The volume of a sphere with radius r can be written as the function V(r)=4/3πr^3 . Determine the inverse of this function and use it to find the radius of a sphere with a volume of 50 cm^3 . Round to two decimal places.
To find the inverse of the function V(r) = (4/3)πr^3, we need to switch the roles of the input and output variables. Let's call the inverse function R(V), where V is the volume and R is the radius.
To find the inverse function, we can start by setting V(r) = V and solving for r.
V(r) = (4/3)πr^3
V = (4/3)πr^3
3V/4π = r^3
Taking the cube root of both sides, we get:
r = (3V/4π)^(1/3)
Now we can substitute 50 cm^3 for V to find the radius R:
R = (3(50)/4π)^(1/3)
R ≈ 2.83 cm
Therefore, the radius of a sphere with a volume of 50 cm^3 is approximately 2.83 cm.
is that correct?
Apologies for the error. Here's the correct calculation:
To find the inverse of the function V(r) = (4/3)πr^3, we need to switch the roles of the input and output variables. Let's call the inverse function R(V), where V is the volume and R is the radius.
To find the inverse function, we can start by setting V(r) = V and solving for r.
V(r) = (4/3)πr^3
V = (4/3)πr^3
To isolate r, we divide both sides by (4/3)π and then take the cube root:
r^3 = V / ((4/3)π)
r = (V / ((4/3)π))^(1/3)
Now we can substitute 50 cm^3 for V to find the radius R:
R = (50 / ((4/3)π))^(1/3)
Calculating this, we get:
R ≈ 2.73 cm
Therefore, the radius of a sphere with a volume of 50 cm^3 is approximately 2.73 cm.
To find the inverse of the function V(r) = 4/3πr^3, we can interchange the role of r and V and solve for r.
Let's start by swapping r and V in the original equation:
r = (4/3πV)^(1/3)
Now, we can solve for r by plugging in V = 50 cm^3 into the equation:
r = (4/3π(50))^(1/3)
r = (200π/3)^(1/3)
To find the numerical value of r, substitute π ≈ 3.14:
r = (200(3.14)/3)^(1/3)
r = (628/3)^(1/3)
Rounding to two decimal places, the radius of the sphere with a volume of 50 cm^3 is approximately 3.64 cm.
To find the inverse of the function V(r)=4/3πr^3, we need to interchange the roles of r and V(r) and solve for r.
So, let's start by rewriting the function as an equation:
V(r) = 4/3πr^3
Now, interchange r and V(r):
r = 4/3πV(r)^3
To find the radius of a sphere with a volume of 50 cm^3, substitute V(r) = 50 into the equation:
r = 4/3π(50)^3 = 4/3π(125000) = 523598.78 cm^3 (rounded to two decimal places)
Therefore, the radius of the sphere with a volume of 50 cm^3 is approximately 523598.78 cm^3 (rounded to two decimal places).