Here's the solution for the instantaneous acceleration: The velocity $v$ of a particle in a time $t$ is given by the equation $v = 10 + 2t^2$. We need to find the instantaneous acceleration after $5$ seconds. Step 1: Find the acceleration function by differentiating the velocity function with respect to time. Acceleration $a(t)$ is the derivative of velocity $v(t)$ with respect to time $t$: $$a(t) = \frac{dv}{dt}$$ Given $v(t) = 10 + 2t^2$: $$a(t) = \frac{d}{dt}(10 + 2t^2)$$ $$a(t) = \frac{d}{dt}(10) + \frac{d}{dt}(2t^2)$$ $$a(t) = 0 + 2(2t^{2-1})$$ $$a(t) = 4t$$ Step 2: Calculate the instantaneous acceleration at $t = 5\text{s}$. Substitute $t = 5$ into the acceleration function: $$a(5) = 4(5)$$ $$a(5) = 20\text{m/s}^2$$ Comparing this result with the given options: A. $60\text{m/s}^2$ B. $15\text{m/s}^2$ C. $20\text{m/s}^2$ D. $10\text{m/s}^2$ The calculated instantaneous acceleration matches option C. The final answer is $\boxed{\text{C. 20 m/s^2}}$.