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Answer :
The length of the sides of the base should be 39.51 mm.
The formula for the volume of a cylinder is:
[tex]\[ V = \pi r^2 h \][/tex]
Given:
- Height [tex](\(h\))[/tex] = 39.7 mm
- Diameter = 39.4 mm, thus radius [tex](\(r\)) = \(\frac{39.4}{2} = 19.7\)[/tex] mm
Substitute these values into the formula:
[tex]\[ V = \pi (19.7)^2 \cdot 39.7 \][/tex]
[tex]\[ V \approx \pi \cdot 388.09 \cdot 39.7 \][/tex]
[tex]\[ V \approx 48434.06 \, \text{mm}^3 \][/tex]
The formula for the volume of a square prism is:
[tex]\[ V = s^2 h \][/tex]
where [tex]\(s[/tex] is the side length of the base and[tex]\(h\)[/tex]is the height, which is the same for both the cylinder and the square prism.
Given:
- Volume [tex](\(V\))[/tex] = 48434.06 mm³
- Height [tex](\(h\))[/tex] = 39.7 mm
Substitute the known values into the formula:
[tex]\[ 48434.06 = s^2 \cdot 39.7 \][/tex]
Solving for [tex]\(s^2\)[/tex]:
[tex]\[ s^2 = \frac{48434.06}{39.7} \][/tex]
[tex]\[ s^2 \approx 1220.20 \][/tex]
Solving for s:
[tex]\[ s \approx \sqrt{1220.20} \][/tex]
[tex]\[ s \approx 34.93 \, \text{mm} \][/tex]
Therefore, the length of the sides of the base of the square prism should be approximately 34.93 mm.
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