Business Calculus Quiz: Interpreting And Checking Answers

What this quiz covers

This quiz focuses on Interpreting And Checking Answers, giving you a quick way to practice the rules, question types, and explanations that matter most for Business Calculus.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Recalculate the annuity formula step-by-step: $(1.06)^{-5} \approx 0.7473$, so $\frac{1-0.7473}{0.06} \approx 4.2124$, giving $PV \approx 15000 \times 4.2124 = \$63,186$
2. Compare to individual present values: $\frac{15000}{1.06} + \frac{15000}{(1.06)^2} + ... + \frac{15000}{(1.06)^5}$ should sum to approximately $63,197
3. Confirm the discount rate of 6% annually is appropriate for the investment risk level, then validate that receiving $63,197 today equals receiving $75,000 over five years
4. Verify that $63,197 < $75,000 makes economic sense since future money is worth less than present money, and check that the 16% discount reflects reasonable time value (correct answer)

Question 2

1. Substitute $t = 0$ into the equation: $P(0) = \frac{5000}{1 + 49e^0} = \frac{5000}{50} = 100$, revealing a calculation error in their original work (correct answer)
2. Verify that 2500 represents exactly half the carrying capacity of 5000, which occurs at the inflection point of the logistic curve where growth rate is maximized
3. Check whether their solution $t = 0$ came from correctly solving $\frac{5000}{1 + 49e^{-0.1t}} = 2500$, which should yield $t = \frac{\ln(49)}{0.1} \approx 38.7$
4. Confirm that the initial population $P(0) = 100$ makes sense for their context, then recognize that $P(t) = 2500$ occurs much later in the growth process

Question 3

1. The model is flawed because the critical point at $x = 4000$ is not achievable within the production constraints.
2. The company should invest in expanding its capacity to produce 4,000 units to achieve the theoretical maximum profit.
3. The maximum profit within the allowed production range occurs at the endpoint, a production level of 3,000 units. (correct answer)
4. There is no production level that maximizes profit within the given constraints because the critical point is out of range.

Question 4

1. Demand is inelastic, so a small increase in price would likely lead to an increase in total revenue.
2. Revenue is already maximized at the current price of $20 because elasticity is not equal to -1.
3. The calculation must be incorrect because price elasticity of demand cannot be negative.
4. Demand is elastic, so a small decrease in price would likely lead to an increase in total revenue. (correct answer)

Question 5

1. The total profit is decreasing for all levels of production, indicating an unprofitable venture.
2. The profit function has a relative minimum value but no relative maximum value.
3. The rate of change of profit is constant, meaning profit grows linearly.
4. The marginal profit is decreasing at a constant rate, indicating diminishing returns for each additional unit produced. (correct answer)

Question 6

1. Since marginal cost is positive, the total cost of production is increasing, which will cause the average cost to also increase.
2. Since marginal cost is below average cost, producing one more unit will cause the average cost per unit to decrease. (correct answer)
3. The average cost is higher than the marginal cost, which means the production process is inefficient and should be changed.
4. To minimize the average cost, the company should reduce production, as the current average cost of $75 is too high.

Question 7

1. Yes, because as long as marginal revenue is positive, increasing production will increase total revenue and therefore increase profit.
2. No, because to maximize profit, the company needs to operate at a point where marginal cost is zero.
3. Yes, because the high marginal cost indicates strong demand for the product, so increasing production is justified.
4. No, because marginal cost exceeds marginal revenue, so producing an additional unit would decrease total profit. (correct answer)

Question 8

1. The conclusion is incorrect because the total leakage must be found by calculating the definite integral $\int_{0}^{3} L'(t) dt$. (correct answer)
2. The methodology is flawed because it uses the rate at the end of the period; it should use the initial rate, $L'(0)$, instead.
3. The conclusion is an overestimate because the rate of the leak is decreasing over time, so the average rate is lower than 250.
4. The methodology is flawed because it does not account for the acceleration of the leak, which is given by $L''(t)$.

Question 9

1. The result is reasonable because continuous compounding increases the value of money over time, so the present value should be higher than the total nominal income.
2. The result is unreasonable because the interest rate is low; a low discount rate should lead to a much smaller present value compared to the nominal payments.
3. The result is unreasonable because the present value must be less than the total undiscounted income stream, which is $50,000 \times 10 = $500,000, making $515,640 impossible. (correct answer)
4. The result is reasonable, as it correctly accounts for the time value of money, which makes future earnings more valuable in today's dollars than their face value.

Question 10

1. The total profit from selling 500 units is negative, resulting in a loss of $2.50.
2. The average profit per unit for the first 500 units sold is -$2.50.
3. Producing and selling the 501st unit will likely decrease the company's total profit by approximately $2.50. (correct answer)
4. The selling price of the product should be increased by $2.50 to make the production of the next unit profitable.

Question 11

1. The company's revenue at the start of 2024 was $2.59 million.
2. The total accumulated revenue increased by approximately $2.59 million between the start of 2022 and the start of 2024. (correct answer)
3. The company's revenue was growing at a rate of $2.59 million per year at the start of 2024.
4. The average rate of revenue growth between the start of 2022 and 2024 was $2.59 million per year.

