A management consultant estimates that the number of hours,h, per day that employees work and their daily pay of p dollars are related by the equation 60h5 = 2,000,000 = p3. Find dh/dp at p = 200 and interpret your answer.
To find dh/dp, we need to differentiate the equation 60h^5 = 2,000,000 = p^3 with respect to both h and p, and then solve for dh/dp.
First, differentiate the equation with respect to h using the power rule of differentiation.
d/dh (60h^5) = d/dh (2,000,000) - d/dh (p^3)
300h^4 = 0 - 0
300h^4 = 0
Now, differentiate the equation with respect to p using the power rule of differentiation.
d/dp (60h^5) = d/dp (2,000,000) - d/dp (p^3)
0 = 3p^2 - 0
0 = 3p^2
Now, solve for dh/dp by rearranging the equations:
300h^4 = 0
h^4 = 0
h = 0
0 = 3p^2
p = 0
Therefore, at p = 200, the slope of the equation is undefined since the value of h is 0. This means that there is no change in h with respect to a change in p at p = 200. In other words, the number of hours worked per day does not change for every increment in the daily pay of 200 dollars.