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Math  /  Calculus

Questiond Help? Read It Watch it Mastor 1 hit Answer 37 Points] DETAILS MY NOTES LARAPCALC10 5.1.070. ASK YOUR TEACHER PRACTICE ANOTHER en planted (t=0)(t=0). a) Find the height after tt years. h(t)=h(t)= (b) How tall are the shrubs when they are sold? □\square Need Help? Read It Watch lit Submit Answer Home My Assignments Request Extension

Studdy Solution

STEP 1

1. The growth of the shrubs is described by the differential equation dhdt=1.2t+9\frac{dh}{dt} = 1.2t + 9.
2. The initial height of the shrubs at t=0t = 0 is 11 centimeters.
3. We need to find the height function h(t)h(t) after tt years.
4. The shrubs are sold after 5 years.

STEP 2

1. Solve the differential equation to find the function h(t)h(t).
2. Determine the height of the shrubs when they are sold.

STEP 3

Solve the differential equation dhdt=1.2t+9\frac{dh}{dt} = 1.2t + 9.
To do this, integrate both sides with respect to tt:
∫dhdt dt=∫(1.2t+9) dt \int \frac{dh}{dt} \, dt = \int (1.2t + 9) \, dt
h(t)=∫(1.2t+9) dt h(t) = \int (1.2t + 9) \, dt
h(t)=1.22t2+9t+C h(t) = \frac{1.2}{2}t^2 + 9t + C
h(t)=0.6t2+9t+C h(t) = 0.6t^2 + 9t + C

STEP 4

Use the initial condition to find the constant CC.
Given that h(0)=11h(0) = 11, substitute into the equation:
11=0.6(0)2+9(0)+C 11 = 0.6(0)^2 + 9(0) + C
C=11 C = 11
Thus, the height function is:
h(t)=0.6t2+9t+11 h(t) = 0.6t^2 + 9t + 11

STEP 5

Determine the height of the shrubs when they are sold after 5 years.
Substitute t=5t = 5 into the height function:
h(5)=0.6(5)2+9(5)+11 h(5) = 0.6(5)^2 + 9(5) + 11
h(5)=0.6×25+45+11 h(5) = 0.6 \times 25 + 45 + 11
h(5)=15+45+11 h(5) = 15 + 45 + 11
h(5)=71 h(5) = 71
The height of the shrubs when they are sold is:
71 cm \boxed{71 \text{ cm}}

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