Question 12

1. The result is nonsensical because the limits of integration are incorrect; they should be from 0 to the price $p_E = 110$.
2. The result is plausible; it represents a market where consumers pay more than their maximum willingness to pay.
3. The result is nonsensical because consumer surplus cannot be negative. The integrand should be $D(q) - p_E$. (correct answer)
4. The result is nonsensical because the supply function was not used. The correct integrand is $D(q) - S(q)$.

Question 13

1. The prediction is unreasonable, as the logistic model shows the number of users approaches a maximum carrying capacity of 500,000. (correct answer)
2. The prediction is reasonable, as the linear model is likely more accurate for short-term forecasts than the logistic model.
3. The prediction is reasonable if the company launches a major marketing campaign, which would alter the parameters of the logistic model.
4. The prediction is unreasonable because the rate of growth $N'(t)$ must be negative by $t=24$, making 600,000 impossible.

Question 14

1. Recalculate using $A = x(50-x)$ where $x$ is width, find $A'(x) = 0$ gives $x = 25$, and verify that $A(24) = 624 < 625 > 624 = A(26)$
2. Explain that the constraint $2x + 2y = 100$ allows squares since $x = y$ is permitted, then test nearby rectangles like 24×26 to confirm area 624 < 625
3. Point out that among all rectangles with fixed perimeter, the square maximizes area, then verify by testing several rectangles: 20×30 gives 600, 15×35 gives 525, both less than 625 (correct answer)
4. Show that the optimization correctly used $A = x(50-x)$, found the critical point at $x = 25$, and confirmed it's a maximum using the second derivative test

Question 15

1. Confirm that $C'(50) = 0$ by calculating $C'(x) = 0.06x^2 - 6x + 150$ and substituting $x = 50$ to get $150 - 300 + 150 = 0$
2. Verify that average cost is minimized when $AC'(x) = 0$, which occurs when $C'(x) = AC(x)$, then check if $C'(50) = 0$ equals $AC(50) = 158$ (correct answer)
3. Check the second derivative $C''(50) = 0.12(50) - 6 = 0$ to confirm this is an inflection point of the cost function, not a minimum
4. Calculate $AC'(x) = \frac{xC'(x) - C(x)}{x^2}$ and verify that $AC'(50) = 0$, then use the second derivative test on $AC(x)$ to confirm a minimum

Question 16

1. Verify that the amplitude of 200 units represents a realistic fluctuation compared to the baseline of 500 units for typical seasonal variation
2. Confirm that the period of 52 weeks matches an annual cycle and that peak summer sales (week 13) and minimum fall sales (week 39) align with business expectations
3. Check that the sine function oscillates between -1 and 1, ensuring the sales range from 300 to 700 units is mathematically correct
4. Validate that week 13 corresponds to late March and week 39 to late September, then assess if these seasonal peaks match the business context (correct answer)

Question 17

1. Check that the antiderivative $\frac{1000e^{0.05t}}{0.05} = 20000e^{0.05t}$ is correct, then verify the definite integral calculation gives the total accumulated profit
2. Compare the result to simple estimates: if profit rate were constant at $1000/month, total would be $6000; since the rate grows exponentially, $6997 is reasonably close but higher (correct answer)
3. Recognize that $6997 represents total accumulated profit while the partner may be thinking of final profit rate $P'(6) = 1000e^{0.3} \approx \1350$$ per month
4. Verify that $e^{0.3} \approx 1.3499$, so $20000(1.3499 - 1) = 20000(0.3499) \approx \$6998$, confirming the integration arithmetic is accurate

Question 18

1. Verify that $P'(30) = 0$ and confirm the profit value by substitution, then check that nearby values like $P(29)$ and $P(31)$ are smaller than 800 (correct answer)
2. Check that the vertex formula $x = -\frac{b}{2a}$ gives $x = 30$, then verify the calculation $P(30) = 800$ is arithmetically correct
3. Confirm that $x = 30$ represents 3000 units and that a profit of $800 per unit seems reasonable for typical manufacturing
4. Verify that the parabola opens downward since $a = -2 < 0$, ensuring a maximum exists, then check that $P(30) > 0$ indicates profitability

Question 19

1. Verify that $I'(10) = 50 - 10(10) = -50$, confirming that inventory is decreasing at 50 units per week when it reaches zero
2. Substitute into the quadratic formula: $50t - 5t^2 + 200 = 0$ gives $t = \frac{-50 \pm \sqrt{2500 + 4000}}{-10} = 10$ or $t = -4$, taking the positive solution
3. Check that $I(5) = 50(5) - 5(25) + 200 = 325$, verifying inventory increases initially, then confirm $I'(5) = 0$ indicates maximum inventory at the halfway point (correct answer)
4. Confirm that the antiderivative $I(t) = 50t - 5t^2 + 200$ satisfies $I'(t) = 50 - 10t$ and $I(0) = 200$, then verify $I(10) = 0$ arithmetically