The overestimate: -2 m/s .
To overestimate the relative velocity of the rising ball, we will use the left Riemann sum.
This means that we will evaluate the acceleration function at the left endpoint of each time interval and sum the products.
We divide the interval [7, 11] into 4 subintervals of equal length:
[7, 8]
[8, 9]
[9, 10]
[10, 11]
The left Riemann sum is then given by:
Σ a(t_i) Δt
where t
i is the left endpoint of the $i$th subinterval and Δt is the common width of the subintervals.
In our case, Δt = (11 - 7) / 4 = 1.
Evaluating the acceleration function at the left endpoints of the subintervals, we get:
a(7) = -5/6 m/s²
a(8) = -1 m/s²
a(9) = -1/6 m/s²
a(10) = 1/6 m/s²
Therefore, the overestimate of the relative velocity is:
Σ a(t_i) Δt = (-5/6 - 1 - 1/6 + 1/6) * 1 m/s = -2 m/s
Therefore, the answer is:
Overestimate: -2 m/s