Question

no additional details were added for this assignment. you are asked to design a rectangular display box for a cylindrical trophy. you need to build a box that minimizes the cost of the box while meeting the design parameters. • the cylindrical trophy has a diameter of 6 inches and a height of 10 inches. • the box needs to allow an extra half inch of space on each side of the trophy and 1 extra inch above the trophy. • the cost of the material for the box is $0.05 per square inch. determine the dimensions of the smallest box that meets the parameters. use the formula for surface area of a rectangular prism is ( sa = 2(ell w+ell h + wh) ) to determine the minimum cost for the box material. move the answers to the boxes to complete the sentences. the smallest size of the box is the lowest cost of the box is

Explanation:

Step1: Calculate the dimensions of the box

• Length (\(l\)) and width (\(w\)): The diameter of the cylinder is \(6\) inches. With an extra half - inch on each side, \(l = w=6 + 2\times0.5=7\) inches.
• Height (\(h\)): The height of the cylinder is \(10\) inches. With an extra \(1\) inch above, \(h = 10+1 = 11\) inches.

Step2: Calculate the surface area of the box

Use the formula \(SA=2(lw+lh+wh)\). Substitute \(l = 7\), \(w = 7\), and \(h = 11\) into the formula:

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Step3: Calculate the cost of the box

The cost per square inch is \(\$0.05\). So the cost \(C=0.05\times SA\). Substitute \(SA = 406\) into the formula: \(C=0.05\times406=\$20.3\)

Answer:

The smallest size of the box is \(7\) inches (length), \(7\) inches (width), and \(11\) inches (height). The lowest cost of the box is \(\$20.3\)