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Date May 2022 Marks available 1 Reference code 22M.1.AHL.TZ2.17
Level Additional Higher Level Paper Paper 1 Time zone Time zone 2
Command term Show that and Hence Question number 17 Adapted from N/A

Question

The cross-section of a beach is modelled by the equation y=0.02x2 for 0x10 where y is the height of the beach (in metres) at a horizontal distance x metres from an origin. t is the time in hours after low tide.

At t=0 the water is at the point (0, 0). The height of the water rises at a rate of 0.2 metres per hour. The point W(x(t), y(t)) indicates where the water level meets the beach at time t.

 

A snail is modelled as a single point. At t=0 it is positioned at (1, 0.02). The snail travels away from the incoming water at a speed of 1 metre per hour in the direction along the curve of the cross-section of the beach. The following diagram shows this for a value of t, such that t>0.

When W has an x-coordinate equal to 1, find the horizontal component of the velocity of W.

[3]
a.

Find the time taken for the snail to reach the point (10, 2).

[4]
b.i.

Hence show that the snail reaches the point (10, 2) before the water does.

[1]
b.ii.

Markscheme

use of chain rule        (M1)

dydt=dydxdxdt

attempt to find dydx at x=1        (M1)

0.2=0.04×dxdt

dxdt=  5m h-1          A1

 

[3 marks]

a.

if the position of the snail is X, Y

from part (a) dXdt=10.04XdYdt

since speed is 1:

finding modulus of velocity vector and equating to 1         (M1)

1=Y˙0.04X2+Y˙2   OR   1=X˙2+0.0016X2X˙2

1=Y˙210.0016X2+1   OR   1=X˙21+0.0016X2

Y˙=110.08Y+1   OR   X˙=11+0.0016X2         (A1)

0.02210.08Y+1dY=0Tdt   OR   1101+0.0016X2dX=0Tdt         (M1)

T=9.26 hours          A1

 

[4 marks]

b.i.

EITHER

time for water to reach top is 20.2=10 hours (seen anywhere)          A1


OR

or at time t=9.26, height of water is 0.2×9.26=1.852          A1


THEN

so the water will not reach the snail          AG

 

[1 mark]

b.ii.

Examiners report

In part (a), a small minority of candidates found the horizontal component of velocity correctly. Few candidates made any significant progress in part (b).

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

Syllabus sections

Topic 5—Calculus » AHL 5.9—Differentiating standard functions and derivative rules
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