A solid is composed of a cone with hemispherical end. If the whole length of the solid is 84 cm and radius of the hemispherical ends is 36 cm, find the cost of polishing the surface at the rate of 70 paise/sq cm.
correct answer is 10,454,40
step by step explanation
correct answer is 10,454,40
step by step explanation
Answers & Comments
[tex]\huge\rm{\color{purple}{{\underline{★An}}}} {\color{orchid}{{\underline{sw}}}}{\pink{{\underline{er★}}}} {\color{lightpink}{}}[/tex]To find the cost of polishing the surface, we need to calculate the total surface area of the solid and then multiply it by the cost per square centimeter.
Step1: Calculate the slant height of the coneThe slant height (l) of the cone can be found using the Pythagorean theorem:
l = √(r^2 + h^2)
Given:
Radius of the hemispherical ends (r) =36 cmTotal length of the solid (l) =84 cmSince the total length is the sum of the height of the cone (h) and the sum of the radii of the hemispheres (2r), we can write:
l = h +2rSubstituting the given values:
84 = h +2(36)
84 = h +72h =84 -72h =12 cmNow we can calculate the slant height:
l = √(36^2 +12^2)
l = √(1296 +144)
l = √1440l ≈37.95 cm (rounded to two decimal places)
Step2: Calculate the total surface area of the solidThe total surface area (A) of the solid can be calculated as the sum of the curved surface area of the cone and the surface area of the two hemispheres.
Curved surface area of the cone:
A_cone = πrlSubstituting the values:
A_cone = π(36)(37.95)
A_cone ≈42667.82 cm^2 (rounded to two decimal places)
Surface area of the hemispheres:
A_hemisphere =2πr^2Substituting the values:
A_hemisphere =2π(36^2)
A_hemisphere ≈4068.41 cm^2 (rounded to two decimal places)
Total surface area:
A = A_cone + A_hemisphereA ≈42667.82 +4068.41A ≈46736.23 cm^2 (rounded to two decimal places)
Step3: Calculate the cost of polishing the surfaceCost = A * RateGiven:
Rate =70 paise/sq cmConverting the rate to rupees:
Rate =70 paise/sq cm =0.70 rupees/sq cmCost =46736.23 *0.70Cost ≈32715.36 rupees (rounded to two decimal places)
Therefore, the cost of polishing the surface is approximately32,715.36 rupees.
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To find the cost of polishing the surface, we need to calculate the total surface area of the solid and then multiply it by the cost per square centimeter.
Step1: Calculate the slant height of the coneThe slant height (l) of the cone can be found using the Pythagorean theorem:
l = √(r^2 + h^2)
Given:
Radius of the hemispherical ends (r) =36 cmTotal length of the solid (l) =84 cmSince the total length is the sum of the height of the cone (h) and the sum of the radii of the hemispheres (2r), we can write:
l = h +2rSubstituting the given values:
84 = h +2(36)
84 = h +72h =84 -72h =12 cmNow we can calculate the slant height:
l = √(36^2 +12^2)
l = √(1296 +144)
l = √1440l ≈37.95 cm (rounded to two decimal places)
Step2: Calculate the total surface area of the solidThe total surface area (A) of the solid can be calculated as the sum of the curved surface area of the cone and the surface area of the two hemispheres.
Curved surface area of the cone:
A_cone = πrlSubstituting the values:
A_cone = π(36)(37.95)
A_cone ≈42667.82 cm^2 (rounded to two decimal places)
Surface area of the hemispheres:
A_hemisphere =2πr^2Substituting the values:
A_hemisphere =2π(36^2)
A_hemisphere ≈4068.41 cm^2 (rounded to two decimal places)
Total surface area:
A = A_cone + A_hemisphereA ≈42667.82 +4068.41A ≈46736.23 cm^2 (rounded to two decimal places)
Step3: Calculate the cost of polishing the surfaceCost = A * RateGiven:
Rate =70 paise/sq cmConverting the rate to rupees:
Rate =70 paise/sq cm =0.70 rupees/sq cmCost =46736.23 *0.70Cost ≈32715.36 rupees (rounded to two decimal places)
Therefore, the cost of polishing the surface is approximately 32,715.36 rupees.