Help please:)) 2. When shipping ice cream, melting is understandably a big concern. You will notice that ice cream is not generally packaged in a cube-shaped container. A standard container of ice cream contains 1 L, or 1000 cm3 of ice cream, a. What would be the optimal dimensions (radius and height) to minimize surface area?

b. What would the surface area be?

C. Suggest at least two reasons why this is different from the ice cream packaging that you see in the stores.​

a. The radius r = 5.42 cm and the height h = 10.84 cm

b. 553.73 cm²

c. i. Beauty ii. Design

Step-by-step explanation:

a. What would be the optimal dimensions (radius and height) to minimize surface area?

The volume of the standard container is a cylinder and its volume is V = πr²h where r = radius of container and h = height of container.

Since V = 1000 cm³,

1000 cm³ = πr²h  (1)

Now, the surface area of a cylinder is A = 2πr² + 2πrh where r and h are the radius and height of the cylinder.

From (1), h = 1000/πr².

Substituting h into A, we have

A = 2πr² + 2πrh

A = 2πr² + 2πr(1000/πr²)

A = 2πr² + 2000/r

To maximize A, we differentiate A with respect to r and equate to zero to find the value of r at which A is maximum.

So, dA/dr = d[2πr² + 2000/r]/dr

dA/dr = d[2πr²]/dr + d[2000/r]/dr

dA/dr = 4πr - 2000/r²

Equating the equation to zero, we have

4πr - 2000/r² = 0

4πr = 2000/r²

r³ = 2000/4π

r = ∛(1000/2π)

r = 10(1/∛(2π))

r = 10(1/∛(6.283))

r = 10/1.8453

r = 5.42 cm

To determine if this value of r gives a minimum for A, we differentiate dA/dr with respect to r.

So, d(dA/dr)/dr = d²A/dr²

= d[4πr - 2000/r²]/dr

= d[4πr]/dr - d[2000/r²]/dr

= 4π  + 4000/r³

Substituting r³ = 2000/4π into the equation, we have

d²A/dr² = 4π  + 4000/r³ = 4π  + 4000/(2000/4π) = 4π  + 2 × 4π = 4π + 8π = 12π > 0

Since d²A/dr² = 12π > 0, then r = 5.42 cm gives a minimum for A.

Since h = 1000/πr²

h = 1000/π(5.42)²

h = 1000/92.288

h = 10.84 cm

So, the radius r = 5.42 cm and the height h = 10.84 cm

b. What would the surface area be?

Since the surface area, A = 2πr² + 2πrh

Substituting the values of r and h into A, we have

A = 2πr² + 2πrh

A = 2πr(r + h)

A = 2π5.42(5.42 + 10.84)

A = 10.84π(16.26)

A = 176.2584π

A = 553.73 cm²

c. Suggest at least two reasons why this is different from the ice cream packaging that you see in the stores.​

i. Beauty

ii. Design

what?

step-by-step explanation: