Question

A storage box with a square base must have a volume of 90 cubic centimeters. The top and bottom cost $0.40 per square centimeter and the sides cost $0.20 per square centimeter. Find the dimensions that will minimize cost. (Let x represent the length of the sides of the square base and let y represent the height. Round your answers to two decimal places.)

          A storage box with a square base must have a volume of 90 cubic centimeters. The top and bottom cost $0.40 per square centimeter and the sides cost $0.20 per square centimeter. Find the dimensions that will minimize cost. (Let x represent the length of the sides of the square base and let y represent the height. Round your answers to two decimal places.)
        
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Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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A storage box with a square base must have a volume of 90 cubic centimeters. The top and bottom cost $0.40 per square centimeter and the sides cost $0.20 per square centimeter. Find the dimensions that will minimize cost. (Let x represent the length of the sides of the square base and let y represent the height. Round your answers to two decimal places.)
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Transcript

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00:01 Hi in the given problem length is let's say the length is x length of the box the width is y and the height is also y so width is x and height is y therefore the volume is given as x square y so that is equal to 90 let's say this is equation number one now the area is two area is two times x plus two times x x into x which is 2 x square so the cost of the top and the bottom of the box is 0 .4 per square centimeter so c1 cost is 0 .4 times 2x square and the area of the side so this for the sides so the cost c2 is 0 .8 times xy which is for the sides of the box and the cost so the total cost function the total cost is c which is 0 .8 x squared plus 0 .8 x y so that's the total cost function now y can be written as c of x so this is c of x or now y can be written as in terms of x so this is 0 .8 x squared plus 0 .8 x and y is written as 90 over x so the cost function comes how to be equal to 0 .8 x squared plus 90 over x now to minimize the cost we will be finding the derivative and put that as equal to 0 so this means 0 .8 this will be 2x minus 90 over x squared is equal to 0 so from here we get x is equal to 3 .556 so the cost at this value and why would be correspondingly 7 .117 so the dimensions therefore so this is for minimum cost for minimum cost so the dimension the required dimensions of the box are dimensions of box would be length is 3 .55 centimeter the width would be also 3 .55 cm and the height is equal to 7 .12 centimeter...
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