IB Mathematics Analysis and Approaches Topic SL…

A hot-air balloon $P$ is anchored by two ropes to points $A$ and $B$ on level ground. The distance $AB$ is 75 m. The balloon is vertically above $P_0$ , and in the horizontal triangle .

(a) a fully-labelled 3-D diagram showing $A,B,P_0,P$ and all the given angles.

(b) Let $AP_0=x$ m and $BP_0=y$ m. Show that .

(e) Hence, find this maximum volume, giving your answer in the form $q\pi \sqrt{\frac{5}{6}}$ , where .

(e) Hence, find this maximum volume, giving your answer in the form $q\pi\sqrt{\frac{2}{3}}$ , where .

(d) Find the height $h$ of the balloon, to the nearest metre.

(e) Now suppose instead that $P_0$ moves horizontally on a circle of radius $75\text{ m}$ centered at $A$ . Find the central angle (radians) and the length of the arc (metres) when the bearing of $P_0$ from $A$ changes from $045^\circ$ to $115^\circ$ .

(f) Using the original position of the balloon, let $T_A=k|AP|$ and $T_B=k|BP|$ , where $k>0$ . Find $\dfrac{T_A}{T_B}$ .

Consider a cylinder of radius $5r$ and height $h$ . A smaller cylinder of radius $r$ is removed from the centre to form a hollow cylinder. This is shown in the following diagram. All lengths are measured in centimetres. The total surface area of the hollow cylinder, in cm $^2$ , is given by $S$ . The volume of the hollow cylinder, in cm $^3$ , is given by $V$ .

(a) Show that $S = 48\pi r^2 + 12\pi rh$ .

(b) The total surface area of the hollow cylinder is $480\pi$ cm $^2$ . Show that $V = 960\pi r - 96\pi r^3$ .

(d) The hollow cylinder has its maximum volume when $r = p \sqrt{\frac{5}{6}}$ , where $p \in \mathbb{Z}^+$ . Find the value of $p$ .

A mall is to be constructed with a concrete slab foundation. To fit this foundation, a rectangular section of earth measuring $10 \text{ m}$ by $20 \text{ m}$ is removed to a depth of $1.5 \text{ m}$ . The removed earth is used to create a hemispherical structure to reduce waste.

(a) Find the diameter of the hemispherical structure. Give your answer correct to $3$ significant figures.

(b) The architect decides to replace the hemispherical structure with a cylindrical structure of height $2 \text{ m}$ . The maximum possible diameter on the site is $12 \text{ m}$ . Find the radius of the cylindrical structure, correct to $3$ significant figures, and show that it is not suitable for the site.

Consider a solid cylinder with a radius of $3r$ and a height of $h$ . From the centre of this cylinder, a smaller cylinder of radius $r$ is removed to create a hollowed-out structure, as shown in the following diagram. All dimensions are in centimetres.

The total surface area of the resulting hollow cylinder is denoted by $S$ (in cm $^2$ ). The volume of the hollow cylinder is denoted by $V$ (in cm $^3$ ).

(a) Show that the expression for the total surface area is $S = 16\pi r^2 + 8\pi rh$ .

(b) Given that the total surface area of the hollow cylinder is fixed at $288\pi$ cm $^2$ , show that its volume can be expressed as $V = 288\pi r - 16\pi r^3$ .

(d) The volume of the hollow cylinder is a maximum when $r = p\sqrt{\frac{2}{3}}$ , where $p \in \mathbb{Z}^+$ . Find the value of $p$ .

IB Mathematics Analysis and Approaches SL 3.1—3d Space, Volume, Angles, Distance, Midpoints Questionbank

IB Mathematics Analysis and Approaches SL 3.1—3d Space, Volume, Angles, Distance, Midpoints Questionbank

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IB Mathematics Analysis and Approaches SL 3.1—3d Space, Volume, Angles, Distance, Midpoints Questionbank

IB Mathematics Analysis and Approaches SL 3.1—3d Space, Volume, Angles, Distance, Midpoints Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 3.1—3d Space, Volume, Angles, Distance, Midpoints Questionbank

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