find the intervals

p=4t²-29t+30, so she makes a profit when p>0.

The quadratic factorises: (4t-5)(t-6)>0.

For this to be true, both factors must either be positive or negative, so t<5/4 (t<1.25 months) or t>6 months.

The intervals are 0 < t < 1.25 or t > 6.

If we are dealing in whole numbers of months, 0 < t ≤ 1 or t > 6. (I haven’t included break-even when p=0.)

The graph below illustrates the solution (ignore negative values of t which have no meaning in this problem). The parabola sits above the axis when the profit is positive. The profit drops at first up to t=1.25 months, then a loss occurs for a few months, and after 6 months rises continually.

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