Question

An area of not more than 42m^2 in a garden, is reserved for the planting of y bushes and t trees.

(a) The client wants a minimum number of 4 bushes planted. Translate this condition into an inequality in terms of y.

(b) For every tree planted, he does not want more than 3 bushes. Interpret this constraint into an inequality in terms of t and y,        making y the subject of the formula.

(c) For aesthetic reasons and for proper growth, one bush requires 2m^2 of soil area and one tree requires 8m^2 of soil area. Interpret this into an inequality in terms of t and y, making y the subject of the formula.

Answer 1

(a) y≥4 At least 4 bushes planted.

(b) If there were 3 bushes for every tree, y=3t; if there were 2 bushes for every tree y=2t, so y≤3t for no more than 3 bushes per tree. Example: if there were 4 trees, there must be no more than 12 bushes.

(c) Space requirements: 2y sq m for all the bushes; 8t sq m for all trees. 2y+8t≤42. So y≤21-4t; and if (a) and (b) are applied: 4≤y≤21-4t. Also y≤3t, therefore 3t≤21-4t, 7t≤21, t≤3, and y≤9. And y≥4 so 4≤y≤9.

CHECK: y={ 4 5 6 7 8 9 }. (a) is satisfied; t≤3 so y≤9, which satisfies (b); the maximum area is when y=9, t=3, area=9*2+3*8=18+24=42.