If a spherical tanl 4m in diameter can be fillef with a liquid for $900, find the cost to fill a tank 12m in diameter. The cost to fill the 12m tank is $__.
To find the cost to fill a tank 12m in diameter, we first need to understand the relationship between the volume of the tanks and their diameters.
The volume of a sphere is given by the formula:
V = (4/3)πr^3
Where V is the volume of the sphere and r is the radius of the sphere.
Given that the first tank has a diameter of 4m, the radius is half of that: r = 4m / 2 = 2m.
We can plug this radius into the volume formula to find the volume of the first tank:
V1 = (4/3)π(2m)^3 = (4/3)π(8m^3) = (32/3)πm^3
Now, we need to find the volume of the second tank. The diameter of the second tank is 12m, so the radius is half of that: r = 12m / 2 = 6m.
We can plug this radius into the volume formula to find the volume of the second tank:
V2 = (4/3)π(6m)^3 = (4/3)π(216m^3) = (288/3)πm^3 = 96πm^3
Now that we have the volumes of both tanks, we can find the cost ratio. We know that filling the first tank costs $900, so the cost per unit volume is given by:
Cost per unit volume = $900 / V1
We can now calculate the cost to fill the second tank:
Cost to fill second tank = Cost per unit volume * V2
Substituting the values we know:
Cost to fill second tank = ($900 / V1) * V2
Now let's calculate it:
Cost to fill second tank = ($900 / (32/3)πm^3) * (96πm^3)
= (900 * 3 * 96) / 32
= 2700 * 3
= $8100
Therefore, the cost to fill a tank 12m in diameter is $8100.