Feb
19
How much should the company spend on advertising to obtain a maximum profit?
Filed Under Mathematics
spazattack99 asked:
The profit “p” (in hundreds $) that a company makes depends on the amount x(in hundreds of $)the company spends on advertising according to the model, p=230+20x-0.5x^2.
The profit “p” (in hundreds $) that a company makes depends on the amount x(in hundreds of $)the company spends on advertising according to the model, p=230+20x-0.5x^2.
How much should the company spend on advertising to obtain a maximum profit?
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p = -0.5x² +20x + 230
This equation would graph as a parabola opening downward. The maximum profit occurs at the vertex. The x-coordinate of the vertex is x=-b/(2a)
Thus, the maximum will occur at x = -20/(2*-0.5) = -20/-1 = 20 (hundred $) or $2000
To help you remember the x=-b/(2a), remember this is part of the quadratic formula. If you move from the vertex an equal distance to the right or left, you will hit the x-axis (the zeros of the equation in question). What you are adding to this in the quadratic formula is sqrt(b²-4ac)/(2a) - which is the zero that is to the right of the vertex. You are subtracting the same thing as well (the formula calls for ±), which would be the zero the the left of the vertex.
Thia holds true for stereotypical TV/Radio/Press/Internet spends. You can also be clever and identify high net worth demographics and include the power of ‘niche’ audiences to maximise ROI.
John @ SKS