Collapse the logarithms into one:
Then write both sides as powers of 3:
Simplify the left side and solve for :
But we're not done yet. Notice that both and
are negative when
. We can't take the logarithm of a negative number (the result isn't real-valued, anyway), so we throw this solution out, and we're left with just
Answer:
The barrel now is more than 1/4 full
Step-by-step explanation:
* A large barrel of oil is 1/4 full.
* Then for 7 1/2 minutes, oil is added to the barrel at a rate of
4.5 gallons per minute.
* Then for 6 1/2 minutes, some of the oil is removed at a rate of
-3 1/2 gallons per minute.
* We need to know The oil in the barrel now is more or less than
1/4 full
* To know that calculate the quantity of oil added in the barrel and the
quantity of oil removed from the barrel.
* <em>If the quantity of oil add is greater than the quantity of oil removed </em>
<em> then, the barrel is more than 1/4 full</em>
* <em>If the quantity of oil removed is greater than the quantity of oil </em>
<em> added then, the barrel is less than 1/4 full</em>
∵ The oil is added to the barrel at a rate of 4.5 gallons per minute
for minutes
∴ The quantity of oil add = 4.5 × 7.5 = 33.75 gallons
∵ The oil is removed from the barrel at a rate of -3.5 gallons per minute
for minutes
∴ The quantity of oil removed = -3.5 × 6.5 = -22.75 gallons
The negative sign to know that this quantity is removed
∵ The quantity of oil add is greater than the quantity of oil removed
∴ The barrel now is more than 1/4 full