University Business Mathematics — Functions for Business Models Worksheet

All 9 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Linear Models for Cost, Revenue, Supply and Demand

   A print shop makes custom notebooks. Its fixed costs are $1 800 a month and each notebook costs $4.50 in paper, binding and labour. Every notebook sells for $12. Let $x$ be the number of notebooks made and sold in a month.

   1. Write the cost $C(x)$, the revenue $R(x)$ and the profit $P(x)$, in dollars.
   2. Find the break-even quantity.
   3. What is the profit or loss in a month when $150$ notebooks are sold?
   4. How many notebooks must be sold in a month for a profit of $1 200?
2. Q2Linear Models for Cost, Revenue, Supply and Demand

   A wholesaler sells insulated water bottles. Market research gives two observations of weekly demand and two of weekly supply, where $p$ is the unit price in dollars and $x$ the number of bottles. At $30, buyers demand $900$ bottles and suppliers offer $600$; at $40, buyers demand $700$ bottles and suppliers offer $1000$. Assume both relations are linear.

   1. Find the price-demand function $p=D(x)$ and the price-supply function $p=S(x)$.
   2. Find the equilibrium quantity and the equilibrium price.
   3. At a price of $40, is there a shortage or a surplus, and of how many bottles?
3. Q3Quadratic Functions and Maximum Revenue

   A small workshop makes leather phone cases. Its price-demand function is $$p=D(x)=90-\frac{1}{50}x,0\le x\le 4500,$$ where $x$ is the number of cases sold a month and $p$ the price in dollars.

   1. Write the revenue function $R(x)$.
   2. Find the quantity and the price that maximize revenue, and the maximum revenue.
   3. Find the monthly revenue at a price of $60, and the one other price that brings in exactly the same revenue.
4. Q4Exponential Functions and Growth Models

   A consultancy buys a laptop for $2 400. Its book value falls by the same percentage every year, and after $2$ years it is worth $1 350.

   1. Find the annual rate of depreciation and write the value $V(t)$ after $t$ years.
   2. Find the value after $3$ years.
   3. Write the same model in the form $V(t)=2400e^{-kt}$, giving $k$ exactly.
   4. After how many full years is the book value first below $500?
5. Q5Logarithmic Functions and the Laws of Logarithms
   1. Write $\ln \left({\displaystyle \frac{x^3\sqrt{y}}{z^2}}\right)$, for $x,y,z>0$, as a sum and difference of multiples of $\ln x$, $\ln y$ and $\ln z$.
   2. Write $3\ln a-\frac{1}{2}\ln b+\ln 4$, for $a,b>0$, as a single logarithm.
   3. Given $\ln 2=u$, $\ln 3=v$ and $\ln 5=w$, express $\ln 360$, $\ln 0.15$ and $\ln \sqrt{1.2}$ in terms of $u$, $v$ and $w$.
6. Q6Logarithmic Functions and the Laws of Logarithms

   A marketing firm models a client's weekly sales $S$, in thousands of dollars, as a function of the weekly advertising budget $x$, in thousands of dollars: $$S(x)=30+8\ln x,x\ge 1.$$

   1. Find the weekly sales on a budget of $1 000.
   2. Use the laws of logarithms to show that doubling any budget raises weekly sales by the same amount, and give that amount exactly.
   3. Find, exactly and then to the nearest dollar, the increase in weekly sales when the budget rises from $5 000 to $15 000. Use $\ln 3\approx 1.0986$.
7. Q7Solving Exponential and Logarithmic Equations

   Solve each equation exactly. Check every candidate against the domain of the original equation.

   1. $5e^{0.2t}=40$
   2. $3^{2x-1}=7^x$
   3. $\ln (x+2)+\ln (x-1)=\ln 10$
8. Q8Solving Exponential and Logarithmic Equations

   A streaming start-up has $2000$ subscribers and an established rival has $6000$. Their subscriber counts $t$ months from now are modelled by $$N_A(t)=2000e^{0.3t}\text{(start-up)},N_B(t)=6000e^{0.1t}\text{(rival)}.$$ When will the two services have the same number of subscribers, and how many will each have then? Give exact answers, then decimals, using $\ln 3\approx 1.0986$ and $\sqrt{3}\approx 1.7321$.

9. Q9Synthesis — drawing on several topics in this unit

   A phone-accessory maker launches a charging stand. Weekly sales $t$ weeks after launch are expected to follow $q(t)=3000e^{-0.1t}$ units. Each stand sells for $15 and costs $5 to make, and the product line carries fixed costs of $7 500 a week.

   1. Write the weekly profit $P(t)$ and find the profit in the launch week ($t=0$).
   2. Find, exactly and to two decimal places, the time at which weekly profit falls to zero. Use $\ln 2\approx 0.6931$.

The 6 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Cost, revenue and profit as linear functions

A linear cost model has two parts: a fixed cost that is paid whether anything is made or not, and a variable cost that grows by the same amount with every unit. Revenue is price times quantity, and profit is always revenue minus cost — never the other way round. Q1(a) asks for all three in dollars as functions of the number of notebooks x.

