Year 10 Interactive Maths - Second Edition

Problem Solving I

Example 15

The rate of a chemical reaction, v, involving the transformation of one compound into another compound is given by

where x is the concentration of the new compound. What value of x results in the maximum value of v, and what is the maximum value?

Solution:

So, the maximum value of v is 9 and it occurs when x = 3.

Example 16

A farmer wants to build a fence around a rectangular paddock using the straight edge of a canal as one side. She has 400 m of fencing and wants to enclose the maximum area.

a.  If the width of the paddock is x m, express the:
(i)   length, l, of the paddock in terms of x
(ii)  area, A, of the paddock in terms of x

b.  Express A in square completion form.
c.  Sketch the graph of A showing all key features.
d.  Determine the length and width of the paddock for its area to be maximised.

Solution:

c. The graph is a parabola opening downwards with axis of symmetry x = 100 and vertex at (100, 20000).

x-intercepts:

A-intercept:

d. The maximum area of the paddock is 20 000 m² and it occurs when width = 100 m and length = 200 m.