From Mechanics

The maximum height is when the \displaystyle \mathbf{j} component of the velocity is zero.

Using \displaystyle \mathbf{v}=\mathbf{u}+\mathbf{a}t \ with \displaystyle \mathbf{u}=8\mathbf{i}+10\mathbf{j}\text{ m}{{\text{s}}^{\text{-1}}} and \displaystyle \ \mathbf{a}=-10\mathbf{j}\ \text{ m}{{\text{s}}^{\text{-2}}} ,

\displaystyle \mathbf{v}=8\mathbf{i}+10\mathbf{j}+(-10\mathbf{j})t \

Thus the ball is at its maximum height when \displaystyle t=1\text{ s}.

We use. the expression for the position vector obtained in part a), \displaystyle \mathbf{r}=(8\mathbf{i}+10\mathbf{j})t+\frac{1}{2}(-10\mathbf{j}){{t}^{\ 2}} .

The maximum height is the \displaystyle \mathbf{j} component of this vector at \displaystyle t=1\text{ s}

\displaystyle 10\times 1+\frac{1}{2}\left( -10 \right)\times {{1}^{2}}=5\ \text{m}