AP Precalculus · Free-response skill

Build and use an exponential model

The move. From two data points, build an exponential model, interpret its parameters in context, and use it and its limits to predict.

How to do it

  1. Check that exponential fits: equal steps in time should multiply the quantity by the same factor.
  2. Find b from the ratio of two outputs: y2y1=bt2−t1\frac{y_2}{y_1} = b^{t_2 - t_1}. Then solve for a with either point. Keep full precision until the end.
  3. Interpret each parameter with units and a time: a is the amount when t is 0; b is the factor per unit of t, and b−1b - 1 as a percent is the rate per unit.
  4. To predict, plug in the input. To find when a target is reached, set the model equal to the target and solve with a log. This question type is in free-response Part A, where a graphing calculator is required.
  5. For limits, compare the model with a fact from the context, such as a maximum size. An exponential model keeps growing forever, so it fails once the real quantity runs out of room.

Practice prompt Try it

A fictional invasive water plant is spreading across Larkin Pond. A survey team measures the area it covers each week. One week after the survey starts, the plant covers 48 square meters. Three weeks after the survey starts, it covers 108 square meters. The surface of the pond is 2,000 square meters in all. The team models the covered area as A(t)=abtA(t) = ab^t, where t is weeks after the survey starts.

Area covered by the plant

Weeks after the survey starts, tArea covered, square meters
148
3108

Using the information above, answer each part. A graphing calculator is allowed.

  1. Work out the values of a and b for the model A(t)=abtA(t) = ab^t. Show how you got them.
  2. Say what a and b each tell you about the plant on Larkin Pond.
  3. Use the model to predict the area covered 6 weeks after the survey starts. Then find the value of t, to the nearest hundredth of a week, at which the model gives an area of 500 square meters.
  4. Give one reason, using a number, that the model should not be used to predict the area 12 weeks after the survey starts.

Write each part, then check yourself against the checklist. Nothing you type here is saved or graded.

Check yourself and see a model answer

Checklist

Our own checklist of what a full answer does, one line per point. Tick each one your response earns.

0 of 7 points

Model answer, annotated

(a) Dividing the two areas gives A(3)A(1)=ab3ab=b2=10848=2.25\frac{A(3)}{A(1)} = \frac{ab^3}{ab} = b^2 = \frac{108}{48} = 2.25, so b=1.5b = 1.5 (b must be positive). Then a⋅1.5=48a \cdot 1.5 = 48, so a=32a = 32. The model is A(t)=32(1.5)tA(t) = 32(1.5)^t.

  • Point 1 Uses the ratio of the outputs, so a cancels and b comes out directly.
  • Point 2 Substitutes b back into one data point to solve for a.

(b) a = 32 means the plant covered 32 square meters when the survey started, at t = 0. b = 1.5 means the covered area is multiplied by 1.5 each week, so it grows by 50 percent per week.

  • Point 3 Gives the starting area with its unit and the moment it describes.
  • Point 4 States the weekly factor and turns it into a percent rate.

(c) At 6 weeks, A(6)=32(1.5)6=364.5A(6) = 32(1.5)^6 = 364.5 square meters. For 500 square meters, 32(1.5)t=50032(1.5)^t = 500, so 1.5t=15.6251.5^t = 15.625 and t=ln⁡15.625ln⁡1.5≈6.78t = \frac{\ln 15.625}{\ln 1.5} \approx 6.78 weeks after the survey starts.

  • Point 5 Plugs the input into the model and keeps the unit.
  • Point 6 Isolates the power, then uses a log to solve for the time.

(d) The model gives A(12)=32(1.5)12≈4152A(12) = 32(1.5)^{12} \approx 4152 square meters, but the pond is only 2,000 square meters. The plant cannot cover more than the whole pond, so its growth must slow before week 12 and the model will overestimate.

  • Point 7 Backs the limit with a computed value and compares it with the size of the pond.