To solve the problem, we need to find the displacement of a particle that starts from rest and moves with a time-varying acceleration given by the equation:
\[ a(t) = 30 - \frac{15}{2} \sqrt{t} \]
where \( t \) is in seconds. The steps to find the displacement in 4 seconds are as follows:
### Step 1: Find the velocity as a function of time
The acceleration \( a(t) \) is the derivative of velocity \( v(t) \) with respect to time \( t \):
\[
a(t) = \frac{dv}{dt}
\]
We can express this as:
\[
dv = a(t) \, dt = \left(30 - \frac{15}{2} \sqrt{t}\right) dt
\]
Now, we integrate both sides from \( t = 0 \) to \( t = 4 \) and from \( v = 0 \) to \( v \):
\[
\int_0^v dv = \int_0^4 \left(30 - \frac{15}{2} \sqrt{t}\right) dt
\]
### Step 2: Perform the integration
First, we calculate the integral on the right-hand side:
1. The integral of \( 30 \, dt \) from 0 to 4:
\[
\int_0^4 30 \, dt = 30t \bigg|_0^4 = 30 \times 4 - 30 \times 0 = 120
\]
2. The integral of \( -\frac{15}{2} \sqrt{t} \, dt \):
\[
\int_0^4 -\frac{15}{2} \sqrt{t} \, dt = -\frac{15}{2} \cdot \frac{2}{3} t^{3/2} \bigg|_0^4 = -\frac{15}{3} (4^{3/2} - 0) = -5 \cdot 8 = -40
\]
Now, combining these results:
\[
v = 120 - 40 = 80 \, \text{m/s}
\]
### Step 3: Find the displacement as a function of time
Now that we have the velocity as a function of time, we can find the displacement \( s \) by integrating the velocity function:
\[
s = \int_0^4 v(t) \, dt
\]
We express \( v(t) \) in terms of \( t \):
\[
v(t) = 30t - \frac{5}{2} t^{3/2}
\]
Now we integrate:
\[
s = \int_0^4 \left(30t - \frac{5}{2} t^{3/2}\right) dt
\]
### Step 4: Perform the integration for displacement
1. The integral of \( 30t \, dt \):
\[
\int_0^4 30t \, dt = 30 \cdot \frac{t^2}{2} \bigg|_0^4 = 15 \cdot (4^2 - 0) = 15 \cdot 16 = 240
\]
2. The integral of \( -\frac{5}{2} t^{3/2} \, dt \):
\[
\int_0^4 -\frac{5}{2} t^{3/2} \, dt = -\frac{5}{2} \cdot \frac{2}{5} t^{5/2} \bigg|_0^4 = -1 \cdot (4^{5/2} - 0) = -32
\]
Now, combining these results for displacement \( s \):
\[
s = 240 - 32 = 208 \, \text{m}
\]
### Final Answer
The displacement of the particle in 4 seconds is:
\[
\boxed{208 \, \text{m}}
\]
