EXAMPLE 6 Solve a multi-step problem TERRARIUM

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EXAMPLE 6 Solve a multi-step problem TERRARIUM A terrarium in the shape of a rectangular prism has a volume of 4608 cubic inches. Its length is more than 10 inches. The dimensions of the terrarium are shown. Find the length, width, and height of the terrarium.

EXAMPLE 6 Solve a multi-step problem SOLUTION STEP 1 Write a verbal model. Then write an equation. 4608 = (36 – w) w (w + 4)

Solve a multi-step problem EXAMPLE 6 Solve a multi-step problem STEP 2 Solve the equation for w. 4608 = (36 – w)(w)(w + 4) Write equation. 0 = 32w2 + 144w – w3 – 4608 Multiply. Subtract 4608 from each side. 0 = (– w3 + 32w2) + (144w – 4608) Group terms. 0 = – w2 (w – 32) + 144(w – 32) Factor each group. 0 = (w – 32)(– w2 + 144) Distributive property 0 = – 1(w – 32)(w2 – 144) Factor –1 from –w2 + 144. Difference of two squares pattern 0 = – 1(w – 32)(w – 12)(w + 12) w – 32 = 0 or w – 12 = 0 or w + 12 = 0 Zero-product property w = 32 w = 12 w = – 12 Solve for w.

EXAMPLE 6 Solve a multi-step problem STEP 3 Choose the solution of the equation that is the correct value of w. Disregard w = – 12, because the width cannot be negative. You know that the length is more than 10 inches. Test the solutions 12 and 32 in the expression for the length. Length = 36 – 12 = 24 or Length = 36 – 32 = 4 The solution 12 gives a length of 24 inches, so 12 is the correct value of w.

EXAMPLE 6 Solve a multi-step problem STEP 4 Find the height. Height = w + 4 = 12 + 4 = 16 ANSWER The width is 12 inches, the length is 24 inches, and the height is 16 inches.

GUIDED PRACTICE for Example 6 10. DIMENSIONS OF A BOX A box in the shape of a rectangular prism has a volume of 72 cubic feet. The box has a length of x feet, a width of (x – 1) feet, and a height of (x + 9) feet. Find the dimensions of the box. The length is 3ft, the width is 2ft, and the height is 12ft.