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Efficient Soup Can Packaging: Dimensions for Minimal Material Usage

Discover the optimal dimensions for a can of soup with a volume of 342 cubic centimeters to minimize material usage. Solve a multi-step problem to find the radius and height of the can. Learn how to write equations, functions, and graph to determine the most efficient packaging. Example calculations provided step by step.

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Efficient Soup Can Packaging: Dimensions for Minimal Material Usage

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  1. A food manufacturer wants to find the most efficient packaging for a can of soup with a volume of 342 cubic centimeters. Find the dimensions of the can that has this volume and uses the least amount of material possible. EXAMPLE 4 Solve a multi-step problem Manufacturing

  2. V= r2h 342= r2h 342 = h r2 EXAMPLE 4 Solve a multi-step problem SOLUTION STEP 1 Write an equation that gives the height hof the soup can in terms of its radius r. Use the formula for the volume of a cylinder and the fact that the soup can’s volume is 342 cubic centimeters. Formula for volume of cylinder Substitute 342 for V. Solve for h.

  3. 684 2 r2 = + r 342 S= 2 r2 ) ( +2 rh r2 π +2 r = 2 r2 Substitute for V. 342 π r2 EXAMPLE 4 Solve a multi-step problem STEP 2 Write a function that gives the surface area Sof the soup can in terms of only its radius r. Formula for surface area of cylinder Solve for h.

  4. You get a minimum value of about 271,which occurs when r3.79 and 342 h 7.58. π (3.79)2 EXAMPLE 4 Solve a multi-step problem STEP 3 Graph the function for the surface area Susing a graphing calculator. Then use the minimum feature to find the minimum value of S.

  5. EXAMPLE 4 Solve a multi-step problem ANSWER So, the soup can using the least amount of material has a radius of about 3.79 centimeters and a height of about 7.58 centimeters. Notice that the height and the diameter are equal for this can.

  6. r 4.42cm ANSWER h 8.86cm for Example 4 GUIDED PRACTICE 5. What If? In Example 4, suppose the manufacturer wants to find the most efficient packaging for a soup can with a volume of 544 cubic centimeters. Find the dimensions of this can.

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