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Difficulty: HardLinear and Exponential Growth

Two startup companies, Company X and Company Y, begin operations at the same time. At the start of operations (m=0m = 0), both companies have a monthly revenue of RR dollars.

* The monthly revenue of Company X increases by a constant amount of 0.12R0.12R dollars every 33 months.
* The monthly revenue of Company Y increases exponentially, growing by a constant percent of 6%6\% every 22 months.

Let X(m)X(m) and Y(m)Y(m) represent the monthly revenue of Company X and Company Y, respectively, after mm months of operation. Which of the following functions represents the ratio Y(m)X(m)\frac{Y(m)}{X(m)} for any positive integer mm that is a multiple of 66?

  1. A
    1+0.03m1+0.04m\frac{1 + 0.03m}{1 + 0.04m}
  2. B
    (1.06)m21+0.12m\frac{(1.06)^{\frac{m}{2}}}{1 + 0.12m}
  3. C
    (1.06)m1+0.04m\frac{(1.06)^m}{1 + 0.04m}
  4. (1.06)m21+0.04m\frac{(1.06)^{\frac{m}{2}}}{1 + 0.04m}Answer

Answer

The expression (1.06)m21+0.04m\frac{(1.06)^{\frac{m}{2}}}{1 + 0.04m}
The correct answer is the expression (1.06)m21+0.04m\frac{(1.06)^{\frac{m}{2}}}{1 + 0.04m}. This is found by representing the linear growth of Company X as X(m)=R(1+0.04m)X(m) = R(1 + 0.04m) because its revenue increases by a constant amount of 0.04R0.04R per month (since 0.12R0.12R every 3 months is 0.12R3=0.04R\frac{0.12R}{3} = 0.04R). The exponential growth of Company Y is represented as Y(m)=R(1.06)m2Y(m) = R(1.06)^{\frac{m}{2}} because its revenue increases by 6%6\% every 2 months, which yields a growth factor of 1.061.06 applied m2\frac{m}{2} times over mm months. The ratio Y(m)X(m)\frac{Y(m)}{X(m)} is then R(1.06)m2R(1+0.04m)\frac{R(1.06)^{\frac{m}{2}}}{R(1 + 0.04m)}, which simplifies to the correct expression after canceling the common factor RR.

Step-by-Step Solution

1
Determine the expression for Company X's monthly revenue, X(m)X(m).
X(m)=R(1+0.04m)X(m) = R(1 + 0.04m)
Since Company X's revenue increases by a constant amount of 0.12R0.12R every 3 months, it grows linearly. The rate of increase per month is 0.12R3=0.04R\frac{0.12R}{3} = 0.04R dollars. Thus, after mm months, the revenue is X(m)=R+0.04R⋅m=R(1+0.04m)X(m) = R + 0.04R \cdot m = R(1 + 0.04m).
2
Determine the expression for Company Y's monthly revenue, Y(m)Y(m).
Y(m)=R(1.06)m2Y(m) = R(1.06)^{\frac{m}{2}}
Since Company Y's revenue grows by a constant percent of 6%6\% every 2 months, it grows exponentially. The growth factor for each 2-month period is 1+0.06=1.061 + 0.06 = 1.06. In mm months, the number of 2-month compounding periods is m2\frac{m}{2}. Therefore, the revenue is Y(m)=R(1.06)m2Y(m) = R(1.06)^{\frac{m}{2}}.
3
Find the ratio of Y(m)Y(m) to X(m)X(m).
Y(m)X(m)=(1.06)m21+0.04m\frac{Y(m)}{X(m)} = \frac{(1.06)^{\frac{m}{2}}}{1 + 0.04m}
Divide the expression for Y(m)Y(m) by the expression for X(m)X(m): Y(m)X(m)=R(1.06)m2R(1+0.04m)\frac{Y(m)}{X(m)} = \frac{R(1.06)^{\frac{m}{2}}}{R(1 + 0.04m)}. The constant initial revenue factor RR cancels out, simplifying to the final ratio.

Key Concept

Linear and Exponential Growth
Estimated Time:2m 30s
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Topics

Linear and Exponential Growth

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