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Difficulty: EasyExponential Functions and Equations

A certain substance decays radioactively such that the mass of the substance, in grams, remaining after tt days is modeled by the function M(t)=80⋅2−t5M(t) = 80 \cdot 2^{-\frac{t}{5}}. After how many days will the mass of the substance be 1010 grams?

Answer: 15 days

Answer

15
To find the number of days after which the mass of the substance is 1010 grams, we substitute 1010 for M(t)M(t) in the given function, yielding 10=80⋅2−t510 = 80 \cdot 2^{-\frac{t}{5}}. Dividing both sides by 8080 gives 1080=2−t5\frac{10}{80} = 2^{-\frac{t}{5}}, which simplifies to 18=2−t5\frac{1}{8} = 2^{-\frac{t}{5}}. Since 18\frac{1}{8} can be written as 2−32^{-3}, the equation becomes 2−3=2−t52^{-3} = 2^{-\frac{t}{5}}. Because the bases are the same, we equate the exponents: −3=−t5-3 = -\frac{t}{5}. Multiplying both sides by −5-5 gives t=15t = 15.

Step-by-Step Solution

1
Set the mass M(t)M(t) equal to 1010 in the given function.
10=80⋅2−t510 = 80 \cdot 2^{-\frac{t}{5}}
We want to find the value of tt when the remaining mass of the substance is 1010 grams.
2
Divide both sides of the equation by 8080.
18=2−t5\frac{1}{8} = 2^{-\frac{t}{5}}
To isolate the exponential expression.
3
Express both sides of the equation with a common base of 22.
2−3=2−t52^{-3} = 2^{-\frac{t}{5}}
Since 8=238 = 2^3, the fraction 18\frac{1}{8} can be written as 2−32^{-3}. Having the same base on both sides allows us to equate the exponents.
4
Set the exponents equal to each other and solve for tt.
t=15t = 15
Because the bases are equal, the exponents must be equal: −3=−t5-3 = -\frac{t}{5}.

Key Concept

Solving exponential equations by finding a common base.
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Topics

Exponential Functions and Equations

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