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A conical tent of given capacity has to be constructed. The ratio of the height to the radius of the base for the minimum amount of canvas required for the tent is
- a)1 : 2
- b)2 : 1
- c)1 : √2
- d)√2 : 1
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
A conical tent of given capacity has to be constructed. The ratio of ...
Curved surface area of cone =πrl
1=
After substituting the value given in options to get two answer.
∴ The least value is √3π, when h = √2 and r = 1
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A conical tent of given capacity has to be constructed. The ratio of ...
To find the ratio of the height to the radius of the base for the minimum amount of canvas required for the tent, we need to minimize the surface area of the tent.
Let's assume the height of the cone is 'h' and the radius of the base is 'r'. The slant height of the cone can be found using the Pythagorean theorem as √(h^2 + r^2).
The surface area of the cone is given by the formula:
A = πr^2 + πr√(h^2 + r^2)
To minimize the surface area, we need to differentiate the above equation with respect to 'r' and set it equal to zero.
dA/dr = 2πr + π√(h^2 + r^2) + πrh(1/r) = 0
2πr + π√(h^2 + r^2) + πh = 0
Now, let's simplify the equation:
2r + √(h^2 + r^2) + h = 0
√(h^2 + r^2) = -2r - h
h^2 + r^2 = 4r^2 + 4rh + h^2
3r^2 + 4rh = 0
r(3r + 4h) = 0
Since the radius cannot be zero, we have 3r + 4h = 0
3r = -4h
r = -4h/3
Since both 'r' and 'h' are positive values, we can ignore the negative sign and write:
r = 4h/3
Therefore, the ratio of the height to the radius is:
h/r = h / (4h/3) = 3/4 = √2 : 1
Hence, the correct answer is option 'D' - √2 : 1.
Let's assume the height of the cone is 'h' and the radius of the base is 'r'. The slant height of the cone can be found using the Pythagorean theorem as √(h^2 + r^2).
The surface area of the cone is given by the formula:
A = πr^2 + πr√(h^2 + r^2)
To minimize the surface area, we need to differentiate the above equation with respect to 'r' and set it equal to zero.
dA/dr = 2πr + π√(h^2 + r^2) + πrh(1/r) = 0
2πr + π√(h^2 + r^2) + πh = 0
Now, let's simplify the equation:
2r + √(h^2 + r^2) + h = 0
√(h^2 + r^2) = -2r - h
h^2 + r^2 = 4r^2 + 4rh + h^2
3r^2 + 4rh = 0
r(3r + 4h) = 0
Since the radius cannot be zero, we have 3r + 4h = 0
3r = -4h
r = -4h/3
Since both 'r' and 'h' are positive values, we can ignore the negative sign and write:
r = 4h/3
Therefore, the ratio of the height to the radius is:
h/r = h / (4h/3) = 3/4 = √2 : 1
Hence, the correct answer is option 'D' - √2 : 1.
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