Question

The equation Upper A left parenthesis t right parenthesis equals 2000 e Superscript 0.055 t gives the balance after t years of an initial investment of 2000 dollars which pays 5.50​% compounded continuously. a. Find a formula for StartFraction dA Over dt EndFraction b. Find and interpret Upper A prime left parenthesis 8 right parenthesis. Include appropriate units. c. Compare the approximation of ​$171 to the actual change. Report your answer to two decimal places.

Answer 1

Answer:

a) Rate of change of amount

$A'(t) =110e^{0.055t}$

b) &170.79

c) 0.21

Step-by-step explanation:

We are given the following in the question:

The balance is given by the equation:

$A(t) =- 2000e^{0.055t}$

where t is the time in years and the initial investment is $2000 when compounded continuously.

a) Rate of change of amount

$\dfrac{d(A(t))}{dt} = \dfrac{d}{dt}(2000e^{0.055t})\\\\\dfrac{d(A(t))}{dt} = 2000e^{0.055t}\times 0.055\\\\\dfrac{d(A(t))}{dt} =110e^{0.055t}$

b) We have to find the value of A'(8)

$A'(t) =110e^{0.055t}\\A'(8) = 110e^{0.055(8)} = 170.79$

Interpretation:

The future value of 9 year investment of $2000 will be $170.79 more than the future value of 8 year investment.

c) Comparison

Approximation = $171

Actual change = $170.79

Difference =

$\text{Approximation - Actual change}\\=171 - 170.79\\=0.21$

Thus, the error is 0.21