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Difficulty: MediumLinear and Exponential Growth

A solar energy farm has two arrays of solar panels, Array A and Array B. At the beginning of 2024 (t=0t = 0), the generating capacity of Array A is 5050 megawatts and increases by 1111 megawatts each year. At the beginning of 2024, the generating capacity of Array B is 3232 megawatts and increases exponentially by a constant percent each year. If the generating capacities of the two arrays are equal at the beginning of 2026 (t=2t = 2), what is the generating capacity, in megawatts, of Array B at the beginning of 2027 (t=3t = 3)?

Answer: 108 megawatts

Answer

The correct answer is 108.
The capacity of Array A at t=2t = 2 is calculated using the linear growth model A(t)=50+11tA(t) = 50 + 11t, which yields A(2)=50+11(2)=72A(2) = 50 + 11(2) = 72 megawatts. Since the capacity of Array B grows exponentially and starts at 3232 megawatts, its model is B(t)=32⋅btB(t) = 32 \cdot b^t, where bb is the annual growth factor. At t=2t = 2, their capacities are equal, so 32⋅b2=7232 \cdot b^2 = 72. Dividing both sides by 3232 yields b2=2.25b^2 = 2.25. Taking the positive square root gives b=1.5b = 1.5. To find the capacity of Array B at t=3t = 3, we compute B(3)=32⋅(1.5)3=108B(3) = 32 \cdot (1.5)^3 = 108 megawatts.

Step-by-Step Solution

1
Write the linear model for the capacity of Array A.
A(t)=50+11tA(t) = 50 + 11t, where tt is the number of years since the beginning of 2024.
Array A increases by a constant number of megawatts each year, which represents linear growth.
2
Calculate the capacity of Array A at t=2t = 2.
A(2)=50+11(2)=72A(2) = 50 + 11(2) = 72 megawatts.
We need to find the capacity of Array A at the beginning of 2026 to determine the capacity of Array B at that same time.
3
Represent the exponential model for the capacity of Array B.
B(t)=32⋅btB(t) = 32 \cdot b^t, where bb is the constant annual growth factor.
Array B increases exponentially by a constant percent each year, which is represented by a geometric sequence or exponential function.
4
Solve for the growth factor bb using the capacity at t=2t = 2.
32⋅b2=72  ⟹  b2=2.25  ⟹  b=1.532 \cdot b^2 = 72 \implies b^2 = 2.25 \implies b = 1.5.
Since the capacities of both arrays are equal at the beginning of 2026 (t=2t = 2), we set B(2)=A(2)=72B(2) = A(2) = 72 to find the positive base bb.
5
Find the capacity of Array B at t=3t = 3.
B(3)=32⋅(1.5)3=108B(3) = 32 \cdot (1.5)^3 = 108 megawatts.
We substitute t=3t = 3 into the exponential model for Array B to find its capacity at the beginning of 2027.

Key Concept

Modeling linear and exponential growth, finding parameters of functions, and calculating values at specific time steps.

Alternative Method

Once the growth factor b=1.5b = 1.5 is found, the capacity of Array B at t=3t = 3 can be calculated by multiplying the capacity at t=2t = 2 by the growth factor: B(3)=B(2)⋅b=72⋅1.5=108B(3) = B(2) \cdot b = 72 \cdot 1.5 = 108 megawatts.
Estimated Time:1m 30s
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Topics

Linear and Exponential Growth

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