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Home Continuous Functions How to Set Up a Problem Examples Example 4: Burned Out Area through a Brush Fire
 
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Example 4: Burned Out Area through a Brush Fire

A brush fire starts along a straight line segment of length 20 ft and expands in all directions at the rate of 2 ft per second. Find the burned out area as a function of time.

 

Step 1

A = total burned out area

A1 = area of left semicircle

A2 = area of central rectangle

A3 = area of right semicircle

s = distance of spread of fire

t = time

The diagram is shown in Figure 3.1.4.

03_continuous_functions-11.gif

Figure 3.1.4

Step 2

s = 2t, t ≥ 0.

A1 = ½πs2,

A2 = 20(2s),

A3 = ½πs2.

A = A1 + A2 + A3.

Step 3

A1 =½π(2t)2 = 2πt2.

A2 =20 · 2 · 2t = 80t.

A3 = ½π (2t)2 = 2πt2.

A = 2πt2 + 80t + 2πt2 = 4πt2 + 80t.

INTERPRET THE SOLUTION

The burned out area is

A = 4πt2 + 80t sq ft, t ≥ 0,

where t is time in seconds. An algebraic identity that comes up frequently in calculus problems is

(a - b)(a + b) = a2 - b2.

Sometimes it occurs in the form

(√a - √b)(√a + √b) = a - b.
Home Continuous Functions How to Set Up a Problem Examples Example 4: Burned Out Area through a Brush Fire

Last Update: 2006-11-15