The problem I have is;
An app developer is creating a 3D snake game using a cylindrical model. Each time the snake grows, which occurs after it eats enough mice to level up and have a growth spurt, it will increase in volume by a certain constant percentage, and in such a manner that the ratio of its length to its cross sectional area remains constant. Ie - L/A= constant (k)
I need help in determining a mathematical model for the length of the snake in terms of the level number
In order to determine the model, we must follow these steps:
Choose your own values (no. of units) for the initial (level 1) dimensions of your snake. State these clearly: length, radius, cross sectional area, volume.
Choose your own growth percentage for the volume.
a well-simplified, precise models are more favorable
Thank you so much, I appreciate any help :)
If the volume increases via a percentage $i$ every level $n$ and starts with a volume $v_0$ (when the snake is level $1$), then it gets a simple exponential model: $$v = v_0(i + 1)^{n-1}$$ Of course, the volume of a cylinder is $v = A\cdot L$: $$AL = v_0(i + 1)^{n-1}$$ Use your constant $k$ such that $A = L / k$: $$\frac{L^2}{k} = v_0(i + 1)^{n-1}$$ Of course, solving that for L: $$L = \sqrt{v_0k}(i + 1)^{\frac{n-1}{2}}$$ Another way of handling this is to find $v$ for any given level, and realize that: $$L = \sqrt{kv}$$