4.
Figure 1
\includegraphics[max width=\textwidth, alt={}, center]{a1b078fe-96e3-4d62-bf0d-415294ba022f-4_588_1008_242_566}
Figure 1 shows a cross-section \(R\) of a dam. The line \(A C\) is the vertical face of the dam, \(A B\) is the horizontal base and the curve \(B C\) is the profile. Taking \(x\) and \(y\) to be the horizontal and vertical axes, then \(A , B\) and \(C\) have coordinates \(( 0,0 ) , \left( 3 \pi ^ { 2 } , 0 \right)\) and \(( 0,30 )\) respectively. The area of the cross-section is to be calculated.
Initially the profile \(B C\) is approximated by a straight line.
- Find an estimate for the area of the cross-section \(R\) using this approximation.
(1)
The profile \(B C\) is actually described by the parametric equations.
$$x = 16 t ^ { 2 } - \pi ^ { 2 } , \quad y = 30 \sin 2 t , \quad \frac { \pi } { 4 } \leq t \leq \frac { \pi } { 2 }$$ - Find the exact area of the cross-section \(R\).
(7) - Calculate the percentage error in the estimate of the area of the cross-section \(R\) that you found in part (a).
(2)