# Question #b15b8

##### 2 Answers

The value is about $233333 (or $230000 if you round to two significant digits).

#### Explanation:

We are assuming that

Integration of both sides leads to

We are given that **decrease** in value is $100000 from time

Solving the first equation for

Therefore,

# V = 1200000 /(t+3) + 100000 #

# V(6) = $233,333 #

#### Explanation:

We must decode the given text to form an appropriate Differential Equation which we must then solve.

Depreciation **reduces** the value of an object.

We are told that the depreciation is **inversely proportional** to the **square** of

# -(dV)/(dt) prop 1/(t+3)^2 => -(dV)/(dt) =K/(t+3)^2#

Where

# int \ dV = - K int \ 1/(t+3)^2 \ dt#

Which we can integrate to get:

# V = K/(t+3) + C #

Note that we have have two unknown constants and two conditions; thus:

"The initial value of the machine was $500,000":

# => V=500000 # when# t=0#

# :. 500000 = K/3 + C # ..... [A]

"its value decreased $100,000 in the first year":

# => V=400000 # when# t=1#

# :. 400000 = K/4 + C # ..... [B]

Eq[A] - Eq[B]:

# 100000 = K/3 - K/4 #

# :. K/12 = 100000 => K = 1200000 #

Subs

# 500000 = 1200000/3 + C #

# :. 500000 = 400000 + C #

# :. C = 100000 #

Thus the solution to the DE is:

# V = 1200000 /(t+3) + 100000 #

We seek the value when

# V = 1200000 /9 + 100000 #

# \ \ = 400000/3 + 100000 #

# \ \ = 400000/3 + 300000/3 #

# \ \ = 700000/3#

# \ \ = 233333.33333 ...#

# \ \ = 233333 \ \ # rounded to the nearest integer