Problem 31 - Calculus of a Single Variable

Question

Surface Area and Volume A shampoo bottle is a right circular cylinder. Because the surface area of the bottle does not change when it is squeezed, is it true that the volume remains the same? Explain.

Step-by-Step Solution

Answer

No, the volume of the bottle does not remain the same when it is squeezed, even if the surface area remains the same.

Step 1: Formulate the problem

Suppose the initial radius and height of the cylinder are r and h respectively. When the bottle is squeezed, let the new radius and height be x and y. Since the surface area of the bottle remains the same when it is squeezed, the formula for the surface area before and after squeezing equates thus: \(2 \pi r (h + r) = 2 \pi x (y + x)\). In another case, to check if the volume remains constant, equate the initial volume formula with the hypothetically changed scenarios’ volume formula. So: \(\pi r^2 h = \pi x^2 y\)

Step 2: Analyze the problem

Observe, it will not always be possible to find a y and x (new height and radius after squeezing) to satisfy both equations for arbitrary values of r and h (initial radius and height). The volume does not remain constant when the bottle is squeezed, even if the surface area remains the same.

Step 3: Conclusion

Regardless of the fact that the surface area remains consistent, the volume of the bottle doesn't. Therefore, it's not accurate to say that in all cases the volume remains the same when a bottle is squeezed.