User interface language: English | Español

Date May 2021 Marks available 2 Reference code 21M.2.SL.TZ1.5
Level Standard Level Paper Paper 2 Time zone Time zone 1
Command term Hence and Find Question number 5 Adapted from N/A

Question

The cross-sectional view of a tunnel is shown on the axes below. The line [AB] represents a vertical wall located at the left side of the tunnel. The height, in metres, of the tunnel above the horizontal ground is modelled by y=-0.1x3+ 0.8x2, 2x8, relative to an origin O.

Point A has coordinates (2, 0), point B has coordinates (2, 2.4), and point C has coordinates (8, 0).

When x=4 the height of the tunnel is 6.4m and when x=6 the height of the tunnel is 7.2m. These points are shown as D and E on the diagram, respectively.

Find dydx.

[2]
a.i.

Hence find the maximum height of the tunnel.

[4]
a.ii.

Use the trapezoidal rule, with three intervals, to estimate the cross-sectional area of the tunnel.

[3]
b.

Write down the integral which can be used to find the cross-sectional area of the tunnel.

[2]
c.i.

Hence find the cross-sectional area of the tunnel.

[2]
c.ii.

Markscheme

evidence of power rule (at least one correct term seen)                 (M1)

dydx=-0.3x2+1.6x                 A1


[2 marks]

a.i.

-0.3x2+1.6x=0                 M1

x=5.33 5.33333, 163                 A1

y=-0.1×5.333333+0.8×5.333332                 (M1)

 

Note: Award M1 for substituting their zero for dydx 5.333 into y.


7.59 m 7.58519                 A1


Note: Award M0A0M0A0 for an unsupported 7.59.
Award at most M0A0M1A0 if only the last two lines in the solution are seen.
Award at most M1A0M1A1 if their x=5.33 is not seen.


[6 marks]

a.ii.

A=12×22.4+0+26.4+7.2                 (A1)(M1)

 

Note: Award A1 for h=2 seen. Award M1 for correct substitution into the trapezoidal rule (the zero can be omitted in working).


=29.6m2                 A1


[3 marks]

b.

A=28-0.1x3+0.8x2dx  OR  A=28ydx                 A1A1

 

Note: Award A1 for a correct integral, A1 for correct limits in the correct location. Award at most A0A1 if dx is omitted.


[2 marks]

c.i.

A=32.4 m2                  A2


Note:
As per the marking instructions, FT from their integral in part (c)(i). Award at most A1FTA0 if their area is >48, this is outside the constraints of the question (a 6×8 rectangle).


[2 marks]

c.ii.

Examiners report

[N/A]
a.i.
[N/A]
a.ii.
[N/A]
b.
[N/A]
c.i.
[N/A]
c.ii.

Syllabus sections

Topic 5—Calculus » SL 5.5—Integration introduction, areas between curve and x axis
Show 85 related questions
Topic 2—Functions » SL 2.5—Modelling functions
Topic 2—Functions
Topic 5—Calculus

View options