1. **State the problem:** We need to find the surface area of a cylinder with radius $r=4$ and diameter $d=15$. 2. **Recall the formula:** The surface area $A$ of a cylinder is given by $$A = 2\pi r^2 + 2\pi r h$$ where $r$ is the radius and $h$ is the height of the cylinder. 3. **Find the height:** The diameter $d$ is twice the radius, so $$d = 2r$$ Given $d=15$, we find $$r = \frac{d}{2} = \frac{15}{2} = 7.5$$ But the problem states $r=4$, so there is a discrepancy. We must clarify which is correct. Since both are given, we assume $r=4$ and $d=15$ means the height $h=15$ (likely the problem meant height $h=15$). 4. **Calculate the surface area:** Using $r=4$ and $h=15$, $$A = 2\pi (4)^2 + 2\pi (4)(15)$$ $$= 2\pi (16) + 2\pi (60)$$ $$= 32\pi + 120\pi$$ $$= 152\pi$$ 5. **Final answer:** The surface area is $$A = 152\pi \approx 477.52$$ This means the total surface area of the cylinder is approximately 477.52 square units.