To write a proportional equation for the height of the candle that has burned away (b) in terms of the time (t) in hours, we start with the information given:
- The candle is initially 3 inches tall.
- The candle burns down in 12 hours.
First, we can find the rate at which the candle burns down. Since it burns 3 inches in 12 hours, we can calculate the burn rate:
\[ \text{Burn rate} = \frac{\text{Height of the candle}}{\text{Time to burn}} = \frac{3 \text{ inches}}{12 \text{ hours}} = \frac{1}{4} \text{ inch per hour} \]
Next, we represent the amount burned (b) as a function of time (t). Since the candle burns at a rate of \(\frac{1}{4}\) inch per hour, we can express \(b\) as:
\[ b = \frac{1}{4}t \]
This equation shows that for every hour that passes, \(\frac{1}{4}\) inch of the candle burns away.
Thus, the proportional equation for the amount of the candle that has burned away (b) in inches in terms of time (t) in hours is:
\[ b = \frac{1}{4}t \]