The tank is completely filled with water and then drains completely into the pool. The volume of the cylindrical tank is V1=36r12, where r1 is the radius of the tank in feet.
The volume of the neighborhood pool is V2=12(43r23), where r2 is the radius of the pool.
Since V1= V2, write an equation that shows r1 as a function of r2.
[Hint: set them equal and solve for r1.]
Write an equation that shows r2 as a function of r1.
[Hint: set them equal and solve for r2.]
If the radius of the pool is 15 feet, what is the radius of the tank?
r1 = (12(43r23))/(36r12)
r2 = (36r12)/(12(43r23))
If the radius of the pool is 15 feet, then the radius of the tank is 10 feet.
To find the equation that shows r1 as a function of r2, we set V1 equal to V2 and solve for r1:
V1 = V2
36r1^2 = 12(4/3πr2^3)
To find the equation that shows r2 as a function of r1, we set V1 equal to V2 and solve for r2:
V1 = V2
36r1^2 = 12(4/3πr2^3)
If the radius of the pool is 15 feet, we can substitute r2 with 15 in the equation and solve for r1:
36r1^2 = 12(4/3π(15)^3)
36r1^2 = 12(4/3π(3375))
36r1^2 = 12(4500π)
36r1^2 = 54000π
r1^2 = 1500π
r1 ≈ 20.346 feet
Therefore, the radius of the tank is approximately 20.346 feet.
To solve this problem, let's first set V1 equal to V2 and write an equation that shows r1 as a function of r2.
V1 = V2
Substituting the given values:
36r1^2 = 12*(4/3)*π*r2^3
We can simplify the equation:
36r1^2 = 16πr2^3
Now, let's solve for r1.
Divide both sides of the equation by 36:
r1^2 = (16πr2^3) / 36
Take the square root of both sides:
r1 = sqrt((16πr2^3) / 36)
This equation shows r1 as a function of r2.
Next, let's write an equation that shows r2 as a function of r1.
V1 = V2
Substituting the given values:
36r1^2 = 12*(4/3)*π*r2^3
We can simplify the equation:
36r1^2 = 16πr2^3
Solving for r2:
r2^3 = (36r1^2) / (16π)
Take the cube root of both sides:
r2 = (cube root of ((36r1^2) / (16π)))
This equation shows r2 as a function of r1.
Finally, if the radius of the pool is given as r2 = 15 feet, we can substitute this value into the equation that shows r2 as a function of r1:
r2 = (cube root of ((36r1^2) / (16π)))
15 = (cube root of ((36r1^2) / (16π)))
Cubing both sides of the equation:
15^3 = (36r1^2) / (16π)
Simplifying:
3375 = (9r1^2) / (4π)
Multiply both sides of the equation by (4π):
3375(4π) = 9r1^2
Divide both sides of the equation by 9:
r1^2 = (3375(4π)) / 9
Taking the square root of both sides:
r1 = sqrt((3375(4π)) / 9)
Simplifying further:
r1 = sqrt(1500π)
Therefore, the radius of the tank is sqrt(1500π) feet.