A Spherical Balloon with Radius $r$ Inches Has Volume $4 \frac { \pi r ^ { 3 } } { 3 }$

question 67

Short Answer

A spherical balloon with radius $r$ inches has volume
$4 \frac { \pi r ^ { 3 } } { 3 }$ .
Find a function that represents the amount of air required to inflate the balloon from a radius of $r$ inches to a radius of $r + 3$ inches.

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Definitions:

β-carotene

β-carotene is a strongly colored red-orange pigment abundant in plants and fruits, a precursor of vitamin A, and an antioxidant.

Conjugated

Refers to a system in organic chemistry where alternating single and multiple bonds result in enhanced stability and unique reactivity.

Carbocation Intermediate

A positively charged carbon ion, often found as a reactive intermediate in organic reactions, characterized by a carbon atom having only six electrons in its valence shell.

Resonance Contributors

Describes the various Lewis structures that represent a molecule's bonding and electron arrangement, highlighting how electrons can delocalize across the molecule.