A company is designing a rectangular aquarium. It will have a base made of slate

Question

A company is designing a rectangular aquarium. It will have a base made of slate, the sides are made of plexiglass, and the top is glass. The company wants the length of the tank to be three times the width, and the tank must hold 12 cubic feet (about 90 gallons) of water. Plexiglass costs $8 per square foot, slate costs $6 per square foot, and glass costs $4 per square foot. Find the dimensions (rounded to 2 decimal places) the company should use for the aquarium to minimize the cost of the materials. What is the cost of the aquarium? Use derivatives to justify your answer. You need to run a cable from point A to point B in on a city block corner (see the drawing).The city allows cable to be run above ground along Main Street but not on First Avenue. You have 3 options: You can run the cable down Main Street (above ground) the full 100 feet to the corner and then bury the cable underground along First Avenue the full 125 feet. Or, you can run the cable all underground straight from A to B. Or you can run the cable along Main Street (above ground) part way to the corner and then cut across underground to B. It costs $25 per foot to run the cable along a street, and $52 per foot to bury it underground. What is the cheapest way to run the cable?

Explanation / Answer

Let the width of the aquarium is x ft, then its length would be 3 times = 3x ft. Its base area would be 3x2 sq ft.

Volume of the aquarium is 12 cu ft, hence its height should be 12/ 3x2 =4/x2 ft

Area of the top of the aquarium would also same as that of the base = 3x2 ft

Area of the sides of aquarium would be 2[(3x)(4/x2) +(x)(4/x2)]= 2[12/x+4/x}]= 32/x

Now cost of Base(made of slate) would be (3x2) ($6)=18x2

cost of the Top(made of glass) would be (3x2)($4)= 12x2

cost of Sides(made o flexiglass) would be    (32/x)( $8)= 256/x      

Thus the total cost C= 18x2 +12x2 +256/x = 30x2 +256/x     

For minimizing the cost dC/dx = 60x- 256/x2     should be equal to 0

Hence 60x-256/x2=0 , Or x3=256/60=64/15    

x=4 /2.466=1.62ft

Thus for minimium cost the dimension of the aquarium should be length= 3(1.62)=4.86 ft

                                                                                          width = 1.62ft

                                                                                          height =4/(1.62)2=1.52 ft

Cost of the aquarium would be 30x2 +256/x = 30(1.62)2 +256/1.62 =78.73 +158.02= $236.75

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