A lemon tree nursery usually sells a certain breed after 4 years of growth and shaping.the growth rate is approximated by: dh/dt =2.4t+3 where t is time in years and h is the height in cm. suppose that seedlings are 10 cm tall when planted (at t=0) a) find the height after t years? b) how tall are the lemon trees when they are sold
To find the height of the lemon trees after t years, we need to integrate the given growth rate equation, which is dh/dt = 2.4t + 3, with the initial condition h(0) = 10.
a) Finding the height after t years:
First, we need to integrate the growth rate equation with respect to t:
∫ dh = ∫ (2.4t + 3) dt
Integrating, we get:
h = 2.4 * (t^2)/2 + 3t + C
To find the value of the constant C, we can use the initial condition h(0) = 10:
10 = 2.4 * (0^2)/2 + 3 * 0 + C
10 = 0 + 0 + C
C = 10
Substituting the value of C back into the equation:
h = 2.4 * (t^2)/2 + 3t + 10
So, the height of the lemon trees after t years is given by the equation h = 1.2t^2 + 3t + 10.
b) To find the height of the lemon trees when they are sold (after 4 years of growth), we can substitute t = 4 into the equation we found in part a:
h = 1.2(4)^2 + 3(4) + 10
h = 1.2(16) + 12 + 10
h = 19.2 + 12 + 10
h = 41.2 cm
Therefore, the lemon trees are approximately 41.2 cm tall when they are sold after 4 years of growth.