When a rectangular vat that is 3 feet deep is filled to 2/3 of its cap

IMO A


volume = area of base *height

let the area be A

then volume = A*3

now this is filled to 2/3 of its capacity

so volume of water = 2/3*(A*3)

now the volume of water = 60(15/2)= 8 cubic feet

so 2/3*(A*3)= 8

So A = 8/2 = 4

award kudos if helpful

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[quote="Bunuel"]When a rectangular vat that is 3 feet deep is filled to 2/3 of its capacity, it contains 60 gallons of water. If \(7\frac{1}{2}\) gallons of water occupies 1 cubic foot of space, what is the area, in square feet, of the base of the vat?

A.  4
B.  8
C.  12
D. 150
E. 225



Here is my take ....

The question is asking,what is the area of the base of the vat= a*b (a=length; b=breath)

Let's say total capacity of the vat= x
2x/3= 60 gallon
x= 90 gallon (total capacity)

If 15/2g of water occupies 1 cubic foot of space
then, 90g of water will occupy= 90*2/15 = 12 cubic foot of space in the vat (Total volume)

We can also write:
a*b*c= 12 cubic foot (c=depth; C=3 feet)
a*b*3= 12
a*b= 4 (which is length * breath= area of the base)

The answer is A.

total fill in feet ; 3*2/3 = 2feet
and 60 gallons in feet have 60*2/15 = 8 ft^3 of water
so base has 8ft^3/2feet = 4ft^2
IMO A

Let's assume V as the volume of our vat. 2/3*X=60 so we can find V=60*3/2=90.
Volume=width*length*height, width*length= area of our base and width*length*3=90 => width*length=90/3=30
Our answer is 30*2/15=4, which is A.

Hello! Snezanelle,

Where does the 15 comes from?

Kind regards!

Unitary method can be applied to this :

If, 2/3 is 60 gallons then 3/3(1) = 90 gallons
As 1 cubic feet = 15/2 ,
12 cubic feet = 15/2*12 (multiplying both sides by 12)
12 cubic = 90 gallons (total volume)

total area = 3*X = 12
Therefore, x = 4
A

Let x be the area sq. foot of base. Volume of the water= x*2 ....(Height of tank filled id 2 ft.)

Now form the proportion 15/2 : 1 :: 60 : 2x
x=4.

Hi,
15/2 = 7*(1/2), which is mentioned in the problem.

Since the vat is 3 feet deep and water is filled to 2/3 of its capacity, the water is ⅔ x 3 = 2 feet deep. In cubic feet, the water is 60/7.5 = 8 cubic feet in volume. Since the water is 2 feet deep, the base of the vat must be 8/2 = 4 square feet.

Answer: A
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60 gallons = (60/7.5)= 8 feet
Total Volume= πr^2h
So, πr^2×(3)×(2/3)=8
πr^2=area of base= 4 sq feet.

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if 2/3 capacity = 60 gallons then 2x/3 = 60 and the capacity = 90 gallons

volume = length*width*height
height here = depth and we know the volume = 90
l*w*3=90
lw=30 gallons

Is there any legitimate way of using the volume to determine the base?

Hey Scott,

I'm a bit unsure how you moved from cubic units to square units here

We are using the formula volume = base area x height, which can be applied to not just cubes or rectangular solids, but to any “right solid”, such as a cylinder or prism. In this formula, the volume is measured in cubic units (cubic feet in our example) and the base area is measured in square units (square feet in our example). All we did to find the base area was to divide the volume by height, which will give the result in square units. The only thing to be careful about is to have all the units agree; in other words, you can’t divide cubic feet by meters and expect any reasonable answer.
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When a rectangular vat that is 3 feet deep is filled to 2/3 of its capacity, it contains 60 gallons of water. If 712712 gallons of water occupies 1 cubic foot of space, what is the area, in square feet, of the base of the vat?


Volume = Base * Height

The height of the water is 2, so base *2 =60, base=30, can someone please explain what did I missed?

We are given 7.5 gallons = 1 cu. ft.

Therefore 60 gallons = (1 x 60)/7.5 = 8 cu. ft.

Now,

2\3 rd volume of the vat = 2/3 (L x B x H)

So, 2\3 x L x B x 3 = 8

2 x L x B = 8

or, L x B = 4 sq. ft.

i.e. option A.

Let Length = L and Width = W

LW = ?

2/3 * 3LW = 60 gallons
LW = 30 gallons/feet
LW = 30 gallons/feet * (1 feet^3/7.5 gallons)
LW = 4 feet^2

ANSWER: A

Well, From the question we know that the VAT is filled 2/3rd capacity with the height of VAT being 3 feet and the number of gallons are 60.

Try not confusing the gallons of water for the volume of the VAT which is in cubic feet (they both have a relation but we have not yet discussed that)

So if 2/3rd the capacity holds 60 gallons, it would mean the total capacity would be 90 gallons ( simply considering 1/3 is 30 gallons, 2/3rd is 60 and 3/3 is 90 gallons)

Also we know that 7.5 (7 1/2) is 1 cubic feet, hence the total volume of the VAT is 90 gallons/ 7.5 (gallons per Cubic feet) = 12 cubic feet (Cancelling gallons)

Now the volume of the VAT is Length * breadth * Height (3 ft) = 12 cubic feet
Length * breadth = 4 (dividing by 3 on both sides)

Answer is 4 (A)

Hope this helps a little

Such a nice explanation thankuuu so much

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When a rectangular vat that is 3 feet deep is filled to 2/3 of its capacity, it contains 60 gallons of water. If \(7\frac{1}{2}\) gallons of water occupies 1 cubic foot of space, what is the area, in square feet, of the base of the vat?

My approach:

A rectangular vat filled to \(\frac{2}{3}\) of its capacity contains 60 gallons of water. Therefore the rectangular vat filled to its capacity contains 90 gallons of water.

We're told 7.5 gallons of water occupies 1 cubic foot of space. As we know the rectangular vat can hold 90 gallons of water, we can conclude the vat takes up 12 cubic foot of space.

We know volume is length * width * height.

\(length * width * 3 = 12\)
\(length * width = 4\\
\)

Answer is A.
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