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Zorluk: ZorLinear and Exponential Growth

At the beginning of 2015, City A and City B each had a daily water consumption of 1,200,0001,200,000 gallons. The daily water consumption of City A decreased linearly by 24,00024,000 gallons each year, and the daily water consumption of City B decreased exponentially by a constant percentage each year. At the beginning of 2025, the daily water consumption of City A was equal to the daily water consumption of City B. If these trends continue, what will be the daily water consumption of City B, in gallons, at the beginning of 2035?

Cevap: 768000 gallons

Cevap

768,000
To find the daily water consumption of City B at the beginning of 2035, we first determine the consumption of both cities at the beginning of 2025 (t=10t = 10 years after the beginning of 2015). For City A, which decreases linearly, the consumption is 1,200,000−10×24,000=960,0001,200,000 - 10 \times 24,000 = 960,000 gallons. Since City B's consumption is equal to City A's at this time and decreases exponentially, we set up the equation 1,200,000⋅b10=960,0001,200,000 \cdot b^{10} = 960,000, where bb is the annual decay factor. This gives b10=0.8b^{10} = 0.8. The consumption of City B at the beginning of 2035 (t=20t = 20) is given by 1,200,000⋅b201,200,000 \cdot b^{20}. Using exponent rules, we can rewrite b20b^{20} as (b10)2(b^{10})^2. Substituting b10=0.8b^{10} = 0.8 gives 1,200,000⋅(0.8)2=1,200,000⋅0.64=768,0001,200,000 \cdot (0.8)^2 = 1,200,000 \cdot 0.64 = 768,000.

Adım Adım Çözüm

1
Calculate the consumption of City A at the beginning of 2025.
960,000960,000 gallons
City A decreases linearly by a constant 24,00024,000 gallons per year for 1010 years starting from 1,200,0001,200,000 gallons.
2
Set up the exponential model for City B at the beginning of 2025.
b10=0.8b^{10} = 0.8
City B's consumption decreases exponentially, so it is modeled by WB(t)=1,200,000⋅btW_B(t) = 1,200,000 \cdot b^t. Since it equals City A's consumption at t=10t = 10, 1,200,000⋅b10=960,0001,200,000 \cdot b^{10} = 960,000.
3
Determine the consumption of City B at the beginning of 2035.
768,000768,000 gallons
At the beginning of 2035 (t=20t = 20), City B's consumption is 1,200,000⋅b201,200,000 \cdot b^{20}. Since b20=(b10)2b^{20} = (b^{10})^2, we substitute 0.80.8 for b10b^{10} to get 1,200,000⋅(0.8)2=1,200,000⋅0.64=768,0001,200,000 \cdot (0.8)^2 = 1,200,000 \cdot 0.64 = 768,000.

Anahtar Kavram

Distinguishing between linear and exponential models and applying exponent properties to solve exponential growth/decay problems.

Alternatif Yöntem

Alternatively, you can solve for the annual decay factor bb directly using a calculator. Since b10=0.8b^{10} = 0.8, we take the 10th root of both sides to get b=0.80.1≈0.97793b = 0.8^{0.1} \approx 0.97793. The water consumption at t=20t = 20 is then calculated as 1,200,000⋅(0.97793)20≈768,0001,200,000 \cdot (0.97793)^{20} \approx 768,000 gallons. Recognizing the algebraic shortcut (b10)2=b20(b^{10})^2 = b^{20} avoids decimal approximations and is much faster.
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Konular

Linear and Exponential Growth

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