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Two pipes are used to fill a swimming pool. If the pipe of the larger diameter is used for $4$ hours and the pipe of the smaller diameter for $9$ hours, only half of the pool can be filled. Find how long it would take for each pipe to fill the pool, separately, if the pipe of smaller diameter takes $10$ hours more than the pipe of larger diameter to fill the pool.
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Sol. Let time taken to fill the pool by larger diameter pipe alone be $x$ hours
Time taken to fill the pool by smaller diameter pipe alone = $(x + 10)$ hours
frac{4}{x} +
frac{9}{x + 10} =
frac{1}{2}$$\begin{aligned}& (2 Marks) \\ & x^2 - 16x - 80 = 0\end{aligned}$$ (1 Mark)
Rightarrow (x + 4)(x - 20) = 0$$\begin{aligned}& (1 Mark) \\ & Rightarrow x = -4, x = 20 \\ & x = -4\end{aligned}$$ (rejected) (1/2 Mark)
therefore x = 20$\Time taken to fill the pool by larger diameter pipe alone = $20$ hours\Time taken to fill the pool by smaller diameter pipe alone = $30$ hours (1/2 Mark)
Time taken to fill the pool by smaller diameter pipe alone = $(x + 10)$ hours
frac{4}{x} +
frac{9}{x + 10} =
frac{1}{2}$$\begin{aligned}& (2 Marks) \\ & x^2 - 16x - 80 = 0\end{aligned}$$ (1 Mark)
Rightarrow (x + 4)(x - 20) = 0$$\begin{aligned}& (1 Mark) \\ & Rightarrow x = -4, x = 20 \\ & x = -4\end{aligned}$$ (rejected) (1/2 Mark)
therefore x = 20$\Time taken to fill the pool by larger diameter pipe alone = $20$ hours\Time taken to fill the pool by smaller diameter pipe alone = $30$ hours (1/2 Mark)