Question

manufacturers often package products in a way that uses the least amount of material. one measure of the efficiency of a package is the ratio of its surface area s to its volume v . the smaller the ratio, the more efficient the packaging. a popcorn company is designing a new tin with the same square base and twice the height of the old tin. a. write an expression for the efficiency ratio sv of each tin.

Answer 1

The surface area of the old tin can be expressed as:

S1 = 4s^2

where s is the length of one side of the square base.

The volume of the old tin can be expressed as:

V1 = s^2h

where h is the height of the old tin.

Using the same square base, the new tin has twice the height of the old tin, so its height can be expressed as:

h = 2h1

The surface area of the new tin can be expressed as:

S2 = 4s^2 + 4sh

Substituting the expression for h, we get:

S2 = 4s^2 + 8s^2h1

Simplifying, we get:

S2 = 4s^2(1 + 2h1)

The volume of the new tin can be expressed as:

V2 = s^2(2h1)

Simplifying, we get:

V2 = 2s^2h1

Therefore, the efficiency ratio sv for the old tin is:

sv1 = S1/V1 = 4s^2/(s^2h1) = 4/h1

The efficiency ratio sv for the new tin is:

sv2 = S2/V2 = [4s^2(1 + 2h1)]/[2s^2h1] = 2(1 + 2h1)/h1 = 2/h1 + 4

So the expressions for the efficiency ratio sv of the old tin and the new tin are:

sv1 = 4/h1

sv2 = 2/h1 + 4

Explanation: