Question

Trees are planted in urban areas for beauty but also to remove air pollution. Trees with a large diameter, which cost $UK £ 100$ to plant, need $80 m^{2}$ of space to grow and remove $1.5$ kg/year of pollutants. Smaller trees, costing $UK £ 30$ , need $50 m^{2}$ of space to grow but remove only $0.5$ kg/year of air pollutants. If your school has a budget of $UK £ 1200$ and $1500 m^{2}$ of space, determine how many of each type of tree should be planted in order to maximize the amount of air pollutants removed.

Accepted Answer

Let the number of large diameter and that smaller trees be $x$ and $y$ respectively.

Number of trees cannot be negative.

Therefore, $x, y \geq 0$

Given that large trees remove $1.5$ kg/year pollutants and smaller trees remove $0.5$ kg/year pollutants.

To maximize : $Z = 1.5 x + 0.5 y$ in kg/year.

The given information can be tabulated as follows:

Space $(m^{2})$ Cost $(UK £)$

Large tree $80$ $100$

Smaller tree $50$ $30$

Availability $1500$ $1200$

Therefore, the constraints are

$80 x + 50 y \leq 1500$

$\Rightarrow 8 x + 5 y \leq 150 . . . . . (i)$

And, $100 x + 30 y \leq 1200$

$\Rightarrow 10 x + 3 y \leq 120 . . . . . . . . . . . . . . . (ii)$

$x, y \geq 0$

The line $8 x + 5 y = 150$ meets the coordinate axes at points $(18.75, 0)$ and $(0, 30)$ .

The line $10 x + 3 y = 120$ meets the coordinate axes at points $(12, 0)$ and $(0, 40)$ .

The above inequalities form a feasible region and is depicted below:

Notice that the feasible region is $O C B A O$ . The corner points are $O (0, 0), C (12, 0), B (5.769, 20.769), A (0, 30)$ .

The values of $Z$ at these corner points are as follows

Corner point Value of $Z = 1.5 x + 0.5 y$ in kg/year

$O (0, 0)$ $0$

$C (12, 0)$ $18$

$B (5.769, 20.769)$ $19.038$

$A (0, 30)$ $15$

Thus, the maximum pollutants removed is $19.038$ kg/year at point $B (5.769, 20.769)$ , but number of trees should be integer, so considering integer value and in feasible region, $x = 6$ and $y = 20$ .

At $(6, 20)$ , $Z = 19$ kg/year.

Hence, $6$ larger trees and $20$ smaller trees should be planted.

Rose Harrison, Clara Huizink, Aidan Sproat Clements and, Marlene Torres Skoumal Solutions for Exercise 19: Review in context

Rose Harrison Mathematics Solutions for Exercise - Rose Harrison, Clara Huizink, Aidan Sproat Clements and, Marlene Torres Skoumal Solutions for Exercise 19: Review in context

Questions from Rose Harrison, Clara Huizink, Aidan Sproat Clements and, Marlene Torres Skoumal Solutions for Exercise 19: Review in context with Hints & Solutions

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