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Zorluk: ZorLinear and Exponential Growth

At an agricultural research station, the nitrogen level of two soil plots, Plot A and Plot B, is monitored over several weeks. The nitrogen level of Plot A increases linearly at a constant rate, and the nitrogen level of Plot B increases exponentially at a constant percentage rate. At week t=0t = 0, Plot A contains 150 grams150\text{ grams} of nitrogen and Plot B contains 80 grams80\text{ grams} of nitrogen. At week t=2t = 2, both plots contain the same amount of nitrogen, which is 180 grams180\text{ grams}. At week t=4t = 4, Plot B contains N gramsN\text{ grams} of nitrogen and Plot A contains A gramsA\text{ grams} of nitrogen. What is the value of N−AN - A?

Cevap: 195 grams

Cevap

195
The correct answer is 195195. At t=0t = 0, Plot A contains 150 grams150\text{ grams} and Plot B contains 80 grams80\text{ grams}. Since Plot A increases linearly and reaches 180 grams180\text{ grams} at t=2t = 2, its weekly rate of change is 180−1502=15 grams per week\frac{180 - 150}{2} = 15\text{ grams per week}. Thus, at week t=4t = 4, Plot A contains 150+15(4)=210 grams150 + 15(4) = 210\text{ grams}. Since Plot B increases exponentially and reaches 180 grams180\text{ grams} at t=2t = 2, its weekly growth factor bb satisfies 80b2=18080b^2 = 180, which gives b2=2.25b^2 = 2.25 and b=1.5b = 1.5. Thus, at week t=4t = 4, Plot B contains 80(1.5)4=405 grams80(1.5)^4 = 405\text{ grams}. The difference N−AN - A is 405−210=195405 - 210 = 195.

Adım Adım Çözüm

1
Find the nitrogen level of Plot A at week t=4t = 4 using a linear model.
A=210A = 210
Plot A grows linearly from an initial 150 grams150\text{ grams} at t=0t = 0 to 180 grams180\text{ grams} at t=2t = 2. The rate of increase is 180−1502=15 grams per week\frac{180 - 150}{2} = 15\text{ grams per week}. Thus, at t=4t = 4, the nitrogen level is 150+15(4)=210 grams150 + 15(4) = 210\text{ grams}.
2
Find the nitrogen level of Plot B at week t=4t = 4 using an exponential model.
N=405N = 405
Plot B grows exponentially from an initial 80 grams80\text{ grams} at t=0t = 0 to 180 grams180\text{ grams} at t=2t = 2. The weekly growth factor bb satisfies 80b2=18080b^2 = 180, so b2=2.25b^2 = 2.25 and b=1.5b = 1.5. Thus, at t=4t = 4, the nitrogen level is 80(1.5)4=80(5.0625)=405 grams80(1.5)^4 = 80(5.0625) = 405\text{ grams}.
3
Calculate the difference between the two nitrogen levels at week t=4t = 4.
195195
Subtracting AA from NN yields N−A=405−210=195N - A = 405 - 210 = 195.

Anahtar Kavram

Modeling linear growth (constant rate of change) and exponential growth (constant percentage rate of change or growth factor) over time.
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Konular

Linear and Exponential Growth

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