L2 Algebra
Question Bank
Solve for $x$ in the equation $3^x = 81$.
A small town has an initial population of 5000 people. The population is growing at a steady rate of 3% each year. Write an expression for the population, $P$, of the town after $t$ years.
The value, $V$, of a piece of equipment in dollars is modelled by the function $V = 80000 \times (0.92)^t$, where $t$ is the number of years since it was purchased. Explain what the $0.92$ represents in the context of this function.
The population, $P$, of a small town is modelled by the function $P = 15000 \times 1.035^t$, where $t$ is the number of years since the model began. Explain what the $1.035$ represents in the context of this function.
Solve the following equation for $x$. Give your answer rounded to three significant figures.
$7^x = 555$Solve $8^{x-1} = (\frac{1}{4})^{2x}$.
A radioactive substance decays at a constant rate of 4% per year. The mass $M$ of the substance after $t$ years can be modelled by the function $M(t) = M_0 r^t$, where $M_0$ is the initial mass. Calculate the half-life of the substance (the time it takes for the mass to reduce to half of its initial amount).
Liam invests \$5,000 into an account with a fixed annual interest rate of 4.5%. The value of his investment, $\$V$, after $t$ years can be modelled by a function of the form $V = Pr^t$. How long will it take for his investment to be worth \$10,000? Assume that the interest gets added on at the end of the year.
Given the equation $6^{2x} + 6^{2x} + 6^{2x} + 6^{2x} + 6^{2x} + 6^{2x} = 6^y$, find an expression for $y$ in terms of $x$.
A car is purchased for \$25,000 and its value depreciates by 15% each year. After how many full years will the car's value first be less than \$10,000?
The temperature, $T$ in degrees Celsius, of a cup of coffee can be modelled by the function $T(t) = 20 + ke^{-0.1t}$, where $t$ is the time in minutes after it is poured. The initial temperature of the coffee was 95°C. How long does it take for the coffee to cool down to 50°C?
At the beginnint of 2025 town A has a population of 12,000 and grows at 2% per year. Town B has 10,000 people and grows at 3.5% per year. During what year will Town B's population be larger than Town A's?
The value of a rare collectible grows according to the model $V(t) = Pk^t$. Two years after it was purchased, its value was \$1,210. Three years after it was purchased, its value was \$1,331. Find the initial price $P$ and the annual growth rate $k$.
Alice and Bob are tracking their savings. Alice's savings grow according to the investment model $A(t) = A_0(1+r)^t$, where $A_0$ is her initial investment and $r$ is a fixed annual growth rate.
Bob starts with triple Alice's initial investment, but his amount is fixed, so $B = 3A_0$.
Prove that the time, $T$, it takes for the value of Alice's savings to exceed the value of Bob's savings is given by:
$T > \frac{\log(3)}{\log(1+r)}$Solve the following system of simultaneous equations for $x$ and $y$.
$3^x \cdot 9^y = 1$ $2^{2x} \cdot 4^{-y} = \frac{1}{8}$A radioactive substance decays over time according to the exponential model $A(t) = A_0e^{-kt}$, where:
- $A(t)$ is the amount of the substance remaining after time $t$
- $A_0$ is the initial amount of the substance
- $k$ is a positive decay constant
Prove that the time, $\tau$, it takes for the substance to decay to half of its original amount (its half-life) is given by the formula:
$\tau = \frac{\ln(2)}{k}$A ball is dropped and bounces up to a height that is 75% of the height from which it was dropped. It then bounces again to a height that is 75% of the previous height and so on.
How many bounces does it make before it bounces up to less than 20% of the original height from which it was dropped?
