A particle is moving with the given data. Find the position of the particle.a(t) = t2 − 9t + 5,    s(0) = 0,    s(1) = 20

To find the position of the particle, we need to integrate the acceleration function twice to get the position function.

Step 1: Find the velocity function, v(t), by integrating the acceleration function, a(t).

∫a(t) dt = ∫(t^2 - 9t + 5) dt = (1/3)t^3 - (9/2)t^2 + 5t + C1

Step 2: Use the initial condition s(0) = 0 to find the constant of integration, C1.

When t = 0, v(0) = 0, so 0 = (1/3)(0)^3 - (9/2)(0)^2 + 5(0) + C1, which gives C1 = 0.

So, v(t) = (1/3)t^3 - (9/2)t^2 + 5t.

Step 3: Find the position function, s(t), by integrating the velocity function, v(t).

∫v(t) dt = ∫[(1/3)t^3 - (9/2)t^2 + 5t] dt = (1/12)t^4 - (3/2)t^3 + (5/2)t^2 + C2

Step 4: Use the initial condition s(1) = 20 to find the constant of integration, C2.

When t = 1, s(1) = 20, so 20 = (1/12)(1)^4 - (3/2)(1)^3 + (5/2)(1)^2 + C2, which gives C2 = 20 - 1/12 + 3/2 - 5/2 = 20 - 1/12 - 2 = 18 - 1/12 = 215/12.

So, the position function is s(t) = (1/12)t^4 - (3/2)t^3 + (5/2)t^2 + 215/12.

This function gives the position of the particle at any time t.