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A given quantity of metal is to be cast into a half cylinder (i.e. with a rectangular base and semicircular ends). If the total surface area is to be minimum, then the ratio of the height of the half cylinder to the diameter of the semicircular ends is
- a)π ∶ (π + 2)
- b)(π + 2) ∶ π
- c)1 : 1
- d)None of the above
Correct answer is option 'A'. Can you explain this answer?
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To find the ratio of the height of the half cylinder to the diameter of the semicircular ends, we need to minimize the surface area.
Let's assume the height of the half cylinder is h, and the diameter of the semicircular ends is d.
The surface area of the half cylinder consists of the area of the rectangular base, the areas of the two semicircular ends, and the area of the curved surface.
The area of the rectangular base is given by A_rectangular = h * d.
The areas of the two semicircular ends are given by A_semicircular = 2 * (π * (d/2)^2)/2 = (π * d^2)/4.
The area of the curved surface is given by A_curved = (π * d * h)/2.
Therefore, the total surface area is A_total = A_rectangular + A_semicircular + A_curved = h * d + (π * d^2)/4 + (π * d * h)/2.
To minimize the surface area, we need to find the values of h and d that minimize A_total.
Taking the derivative of A_total with respect to h, we get d + (π * d)/2 = 0.
Simplifying this equation, we have d(1 + π/2) = 0.
Since d cannot be equal to zero, we have 1 + π/2 = 0.
Solving for π, we get π = -2.
However, this is not a valid value for π, so there is no minimum surface area.
Therefore, we cannot determine the ratio of the height of the half cylinder to the diameter of the semicircular ends.
Let's assume the height of the half cylinder is h, and the diameter of the semicircular ends is d.
The surface area of the half cylinder consists of the area of the rectangular base, the areas of the two semicircular ends, and the area of the curved surface.
The area of the rectangular base is given by A_rectangular = h * d.
The areas of the two semicircular ends are given by A_semicircular = 2 * (π * (d/2)^2)/2 = (π * d^2)/4.
The area of the curved surface is given by A_curved = (π * d * h)/2.
Therefore, the total surface area is A_total = A_rectangular + A_semicircular + A_curved = h * d + (π * d^2)/4 + (π * d * h)/2.
To minimize the surface area, we need to find the values of h and d that minimize A_total.
Taking the derivative of A_total with respect to h, we get d + (π * d)/2 = 0.
Simplifying this equation, we have d(1 + π/2) = 0.
Since d cannot be equal to zero, we have 1 + π/2 = 0.
Solving for π, we get π = -2.
However, this is not a valid value for π, so there is no minimum surface area.
Therefore, we cannot determine the ratio of the height of the half cylinder to the diameter of the semicircular ends.
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A given quantity of metal is to be cast into a half cylinder (i.e. with a rectangular base and semicircular ends). If the total surface area is to be minimum, then the ratio of the height of the half cylinder to the diameter of the semicircular ends isa)π (π + 2)b)(π + 2) πc)1 : 1d)None of the aboveCorrect answer is option 'A'. Can you explain this answer?
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