1. **State the problem:** We need to find the exact volume of a cylinder with diameter 3 inches and height 5 inches. 2. **Formula for the volume of a cylinder:** $$V = \pi r^2 h$$ where $r$ is the radius and $h$ is the height. 3. **Find the radius:** The diameter is 3 inches, so the radius is half of that: $$r = \frac{3}{2} = 1.5$$ inches. 4. **Substitute values into the formula:** $$V = \pi (1.5)^2 \times 5$$ 5. **Calculate the square of the radius:** $$1.5^2 = 2.25$$ 6. **Calculate the volume:** $$V = \pi \times 2.25 \times 5 = \pi \times 11.25$$ 7. **Simplify the volume:** $$V = 11.25\pi$$ cubic inches. 8. **Express as a fraction:** $$11.25 = \frac{45}{4}$$ so $$V = \frac{45}{4} \pi$$ cubic inches. 9. **Check the options:** The closest exact option is $45\pi$ cubic inches, but since our exact volume is $\frac{45}{4}\pi = 11.25\pi$, none of the options match exactly. 10. **Re-examine the problem:** The options are multiples of $\pi$ with integers: 15, 30, 45, 60. Since $11.25\pi$ is not listed, check if the diameter was confused with radius. If radius is 3 inches (diameter 6), volume would be: $$V = \pi \times 3^2 \times 5 = 45\pi$$ cubic inches. But the problem states diameter 3 inches, so radius 1.5 inches is correct. Therefore, the exact volume is: $$V = \frac{45}{4} \pi = 11.25\pi$$ cubic inches. Since none of the options match exactly, the closest is $15\pi$ cubic inches, but the exact volume is $11.25\pi$ cubic inches. **Final answer:** The exact volume is $$\boxed{\frac{45}{4} \pi}$$ cubic inches or $$11.25\pi$$ cubic inches.