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Zorluk: OrtaLinear and Exponential Growth

A scientist is tracking the growth of a cell culture. The cell culture's area, in square millimeters, can be modeled by either a linear function or an exponential function. The table below shows the area of the cell culture at the end of Day 1 and Day 2.

DayArea (square millimeters)
1150
2180

If the area of the cell culture grows exponentially, the area on Day 4 would be EE square millimeters. If the area grows linearly, the area on Day 4 would be LL square millimeters. What is the value of E−LE - L?

Cevap: 19.2 square millimeters

Cevap

19.2
For the linear model, the growth is 3030 square millimeters per day, so the area on Day 4 is 150+3(30)=240150 + 3(30) = 240 square millimeters. For the exponential model, the growth factor is 1.21.2 per day, so the area on Day 4 is 150×(1.2)3=259.2150 \times (1.2)^3 = 259.2 square millimeters. The difference E−LE - L is 259.2−240=19.2259.2 - 240 = 19.2.

Adım Adım Çözüm

1
Determine the linear growth model and calculate the area on Day 4.
L=240L = 240
Under a linear growth model, the area increases by a constant amount each day. The difference between Day 1 and Day 2 is 180−150=30180 - 150 = 30 square millimeters. Extending this pattern, the area on Day 3 is 180+30=210180 + 30 = 210 square millimeters, and the area on Day 4 is 210+30=240210 + 30 = 240 square millimeters.
2
Determine the exponential growth model and calculate the area on Day 4.
E=259.2E = 259.2
Under an exponential growth model, the area increases by a constant multiplier each day. The ratio of the area on Day 2 to Day 1 is 180150=1.2\frac{180}{150} = 1.2. Extending this pattern, the area on Day 3 is 180×1.2=216180 \times 1.2 = 216 square millimeters, and the area on Day 4 is 216×1.2=259.2216 \times 1.2 = 259.2 square millimeters.
3
Calculate the difference between the two models.
19.219.2
Subtract the linear model value from the exponential model value: E−L=259.2−240=19.2E - L = 259.2 - 240 = 19.2.

Anahtar Kavram

Distinguishing between linear growth (constant additive change) and exponential growth (constant multiplicative change).
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Konular

Linear and Exponential Growth

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