# Thread: Hourly Decay Rate for Salt

1. ## Hourly Decay Rate for Salt

The question:

A tank of water is contaminated with 65 pounds of salt. In order to bring the salt concentration down to a level consistent with EPA standards, clean water is being piped into the tank, and the well-mixed overflow is being collected for removal to a toxic-waste site. The result is that at the end of each hour there is 19% less salt in the tank than at the beginning of the hour. Let S = S(t) denote the number of pounds of salt in the tank t hours after the flushing process begins.

(a.1) Explain why S is an exponential function.

My answer: The amount of salt is being decreased by 19% each hour, so the decay is at a constant proportional rate.

(a.2) Find its hourly decay factor.

____

(b) Give a formula for S.

S =

(I thought this would be in a $\displaystyle a(b)^x$ format... but I'm not really sure at all.)

2. Originally Posted by MathBane
The question:

A tank of water is contaminated with 65 pounds of salt. In order to bring the salt concentration down to a level consistent with EPA standards, clean water is being piped into the tank, and the well-mixed overflow is being collected for removal to a toxic-waste site. The result is that at the end of each hour there is 19% less salt in the tank than at the beginning of the hour. Let S = S(t) denote the number of pounds of salt in the tank t hours after the flushing process begins.

(a.1) Explain why S is an exponential function.

My answer: The amount of salt is being decreased by 19% each hour, so the decay is at a constant proportional rate.

(a.2) Find its hourly decay factor.

____

(b) Give a formula for S.

S =

(I thought this would be in a $\displaystyle a(b)^x$ format... but I'm not really sure at all.)
1. In general the equation

$\displaystyle A(t)=A_0 \cdot e^{k \cdot t}$

determines the amount in an exponential process, where k is a constant, t denotes the time and $\displaystyle A_0$ is the initial amount.

$\displaystyle S_0 = 65 \ lbs$

After one hour you have $\displaystyle S(1)=0.81 \cdot 65$ . Use this value to get the constant k:

$\displaystyle 0.81 \cdot 65 = 65 \cdot e^{k \cdot 1}~\implies~k = \ln(0.81) \approx - 0.21072...$

3. Combining all results you get:

$\displaystyle S(t) = 65 \cdot e^{\ln(0.81) \cdot t} ~\implies~\boxed{S(t) = 65 \cdot 0.81^t}$

t measured in hours.

3. Originally Posted by MathBane
The question:

A tank of water is contaminated with 65 pounds of salt. In order to bring the salt concentration down to a level consistent with EPA standards, clean water is being piped into the tank, and the well-mixed overflow is being collected for removal to a toxic-waste site. The result is that at the end of each hour there is 19% less salt in the tank than at the beginning of the hour. Let S = S(t) denote the number of pounds of salt in the tank t hours after the flushing process begins.

(a.1) Explain why S is an exponential function.

My answer: The amount of salt is being decreased by 19% each hour, so the decay is at a constant proportional rate.

(a.2) Find its hourly decay factor.

____

(b) Give a formula for S.

S =

(I thought this would be in a $\displaystyle a(b)^x$ format... but I'm not really sure at all.)
You can also derive $\displaystyle S_t$ from the recurrence relation

$\displaystyle S_{t+1} = 0.81S_t$

$\displaystyle S_{t+2} = 0.81S_{t+1} = 0.81(0.81S_t)$

and so on until it gets to the nth stage

$\displaystyle S_{n(t)} = t_0 \cdot 0.81^t$

As $\displaystyle t_0=65$ we get $\displaystyle S_{n(t)} = 65 \cdot 0.81^t$