An architect designs a flower bed for a garden. The flower bed is enclosed by the curve $C$ with e

Question

An architect designs a flower bed for a garden. The flower bed is enclosed by the curve $C$ with equation $\left( \frac{x+y}{2} \right)^2 + \left( \frac{y-x}{4} \right)^2 = 1$. The region in the first quadrant bounded by $C$, the $x$-axis and the lines $x = 0$ and $x = 0.5$, is to be planted with tulips. By finding the Maclaurin series expansion of $y$ up to and including the term in $x^2$, where $y > 0$, find an approximation for the area to be planted with tulips.[7] The curve $C$ is obtained by rotating an ellipse $S$ about the origin in the $x-y$ plane. It is known that the curve $C$ can be expressed by the following parametric equations $$x = \cos t - 2\sin t, y = \cos t + 2\sin t,$$ where $-\pi < t \le \pi$. Let $L$ be the distance between the origin and a point on $C$. By using differentiation, determine the exact maximum and minimum values of $L$. You do not need to show that these values are maximum and minimum.[5] Write down a possible cartesian equation of $S$. Hence, determine the total area of the flower bed.[3]

Accepted Answer

(a)0.812m$^2$ (b)$2\sqrt 2,\sqrt 2$ (c)12.6m$^3$

2025 TMJC P1 Q11

2025 TMJC P1 Q11