The number of units of weekly output of a certain product is represented by the equation shown below, where x is the number of workers on the assembly line. There are presently 68 workers on the line.
Q(x) = 200x + 4x2
(a) Find Q'(x) and estimate the change in the weekly output caused by the addition of one worker.
(b) Calculate Q(69) − Q(68) to see the actual change in the weekly output.
To find Q'(x), we need to take the derivative of Q(x) with respect to x. In this case, Q(x) = 200x + 4x^2, so we have:
Q'(x) = dQ/dx = d(200x + 4x^2)/dx
To find the derivative, we apply the power rule and the constant multiple rule:
Q'(x) = 200 + 8x
Now let's estimate the change in the weekly output caused by the addition of one worker. Since there are currently 68 workers on the line, we can substitute x = 68 in the equation for Q'(x):
Q'(68) = 200 + 8(68)
= 200 + 544
= 744
So the estimated change in the weekly output caused by the addition of one worker is 744 units.
Now let's calculate Q(69) - Q(68) to see the actual change in the weekly output. We have:
Q(69) = 200(69) + 4(69)^2
= 13,800 + 16,044
= 29,844
Q(68) = 200(68) + 4(68)^2
= 13,600 + 14,624
= 28,224
Q(69) - Q(68) = 29,844 - 28,224
= 1,620
So the actual change in the weekly output caused by the addition of one worker is 1,620 units.