Every later part of Q1 is one of these functions read in a particular direction.

Supply, demand and equilibrium

Q2 gives two observations of demand and two of supply and asks for each as a line of the form p = D(x) and p = S(x). The convention matters: price is the output and quantity the input, so the points are written (x, p) — quantity first — before any slope is computed.

Q2(c) fixes a price and compares the two quantities at that price. Read each quantity off its own line at that price: when buyers want more than suppliers offer there is a shortage, when suppliers offer more than buyers want there is a surplus, and the size is the difference between the two quantities. Which one it is comes from where the price sits relative to the equilibrium price.

Quadratic revenue and its maximum

When the price depends on how many units are sold, revenue is no longer a line. In Q3 the price-demand function gives p in terms of x, and revenue is x times that price — a quadratic in x that opens downward.

Q3(b) asks for the maximum without any calculus: the maximum of a downward parabola is its vertex. Put R(x) in the form ax² + bx, take x = −b/(2a), and then the question wants three things — the quantity, the price that produces it (from the demand function, not from the revenue function), and the revenue itself. Check that the quantity lies inside the stated domain.

Q3(c) starts from a price instead of a quantity: find the quantity that price sells from the demand function, then the revenue. The one other price with the same revenue follows from symmetry — a parabola takes each value below its maximum at two points equally far from the vertex. Find the partner quantity on the other side of the vertex, then convert it to a price. Solving R(x) equal to the given revenue as a quadratic equation gives the same pair and is the check.

Exponential models: a percentage and the base e

"Falls by the same percentage every year" means each year's value is the previous one times a fixed factor 1 − r. After t years the starting value has been multiplied by that factor t times.

Q4(a) gives the starting value and the value after two years. Divide one by the other to get (1 − r)², take the square root for the yearly factor, and read the rate off it. Q4(b) is then an evaluation.

Q4(c) rewrites the same model with the base e. Two forms describe the same function when their bases agree, so set e⁻ᵏ equal to the yearly factor and take the natural logarithm of both sides. The question asks for k exactly, so leave it as a logarithm rather than a decimal.

Q4(d) asks when the value first drops below a threshold. Write the inequality, isolate the power, take logarithms and solve for t. Two things decide the final answer: dividing by the logarithm of a factor smaller than 1 divides by a negative number, which reverses the inequality; and "after how many full years" means the first whole number of years that satisfies it, which is a rounding up, not a rounding to the nearest.

The laws of logarithms

Three laws do all the work, and each one is an exponent law read backwards.

Q5(a) expands: split the quotient first, then the product, then bring each power down — a square root is the power one half. Q5(b) runs the laws the other way: move each coefficient back up as a power, then combine sums into products and differences into quotients. Q5(c) asks for three logarithms in terms of the logarithms of 2, 3 and 5. Factor each number into powers of 2, 3 and 5; a decimal becomes a fraction first, and a root becomes a power of one half.

Q6 puts the same laws to work in a sales model with a logarithm in it. Q6(a) is a substitution; watch the units, since both the budget and the sales are measured in thousands of dollars. Q6(b) asks you to show that doubling any budget adds the same amount: write S(2x) − S(x) and use the product law on ln(2x). The x has to disappear from the difference — that disappearance is the proof. Q6(c) is the same move with a different ratio: the difference of two logarithms is the logarithm of their quotient, and the approximation given in the question is used only at the end.

Solving exponential and logarithmic equations

Q8 asks when two exponential models are equal. Set them equal and gather the exponentials on one side and the constants on the other, so that one exponential equals one number; then take the natural logarithm. The question wants an exact time first, then a decimal from the values it supplies, and then the subscriber count — which should come out the same from both models, so compute it from both as the check. The exact form is easiest to evaluate if you use the exponent laws before reaching for decimals.

The synthesis question

Q9 puts Q1 and Q4 together. Weekly sales now decay exponentially over time, so profit becomes a function of time: the profit on each unit times the number of units sold that week, minus the fixed costs. Q9(a) writes that function and evaluates it in the launch week. Q9(b) sets it to zero, which isolates an exponential equal to a number — the Q7(a) method — and asks for the exact time and a decimal to two places using the approximation supplied.

Before solving, notice what the profit function does: the exponential term shrinks and the fixed cost does not, so profit only decreases. That tells you the zero is a single crossing and what the sign of profit is on each side of it.

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Getting the most out of it

Name the input and the output before writing a formula

Quantity or price? Months or years? Dollars or thousands of dollars? Q2 and Q6 both turn on this. Write "x is … and p is …" at the top of the question, as Q1 does for you.

Answer in the question's words

A break-even quantity is a number of units; "profit or loss" is answered with the word; "shortage or surplus" is answered with the word and a size. A bare number does not answer a business question.

Stay exact until the last line

When a question says "exactly", leave logarithms as logarithms. Round once, at the end, and round in the direction the context demands — up for "at least" and "first below".

Check with a second route

Substitute an equilibrium quantity into both lines, compute a crossing time's value from both models, confirm a symmetric price by solving the quadratic. Every question on this sheet has a second route, and it is usually one line long.

Common questions