The volume of a spherical cancerous tumor is given by the following equation.

V(r) = (4/3)pi r^3
If the radius of a tumor is estimated at 1.4 cm, with a maximum error in measurement of 0.003 cm, determine the error that might occur when the volume of the tumor is calculated.
cm3

Well, it seems like our tumor likes to add a little uncertainty to the equation, doesn't it? Calculating the error in the volume of the tumor is as fun as juggling chainsaws! Let's get started.

First, let's find the maximum possible volume by plugging in the maximum value of the radius into our equation:

V(max) = (4/3) * pi * (1.4 + 0.003)^3

Now, let's calculate the minimum possible volume by plugging in the minimum value of the radius:

V(min) = (4/3) * pi * (1.4 - 0.003)^3

Subtracting the minimum volume from the maximum volume, we can determine the error that might occur:

Error = V(max) - V(min)

By performing some algebraic magic and simplifying, we get:

Error = (4/3) * pi * [(1.4 + 0.003)^3 - (1.4 - 0.003)^3]

Now that we have the expression, let's do some math and find out the error that might occur. Just make sure you keep an eye out for any sneaky clowns trying to mess with your calculations!

To determine the error that might occur when calculating the volume of the tumor, we can use the formula for error propagation.

The formula for error propagation when a value is raised to a power is:
δz = |dz/dr| * Δr

Where:
δz is the error in the result (volume, in this case)
dz/dr is the derivative of the function with respect to r
Δr is the maximum error in the measurement of r

Given the volume formula V(r) = (4/3)πr^3, we can differentiate it with respect to r:
dV/dr = 4πr^2

Now, we can substitute the values into the formula for error propagation:
δz = |4πr^2| * Δr

Let's substitute the values we have:
r = 1.4 cm
Δr = 0.003 cm

δz = |4π(1.4)^2| * 0.003 cm
= |4π * 1.96| * 0.003 cm
= 0.024608... π * 0.003 cm
≈ 0.073 cm^3 (rounded to three decimal places)

Therefore, the error that might occur when calculating the volume of the tumor is approximately 0.073 cm^3.

v = 4/3 pi r^3

dv = 4pi r^2 dr
So, plug in your values:
r=1.4
dr = 0.003