Amar Singh has a cold storage with dimensions in the ratio [tex]4:3:2[/tex]. He can keep 24,000 cubical fruit boxes, each with an edge of 25 cm, in it.

Now, answer the following questions:

14. What is the volume of each fruit box?
- (a) [tex]15625 \, \text{cm}^3[/tex]
- (b) [tex]16525 \, \text{cm}^3[/tex]
- (c) [tex]15675 \, \text{cm}^3[/tex]
- (d) [tex]16575 \, \text{cm}^3[/tex]

15. What is the volume of the storage (in [tex]m^3[/tex])?
- (a) [tex]325 \, \text{m}^3[/tex]
- (b) [tex]375 \, \text{m}^3[/tex]
- (c) [tex]625 \, \text{m}^3[/tex]
- (d) [tex]475 \, \text{m}^3[/tex]

16. What are the dimensions of the storage?
- (a) [tex]4 \, \text{m} \times 3 \, \text{m} \times 2 \, \text{m}[/tex]
- (b) [tex]6 \, \text{m} \times 4.5 \, \text{m} \times 3 \, \text{m}[/tex]



Answer :

Let's solve the given problems step-by-step:

1. Volume of each fruit box:
The edge of each cubical fruit box is 25 cm. To find the volume of a cubical box, we use the formula for the volume of a cube, which is [tex]\( \text{side}^3 \)[/tex].

[tex]\[ \text{Volume of each fruit box} = 25 \, \text{cm} \times 25 \, \text{cm} \times 25 \, \text{cm} = 15625 \, \text{cm}^3 \][/tex]

So, the correct answer is:
(a) [tex]\( 15625 \, \text{cm}^3 \)[/tex]

2. Volume of the storage:
To find the total volume required to store all the boxes, we first calculate the total volume by multiplying the volume of one box by the total number of boxes.

[tex]\[ \text{Total volume required} = 15625 \, \text{cm}^3 \times 24000 = 375000000 \, \text{cm}^3 \][/tex]

To convert the total volume from cubic centimeters to cubic meters, we use the conversion factor where [tex]\(1 \, \text{m}^3 = 1 \times 10^6 \, \text{cm}^3\)[/tex]:

[tex]\[ \text{Volume in cubic meters} = \frac{375000000 \, \text{cm}^3}{1 \times 10^6} = 375 \, \text{m}^3 \][/tex]

So, the correct answer is:
(b) [tex]\( 375 \, \text{m}^3 \)[/tex]

3. Dimensions of the storage:
The dimensions of the storage are in the ratio [tex]\(4:3:2\)[/tex]. Let the dimensions be [tex]\(4k\)[/tex], [tex]\(3k\)[/tex], and [tex]\(2k\)[/tex].

The volume of the storage can be expressed in terms of [tex]\(k\)[/tex] as:

[tex]\[ \text{Volume} = 4k \times 3k \times 2k = 24k^3 \][/tex]

Given that the volume of the storage is [tex]\(375 \, \text{m}^3\)[/tex]:

[tex]\[ 24k^3 = 375 \, \text{m}^3 \][/tex]

Solving for [tex]\(k^3\)[/tex],

[tex]\[ k^3 = \frac{375}{24} = 15.625 \][/tex]

Taking the cube root of both sides, we get:

[tex]\[ k = \sqrt[3]{15.625} = 2.5 \][/tex]

Now, substituting [tex]\(k = 2.5\)[/tex] back into the dimensions:

[tex]\[ \text{Dimension} \, \text{a} = 4k = 4 \times 2.5 = 10 \, \text{m} \][/tex]
[tex]\[ \text{Dimension} \, \text{b} = 3k = 3 \times 2.5 = 7.5 \, \text{m} \][/tex]
[tex]\[ \text{Dimension} \, \text{c} = 2k = 2 \times 2.5 = 5 \, \text{m} \][/tex]

So, the correct dimensions of the storage are:
[tex]\( 10 \, \text{m} \times 7.5 \, \text{m} \times 5 \, \text{m} \)[/tex], which is not listed among the options given.

To summarize:
1. The volume of each fruit box is [tex]\(15625 \, \text{cm}^3\)[/tex] (Option (a)).
2. The volume of storage is [tex]\(375 \, \text{m}^3\)[/tex] (Option (b)).
3. The dimensions of the storage are [tex]\(10 \, \text{m} \times 7.5 \, \text{m} \times 5 \, \text{m}\)[/tex], which are not listed among the given options.