Question

20.

Relief Donations The number of private donations

received by nongovernment disaster relief organizations can

be modeled as

f(x) = 0.3xe -0.03x thousand donations

where x is the number of hours since a major disaster has

struck

a. 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?

Answer 1

a) e-0.03x(0.3-0.009x)

b) 33 hrs 20 min

c) 17.182

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'u

hence

f'(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-0.03x

hence

a) rate of change in donations = e-0.03x(0.3-0.009x) taking exponent common

b) f'(x) = 0 according to statement

hence 0 = e-0.03x(0.3-0.009x)

0 = 0.3 - 0.009 x

which 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 by

f(x) = 0.3xe-0.03x

f(33.33) = 0.3 (33.33)e-0.03(33.33)

= 17.182

That is 17.182 thousand donations were made at that time.