SOLUTION: ***I NEED THE STEPS TO SOLVE THIS PROBLEM SO THAT I CAN BETTER UNDERSTAND IT...PLEASE HELP*** ANSWER: (a)2sqroot2 ft (b) 2ft PROBLEM: DIMENSIONS OF AN AQUARIUM A

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Question 384661: ***I NEED THE STEPS TO SOLVE THIS PROBLEM SO THAT I CAN BETTER UNDERSTAND IT...PLEASE HELP***
ANSWER: (a)2sqroot2 ft
(b) 2ft
PROBLEM: DIMENSIONS OF AN AQUARIUM
An open-topped aquarium is to be constructed with 6ft-long sides and square ends.
(a) Find the height of the aquarium if the volume is to be 48ft^3.
(b) Find the height if 44ft^2 of glass is to be used.
THANK YOU!!!!!!!!
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
6 foot long sides.
square ends.

find height if volume is 48 cubic feet

Volume of rectangular prism is h*l*w (height * length * width)

length is 6 feet.

height and width (form the ends) are the same measure, call it s.

volume is therefore h*s^2 (height * area of the base)

formula for volume is v = h*s^2

height is 6 feet, so formula becomes v = 6*s^2

volume is 48 cubic feet, so formula becomes 48 = 6*s^2

divide both sides of equation by 6 to get 8 = s^2

take square root of both sides of equation to get s = sqrt(8).

length is 6
width is sqrt(8)
height is sqrt(8)

volume is l*w*h = 6*sqrt(8)*sqrt(8) = 6*8 = 48 cubic feet.

sqrt(8) = sqrt(2*2*2) = 2*sqrt(2)

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(b) Find the height if 44 square feet of glass is to be used.

aquarium is open topped.

length is still 6 feet.

ends are still square.

aquarium is open on top.

it therefore has 2 sides (one on each end) and 3 sides along the length.

since the ends are still square, then the width equal the height and we can still call them s rather than w and h.

the area of each end is still s^2.

2 times the area of each end is therefore equal to 2*s^2.

the area of each side along the length will be s*l.

we have 3 of them.

the total area of the sides along the length is therefore 3*s*l

the side along the length on the top of the aquarium is open and so its area is not counted for total area of glass.

total area of glass is the area of the 2 ends plus the area along the length for 3 of the 4 sides along the length..

that equals 2*s^2 + 3*s*l which equals 44 square feet.

formula becomes 44 = 2*s^2 + 3*s*l

we know that l = 6, so we substitute 6 for l to get:

44 = 2*s^2 + 3*s*6 which becomes 44 = 2*s^2 + 18*s

subtract 44 from both sides of this equation to get the standard form of a quadratic equation of:

2*s^2 + 18*s -44 = 0

divide both sides of this equation by 2 to get:

s^2 + 9*s - 22 = 0

factor this equation to get:

(s+11) * (s-2) = 0

solve for s to get s = -11 or s = 2

s can't be negative, so s has to be equal to 2.

that's your answer.

total surface area = 2*4 + 3*2*6 = 8 + 36 = 44 square inches.

a picture of your aquarium that might help is shown here: