Exponential model for water hyacinth mat diameter D(t)=a(b)^t — find a and b from D(1)=4.2 and D(2)=4.41

Step-by-step solution to find the exponential function D(t)=a(b)^t for the water hyacinth mat given D(1)=4.2 and D(2)=4.41. Result: D(t)=4(1.05)^t.

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We are given D(t)=a(b)^t and two values: D(1)=4.2 and D(2)=4.41.

  1. Use t=1: a·b = 4.2.
  2. Use t=2: a·b^2 = 4.41.
  3. Divide the second equation by the first to eliminate a: b = 4.41 / 4.2 = 1.05.
  4. Substitute b back into a·b = 4.2: a = 4.2 / 1.05 = 4.

Therefore, the model is D(t) = 4(1.05)^t. (Check: D(1)=4·1.05=4.2, D(2)=4·1.05^2=4·1.1025=4.41.)

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