Question

According to a plan, a team of woodcutters had to cut $216\text{ }{{\text{m}}^{3}}$ of wood in several days. The first three days, the team fulfilled the daily assignment, and then it cut $8\text{ }{{\text{m}}^{3}}$ of wood over and above the plan every day. Therefore, a day before the planned date, they cut $232\text{ }{{\text{m}}^{3}}$ of wood. How many cubic metres of wood a day did the team have to cut according to the plan?

Answer 1

Hint: We will assume the number of total days for cutting the given quantity of wood. Using that, we will calculate the amount of wood needed to be cut everyday by the team. Then we will form an equation with the given information. We will simplify this equation and solve it to find the number of total days. Then we will calculate the amount of wood that the team had to cut per day according to the plan.

Complete step by step answer:
Let us assume that the plan requires $n$ days to cut $216\text{ }{{\text{m}}^{3}}$ of wood. So, according to the plan, the team needs to cut $\dfrac{216}{n}\text{ }{{\text{m}}^{3}}$ per day. Now, we know that for the first three days, the team fulfilled the daily assignment. So, for the first three days, the amount of wood that was cut is $3\times \dfrac{216}{n}\text{ }{{\text{m}}^{3}}$. Then, for the remaining days in the schedule, they cut $8\text{ }{{\text{m}}^{3}}$ of wood over and above the plan. So, the quantity of wood cut in one day out of the remaining days is $\dfrac{216}{n}+8$. Now, a day before the planned date, they cut $232\text{ }{{\text{m}}^{3}}$ of wood. So, we will exclude the first three days and the last day to get the quantity of wood cut in remaining days by the team. So, we will get the following equation,
$3\times \dfrac{216}{n}+\left( n-4 \right)\left( \dfrac{216}{n}+8 \right)=232$
Simplifying this equation, we get
$\begin{align}
  & 3\times \dfrac{216}{n}+216-4\times \dfrac{216}{n}+8n-32=232 \\
 & \Rightarrow \dfrac{216}{n}\left( 3-4 \right)+184+8n=232 \\
 & \therefore 8n-\dfrac{216}{n}=48 \\
\end{align}$
We will multiply the equation by $n$ as follows,
$\begin{align}
  & 8\left( {{n}^{2}}-27 \right)=48n \\
 & \Rightarrow {{n}^{2}}-27=6n \\
 & \therefore {{n}^{2}}-6n-27=0 \\
\end{align}$
Now, we have a quadratic equation. We will compare this equation with the standard quadratic equation $a{{x}^{2}}+bx+c=0$ and obtain $a=1$, $b=-6$ and $c=-27$. We will use the quadratic formula to find the value of $n$. The quadratic formula is $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$. Substituting the values, we get
$\begin{align}
  & n=\dfrac{-\left( -6 \right)\pm \sqrt{{{\left( -6 \right)}^{2}}-4\left( 1 \right)\left( -27 \right)}}{2\times 1} \\
 & \Rightarrow n=\dfrac{6\pm \sqrt{36+108}}{2} \\
 & \Rightarrow n=\dfrac{6\pm \sqrt{144}}{2} \\
 & \Rightarrow n=\dfrac{6\pm 12}{2} \\
 & \therefore n=3\pm 6 \\
\end{align}$
Therefore, we have $n=9$ or $n=-3$. As the number of days cannot be negative, we discard $n=-3$ and obtain the number of total days as 9.

Now, the quantity of wood a day did the team have to cut according to the plan is $\dfrac{216}{9}=24$ days.

Note: It is important to form the correct expression by interpreting the given information from the question. In the question, it is mentioned ‘a day before the planned date’. So, we should not forget to exclude the last day while calculating the quantity of wood cut in the remaining days after the first three days. It is useful to do the calculations explicitly so that errors in calculations can be avoided.