the radioactive substance uranium - 240 has a...
the radioactive substance uranium - 240 has a half - life of 14 hours. the amount a(t) of a sample of uranium - 240 remaining (in grams) after t hours is given by the following exponential function. a(t)=4700(1/2)^(t/14) find the initial amount in the sample and the amount remaining after 50 hours. round your answers to the nearest gram as necessary. initial amount: grams amount after 50 hours: grams
Answer
# Explicación:
## Paso 1: Encontrar la cantidad inicial
Para encontrar la cantidad inicial, sustituimos \(t = 0\) en la función \(A(t)=4700\left(\frac{1}{2}\right)^{\frac{t}{14}}\).
$$A(0)=4700\left(\frac{1}{2}\right)^{\frac{0}{14}}=4700\left(\frac{1}{2}\right)^0$$
Como cualquier número (excepto 0) elevado a la potencia 0 es 1, entonces \(A(0) = 4700\).
## Paso 2: Encontrar la cantidad después de 50 horas
Sustituimos \(t = 50\) en la función \(A(t)=4700\left(\frac{1}{2}\right)^{\frac{t}{14}}\).
$$A(50)=4700\left(\frac{1}{2}\right)^{\frac{50}{14}}=4700\left(\frac{1}{2}\right)^{\frac{25}{7}}$$
Calculamos \(\left(\frac{1}{2}\right)^{\frac{25}{7}}\approx0.057\).
Luego \(A(50)=4700\times0.057\approx268\).
# Respuesta:
Initial amount: 4700 grams
Amount after 50 hours: 268 grams