A sphere of copper is subjected to 89.7 MPa of pressure. The bulk modulus is 130 GPa. a) By what fraction does the volume of the sphere change? b) By what fraction does the radius of the sphere change?

for a) I have dV/V = 0.00069 - this is correct.

for b) the area I have 4/3 x pi x r^3.

I did the following:

Try 1 - 0.00069 = 4/3 x pi x dr^3 - wrong

Try 2 - then I took the cube root of 0.00069 and got 0.088 - wrong.

Try 3 - I used the same factor of 0.00069 since the radius is all that is changing in the equation - wrong.

Try 4 - I tried factoring dr = rfinal - rinitial and I'm left with something I don't know how to deal with.

HELP! Is it 4 and I need to use some algebra technique I can't remember how to do???

To find the change in the radius of the sphere, we can start by using the formula for the fractional change in volume:

dV/V = -B dP

Where dV is the change in volume, V is the original volume, B is the bulk modulus, and dP is the change in pressure.

From the given information:
Pressure (P) = 89.7 MPa = 89.7 x 10^6 Pa
Bulk modulus (B) = 130 GPa = 130 x 10^9 Pa

Let's solve for dV/V first:
dV/V = -B dP
dV/V = - (130 x 10^9 Pa) * (89.7 x 10^6 Pa)^-1

To make the calculations easier, let's convert the pressure values to the same unit:
dV/V = - (130 x 10^9) * (89.7 x 10^6)^-1

Using a calculator, we find:
dV/V = -1.45 x 10^-3

Now, let's find the change in the radius (dr) using the volume formula of a sphere:

V = (4/3) π r^3

Taking the derivative of this equation, we get:
dV = 4π r^2 dr

Dividing both sides by V = (4/3) π r^3, we have:
dV/V = (4π r^2 dr) / ((4/3) π r^3)
dV/V = (3 dr) / r

We already have the value of dV/V as -1.45 x 10^-3. Substituting this value, we can solve for dr:

-1.45 x 10^-3 = (3 dr) / r

To isolate dr, we multiply both sides by r and divide by 3:
dr = (-1.45 x 10^-3 r) / 3

Now we can calculate the change in the radius using the given radius of the sphere. Let's assume the initial radius is r_initial:

dr = (-1.45 x 10^-3 r_initial) / 3

With this equation, you can plug in the value of r_initial and solve for dr, which will give you the fractional change in the radius of the sphere.