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A large brine tank containing a solution of salt and water is being diluted with fresh water.
The relationship between the elapsed time,
t, in hours, after the dilution begins, and the concentration of salt in the tank,
S(t), in grams per liter
(g//l), is modeled by the following function.

S(t)=500*e^(-0.25 t)
What will the concentration of salt be after 10 hours?
Round your answer, if necessary, to the nearest hundredth.

g//1

A large brine tank containing a solution of salt and water is being diluted with fresh water. $\newline$ The relationship between the elapsed time, $t$ , in hours, after the dilution begins, and the concentration of salt in the tank, $S(t)$ , in grams per liter $(\mathrm{g} / \mathrm{l})$ , is modeled by the following function. $\newline$ $S(t)=500 \cdot e^{-0.25 t}$ $\newline$ What will the concentration of salt be after $10$ hours? $\newline$ Round your answer, if necessary, to the nearest hundredth. $\newline$ g/ $1$

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Weighted averages: word problems

Full solution

Q. A large brine tank containing a solution of salt and water is being diluted with fresh water.

\newline

The relationship between the elapsed time,

t

, in hours, after the dilution begins, and the concentration of salt in the tank,

S(t)

, in grams per liter

(\mathrm{g} / \mathrm{l})

, is modeled by the following function.

\newline

S(t)=500 \cdot e^{-0.25 t}

\newline

What will the concentration of salt be after

10

hours?

\newline

Round your answer, if necessary, to the nearest hundredth.

\newline

g/

1

Identify Function and Value: Identify the given function and the value of $t$ for which we need to find the concentration of salt. $\newline$ The function given is $S(t) = 500 \cdot e^{(-0.25 \cdot t)}$ , and we need to find $S(10)$ .
Substitute Value and Calculate: Substitute the value of $t$ into the function to calculate the concentration of salt after $10$ hours. $S(10) = 500 \cdot e^{(-0.25 \cdot 10)}$
Calculate Exponent: Calculate the exponent part of the function. $\newline$ $-0.25 \times 10 = -2.5$
Calculate e Value: Calculate the value of $e$ raised to the power of $-2.5$ . $\newline$ $e^{-2.5} \approx 0.082085$
Multiply by $500$ : Multiply the result from Step $4$ by $500$ to find the concentration of salt after $10$ hours. $\newline$ $S(10) = 500 \times 0.082085$
Perform Multiplication: Perform the multiplication to get the final result. $\newline$ $S(10) \approx 500 \times 0.082085 \approx 41.0425$
Round to Nearest Hundredth: Round the answer to the nearest hundredth as instructed. $\newline$ $S(10) \approx 41.04$ g/l

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