Problem 45 - Calculus - AP Edition

Question

A meteor enters the Earth's atmosphere and burns up at a rate that, at each instant, is proportional to its surface area. Assuming that the meteor is always spherical, show that the radius decreases at a constant rate.

Step-by-Step Solution

Answer

The radius of the meteor decreases at a constant rate \( \frac{dr}{dt} = -\frac{k}{2} \).

Step 1: Define the Problem Mathematically

Let the radius of the meteor at time \(t\) be \( r(t) \). Since the meteor is spherical, its surface area \(A\) at time \(t\) is \( A(t) = 4 \pi r(t)^2 \). The rate at which the meteor's surface area burns up is proportional to \(A(t)\), which can be expressed as \( \frac{dA}{dt} = -k A(t) \), where \(k\) is a positive constant.

Step 2: Differentiate the Surface Area Equation

Utilize the formula for the surface area of a sphere, \( A = 4 \pi r^2 \), and differentiate it with respect to \(t\). This gives \( \frac{dA}{dt} = 8 \pi r \frac{dr}{dt} \).

Step 3: Set up the Proportional Relation

By combining the differentiated surface area equation with the proportionality condition, we set \( 8 \pi r \frac{dr}{dt} = -k (4 \pi r^2) \). Simplifying, we have \( 2 r \frac{dr}{dt} = -kr^2 \).

Step 4: Solve the Differential Equation

Rearrange the equation from Step 3: \( \frac{dr}{dt} = -\frac{k}{2}r \). This indicates the rate of change of the radius is constant, i.e., \( \frac{dr}{dt} \) is a constant value.

Key Concepts

Differential Equations

Differential equations are mathematical equations that involve functions and their derivatives. These equations are essential in modeling how a quantity changes over time. In this problem, the quantity changing is the meteor's radius, as it burns up in the Earth's atmosphere. We start by defining the surface area of the meteor in terms of the radius: the formula is given by \( A(t) = 4 \pi r(t)^2 \). Here, \( r(t) \) is the radius of the meteor at time \( t \), and \( A(t) \) is the surface area. Differential equations help us understand how these quantities change with respect to time. By differentiating the surface area formula with respect to time, we have \( \frac{dA}{dt} = 8 \pi r \frac{dr}{dt} \). This expression gives the rate of change of the surface area concerning time, which is crucial for solving the problem of the meteor's shrinking radius. By solving the differential equation, we discovered that the radius decreases at a constant rate, indicating a uniform rate of burning.

Proportional Relationships

Proportional relationships describe a situation where two quantities increase or decrease at the same rate. In this exercise, the rate at which the meteor burns up is proportional to its surface area. This relationship is expressed mathematically as \( \frac{dA}{dt} = -k A(t) \), where \( k \) is a positive constant. The negative sign indicates that the surface area decreases over time. Proportional relationships are crucial in describing how the meteor's properties interact with its environment. With this relationship in place, the change in surface area over time is tied directly to the initial surface area at each moment. The solution utilizes this proportionality to eventually deduce that the radius of the meteor decreases at a constant rate, showcasing a fundamental concept in calculus where the rate of change of a variable is proportional to its current value or some power of it.

Spherical Geometry

Spherical geometry deals with the properties and dimensions of spheres, making it highly relevant when discussing naturally occurring spherical objects like meteors. In this problem, the meteor is assumed to remain spherical as it enters and burns up in the Earth's atmosphere. When dealing with spherical objects, the geometry provides specific formulas, such as the surface area formula \( A = 4 \pi r^2 \) and the volume formula \( V = \frac{4}{3} \pi r^3 \). These formulas help in understanding how the sphere's properties like surface area and volume change as its size, particularly its radius, changes. By understanding spherical geometry, we can correctly model how the surface area (and consequently, the burning rate) changes as the meteor decreases in size. This understanding allows us to frame and solve the differential equations accordingly, illustrating the decrease in radius is at a constant rate, as derived mathematically.