it has been observed that a particular plant's growth is directly proportional to time. it measured 2 cm when arrived at the nursery and 2.5 cm exactly one week later. if the plant continues to grow at this rate, determine the function that represents the plants growthand graph it.
growth in one week = .5 cm
rate of growth in that week = .5/2 = .25
f(t) = 2(1 + .25)^t, where t is in days
f(t) = 2(1.25)^t
check: when t = 1, f(1) = 2(1.25) = 2.5 cm
after 4 weeks, f(4) = 2(1.25)^4 = 4.88 cm
Of course t will have to have some logical upper value, and the rate of
growth cannot continue like that.
dh/dt = 2kt
dh = 2kt dt
h = kt^2 + C
...
2 cm , 2.5 cm
well 2.5 cm = 2 cm * 1.25
so every week multiply by 1.25
2 , 2.5 ,
2.5*1.25 ,
2.5*1.25*1.25 ,
2.5*1.25*1.25 , *1.25 ....
that is a geometric sequence (Google math is fun geometric sequence)
at week n, height = a r^k
where a = 2.5 and r = 1.25
height zero = a = 2.5
height after 1 wk = a r^1 = 2.5 * 1.25
etc
exponential growth occurs when growth is proportional to the height ...
I think they mean the growth dH/dt is proportional to time, not the height R_Scott , but indeed the wording is unclear.
That's good
Thank you
This answer is good but your solution is really annoying because i din't really understand it but thank you i have a idea on how it
To find the function that represents the plant's growth, we need to determine the relationship between the growth and time.
Since it is mentioned that the plant's growth is directly proportional to time, we can assume that the growth of the plant is linear. This means that the plant's growth can be represented by a linear equation of the form y = mx + b, where y represents the plant's growth (in centimeters), x represents time (in weeks), m represents the rate of growth, and b represents the initial growth of the plant.
Given that the plant measured 2 cm when it arrived at the nursery (at t = 0) and 2.5 cm exactly one week later (at t = 1), we can use these data points to determine the values of m and b.
Let's substitute the values into the equation:
When t = 0 (initial growth): y = 2
When t = 1 (growth after one week): y = 2.5
Substituting these values into the equation, we get two equations:
2 = m(0) + b
2.5 = m(1) + b
The first equation can be simplified to: 2 = b
Substituting b = 2 into the second equation: 2.5 = m(1) + 2
Now, solve for m:
0.5 = m(1)
m = 0.5
So, the equation representing the plant's growth is:
y = 0.5x + 2
To graph this equation, plot the points that represent the growth of the plant (x, y) using the values of x from 0 to whatever time period you want to consider. Then, connect the dots to form a straight line.
h = plant height
h = 2 + .5t