Question

The owner of a large house decided to spend some money making it more secure. An expert analysed the situation and said that for $5000 the security rating, R, of the property would rise from its current score of 30 points to 100 points. Every $500 spent after that would lift the rating by 5 points.
The owner feels that the more security devices he pays for the more tedious he is going to find it to enter and leave his property with all that he will have to remember lock/unlock. He feels there is an owner convenience rating, C, which will go down by 2 points, from an initial 100 points, for every $500 he spends over the $5000 that he accepts is necessary.
As both the security rating and the convenience rating interest him he decides to multiply them together to form the secure but not inconvenient rating, Z.
Clearly showing your use of calculus, determine how much the owner should spend on security to maximise Z.

Answer 1

Each rating is described by the straight line with a constant slope. So, the straight line equation:

$y = kx + b,\\$ k = slope, b = the y-intercept.

We can determine slope from the condition: 5 points per $500 for R and -2 points per $500 for C. So for R: $k = 5/500 = 0.01$ and for C: $k = -2/500 = 0.004\\$

Now let's determine b. As far as $R = 100$ for $x = 0$ , we can write:

$100 = 0.01\cdot0 + b \Rightarrow b = 100\\$ .

The case of C is similar. So we finally get our expressions:

$\\R = 0.01x + 100;\\$

$C = -0.004x + 100\\$ , where $x$ is extra cost (over $5000).

Multiplication rating:

$Z = R\cdot C=(0.01x + 100)(-0.004x+100) =-4\cdot10^{-5}x^2 + 0.6x + 10^4\\$

Z is downward parabola, so it has maximum value. The maximum condition: $\dfrac{dZ}{dx}=0\\$

$\dfrac{dZ}{dx}=-8\cdot10^{-5}x_{max}+0.6 = 0\\ x_{max} = -0.6/(-8\cdot10^{-5}) = \$7500$

So the additional acceptable cost is $7500. Together with initial $5000 it will be $12500.