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A company's profit is modelled by the fraction 2x/(x-5) where x represents units sold. i. Explain what happens to the profit when x=5. Why is this significant? ii. Determine for which values of x the profit is positive.

Mathematics

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3. c. A company's profit is modelled by the fraction 2xx−5\frac{2x}{x-5} where xx represents units sold.

  • i. Explain what happens to the profit when x=5x=5. Why is this significant? (2 marks) When x=5x=5, the denominator x−5x-5 becomes 5−5=05-5=0. This makes the profit expression 2(5)0=100\frac{2(5)}{0} = \frac{10}{0}, which is undefined. This is significant because it indicates a critical point where the mathematical model breaks down, possibly representing a scenario where the profit cannot be determined or a point of instability in the company's operations.

  • ii. Determine for which values of xx the profit is positive. (2 marks)

  1. For the profit to be positive, 2xx−5>0\frac{2x}{x-5} > 0.

    This occurs when both the numerator and denominator have the same sign.
    Case 1: Both are positive.
    2x>0  ⟹  x>02x > 0 \implies x > 0
    x−5>0  ⟹  x>5x-5 > 0 \implies x > 5
    For both conditions to be true, x>5x > 5.
    Case 2: Both are negative.
    2x<0  ⟹  x<02x < 0 \implies x < 0
    x−5<0  ⟹  x<5x-5 < 0 \implies x < 5
    For both conditions to be true, x<0x < 0.
    The profit is positive when x<0 or x>5\boxed{x < 0 \text{ or } x > 5}.

    4. a. Make the indicated variable the subject: (4 marks)

    • i. Make yy the subject of 3x+2y=123x + 2y = 12

  2. Subtract $3x$ from both sides.

    $$
2y = 12 - 3x
$$

  3. Divide both sides by 2.

    $$
y = \frac{12 - 3x}{2}
$$
The subject is $\boxed{y = \frac{12 - 3x}{2}}$.
    • ii. Make rr the subject of A=πr2+2πrhA = \pi r^2 + 2\pi rh

  4. Rearrange the equation into the standard quadratic form ar2+br+c=0ar^2 + br + c = 0.

    $$
\pi r^2 + 2\pi hr - A = 0
$$

  5. Use the quadratic formula r=−b±b2−4ac2ar = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where a=πa=\pi, b=2πhb=2\pi h, and c=−Ac=-A.

    $$
r = \frac{-(2\pi h) \pm \sqrt{(2\pi h)^2 - 4(\pi)(-A)}}{2(\pi)}
$$

  6. Simplify the expression.

    $$
r = \frac{-2\pi h \pm \sqrt{4\pi^2 h^2 + 4\pi A}}{2\pi}
$$
$$
r = \frac{-2\pi h \pm \sqrt{4\pi(\pi h^2 + A)}}{2\pi}
$$
$$
r = \frac{-2\pi h \pm 2\sqrt{\pi(\pi h^2 + A)}}{2\pi}
$$

  7. Divide all terms by 2π2\pi.

    $$
r = -h \pm \frac{\sqrt{\pi(\pi h^2 + A)}}{\pi}
$$
The subject is $\boxed{r = -h \pm \frac{\sqrt{\pi(\pi h^2 + A)}}{\pi}}$.

    4. b. Solve the following equations: (6 marks)

    • i. 5x−3=2x+95x - 3 = 2x + 9

  8. Subtract $2x$ from both sides.

    $$
5x - 2x - 3 = 9
$$
$$
3x - 3 = 9
$$

  9. Add 3 to both sides.

    $$
3x = 9 + 3
$$
$$
3x = 12
$$

  10. Divide by 3.

    $$
x = \frac{12}{3}
$$
$$
x = 4
$$
The solution is $\boxed{x = 4}$.
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