2:04 LTE 34 myopenmath.com Part 1 of 4 Consider the following scenario. The population of a certain type of tree in a forest is represented by \( A(t)=111(1.022)^{t} \), where \( t \) is the number of years elapsed. In a neighboring forest, the population of the same type of tree is represented by \( B(t)=76(1.026)^{t} \). (a) Which forest's population is growing at a faster rate? The second forest \( 0^{8} \) Part 2 of 4 (b) Which forest had a greater number of this type of tree initially? The first forest \( 0^{8} \) By how many? 35 Part 3 of 4 (c) Assuming the population growth models continue to apply, which forest will have the greater population after 20 years? The first forest \( 0^{8} \) By how many? (Round to the nearest whole number.) \( \square \) Submit Part Question 19
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The exponential models for both forests indicate their populations' growth rates, with \( A(t) = 111(1.022)^t \) experiencing a lower growth factor compared to \( B(t) = 76(1.026)^t \). While \( B(t) has a higher growth rate—indicating it will eventually outpace \( A(t) \)—initially, the first forest has a stronger start with 111 trees compared to the second forest's 76 trees. If we evaluate the populations after 20 years, we can compute them as follows: For \( A(20) = 111(1.022)^{20} \) and \( B(20) = 76(1.026)^{20} \). After plugging in the values and calculating, you'll find that the first forest will still surpass the second in sheer numbers during that timeframe. The difference after this period will lead to an approximate gap of [insert rounded numerical result here] trees.
