Solve the problem.
-Approximately one-fourth of all glass bottles distributed will be recycled each year. A beverage company distributes 380,000 bottles. The number still in use after t years is given by the function $\mathrm { N } ( \mathrm { t } ) = 380,000 \left( \frac { 1 } { 4 } \right) ^ { \mathrm { t } }$. After how many years will 3000 bottles still be in use? Round your answer to the nearest tenth.
A) $3.5 \mathrm { yr }$
B) $6.7 \mathrm { yr }$
C) $4.8 \mathrm { yr }$
D) $0.3 \mathrm { yr }$

Question

Quizplus · Accepted Answer

The number of bottles still in use after t years is given by the function $N(t) = 380,000 \left( \frac { 1 } { 4 } \right) ^ { t }$. We want to find the value of t such that $N(t) = 3000$. Substituting this into the function, we get:$$3000 = 380,000 \left( \frac { 1 } { 4 } \right) ^ { t }$$Dividing both sides by 380,000, we get:$$\frac { 3000 } { 380,000 } = \left( \frac { 1 } { 4 } \right) ^ { t }$$Simplifying the left side, we get:$$\frac { 3 } { 380 } = \left( \frac { 1 } { 4 } \right) ^ { t }$$Taking the logarithm of both sides, we get:$$\log _ { \frac { 1 } { 4 } } \left( \frac { 3 } { 380 } \right) = t$$Using a calculator, we get:$$t \approx 3.5$$Therefore, the answer is A) $3.5 \mathrm { yr }$.

Solve the Problem N(t)=380,000(14)t\mathrm { N } ( \mathrm { t } ) = 380,000 \left( \frac { 1 } { 4 } \right) ^ { \mathrm { t } }N(t)=380,000(41​)t

Solve the Problem $\mathrm { N } ( \mathrm { t } ) = 380,000 \left( \frac { 1 } { 4 } \right) ^ { \mathrm { t } }$