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Sagot :
An extremum is a point where the function has its highest or lowest value, and at which the slope is zero.
The dimensions to be used if the volume is fixed and the cost of construction is to be kept to a minimum is as follows;
- Height of cylinder = 2 × Radius of cylinder
Reason:
The given parameters are;
Form of silo = Cylinder surmounted by a hemisphere
Cost of construction of hemisphere per square unit = 2 × The cost of of construction of the cylindrical side wall
Required:
The dimensions to be used if the volume is fixed and the cost of construction is to be kept to a minimum
Solution:
The fixed volume of the silo, V, can be expressed as follows;
[tex]V = \pi \cdot r^2 \cdot h + \dfrac{2}{3} \cdot \pi \cdot r^3[/tex]
- [tex]h = \dfrac{V - \dfrac{2}{3} \cdot \pi \cdot r^3 }{\pi \cdot r^2 }[/tex]
Where;
h = The height of the cylinder
r = The radius of the cylinder
The surface area is, Surface Area = 2·π·r·h + 2·π·r²
The cost, C = 2·π·r·h + 2×2·π·r² = 2·π·r·h + 4·π·r²
Therefore;
[tex]C = 2 \times \pi \times r\times \dfrac{V - \dfrac{2}{3} \cdot \pi \cdot r^3 }{\pi \cdot r^2 } + 4 \cdot \pi \cdot r^2 = \dfrac{8 \cdot r^3 \cdot \pi + 6 \cdot V}{3 \cdot r}[/tex]
- [tex]C = \dfrac{8 \cdot r^3 \cdot \pi + 6 \cdot V}{3 \cdot r}[/tex]
At the minimum value, we have;
- [tex]\dfrac{dC}{dr} =0 = \dfrac{d}{dr} \left(\dfrac{8 \cdot r^3 \cdot \pi + 6 \cdot V}{3 \cdot r} \right) = \dfrac{16 \cdot r^3 \cdot \pi - 6 \cdot V}{3 \cdot r^2}[/tex]
Which gives;
16·π·r³ = 6·V
- [tex]V = \dfrac{16 \cdot \pi \cdot r^3}{6} = \dfrac{8 \cdot \pi \cdot r^3}{3}[/tex]
Which gives;
[tex]h = \dfrac{\dfrac{8 \cdot \pi \cdot r^3}{3} - \dfrac{2}{3} \cdot \pi \cdot r^3 }{\pi \cdot r^2 } = 2 \cdot r[/tex]
h = 2·r
The height of the silo, h = 2 × The radius, 2
Therefore, the dimensions to be used if the volume is fixed and the cost is to be kept to a minimum is, height, h = 2 times the radius
Learn more about extremum point here:
https://brainly.com/question/2869450
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