r² (r² + 750² / (π² r⁴))

A² = π² (r⁴ + 750² / (π² r²))

Take derivative of both sides with respect to r:

2A dA/dr = π² (4r³ - 2 * 750² / (π² r³))

Set dA/dr to 0 and solve for r.

0 = π² (4r³ - 2 * 750² / (π² r³))

0 = 4r³ - 2 * 750² / (π² r³)

4r³ = 2 * 750² / (π² r³)

r⁶ = 750² / (2π²)

r³ = 750 / (π√2)

r³ = 375√2 / π

r = ∛ (375√2 / π)

r ≈ 5.527

Now solve for h.

h = 750 / (π r²)

h = 750 / (π (375√2 / π) ^⅔)

h = 750 ∛ (375√2 / π) / (π (375√2 / π))

h = 2 ∛ (375√2 / π) / √2

h = √2 ∛ (375√2 / π)

h ≈ 7.816

Notice that at the minimum area, h = r√2.