The cross-section of a designer architectural arch is modelled by the curve shown in a coordinate plane, with the equation

16\sin^2 y + 3, \quad 0 \leqslant y \leqslant \frac{\pi}{2}

where $x\,$ and $y\,$ are spatial coordinates measured in decimetres. The point $P(k,π6)P\left(k, \frac{\pi}{6}\right)$ lies on the curve.

Verify that $k = 7$ .

(i) Find $dxdy\frac{dx}{dy}$ in terms of $y$ .

(ii) Hence show that $dydx=12(x−3)(19−x)\frac{dy}{dx} = \frac{1}{2\sqrt{(x-3)(19-x)}}$ .

The normal to the curve at $P$ intersects the $x$ -axis at the point $N$ .

Determine the exact area of triangle $O P N$ , where $O$ is the origin. Give your answer in the form $aπ+bπ2a\pi + b\pi^2$ where $a$ and $b$ are constants to be found.

The cross-section of a designer architectural arch is modelled by the curve shown in a coordinate plane, with the equation

16\sin^2 y + 3, \quad 0 \leqslant y \leqslant \frac{\pi}{2}

where $x\,$ and $y\,$ are spatial coordinates measured in decimetres. The point $P(k,π6)P\left(k, \frac{\pi}{6}\right)$ lies on the curve.

Verify that $k = 7$ .

(i) Find $dxdy\frac{dx}{dy}$ in terms of $y$ .

(ii) Hence show that $dydx=12(x−3)(19−x)\frac{dy}{dx} = \frac{1}{2\sqrt{(x-3)(19-x)}}$ .

The normal to the curve at $P$ intersects the $x$ -axis at the point $N$ .

Determine the exact area of triangle $O P N$ , where $O$ is the origin. Give your answer in the form $aπ+bπ2a\pi + b\pi^2$ where $a$ and $b$ are constants to be found.

Differentiation

Question 35

Markscheme

Differentiation

Question 35

Markscheme

Differentiation

Question 35

Differentiation Questions

Differentiation

Question 35

Differentiation Questions