Answer:
It takes the younger gardener about 49.11 minutes while it takes the older gardener about 77.11 minutes.
Explanation:
Let the time taken by the younger gardener = x minutes
[tex]\text{The rate at which the young gardener will work}=\frac{1}{x}[/tex]The older gardener takes 28 minutes more than the younger gardener, therefore:
The time taken by the older gardener = (x+28) minutes
[tex]\text{The rate at wh}\imaginaryI\text{ch the older gardener w}\imaginaryI\text{ll work}=\frac{1}{x+28}[/tex]If they work together, it takes 30 minutes.
[tex]\text{The rate working together}=\frac{1}{30}[/tex]Therefore:
[tex]\frac{1}{x}+\frac{1}{x+28}=\frac{1}{30}[/tex]We solve the equation for x:
[tex]\begin{gathered} \frac{(x+28)+x}{x(x+28)}=\frac{1}{30} \\ \frac{2x+28}{x(x+28)}=\frac{1}{30} \\ \text{ Cross multiply} \\ 30(2x+28)=x(x+28) \\ \text{ Open the brackets} \\ 60x+840=x^2+28x \\ x^2+28x-60x-840=0 \\ x^2-32x-840=x \end{gathered}[/tex]We then solve the quadratic equation for x using the quadratic formula:
[tex]$x=\dfrac{-b\pm\sqrt{b^2-4ac} }{2a}$[/tex]In our equation:a=1, b=-32, and c=-840
[tex]\begin{gathered} $ x=\dfrac{-(-32)\pm\sqrt{(-32)^2-4(1)(-840)}}{2\times1} $ \\ =\dfrac{32\pm\sqrt{(-32)^2-4(1)(-840)}}{2} \\ =\dfrac{32\pm\sqrt{4384}}{2} \\ \implies x=\dfrac{32+\sqrt{4384}}{2}\text{ or }x=\dfrac{32-\sqrt{4384}}{2}\text{ } \\ x=49.11\text{ or x=-17.11} \end{gathered}[/tex]Since time cannot be negative: x=49.11 minutes.
Thus, the time it takes:
• The younger gardener = 49.11 minutes
,• The older gardener = 77.11 minutes
It takes the younger gardener about 49.11 minutes while it takes the older gardener about 77.11 minutes (correct to 2 decimal places).
