[It is given that the volume and surface area of a sphere with radius $r$ is \(\frac{4}{3}\pi r^3\

Question

[It is given that the volume and surface area of a sphere with radius $r$ is $\frac{4}{3}\pi r^3$ and $4\pi r^2$ respectively.] Jan owns a small company that produces and sells a particular type of container. Figure $1$ shows the cross section of the container. It consists of a cylinder of radius $x$ cm and height $h$ cm between the open top and the circular base. A hemisphere of radius $x$ cm is removed from the bottom of the container. The volume of the container is fixed at $375$ cm$^3$. Show that $h = \frac{375}{\pi x^2} + \frac{2}{3}x$.[1] Hence show that the total surface area, $A$ cm$^2$, of material used for the container is modelled by $A = \frac{750}{x} + \frac{10}{3}\pi x^2$.[2] Use a non-calculator method to find the value of $x$ that minimizes $A$. Hence, find the minimum value of $A$ correct to $3$ significant figures.[4] The cost of producing $n$ containers using the least amount of material is $(\frac{1}{3}n^2 + 14n + 12)$. Jan sells each container for $\mathit{P}$ per container. Show that $\mathit{P} = 8 - \frac{1}{3}n - \frac{12}{n}$.[1] Sketch the graph of $\mathit{P}$ against $n$, stating the coordinates of the turning point. State the meaning of this turning point in the context of the question.[3] Hence find the set of possible values of $n$ for which Jan makes a profit of at least $\$2$ per container.[3]

Accepted Answer

(c)$x=\sqrt[3]{\frac{225}{2\pi}}^2$, min A = 341 (f)$ {{n \in \mathbb {Z} : 3 \leq n \leq 15}}$

2024 NYJC H1 Q5

2024 NYJC H1 Q5