Question

Rate of change problem

Answer 1

Wood in a pine plantation is valued at 2 million Rand per square km. If the radius
of a circular fire in the plantation is 2km and is increasing at 0.1 km/hour, find the
rate at which money is being lost. (Give your answer in millions of Rands per hour.)

Answer 2

(I'm South African so the problem is in Rands. Can replace Rands with Dollars if need be)

Answer 3

\[\frac{ dR }{ dt } = \frac{dR}{dA}\frac{dA}{dt}\]

Answer 4

so above i have chain rule, R standing for Rands lost. For my area w.r.t. time i have:
\[A(t)=\pi(2 + 0.1t)^{2}\]

Answer 5

so, \[\frac{ dA }{ dt } = 0.4\pi + 0.02\pi t\]

Answer 6

@remnant as the rate is of loosing Rands so sign must be negative:)

Answer 7

so, \[\frac{dR}{dt} = 2(0.4\pi + 0.02\pi t)\]
so, \[\frac{dR}{dt} = 0.8\pi + 0.04\pi t\]

Answer 8

the answer is \[0.8\pi\] so i'm close but don't know what i've done wrong

Answer 9

@TuringTest

Answer 10

@hartnn @phi

Answer 11

I would start with
A= pi r^2

take the derivative with respect to time t
\[\frac{dA}{dt}= 2\pi r \frac{dr}{dt} \]
now plug in for r , dr/dr
multiply by 2 (for 2 million rands )

Answer 12

*dr/dt = 0.1 km/hr (not dr/dr)