Surface area and volume of 3D solidsIB MYP Maths Standard: Subtopic test

1. 1
   A closed cylindrical water tank has radius $0.8$ m and height $1.5$ m. Volume of a cylinder $=\pi r^2h$. Curved surface area of a cylinder $=2\pi rh$.

   (a)

   Find the volume of the tank.

   [1 mark]

   • A$3.02$ m$^3$
   • B$1.01$ m$^3$
   • C$5.65$ m$^3$
   • D$7.54$ m$^3$

   (b)

   Find the capacity of the tank in litres, given that $1$ m$^3=1000$ litres.

   [1 mark]

   • A$3.02$ litres
   • B$302$ litres
   • C$30\,200$ litres
   • D$3020$ litres

   (c)

   Find the total outer surface area of the closed tank.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A solid metal prism has a right-angled triangle as its cross-section, with perpendicular sides $6$ cm and $8$ cm and hypotenuse $10$ cm. The prism is $15$ cm long.

   (a)

   Find the volume of the prism.

   [1 mark]

   • A$720$ cm$^3$
   • B$360$ cm$^3$
   • C$24$ cm$^3$
   • D$480$ cm$^3$

   (b)

   Find the total surface area of the prism.

   [1 mark]

   • A$360$ cm$^2$
   • B$258$ cm$^2$
   • C$408$ cm$^2$
   • D$816$ cm$^2$

   (c)

   Write the volume of the prism in mm$^3$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A square-based pyramid has a base of side $10$ cm and a perpendicular height of $12$ cm. Each of its four triangular faces has a slant height of $13$ cm. Volume of a pyramid $=\frac13\times\text{base area}\times\text{height}$.

   (a)

   Find the volume of the pyramid. A cuboid box has the same base and the same height. What fraction of the box does the pyramid fill?

   [3 marks]

   (b)

   Find the total surface area of the pyramid, including its base.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A cone-shaped cup has a circular top of radius $5$ cm, a perpendicular height of $12$ cm and a slant height of $13$ cm. Formulae: volume of a cone $=\frac13\pi r^2h$; curved surface area of a cone $=\pi rl$; volume of a sphere $=\frac43\pi r^3$.

   (a)

   (i) Find the volume of the cup.
   (ii) The cup is filled with orange juice, which is then poured into a cylindrical glass of radius $4$ cm. Find the depth of juice in the glass.
   (iii) Find the curved surface area of the cup.

   [6 marks]

   (b)

   A shop fills each cup level with ice cream and puts a spherical scoop of radius $3$ cm on top. Ice cream is bought in $5$ litre tubs ($1$ litre $=1000$ cm$^3$).
   (i) Find the volume of the spherical scoop.
   (ii) Find the total volume of ice cream in one serving.
   (iii) Find the number of complete servings that can be made from one tub, justifying your answer.

   [6 marks]

   Total for question 4: 12 marks

End of questions