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gladu [14]
3 years ago
5

A rectangular box with a square base is to be constructed from material that costs $9 per ft2 for the bottom, $4 per ft2 for the

top, and $3 per ft2 for the sides. Find the box of greatest volume that can be constructed for $193. Round your answer to 2 decimals.
Mathematics
1 answer:
mafiozo [28]3 years ago
6 0

Answer:

V = 23.85 ft³

Step-by-step explanation:

The dimension of the square base = x ft by x ft

Height of box = y

Volume of the box, V = x^{2} y

Cost of top = $4 * x^{2}

Cost of the bottom =  $9 * x^{2}

Cost of the sides = $3 * 4xy

Total cost = 4x² + 9x² + 12xy

Total cost = 13x² + 12xy

Total cost = $193

193 = 13x² + 12xy

y = \frac{193 - 13x^{2} }{12x}

But volume, V = x²y

V = \frac{x^{2} (193 - 13x^{2}) }{12x}

V = \frac{x (193 - 13x^{2}) }{12}...................(1)

At maximum value, V' = 0

0 = \frac{193 - 39x^{2} }{12}

39x^{2}  = 193\\x^{2} = \frac{193}{39}

x = 2.23

Put the value of x into (1)

V = \frac{2.23 (193 - 13*2.23^{2}) }{12}

V = 23.85 ft³

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