Math Snap

STEP 1

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

STEP 2

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

STEP 3

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

STEP 4

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

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