Question

a forest fire has been burning for several days. the burned area, in acres, is given by the equation $y = (4,800) \cdot 2^d$, where $d$ is the number of days since the area of the fire was first measured.

part a
complete the table

d, days since first measurment	y, acres burned since fire started
-1
-2
-3
-5

part b
how much of the area had the fire burned a week before it measured 4,800 acres?
\boxed{} acres had burned the week before.

Explanation:

PART A

Step1: For \( d = 0 \)

Substitute \( d = 0 \) into \( y=(4800)\cdot2^{d} \). Since \( 2^{0}=1 \), we have \( y = 4800\times1 = 4800 \).

Step2: For \( d=- 1 \)

Substitute \( d=-1 \) into \( y=(4800)\cdot2^{d} \). Using the rule \( a^{-n}=\frac{1}{a^{n}} \), \( 2^{-1}=\frac{1}{2} \), so \( y = 4800\times\frac{1}{2}=2400 \).

Step3: For \( d = - 2 \)

Substitute \( d=-2 \) into \( y=(4800)\cdot2^{d} \). \( 2^{-2}=\frac{1}{2^{2}}=\frac{1}{4} \), so \( y = 4800\times\frac{1}{4} = 1200 \).

Step4: For \( d=-3 \)

Substitute \( d = - 3 \) into \( y=(4800)\cdot2^{d} \). \( 2^{-3}=\frac{1}{2^{3}}=\frac{1}{8} \), so \( y=4800\times\frac{1}{8}=600 \).

Step5: For \( d=-5 \)

Substitute \( d=-5 \) into \( y=(4800)\cdot2^{d} \). \( 2^{-5}=\frac{1}{2^{5}}=\frac{1}{32} \), so \( y = 4800\times\frac{1}{32}=150 \).

PART B

A week before the first measurement, \( d=-7 \) (since a week has 7 days). Substitute \( d = - 7 \) into the equation \( y=(4800)\cdot2^{d} \). We know that \( 2^{-7}=\frac{1}{2^{7}}=\frac{1}{128} \). Then \( y = 4800\times\frac{1}{128}=\frac{4800}{128}=\frac{375}{10}=37.5 \)? Wait, no, wait: Wait, \( 4800\div128 = 37.5 \)? Wait, \( 128\times37 = 4736 \), \( 4800 - 4736=64 \), \( 64\div128 = 0.5 \), so \( 37.5 \). Wait, but let's check the formula again. The formula is \( y = 4800\cdot2^{d} \), where \( d \) is the number of days since the first measurement. So a week before, \( d=-7 \), so \( y = 4800\times2^{-7}=4800\times\frac{1}{128}=\frac{4800}{128}=37.5 \). Wait, but maybe I made a mistake. Wait, \( 2^{7}=128 \), so \( 4800\div128 = 37.5 \). So the area burned a week before is 37.5 acres? Wait, but let's check with the pattern from part A. When \( d=-1 \), \( y = 2400 \) (half of 4800), \( d=-2 \), \( 1200 \) (half of 2400), \( d=-3 \), 600 (half of 1200), \( d=-4 \), 300, \( d=-5 \), 150, \( d=-6 \), 75, \( d=-7 \), 37.5. Yes, that's the pattern of dividing by 2 each day before the measurement. So each day before, the area is half of the next day. So from \( d = 0 \) (4800), \( d=-1 \) (2400), \( d=-2 \) (1200), \( d=-3 \) (600), \( d=-4 \) (300), \( d=-5 \) (150), \( d=-6 \) (75), \( d=-7 \) (37.5). So the area burned a week before is 37.5 acres.

Answer:

(PART A Table):

d, days since first measurement	y, acres burned since fire started
-1	2400
-2	1200
-3	600
-5	150