Question 1

Paper 2Medium13 marks

A landscape designer models the vertical profile of a section of a footpath by

h(x)=0.009x3−0.25x2+1.7x+2.5,0≤x≤20,h(x)=0.009x^3-0.25x^2+1.7x+2.5,\qquad 0\leq x\leq20,

where xx is the horizontal distance, in metres, from the start of the section and h(x)h(x) is the height, in metres, above a reference level.

Consider the point on the profile where x=8x=8

(a)

Calculate the height of the profile at this point.

[2]
(b)

Show that

h′(x)=0.027x2−0.50x+1.7.h'(x)=0.027x^2-0.50x+1.7.
[2]
(c)

Determine the equation of the tangent to the profile at x=8x=8 giving your answer in the form h−h(8)=m(x−8)h-h(8)=m(x-8)

[3]
(d)

Determine the equation of the normal to the profile at x=8x=8 in the same form. Give its gradient to 3 significant figures.

[3]
(e)

Use your GDC to find the coordinates of the other point, different from the point where x=8x=8 at which the tangent in part (c) meets the profile. Give your coordinates to 3 significant figures.

[3]

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