20.Relief Donations The number of private donationsreceived by nongovernment disaster relief organizations canbe modeled asf(x) = 0.3xe -0.03x thousand donationswhere x is the number of hours since a major disaster hasstrucka. Write an expression for the rate of change in donations.b. At what time is the rate of change of donations zero?c. What is the donation level at the time found in part b?

Accepted Solution

Answer:a) e-0.03x(0.3-0.009x)b) 33 hrs 20 minc) 17.182Step-by-step explanation:Taking derivative of the above equation would give us the rate for change in donations :hence f'(x) = d/dx (0.3xe-0.03x)using product rule of derivation where (uv)' = u'v +v'uhencef'(x) = (e-0.03x)d/dx(0.3x) + 0.3x*d/dx(e-0.03x)        = 0.3e-0.03x + 0.3x(-0.03)(e-0.03x)   f'(x)   =0.3e-0.03x -0.009xe-0.03xhencea) rate of change in donations = e-0.03x(0.3-0.009x) taking exponent commonb) f'(x) = 0 according to statementhence 0 = e-0.03x(0.3-0.009x)      0 = 0.3 - 0.009 xwhich gives x = 33.33 hrs i.e after 33 hrs and 20 minutes rate of change of donations is 0.c) donation level at the time is given byf(x) = 0.3xe-0.03xf(33.33) = 0.3 (33.33)e-0.03(33.33)               = 17.182That is 17.182 thousand donations were made at that time.