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Sagot :
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.
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.
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