During the early growth of a super-fast growing bamboo ‘Guadua Angustifolia’, found in Southern America andoften referred to as ‘Vegetal Steel’, the average height was measured (in mm) against the time (in days) afterobservations began. This was then modelled with the equation:H = 3e1.3t.Interpret the meaning of the ’3’ in the model’s equation.a)Calculate the height of the bamboo after 5 days.b)Calculate how much the plant grows during day 6, giving your answer to the nearest metre.c)Calculate the rate of increase of the plant at the start of day 6. Additionally, explain why this is differentto your answer to part (c).d)When graphing the data, explain why the researchers would have plotted ln(H) against t, rather thanln(H) against ln(t)e)Explain why the model may not be suitable for the lifetime of the bamboo. Use a suitable calculation tohelp

a) The '3' in the model's equation represents the initial height of the bamboo when the time (t) is 0.

b) To calculate the height of the bamboo after 5 days, we substitute t=5 into the equation: H = 3e^(1.3*5) = 3e^6.5 = 1468.26 mm or approximately 1.47 meters.

c) To calculate how much the plant grows during day 6, we need to find the difference in height between day 6 and day 5. Height on day 6: H = 3e^(1.36) = 3e^7.8 = 2414.57 mm Height on day 5: H = 3e^(1.35) = 3e^6.5 = 1468.26 mm So, the growth on day 6 is: 2414.57 mm - 1468.26 mm = 946.31 mm or approximately 0.95 meters.

d) The rate of increase of the plant at the start of day 6 can be calculated by taking the derivative of the equation with respect to time (t) and then substituting t=6: dH/dt = 31.3e^(1.36) = 3.9e^7.8 = 3137.94 mm/day or approximately 3.14 meters/day. This is different from the answer to part (c) because the rate of increase is not constant but increases exponentially with time.

e) When graphing the data, the researchers would have plotted ln(H) against t, rather than ln(H) against ln(t) because the original equation is an exponential function of t, not a power function of t. Taking the natural logarithm of both sides of the equation gives a linear relationship between ln(H) and t, which is easier to analyze and interpret.

f) The model may not be suitable for the lifetime of the bamboo because it assumes that the growth rate is constant and does not take into account factors such as nutrient availability, environmental conditions, and the natural limit to growth. For example, if we use the model to predict the height of the bamboo after 100 days, we get a height of 3e^(1.3*100) = 3e^130, which is astronomically large and clearly unrealistic.