**Toad Population Growth in a Wetland: Problem Set**The size of a population, \( P \), of toads \( t \) years after they are introduced into a wetland is given by the equation:\[P = \frac{8000}{1 + 49(1/2)^t}\]### Tasks:**(a) How many toads are there in year \( t = 0 \)? \( t = 5 \)? \( t = 10 \)?**Round your answers to the nearest toad.- In the year \( t = 0 \), there are [ ] toads.- In the year \( t = 5 \), there are [ ] toads.- In the year \( t = 10 \), there are [ ] toads.**(b) How long does it take for the toad population to reach 2100? 3150?**Round your answers to one decimal place.- The toad population reaches 2100 after [ ] years.- The toad population reaches 3150 after [ ] years.**(c) What is the maximum number of toads that the wetland can support?**Round your answer to the nearest toad.- The population levels off at about [ ] toads, so the maximum population is about [ ] toads.---This problem set requires students to apply a given mathematical model to determine specific population numbers at certain times, predict when the population reaches particular sizes, and ascertain the carrying capacity of the environment.

The size of population of toads is given by P(t)=80001+491/2t. To Find: (a) The number of toads in the year t=0, t=5, t=10. (b) Time taken by toad population to reach 2100, and 3150. (c) Maximum number of toads that wetland can support. (a)P(t)=80001+491/2t. At t=0 , P(0)=160. At t=5, P(5)=3160.494. At t=10, P(10)=7634.669. So, in the year t=0, there are 160 toads. So, in the year t=5, there are 3160.494 toads. So, in the year t=10, there are 7634.669 toads. (b) when P(t)=2100. 80001+491/2t=2100. ⇒1+491/2t=80002100⇒491/2t=80002100-1⇒491/2t=2.8571⇒1/2t=2.857149.⇒t ln(1/2)=0.05830.⇒t=4.124 thus, The population reaches 2100 after 4.124 years.(approx.) Now, when P(t)=3150, ⇒1+491/2t=80003150⇒491/2t=80003150-1⇒491/2t=1.5714⇒1/2t=1.571449.⇒t ln(1/2)=0.03206⇒t=4.992 Thus, the population reaches 3150 after 4.992 years (approx.).(c) To find maximize number of toads that wetland can support. We need to find maximum possible value of the function P(t)=80001+491/2t. take the limit t→∞, limt→∞80001+491/2t=8000 (since 12t→0 as t→∞). So, the maximum number of toads that wetland can support is 8000. The population levels off at bout 8000 toads, so the maximum population is about 8000 toads.