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
dh/dt = 2.4t + 3
h = 1.2 t^2 + 3t + c
h(0) = 10, so c=10 and
h(t) = 1.2 t^2 + 3t + 10
h(4) = 41.2
To find the height of the lemon trees after t years, we need to integrate the given growth rate equation.
a) To find the height after t years, we integrate the given equation: dh/dt = 2.4t + 3.
∫ dh = ∫ (2.4t + 3) dt
Integrating both sides gives us:
h = ∫ (2.4t + 3) dt
Integrating the equation on the right side yields:
h = 1.2t^2 + 3t + C
Where C is the constant of integration. Since we know that the seedlings are 10 cm tall when planted (at t=0), we can plug in this information to find the value of C:
h = 1.2t^2 + 3t + C
10 = 1.2(0)^2 + 3(0) + C
10 = C
So the equation for the height after t years is:
h = 1.2t^2 + 3t + 10
b) To find the height of the lemon trees when they are sold, we substitute the value of t into the equation for height:
h = 1.2t^2 + 3t + 10
Since the nursery sells the breed after 4 years of growth and shaping, we substitute t = 4 into the equation:
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.