问题1082: 指数函数

Solution to Problem 5

a) Equation for Exponential Decay

The general formula for exponential decay is:
\[
A(t) = A_0 \left(\frac{1}{2}\right)^{\frac{t}{h}}
\]
where:

• \( A(t) \) = amount remaining after time \( t \),
• \( A_0 \) = initial amount,
• \( h \) = half-life,
• \( t \) = time elapsed.

Given:

• Initial mass \( A_0 = 25 \) mg,
• Half-life \( h = 2 \) days.

Substitute the given values into the formula:
\[
A(t) = 25 \left(\frac{1}{2}\right)^{\frac{t}{2}}
\]

Final Equation:
\[
\boxed{A(t) = 25 \left(\frac{1}{2}\right)^{\frac{t}{2}}}
\]

b) Amount Remaining After 7 Days

Substitute \( t = 7 \) into the equation from part (a):
\[
A(7) = 25 \left(\frac{1}{2}\right)^{\frac{7}{2}}
\]

Simplify the exponent:
\[
\frac{7}{2} = 3.5
\]

Calculate \( \left(\frac{1}{2}\right)^{3.5} \):
\[
\left(\frac{1}{2}\right)^{3.5} = \left(\frac{1}{2}\right)^3 \times \left(\frac{1}{2}\right)^{0.5} = \frac{1}{8} \times \frac{1}{\sqrt{2}} = \frac{1}{8 \times 1.414} \approx \frac{1}{11.312} \approx 0.0884
\]

Multiply by the initial amount:
\[
A(7) = 25 \times 0.0884 \approx 2.21 \text{ mg}
\]

Final Answer:
\[
\boxed{2.21 \text{ mg}}
\]

Note: The calculation can also be done using logarithms or a calculator for more precision. The approximate value is rounded to two decimal places.

添加微信可以更快获取解答