Algebra 2 Quiz: Rearranging Formulas To Highlight Quantities

In geometry, the volume of a cone is $V = \dfrac{1}{3}\pi r^2 h$, where $r$ is radius and $h$ is height. Solve $V = \dfrac{1}{3}\pi r^2 h$ for $r$.

In optics, the thin lens equation is $\dfrac{1}{f}=\dfrac{1}{a}+\dfrac{1}{b}$, where $f$ is focal length, $a$ is object distance, and $b$ is image distance. Solve for $f$ in terms of $a$ and $b$.

A projectile's vertical displacement can be modeled by the kinematic equation $s=ut+\tfrac{1}{2}at^2$, where $s$ is displacement, $u$ is initial velocity, $a$ is acceleration, and $t$ is time. Solve $s=ut+\tfrac{1}{2}at^2$ for $t$ (a quadratic in $t$).

In physics, kinetic energy is $KE=\tfrac{1}{2}mv^2$, where $m$ is mass and $v$ is speed. Solve $KE=\tfrac{1}{2}mv^2$ for $v$ (include $\pm$ where appropriate).

In physics, the kinematic equation $v^2 = u^2 + 2as$ relates final speed $v$, initial speed $u$, acceleration $a$, and displacement $s$. Solve $v^2 = u^2 + 2as$ for $a$ (treat the other variables as constants, using the same inverse-operation reasoning as solving a numeric equation).

In physics, the kinematic equation $v^2 = u^2 + 2as$ relates final speed $v$, initial speed $u$, acceleration $a$, and displacement $s$. Solve $v^2 = u^2 + 2as$ for $a$ (treating the other variables as constants, as in solving a numeric equation).

In chemistry, a simplified relationship can be written as $P=\dfrac{nRT}{V}$, where $P$ is pressure, $n$ is moles, $R$ is a constant, $T$ is temperature, and $V$ is volume. Solve $P=\dfrac{nRT}{V}$ for $T$.

In geometry, the volume of a cone is $V=\dfrac{1}{3}\pi r^2h$, where $r$ is radius and $h$ is height. Solve $V=\dfrac{1}{3}\pi r^2h$ for $r$.

In algebra-based modeling, suppose $y = \dfrac{kx}{x+a}$ where $a$ and $k$ are constants and $x\neq -a$. Solve $y = \dfrac{kx}{x+a}$ for $x$ in terms of $y, a, k$.

In chemistry, the ideal gas law is $PV = nRT$, where $P$ is pressure, $V$ is volume, $n$ is number of moles, $R$ is a constant, and $T$ is temperature. Solve $PV = nRT$ for $T$.

In algebraic modeling, suppose a quantity is defined by $y=\dfrac{x-a}{b}+c$, where $a$, $b$, and $c$ are constants and $b\ne 0$. Isolate $x$ in $y=\dfrac{x-a}{b}+c$.

A physics student needs to determine the acceleration of an object given the formula $F = ma + kv^2$, where $F$ is the total force, $m$ is mass, $a$ is acceleration, $k$ is a drag coefficient, and $v$ is velocity. Which expression correctly isolates the acceleration?

The formula for the period of a pendulum is $T = 2\pi\sqrt{\frac{L}{g}}$, where $T$ is period, $L$ is length, and $g$ is gravitational acceleration. An engineer needs to determine what length produces a specific period. Which expression correctly gives $L$?

The compound interest formula $A = P(1 + \frac{r}{n})^{nt}$ can be rearranged to find the principal. If an investment grows to $8000 after 3 years with 6% annual interest compounded quarterly, which setup correctly finds the initial principal?

The surface area formula for a cylinder is $S = 2\pi r^2 + 2\pi rh$. A manufacturer wants to express the height in terms of the other variables to optimize production costs. Which rearrangement correctly isolates $h$?

The formula relating pressure, volume, and temperature for an ideal gas is $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$. A chemistry student needs to find the final temperature $T_2$ when the pressure doubles and volume is halved. Which setup is correct?

The relativistic energy formula is $E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}}$, where $E$ is energy, $m$ is rest mass, $c$ is speed of light, and $v$ is velocity. A physics student needs to solve for velocity $v$. Which expression is correct?

In economics, the break-even formula is $R = C + F$, where $R$ is revenue, $C$ is variable costs, and $F$ is fixed costs. If $R = px$ and $C = vx$ (where $p$ is price per unit, $v$ is variable cost per unit, and $x$ is quantity), which expression gives the break-even quantity?

The lens equation in optics is $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$, where $f$ is focal length, $d_o$ is object distance, and $d_i$ is image distance. A student needs to find the object distance. Which rearrangement is correct?

In geometry, the volume of a sphere is $V=\dfrac{4}{3}\pi r^3$. Solve for the radius $r$ in terms of $V$.