O-LevelAdditional MathematicsDifferentiation and integration4037/21 · May/June 2018 · Q129 Marks
In this question all lengths are in metres. A water container is in the shape of a triangular prism. The diagrams show the container and its cross-section. The cross-section of the water in the container is an isosceles triangle ABC, with angle ABC = angle BAC = 30°. The length of AB is x and the depth of water is h. The length of the container is 5. [Figure 12.1]
- (i)Show that x = 2√3h and hence find the volume of water in the container in terms of h.[3]
- (ii)The container is filled at a rate of 0.5 m³ per minute. At the instant when h is 0.25 m, find
- (ii)(a)the rate at which h is increasing,[4]
- (ii)(b)the rate at which x is increasing.[2]
This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.
Mark scheme
Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.
- [1] (i) tan 30 = h / (x/2) oe
- [1] (i) Correct completion to given answer
- [1] (i) V = 5√3h² isw
- [1] (ii)(a) dV/dh = their 10√3h or 5√3 * 2h
- [1] (ii)(a) FT their V = kh²
- [1] (ii)(a) dh/dt = dh/dV × dV/dt soi
- [1] (ii)(a) dh/dt = (1 / (their dV/dh)) × 0.5
- [1] (ii)(a) 0.115 or 0.11547 to 0.1155 oe
- [1] (ii)(b) (dx/dt = dx/dh × dh/dt =) 2√3 × their (1 / (5√3 * 0.25)) * 0.5
- [1] (ii)(b) 2/5 or 0.4
Write your answer and get it marked
On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.
Sign up free →