Math topic中文

Local maxima minima problems

Local maxima and minima problems (optimisation problems).

Practice questions

1[The volume of a cone of base radius $r$ and height $h$ is given by $V = \frac{1}{3}\pi r^2 h$ .] A manufacturer makes a…
2A sector of angle $\theta$ radians and radius $4\pi\text{ cm}$ is used to form the curved surface of a right circular co…
3As part of an Art Project, a student designed a lantern in the shape of an octahedron as shown in the diagram. The frame…
4The following diagram shows the dimensions of a trapezoidal prism with fixed volume $4k\sqrt{3}$ units $^{3}$ , with varia…
5[The volume of a right circular cone of radius $r$ and height $h$ is $\frac{1}{3}\pi r^2 h$ .] A student makes a printer…
6Figure 1 shows an open cylindrical tank. Figure 2 shows the cross-sectional view of the tank. The external radius of the…
7A Hwa Chong student taking part in Project Day needs to design a rectangular poster to showcase his project. According t…
8The diagram shows an ellipse, centred at the origin with semi-minor axis $a$ and semi-major axis $2a$ , where $a > 0$ . Th…
9[The surface area and volume of a sphere are given by $4\pi r^2$ and $\dfrac{4}{3}\pi r^3$ respectively.] An aquarist wh…
10Fig. 1 shows a piece of card, $ABCD$ , in the form of a rhombus of sides $a$ cm. The card is made up of 2 identical equil…
11[Cambridge/J78/II/6] An isosceles triangle $ABC$ with $AB = AC$ is inscribed in a fixed circle, centre $O$ , whose ra…
12A Dunman High alumnus plans to donate money to build two koi ponds in the middle of Zheng Xin Yuan. The ponds are in the…
13In the diagram, point $Q$ is $400\text{ m}$ vertically above point $A$ . Initially, a balloon is at point $Q$ and moves h…
14The diagram below shows a parabola which cuts the $x$ -axis at $(-2a, 0)$ , $(2a, 0)$ and has a maximum point at $(0, 4a^2)$ …
15The diagram shows a shot put being projected with a velocity $v\text{ ms}^{-1}$ from the point $O$ at an angle $\theta$ …
16The following diagram shows the graphs of two differentiable functions $y=f(x)$ and $y=g(x)$ . At $x = c$ , the distance b…
17In a particular building, there is an $L$ -shaped corridor (see diagram below). Corridor 1 has a fixed width of 3 m and C…
18The cross section PQRS of a water passageway of depth d is as shown in the figure below. It is given that $PQ+QR+RS=2a$ …
19Sketch the curve with equation $\frac{x^2}{400} + \frac{(y-10)^2}{900} = 1.$ [There is no need to indicate the x-inter…
20It is given that $f(x) = \begin{cases} \tfrac{1}{2}x^2, & \text{for } 1 \le x < 2, \\ \tfrac{1}{4}(-3x+14), & \text{for } 2 \le x < 4, \end{cases}$ …
21Fig. 1 shows the net of a triangular prism cut from a square cardboard of side length 20 cm. The net consists of one rec…
22A farmer has a plot of the land, $OAB$ , with adjacent sides $OA$ and $OB$ . With reference to $O$ as the origin, $OA$ and…
23In the isosceles triangle $ABC$ , $AC = BC$ , $AB = 20\text{ cm}$ and $\angle BAC = 30^\circ$ . A rectangle $PQRS$ is inscr…
24[A sphere of radius $r$ has surface area $4\pi r^2$ and volume $\frac{4}{3}\pi r^3$ .] A water fountain is to be construc…
25The diagram shows a parabola formed by the intersection of a right circular cone and a plane parallel to the slanted edg…
26Projectile motion is the motion of a launched object, subject to only the constant acceleration of gravity, $g\text{ ms}^{-2}$ …
27The following diagram shows the dimensions of a trapezoidal prism with fixed volume $4k\sqrt{3}$ units $^3$ , with variabl…
28The plane $\pi_1$ has equation $\mathbf{r}\cdot\begin{pmatrix}2 \\ 5 \\ 1\end{pmatrix}=11.$ The plane $\pi_2$ passes t…
29Fig. 1 shows a net of a hexagonal pyramid folded from a star-shaped cardboard of equal edge length $a$ cm. The net consi…
30A piece of vanguard sheet, $ABCDEF$ , is in the form of a regular hexagon of side $a\text{ cm}$ . A kite shape is cut out…
