Finite Mathematics Quiz: Working With Formulas

The formula for the focal length of a thin lens 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 lens with focal length 15 cm creates an image that is twice as far from the lens as the object. What is the object distance?

In the compound interest formula $A = P(1 + r)^t$, if an investment triples in value over 8 years, what is the annual interest rate $r$ expressed as a percentage?

The break-even point formula is $x = \frac{F}{p - v}$, where $x$ is the number of units, $F$ is fixed costs, $p$ is price per unit, and $v$ is variable cost per unit. A company wants to reduce its break-even point from 500 units to 300 units. If fixed costs remain at $15,000 and the variable cost per unit remains at $12, what should be the new price per unit?

The formula for the period of a pendulum is $T = 2\pi\sqrt{\frac{L}{g}}$, where $T$ is the period, $L$ is length, and $g$ is gravitational acceleration. On a planet where $g = 4$ m/s², a pendulum has a period of $\pi$ seconds. If the same pendulum were moved to Earth where $g = 10$ m/s², what would be its new period?

The relationship between pressure, volume, and temperature for an ideal gas is given by $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$. A gas sample initially at 2 atm, 300 K, and 5 L undergoes a process where the pressure is tripled and the temperature increases to 450 K. The final volume is then used as the initial volume for a second process where the pressure returns to 2 atm and the temperature drops to 250 K. What is the final volume after both processes?

The sum of the first $n$ terms of an arithmetic sequence is given by the formula $S_n = \frac{n}{2}(2a_1 + (n-1)d)$, where $a_1$ is the first term and $d$ is the common difference. Which of the following correctly expresses the common difference $d$ in terms of $S_n$, $n$, and $a_1$?

The present value, $PV$, of an ordinary annuity is given by $PV = PMT \frac{1 - (1+i)^{-n}}{i}$, where $PMT$ is the regular payment, $i$ is the interest rate per period, and $n$ is the number of periods. Which of the following gives a correct expression for $n$?

The formula for the number of permutations of $n$ objects taken $k$ at a time, $P(n,k)$, can be defined recursively. For $k > 1$, $P(n, k)$ can be related to $P(n, k-1)$ by the formula $P(n, k) = (n - k + 1) \cdot P(n, k-1)$. How can $k$ be expressed in terms of $n$, $P(n, k)$, and $P(n, k-1)$?

A company's production cost, $C$, is modeled by the formula $C = F + Vx\left(1 - \frac{x}{E}\right)$, where $F$ is the fixed cost, $V$ is the variable cost per unit, $x$ is the number of units produced, and $E$ is an efficiency constant. Which of the following correctly expresses the efficiency constant $E$ in terms of the other variables?

The total surface area, $A$, of a right circular cylinder with radius $r$ and height $h$ is given by the formula $A = 2\pi r^2 + 2\pi rh$. Which of the following correctly expresses the height $h$?

The monthly payment $M$ for a loan with principal $P$ at a periodic interest rate $i$ over $n$ periods can be expressed as $M = \frac{Pi}{1 - (1+i)^{-n}}$. Which of the following correctly expresses the term $(1+i)^{-n}$ in terms of $M$, $P$, and $i$?

In financial analysis, the debt-to-assets ratio, $R_{DA}$, is related to the debt-to-equity ratio, $R_{DE}$, by the formula $R_{DA} = \frac{R_{DE}}{1 + R_{DE}}$. Which of the following formulas correctly expresses the debt-to-equity ratio, $R_{DE}$, in terms of the debt-to-assets ratio, $R_{DA}$?

In an economic model, the quantity demanded, $Q_d$, and quantity supplied, $Q_s$, for a product are given by $Q_d = a - bP$ and $Q_s = c + dP$, where $P$ is the price. When the government imposes a per-unit tax, $t$, collected from producers, the new supply equation becomes $Q_s = c + d(P-t)$. At the new market equilibrium, $Q_d = Q_s$. What is the new equilibrium price $P$ in terms of the constants $a, b, c, d,$ and $t$?

In a statistical dataset, two data points $x_1$ and $x_2$ have corresponding z-scores of $z_1$ and $z_2$. The z-score formula is $z = \frac{x - \mu}{\sigma}$, where $\mu$ is the mean and $\sigma$ is the standard deviation. Which of the following expressions correctly represents the mean $\mu$ of the dataset in terms of $x_1$, $x_2$, $z_1$, and $z_2$?

The future value, $FV$, of an ordinary annuity is given by the formula $FV = PMT \frac{(1+i)^n - 1}{i}$, where $PMT$ is the payment amount, $i$ is the interest rate per period, and $n$ is the number of periods. Which of the following formulas correctly represents the number of periods, $n$, in terms of $FV$, $PMT$, and $i$?

The sum $S$ of an infinite geometric series with first term $a_1$ and common ratio $r$ (where $|r| < 1$) is given by the formula $S = \frac{a_1}{1-r}$. Which of the following correctly expresses the common ratio $r$ in terms of $S$ and $a_1$?

A manufacturing company uses the economic order quantity (EOQ) model to minimize inventory costs. The total cost function includes both ordering costs and carrying costs.

The EOQ formula is $Q = \sqrt{\frac{2DS}{H}}$, where $Q$ is the optimal order quantity, $D$ is annual demand, $S$ is cost per order, and $H$ is annual holding cost per unit. If the annual demand increases by 44% and the cost per order decreases by 19%, by what percentage should the annual holding cost per unit change to keep the optimal order quantity unchanged?

The resistance $R$ of a wire is given by $R = \rho \frac{L}{A}$, where $\rho$ is resistivity, $L$ is length, and $A$ is cross-sectional area. If the resistance must be reduced to 60% of its original value by changing only the length and keeping the wire's volume constant, by what factor should the length be changed?

In a system of linear equations $AX = B$, where $A = \begin{pmatrix} k & 1 \\ 3 & 2 \end{pmatrix}$, $X = \begin{pmatrix} x \\ y \end{pmatrix}$, and $B = \begin{pmatrix} p \\ q \end{pmatrix}$, the solution for $X$ is given by $X = A^{-1}B$. Assuming $A$ is invertible, which of the following gives the correct expression for the variable $x$?