Given the function $C(r)=3 r^4+12 r^3-36 r^2$:
Its $C$-intercept is □
Its $r$-intercepts are □
To find the C-intercept of the function \(C(r) = 3r^4 + 12r^3 - 36r^2\), we evaluate the function at \(r = 0\). Substituting \(r = 0\) into the expression, we get \(C(0) = 3(0)^4 + 12(0)^3 - 36(0)^2 = 0\). Therefore, the C-intercept is 0.
To find the r-intercepts, we set \(C(r) = 0\) and solve for \(r\):
\[3r^4 + 12r^3 - 36r^2 = 0\]
We can factor out the common term \(3r^2\):
\[3r^2(r^2 + 4r - 12) = 0\]
This gives us two main components to solve:
To solve the quadratic equation \(r^2 + 4r - 12 = 0\), we look for two numbers that multiply to -12 and add to 4. These numbers are 6 and -2. Thus, the equation factors as:
\[(r + 6)(r - 2) = 0\]
This gives us the roots \(r = -6\) and \(r = 2\). Combining all these, the r-intercepts are \(-6, 0,\) and \(2\).
We have determined that the C-intercept is 0 and the r-intercepts are \(-6\), \(0\), and \(2\). Thus, the C-intercept is \(\boxed{0}\) and the r-intercepts are \(\boxed{-6, 0, 2}\).
In physics, modeling the trajectory or energy output of a system often involves finding intercepts to identify critical points (like when a projectile hits the ground or when energy levels change). For example, if a machine's energy efficiency is modeled by \(E(t) = 3t^4 + 12t^3 - 36t^2\), finding the time intercepts (r-intercepts) would help determine the specific moments when the net energy output is zero, which is crucial for analyzing energy cycles or system operation windows.