Question

The cost C (in dollars) of building a house is related to the number k of carpenters used and the number e of electricians used by the formula† C = 15,000 + 50k2 + 60e2. If the cost of the house is fixed at $100,000, find dk/de e = 15

Answer 1

$C = 15000 + 50k^2 + 60e^2 \Rightarrow \\ \frac{dC}{de} = \frac{d}{de} 50k^2 + \frac{d}{de} 60e^2 \Rightarrow \\ \frac{dC}{de} = 100k\frac{dk}{de} + 120e\\ Cost\ of\ house\ is\ fixed\ so\ \frac{dC}{de} = 0 \\ 0 = 100k\frac{dk}{de} + 120e\ \Rightarrow\ -120e = 100k\frac{dk}{de} \ \Righarrow \\ \frac{dk}{de} = -\frac{6e}{5k}$

find k when e = 15

$100000 = 15000 + 50k^2 + 60(15)^2 \Rightarrow k = \sqrt{1430}$

$\frac{dk}{de} = -\frac{6e}{5k} = -\frac{6(15)}{5\sqrt{1430}} = -0.476$

inst rate of carpenters with 15 electricians: decreasing at 0.476 carpenters per electrician