Question

The length, breadth and height of a closed rectangular box are 18cm, 10cm and 6cm respectively and thickness of wood is 0.5cm. When the box is empty, its weight is 15kg and when filled with sand its weight is 100kg. Find weight of $1c{m^3}$ of wood and sand.

Answer 1

Hint: Find the volume of the box by using the formula, $V = l \times b \times h$, where $l$ is the length, \[b\] is the breadth and $h$ is the height of the rectangular box. Then, calculate the dimensions of the inner space of the box by subtracting thickness from each dimension of the given length. Then, the volume of the wooden box can be calculated by subtracting the volume of hollow space from the volume of the box. Next, use the given condition to find the weight of $1c{m^3}$ of wood and sand.

Complete step-by-step answer:
We are given that the length, breadth and height of a closed rectangular box are 18cm, 10cm and 6cm.
We will calculate the volume of the box by taking the product of length, breadth and height of the box.
$18 \times 10 \times 6 = 1080c{m^3}$
Now, we will calculate the volume of the hollow space of the rectangular box.
The thickness of the box is 0.5cm.
Then, the length of the box will be $18 - \left( {0.5 + 0.5} \right) = 17cm$, breadth will be $10 - \left( {0.5 + 0.5} \right) = 9cm$ and $6 - \left( {0.5 + 0.5} \right) = 5cm$
Then the volume of the hollow space is $17 \times 9 \times 5 = 765c{m^3}$
The volume of the rectangular wood can be calculated by subtracting the volume of hollow space from the volume of the box.
$\left( {1080 - 765} \right)c{m^3} = 365c{m^3}$
We are given that the weight of the wood is 15kg.
That is, the weight of $365c{m^3}$ is 15kg.
Then, weight of $1c{m^3}$ can be calculated by dividing the weight by 365
$\dfrac{{15}}{{365}} = \dfrac{1}{{21}}kg$
Similarly, we are given that the weight of the box with sand is 100kg and also the weight of the box is 15kg.
Then, the weight of sand can be calculated by subtracting the weight of the box from the weight of the box with sand.
$\left( {100 - 15} \right)kg = 85kg$
That is, the weight volume of the hollow box is equal to 85kg.
Which is weight of $765c{m^3}$ is 85kg
Then, weight of $1c{m^3}$ of sand is $\dfrac{{85}}{{765}} = \dfrac{1}{9}kg$
Hence, the weight of $1c{m^3}$ of wood is $\dfrac{1}{{21}}kg$ and weight of $1c{m^3}$ sand is $\dfrac{1}{9}kg$
Note: Many students make mistakes in calculating the dimensions of inner space of the box. We have to subtract the thickness two times and not the one time to find the dimensions of the inner box.

Also, the volume is measured in cubic units.