31MI PU2 Promo 9758/2019/02/Q1 Town A, Town B and Town C are located in Wakandi Country. The distance between Town A and T…
32A company manufactures a closed container made of glass as shown in the figure below. The closed container, of negligibl…
33Fig. 1 shows the net of a right regular hexagon-based prism cut from a rectangular cardboard of one side length 30 cm. T…
34Figure 1 shows the net of an open square-based pyramid cut from a square of cardboard of fixed side length d cm. The net…
35A popular toy company is designing a new water play feature for children. The toy consists of a cylindrical water contai…
36[It is given that the volume of a pyramid is $\frac{1}{3} \times (\text{base area}) \times (\text{height})$ .] A right py…
37A lighting consultant intends to construct a giant light structure for Christmas that is octahedral in shape. The edges…
38已知函数 $f(x)= \dfrac {ax}{e^{x}}$ 在 $x=0$ 处的切线方程为 $y=x$ ． (1)求 $a$ 的值； (2)若对任意的 $x∈(0,2)$ ，都有 $\dfrac {1}{f(x)} > k+2x-x^{2}$ …
39已知函数 $f(x)=m\ln x$ ． (1)讨论函数 $F(x)=f(x)+ \dfrac {1}{x}-1$ 的单调性； (2)定义：“对于在区域 $D$ 上有定义的函数 $y=f(x)$ 和 $y=g(x)$ ，若满足 $f(x)\leqslant g(x)$ …
40The closed container shown below consists of a cylinder of height $h\text{ cm}$ fixed to a cone. The cylinder and cone h…
41[It is given that a cone with radius $r$ , height $h$ and slant height $l$ has curved surface area $\pi r l$ .] A drinking…
42A metal tank with an open top is in the form of a right circular cylinder. The total cost of metal used to make the tank…
43An ellipse has equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ , where $a$ and $b$ are positive real constants. Show tha…
44In a futuristic movie, a hemispherical force field dome (an open hemisphere) is created to protect a cylindrical silo un…
45A cone-shaped cup is made of paper of negligible thickness to hold $20\pi\text{ cm}^3$ of liquid. The open inverted cone…
46An offshore oil well is located in the ocean at point $W$ , which is 8 km from point $A$ on land as shown in the figure a…
47A curve $C$ has parametric equations $x = \frac{\lambda}{1 + \lambda^3} \quad \text{and} \quad y = \frac{\lambda^2}{1 + \lambda^3}, \quad \text{for } \lambda > 0.$ …
48The diagram shows a shot put being projected with a velocity $v\ \mathrm{ms}^{-1}$ from the point $O$ at an angle $\theta$ …
49[ The volume of a cone of base radius $r$ and height $h$ is given by $V = \frac{1}{3}\pi r^2 h$ . ] A disposable cup for…
50Fig. 1 shows a net of a hexagonal pyramid folded from a star-shaped cardboard of equal edge length $a\text{ cm}$ . The ne…
51Fig. 1 shows a piece of card, $ABCD$ , in the form of a rhombus of sides $a$ cm. The card is made up of 2 identical equil…
52A circular sector has radius $r$ cm and angle $\theta$ radians. This sector has area $A\text{ cm}^2$ and fixed perimeter…
53Sketch the curve with equation $\displaystyle\frac{x^2}{400}+\frac{(y-10)^2}{900}=1.$ [There is no need to indicate th…
54MI PU2 Promo 9758/2019/02/Q1 Town A, Town B and Town C are located in Wakandi Country. The distance between Town A and T…
55In a futuristic movie, a hemispherical force field dome (an open hemisphere) is created to protect a cylindrical silo un…
56Fig. 1 shows the net of a right regular hexagon-based prism cut from a rectangular cardboard of one side length 30 cm. T…
57[It is given that the area of a sector of a circle with radius $r$ and angle $\theta$ is given by $\frac{1}{2}r^2\theta$ …
58A Dunman High alumnus plans to donate money to build two koi ponds in the middle of Zheng Xin Yuan. The ponds are in the…
59Candy is being stored in a closed container in the form of a regular hexagonal prism, with sides $x\text{ cm}$ and heigh…
60A closed right circular cone of base radius $r$ and height $h$ has a fixed total surface area $k\pi$ units $^2$ , where $k$ …