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Zorluk: ZorExponential Functions and Equations

An environmental study monitors the populations of two different plant species in a conservation area. The population of Species A is modeled by a linear function, A(t)=120+15tA(t) = 120 + 15t, where tt represents the number of years since the start of the study. The population of Species B is modeled by an exponential function, B(t)=c⋅dtB(t) = c \cdot d^t, where cc and dd are constants. At the start of the study (t=0t = 0), the population of Species A is 44 times the population of Species B. After 22 years, the population of Species A is equal to the population of Species B. What is the population of Species B after 44 years?

Cevap: 750

Cevap

The population of Species B after 44 years is 750750.
To find the population of Species B after 44 years, we evaluate the models at the given points. At t=0t = 0, A(0)=120A(0) = 120. Since the population of Species A is 44 times that of Species B at t=0t = 0, the initial population of Species B is 3030, which gives c=30c = 30. At t=2t = 2, A(2)=120+15(2)=150A(2) = 120 + 15(2) = 150. Since the populations are equal at t=2t = 2, we have B(2)=30⋅d2=150B(2) = 30 \cdot d^2 = 150, which simplifies to d2=5d^2 = 5. The population of Species B at t=4t = 4 is given by B(4)=30⋅d4=30⋅(d2)2=30⋅52=750B(4) = 30 \cdot d^4 = 30 \cdot (d^2)^2 = 30 \cdot 5^2 = 750.

Adım Adım Çözüm

1
Find the population of Species A at the start of the study (t=0t = 0)
A(0)=120A(0) = 120
This establishes the baseline population of Species A to find the corresponding initial population of Species B.
2
Determine the constant cc, which represents the initial population of Species B
c=30c = 30
Since the population of Species A is 44 times that of Species B at t=0t = 0, we solve 120=4c120 = 4c.
3
Calculate the population of Species A after 22 years (t=2t = 2)
A(2)=150A(2) = 150
This value is needed because the population of Species B equals the population of Species A at t=2t = 2.
4
Solve for the growth factor term d2d^2
d2=5d^2 = 5
Using the equality B(2)=150B(2) = 150, we solve 30⋅d2=15030 \cdot d^2 = 150.
5
Calculate the population of Species B after 44 years (t=4t = 4)
B(4)=750B(4) = 750
Using the exponential model B(t)=30⋅dtB(t) = 30 \cdot d^t, we find B(4)=30⋅d4=30⋅(d2)2=30⋅52B(4) = 30 \cdot d^4 = 30 \cdot (d^2)^2 = 30 \cdot 5^2.

Anahtar Kavram

Solving systems involving linear and exponential models using initial conditions and key points.
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Konular

Exponential Functions and Equations

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