The concentration of a bioactive compound, σ(z)\sigma(z)σ(z), in mg/L, at a depth zzz meters below the surface of a lake is modeled by the function:
σ(z)=z4−z3−2z2+z−14z2−z−6z>3 \sigma(z) = \frac{z^4 - z^3 - 2z^2 + z - 14}{z^2 - z - 6} \quad z > 3 σ(z)=z2−z−6z4−z3−2z2+z−14z>3Given that
σ(z)≡z2+P+Qz−3z>3 \sigma(z) \equiv z^2 + P + \frac{Q}{z - 3} \quad z > 3 σ(z)≡z2+P+z−3Qz>3find the value of the constant PPP and show that Q=5Q = 5Q=5.
Find the equation of the tangent to the concentration curve at the point where z=4z = 4z=4. Give your answer in the form σ=mz+c\sigma = mz + cσ=mz+c, where mmm and ccc are constants to be found.
A researcher calculates the total mass potential between depths z=4z = 4z=4 and z=5z = 5z=5, which is represented by the area RRR bounded by the curve σ(z)\sigma(z)σ(z), the zzz-axis, and the vertical lines z=4z = 4z=4 and z=5z = 5z=5. Calculate the exact value of this area, writing your answer in the form a+bln2a + b \ln 2a+bln2, where aaa and bbb are constants to be found.