SOLUTION: A botanist starts with 3 plants. She takes 5 cuttings from each plant to start new plants. Later, she takes 5 cuttings from each new plant, and so on. a) Draw a graph of the numb

Question 1171575: A botanist starts with 3 plants. She takes 5 cuttings from each plant
to start new plants. Later, she takes 5 cuttings from each new plant, and so on.
a) Draw a graph of the number of new plants in each of the first 5 rounds of cuttings.
b) Write an equation to model the number of new plants, P, in the nth round of cuttings.
c) How would the graph and equation change in each scenario? Explain.
i) The botanist starts with 10 plants.
ii) In each round, the botanist takes 2 cuttings from each plant.
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Let's break this down step by step:
**a) Graph of New Plants**
* **Round 1:**
* Starts with 3 plants.
* Takes 5 cuttings from each: 3 plants * 5 cuttings/plant = 15 new plants.
* **Round 2:**
* Starts with 15 plants.
* Takes 5 cuttings from each: 15 plants * 5 cuttings/plant = 75 new plants.
* **Round 3:**
* Starts with 75 plants.
* Takes 5 cuttings from each: 75 plants * 5 cuttings/plant = 375 new plants.
* **Round 4:**
* Starts with 375 plants.
* Takes 5 cuttings from each: 375 * 5 = 1875
* **Round 5:**
* Starts with 1875 plants.
* Takes 5 cuttings from each: 1875 * 5 = 9375.
Graph:
* X-axis: Round number (1, 2, 3, 4, 5)
* Y-axis: Number of new plants (15, 75, 375, 1875, 9375)
The graph will show an exponential growth curve.
**b) Equation to Model New Plants (P)**
* We see a pattern:
* Round 1: 3 * 5^1 = 15
* Round 2: 3 * 5^2 = 75
* Round 3: 3 * 5^3 = 375
* Round 4: 3*5^4 = 1875
* Round 5: 3*5^5 = 9375
* Therefore, the equation is: P = 3 * 5^n, where P is the number of new plants and n is the round number.
**c) Changes in Graph and Equation**
**i) Starting with 10 Plants:**
* **Equation:**
* The initial factor changes from 3 to 10.
* P = 10 * 5^n
* **Graph:**
* The y-intercept (starting point) of the graph will be higher.
* The exponential growth will be steeper, as there are more plants at each round.
**ii) Taking 2 Cuttings from Each Plant:**
* **Equation:**
* The base of the exponent changes from 5 to 2.
* P = 3 * 2^n
* **Graph:**
* The graph will still show exponential growth, but it will be less steep.
* The number of new plants will increase at a slower rate compared to 5 cuttings per plant.

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