EXAMPLE 4 Solve a multi-step problem

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EXAMPLE 4 Solve a multi-step problem 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. Manufacturing

Solve a multi-step problem EXAMPLE 4 Solve a multi-step problem SOLUTION STEP 1 Write an equation that gives the height h of 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. V = r2h Formula for volume of cylinder 342 = r2h Substitute 342 for V. 342 r2 = h Solve for h.

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

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

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.

GUIDED PRACTICE for Example 4 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. ANSWER r 4.42cm h 8.86cm