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A huge ice glacier in the Himalayas initially covered an area of 45 square kilometers. Because of changing weather patterns, this glacier begins to melt, and the area it covers begins to decrease exponentially.
The relationship between
A, the area of the glacier in square kilometers, and
t, the number of years the glacier has been melting, is modeled by the following equation.

A=45e^(-0.05 t)
How many years will it take for the area of the glacier to decrease to 15 square kilometers?
Give an exact answer expressed as a natural logarithm.
years

A huge ice glacier in the Himalayas initially covered an area of $45$ square kilometers. Because of changing weather patterns, this glacier begins to melt, and the area it covers begins to decrease exponentially. $\newline$ The relationship between $A$ , the area of the glacier in square kilometers, and $t$ , the number of years the glacier has been melting, is modeled by the following equation. $\newline$ $A=45 e^{-0.05 t}$ $\newline$ How many years will it take for the area of the glacier to decrease to $15$ square kilometers? $\newline$ Give an exact answer expressed as a natural logarithm. $\newline$ years

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Write exponential functions: word problems

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Q. A huge ice glacier in the Himalayas initially covered an area of

45

square kilometers. Because of changing weather patterns, this glacier begins to melt, and the area it covers begins to decrease exponentially.

\newline

The relationship between

A

, the area of the glacier in square kilometers, and

t

, the number of years the glacier has been melting, is modeled by the following equation.

\newline

A=45 e^{-0.05 t}

\newline

How many years will it take for the area of the glacier to decrease to

15

square kilometers?

\newline

Give an exact answer expressed as a natural logarithm.

\newline

years

Rephrase the Question: First, let's rephrase the ＂How many years will it take for the area of the glacier to decrease to $15$ square kilometers?＂
Given Exponential Decay Model: We are given the exponential decay model for the area of the glacier as a function of time: $A = 45e^{-0.05t}$ . We need to find the value of $t$ when $A = 15$ square kilometers.
Set $A$ equal to $15$ : To find $t$ , we set $A$ equal to $15$ and solve for $t$ : $15 = 45e^{(-0.05t)}$
Isolate Exponential Term: Divide both sides of the equation by $45$ to isolate the exponential term: $\newline$ $\frac{15}{45} = e^{-0.05t}$
Simplify Left Side: Simplify the left side of the equation: $\frac{1}{3} = e^{-0.05t}$
Take Natural Logarithm: Take the natural logarithm of both sides to solve for $t$ : $\ln(\frac{1}{3}) = \ln(e^{-0.05t})$
Simplify Right Side: Use the property of logarithms that $\ln(e^x) = x$ to simplify the right side of the equation: $\ln(\frac{1}{3}) = -0.05t$
Divide by $-0.05$ : Divide both sides by $-0.05$ to solve for $t$ : $t = \frac{\ln(\frac{1}{3})}{-0.05}$

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