Solved on Feb 07, 2024

A lake covers 11 square km, decreasing exponentially by 2% per year: A(t)=11(0.98)tA(t)=11 \cdot(0.98)^{t}. By what factor does the area decrease in 10 years?

STEP 1

Assumptions
1. The initial area of the lake is 11 square kilometers.
2. The area decreases exponentially at a rate of 2% each year.
3. The exponential decay can be modeled by the function A(t)=11(0.98)tA(t)=11 \cdot(0.98)^{t}, where A(t)A(t) is the area after tt years.
4. We want to find the factor by which the area decreases after 10 years.

STEP 2

To find the factor by which the area decreases after 10 years, we need to evaluate the function A(t)A(t) at t=10t=10.
A(10)=11(0.98)10A(10)=11 \cdot(0.98)^{10}

STEP 3

Calculate the exponent part of the function by raising 0.98 to the power of 10.
0.98100.98^{10}

STEP 4

Use a calculator or any computational tool to compute the value of 0.98100.98^{10}.
0.98100.81790693740.98^{10} \approx 0.8179069374

STEP 5

Now, multiply the initial area of the lake by the calculated value to find the area after 10 years.
A(10)=110.8179069374A(10)=11 \cdot 0.8179069374

STEP 6

Perform the multiplication to find the area after 10 years.
A(10)110.81790693748.9969763104A(10) \approx 11 \cdot 0.8179069374 \approx 8.9969763104

STEP 7

To find the factor by which the area decreases, we divide the area after 10 years by the initial area.
Decrease factor=A(10)11\text{Decrease factor} = \frac{A(10)}{11}

STEP 8

Plug in the values to calculate the decrease factor.
Decrease factor=8.996976310411\text{Decrease factor} = \frac{8.9969763104}{11}

STEP 9

Calculate the decrease factor.
Decrease factor8.9969763104110.8179069374\text{Decrease factor} \approx \frac{8.9969763104}{11} \approx 0.8179069374
The area decreases by a factor of approximately 0.8179 in 10 years.

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