Question

A worker uses a forklift to move boxes that weigh either 40 pounds or 65 pounds each. Let $x$ be the number of 40-pound boxes and $y$ be the number of 65- pound. The forklift can carry up to either 45 boxes or a weight of 2400 pound. Which of the following systems of inequalities represents this relationship?
(a)\left\\{ \begin{aligned} & 40x+65y\le 2,400 \\\ & x+y\le 45 \\\ \end{aligned} \right.
(b) \left\\{ \begin{aligned} & \dfrac{y}{40}+\dfrac{y}{65}\le 2,400 \\\ & x+y\le 45 \\\ \end{aligned} \right.
(c) \left\\{ \begin{aligned} & 40x+65y\le 45 \\\ & x+y\le 2,400 \\\ \end{aligned} \right.
(d) \left\\{ \begin{aligned} & x+y\le 2,400 \\\ & 40x+65y\le 2,400 \\\ \end{aligned} \right.

Answer 1

Hint: The method to solve this to find the equation of the total number of boxes which is the sum of the total number of 40-pound boxes and 65-pound boxes. Also, we need to find the total weight that the forklift then carry and compare it with the condition given in the question.

Complete step-by-step answer:
There are two types of boxes to be lifted by a forklift. One is 40-pound boxes and the other ones are 65-pound boxes.
Let us assume that the total weight for 40-pound boxes be$a$pounds and for 65-pound boxes be 65 pounds.
Since $x$ is the number of 40-pound boxes, the total weight of the weight of 40-pound boxes (a)$=\text{Total number of 40 pound boxes }\\!\\!\times\\!\\!\text{ Weight of each box}$.
Therefore, $a=40\times x=40x.......................(i)$
Since $y$ is the number of 65-pound boxes, the total weight of the weight of 65-pound boxes (b)$=\text{Total number of 65 pound boxes }\\!\\!\times\\!\\!\text{ Weight of each box}$.
Therefore, $b=65\times y=65y.......................(ii)$
The combined weight of the boxes is going to be $a+b$.
From (i) and (ii),
We get the combined weight of the boxes $=a+b=40x+65y..............(iii)$.
Since $x$ is the number of 40-pound boxes and $y$ is the number of 65-pound boxes the total number of boxes $=x+y............(iv)$
From the condition given in the question the forklift can carry up to either 45 boxes or weight of 2400 pounds, we get,
The combined weight of boxes $\le 2400........................(v)$
And, the total no of boxes $\le 45..........................(vi)$
From (iii) and (iv), we get,
$40x+65y\le 2400$ and $x+y\le 45$
Hence, the correct option is (a).

Note: When it is asked whether condition 1 or condition 2 has to satisfy then we have to take less than equal to ( $\le$ ) sign because if we used greater than equal to side then at least one condition is bound to violate. Some students get confused and end up forming the wrong inequalities, they may for it as $40x+65y\le 45$ and $x+y\le 2,400$ and choose option (c) as the correct answer. So, they must read the question carefully and then form the right inequalities.