$7\mathrm{.}12.$ A $915-$mm$-$diameter concrete culvert is $15m$ long and is laid on a horizontal slope, and entrance conditions are such that the entrance$-$loss coefficient is $0\mathrm{.}2.$ The design flow rate and tailwater depth are $1\mathrm{.}5\frac{m^3}{s}$ and $0\mathrm{.}4m,$ respectively. If Type $2$ flow conditions are assumed, estimate and compare the headwater depth for the following cases: $($a$)$ the culvert flows full for the entire length, and $($b$)$ the culvert flows full for only part of the length, as shown in Figure $7\mathrm{.}4.$