Let's call the volume of oil in the tank "o" and the volume of water "w". We know that:
o + w = total volume of tank - 9650
We also know that the ratio of oil to water is 3:4, which means that:
o/w = 3/4
We can use this information to create a system of equations. First, we can solve for o in the second equation:
o = (3/4)w
Next, we can substitute this expression for o into the first equation:
(3/4)w + w = total volume of tank - 9650
Simplifying this equation, we get:
(7/4)w = total volume of tank - 9650
Now we have an equation that relates the volume of water in the tank to the total volume and the amount of unfilled space. But we still need another equation to solve for the unknowns.
We're told that the tank contains 420ml more water than oil. Since we know that o = (3/4)w, we can write:
w = o + 420
Substituting the expression for o from above, we get:
w = (3/4)w + 420
Solving for w, we get:
w = 1680
Now we can use this value of w to find o:
o = (3/4)w
o = (3/4) * 1680
o = 1260
So the tank contains 1260ml of oil and 1680ml of water. To find the base area of the tank, we need to know the height of the tank. Let's call this "h". We can use the formula:
volume = base area * height
We know that the total volume of the tank is:
total volume = o + w + 9650
total volume = 1260 + 1680 + 9650
total volume = 12590
So we can write:
12590 = base area * h
We don't know the value of h, but we do know that the ratio of oil to water in the tank is 3:4, which means that the height of the oil layer is:
h_oil = (3/7) * h
And the height of the water layer is:
h_water = (4/7) * h
We also know that the base area of the tank is the same for both the oil and water layers. So we can write:
base area * h_oil = 1260
base area * h_water = 1680
Substituting the expressions for h_oil and h_water into these equations, we get:
base area * (3/7)h = 1260
base area * (4/7)h = 1680
Dividing these equations, we get:
(3/4) * base area = 1260/1680
(3/4) * base area = 3/4
base area = 3/4 * (1680/1260)
base area = 1
So the base area of the tank is 1 cm^2.