/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 31 Surface Area and Volume A shampo... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 31

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.

Short Answer

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

Step by step solution

01

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鈥� volume formula. So: \(\pi r^2 h = \pi x^2 y\)
02

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.
03

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.

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