Updating the suspension of a car

[KorvusKoraks]

Homework Statement

A car driver updates the springs of a car by replacing the old springs with stiffer ones. The old springs give an amount of 8 cm when under the car, and their length when not under the car is 29 cm.
The spring constant of the new springs is 30 % greater than that of the old springs.
How long do the new springs have to be in order the car to be located the same distance from the ground with the new springs as with the old springs?

k = the spring constant of the old springs
x = the length of the new spring(s) when under the car
x_norm = the normal ("unloaded") length of the new spring(s)
x_req = the required length of the new spring(s)

Homework Equations

Hooke's law: F = kx

The Attempt at a Solution

I have no idea. I have trouble formulating the equation.

k * (29 cm - 8 cm) = 1.3k * x
k * 21 cm = 1.3k * x || / 1.3k
k * 21 cm / 1.3k = x
x ≈ 16.153 cm

29 cm / 21 cm = x_norm / 16.153 cm
1.381 = x_norm / 16.153 cm || * 16.153 cm
x_norm ≈ 22.307 cm

22.307 cm / 16.153 cm = x_req / 21 cm
1.381 = x_req / 21 cm || * 21 cm
x_req = 29.001 cm ? It's the same as the original!

 

[BvU]

Well, this looks like a beauty contest where I can vote ...

You can discard the first attempt, because you know for sure that the answer must be more than 21 cm, right ?
Ditto number three, for expectation is less than 29 cm.

Physics isn't a voting contest, though. So the one left standing doesn't have to be the winner.

Can you see what's wrong with your first shot ? What about the x at the end ?
And what about the 8 on the left ? They say 'gives 8 cm' they don't mean it ends up being 8 cm long ...

 

[KorvusKoraks]

Well, this looks like a beauty contest where I can vote ...

You can discard the first attempt, because you know for sure that the answer must be more than 21 cm, right ?
Ditto number three, for expectation is less than 29 cm.

Physics isn't a voting contest, though. So the one left standing doesn't have to be the winner.

Can you see what's wrong with your first shot ? What about the x at the end ?
And what about the 8 on the left ? They say 'gives 8 cm' they don't mean it ends up being 8 cm long ...

I don't know... at least in the last (third) group of equations I now think I was calculating the length of a spring of a different length with the same spring constant...

 

[haruspex]

I don't know... at least in the last (third) group of equations I now think I was calculating the length of a spring of a different length with the same spring constant...

None of the attempts are correct. You need to have a clear idea what each step of the calculation is supposed to be calculating.
In applying equations, you need to be clear what roles the variables play and how those match up to the given data.
In all of the numbers given in the question, which is suitable for use as the 'x' in Hooke's F=kx?