How many gallons of water, on average,does each exhibit in the aquarium have?

Question

The Tennessee Aquarium in Chattanooga, Tennessee is the largest freshwater aquarium in the world with 450,000 gallons of water in 24 exhibits. The 130,000 square feet of the aquarium are home to more than 9000 creatures. An interesting fact is that 300,000 crickets, about 400,000 worms, and 15,000 pounds of seafood are devoured by the aquariums wildlife each year.

 

1. How many gallons of water,on average, does each exhibit in the aquarium have?

 

2. Suppose the tennessee Aquarium is opening a new saltwater exhibit that will have tropical fish from around the world. Three of the new rectangular aquariums being ordered are described below:

 

Aquarium one(180-gallon tank): The aquarium has a length of x inches, a height and width that are both 48 inches less than the length, and a volume of 41,472 cubic inches. Write a polynomial in standard form to describe the volume of the aquarium. The find its dimensions.

 

Aquarium two (300-gallon tank): The aquarium has dimensions of x feet long,(x-6)feet wide, and (x-5.5)feet high. The volume of the aquarium is 40 cubic feet. write a polynomial in standard form to describe the volume of the aquarium. Then find its dimensions.

 

Aquarium three(125- gallon tank): The length of this aquarium is three times its height and its width is 4 inches more than the height. The volume of the aquarium is 28,800 cubic inches. Write a polynomial in standard form to describe the volume of the aquarium. then find its dimensions.

Answer 1

1) 450000 / 24 = 18750 gallons per exhibit, on average.

2)

a) 41472 = x(x-48)^2 simplifies to:

x³ - 96x² + 2304x - 41472 = 0 , or

(x-72)(x² -24x +576) = 0

which only has one real root: x=72 , and 72 - 48 = 24.

So the dimensions are 72 inches by 24 inches by 24 inches. (72 * 24 * 24 = 41472)

b) 40 = x(x-6)(x-5.5) simplifies to:

x³ -11.5x² + 33x - 40= 0

which only has one real root: x=8, and 8 - 6 = 2, and 8 - 5.5 = 2.5.

So the dimensions are 8 feet by 2 feet by 2.5 feet. (8 * 2 * 2.5 = 40)

c) 28800 = x(3x)(x+4) simplifies to:

3x³ + 12x² - 28800 = 0

or

3(x-20)(x² + 24x + 480) = 0

Which only has one real root, x=20. and 3 * 20 = 60, and 20 + 4 = 24.

So, the dimensions are 20 inches by 60 inches by 24 inches. ( 80 * 60 * 24 = 28800)

(Note: The method of finding the roots of the cubics varies from class to class. You might need to use the Rational Roots theorem to generate a list of possible rational solutions to check, or you might be able to use a TI-84 to find an x-intercept. In extreme cases, you could use Newton's Method. I used Symbolab to both solve and factor the cubics. Regardless of how you find the roots of the cubic polynomials above, the general solution is the same.)