The value of the total assets of Safeway, Inc., can be modeled by the following
ID: 2893205 • Letter: T
Question
The value of the total assets of Safeway, Inc., can be modeled by the following function: A(t) = 280.2t^4 - 2238.3t^3 + 4970.3t^2 - 1947.2t + 14900.3, where A(t) is given in millions of dollars and t is time in years since 1998. (Model based on data from Safeway, Inc., 2003 Annual Report, page 17.) Fill in the tables below to estimate the instantaneous rate of change in the value of Safeway's assets at t = 3. F(3 + h) - F(3)/h (3 + h)^2 - 3^2/h. What is your estimate for the instantaneous rate of change? Interpret A'(3), if describing the graph of the function. interpret the meaning of A'(3) f describing it to a Safeway manager (this means no complicated math terms).Explanation / Answer
A(t) = 280.2t4 -2238.3t3 + 4970.3t2 - 1947.2t + 14900.3
A(3) = 280.2(3)4 -2238.3(3)3 + 4970.3(3)2 - 1947.2(3) + 14900.3 = 16053.5
Average Rate of change = [A(t + h) - A(t)]/h
@ t = 3 ==> Average Rate of change = [A(3 + h) - A(3)]/h
(i) h = 1
==> Average Rate of change = [A(4) - A(3)]/1
A(4) = 280.2(4)4 -2238.3(4)3 + 4970.3(4)2 - 1947.2(4) + 14900.3 = 15116.3
==> Average Rate of change = [15116.3 - 16053.5] = -937.2
(ii) h = 0.1
==> Average Rate of change = [A(3.1) - A(3)]/(0.1)
A(3.1) = 280.2(3.1)4 -2238.3(3.1)3 + 4970.3(3.1)2 - 1947.2(3.1) + 14900.3 = 15824.43
==> Average Rate of change = [15824.43 - 16053.5]/(0.1) = -2290.7
(iii) h = 0.01
==> Average Rate of change = [A(3.01) - A(3)]/(0.01)
A(3.01) = 280.2(3.01)4 -2238.3(3.01)3 + 4970.3(3.01)2 - 1947.2(3.01) + 14900.3 = 16030.52
==> Average Rate of change = [16030.52 - 16053.5]/(0.01) = -2298
(iii) h = 0.001
==> Average Rate of change = [A(3.001) - A(3)]/(0.001)
A(3.001) = 280.2(3.001)4 -2238.3(3.001)3 + 4970.3(3.001)2 - 1947.2(3.001) + 14900.3 = 16051.20
==> Average Rate of change = [16051.20 - 16053.5]/(0.01) = -2299.99
(iv) h = 0.0001
==> Average Rate of change = [A(3.0001) - A(3)]/(0.0001)
A(3.0001) = 280.2(3.0001)4 -2238.3(3.0001)3 + 4970.3(3.0001)2 - 1947.2(3.0001) + 14900.3 = 16053.27
==> Average Rate of change = [16053.27 - 16053.5]/(0.0001) = -2299.99
(v) h = 0.00001
==> Average Rate of change = [A(3.00001) - A(3)]/(0.00001)
A(3.00001) = 280.2(3.00001)4 -2238.3(3.00001)3 + 4970.3(3.00001)2 - 1947.2(3.00001) + 149000.3 = 16053.48
==> Average Rate of change = [16053.477 - 16053.5]/(0.00001) = -2299.99
b) Instantaneous rate of change A'(3)
A'(t) = 280.2(4)t4-1 -2238.3(3)t3-1 + 4970.3(2)t2-1 - 1947.2(1) + 0 ; since d/dx xn = nxn-1
==> A '(t) = 1120.8t3 - 6714.9t2 + 9940.6t - 1947.2
==> A '(3) = 1120.8(3)3 - 6714.9(3)2 + 9940.6(3) - 1947.2 = -2297.9
c) as A'(3) is negative the graph A(t) is decreasing at the instant t = 3
d) Describing to a Safeway manager:
t = 3 ==> year 2001
the total assets are decreasing in the year 2001
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