Question

A hotel is building a large new aquarium for their lobby. They know the length from the top corner to the opposite bottom corner is 13 feet long, the height is 12 feet and the length is 4 feet. Solve for the width and then determine the volume in cubic feet of water it will take to fill the aquarium

Answer 1

width = 3ft

volume = 144 ft³

Explanation:

Pythagoras Theorem

`\sf a^2+b^2=c^2`

(where a and b are the legs, and c is the hypotenuse, of a right triangle)

The diagonal of the base of the aquarium, its height, and the length from the top corner to the opposite bottom corner forms a right triangle. As we know two of the lengths, we can use Pythagoras Theorem to calculate the unknown length (diagonal of base).

Given:

• a = height of aquarium = 12 ft
• b = diagonal of base
• c = top corner to opposite corner = 13 ft

`\implies \sf 12^2+b^2=13^2`

`\implies \sf b=√(13^2-12^2)`

`\implies \sf b=5`

Now we know the length of the diagonal of the base, we can again use Pythagoras Theorem to find the width of the aquarium.

Given:

• a = length of aquarium = 4 ft
• b = width of aquarium
• c = diagonal of base = 5 ft

`\sf \implies 4^2+b^2=5^2`

`\sf \implies b=√(5^2-4^2)`

`\sf \implies b=3`

Therefore, the width of the aquarium is 3 ft.

As the aquarium is modeled as a rectangular prism:

`\begin{aligned}\textsf{Volume of a rectangular prism} & = \sf width * length * height\\ \implies \textsf{Volume of aquarium} & = \sf 3\:ft * 4\:ft * 12\:ft\\& = \sf 144 \:\:ft^3\end{aligned}`

Answer 2

• 3 ft, 144 ft³

Explanation:

Find the width as below

• l² + w² + h² = d², where l- length, w- width, h- height, d- the distance between top and bottom corners

Substitute the values and solve for w

• 4² + w² + 12² = 13²
• w² + 160 = 169
• w² = 9
• w = 3 feet

Now find the volume

• V = lwh
• V = 4*3*12 = 144 ft³