To find the new volume of the cylinder after increasing the radius by 3 cm, we need to understand the formula for the volume of a cylinder, which is V = πr²h, where V is the volume, r is the radius, and h is the height.
1. Initially, the volume of the cylinder is 150 cm³, and we are increasing the radius by 3 cm. Let's assume the initial radius is r₁ and the new radius after the increase is r₂ = r₁ + 3.
2. Since the height remains constant, the only change affecting the volume is the increase in the radius. Therefore, the new volume can be calculated as V₂ = π(r₂)²h.
3. Substituting r₂ = r₁ + 3 into the formula, we get V₂ = π(r₁ + 3)²h.
4. Expanding the equation, V₂ = π(r₁² + 6r₁ + 9)h.
5. Simplifying further, V₂ = πr₁²h + 6πr₁h + 9πh.
6. We know that the initial volume V₁ is 150 cm³, so V₁ = πr₁²h.
7. Therefore, V₂ = V₁ + 6πr₁h + 9πh.
8. Substituting the initial volume V₁ = 150 cm³, we get V₂ = 150 + 6πr₁h + 9πh.
9. This gives us the new volume of the cylinder after increasing the radius by 3 cm while keeping the height constant.
