The growth of the size of a wild fire can be estimated by the equation S(t) = 520(1.3)t, where represents the size in acres after t hours. How long will it take for the fire to cover 1200 acres? Round to the nearest hundredth.

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Answer 1

To find out how long it will take for the fire to cover 1200 acres, we need to solve the equation S(t) = 1200 for t.

S(t) = 520(1.3)^t
1200 = 520(1.3)^t

Divide both sides of the equation by 520:
(1.3)^t = 1200/520
(1.3)^t = 2.3077

To solve for t, we need to take the logarithm of both sides of the equation. Let's use the natural logarithm (ln):
ln((1.3)^t) = ln(2.3077)

Using the logarithmic property ln(a^b) = b ln(a), we can simplify the left side of the equation:
t ln(1.3) = ln(2.3077)

Divide both sides of the equation by ln(1.3):
t = ln(2.3077) / ln(1.3)

Using a calculator, we can find that ln(2.3077) ≈ 0.8364 and ln(1.3) ≈ 0.2624.

So, t ≈ 0.8364 / 0.2624 ≈ 3.1852.

Rounded to the nearest hundredth, it will take approximately 3.19 hours for the fire to cover 1200 acres.