Mathematics Solution

बेलनाको अर्धव्यास र अर्धगोलाको अर्धव्यास बराबर भएको एउटा ठोस बस्तु छ, जसको पूरा आयतन 2707 घन से.मि. छ । यदि वेलनाको मात्र उचाइ 80 से.मि. भए उक्त्त ठोस वस्तुको पूरा सतहको क्षेत्रफल पत्तालगाउनुहोस् ।
A solide is made by hemisphare and cylinder having equal radii. The volume of the solide is 2707 $cm^3$. If the height of the cylinder is 80 cm, find the total surface area of the solide.
1) बेलनाको अर्धव्यास र अर्धगोलाको अर्धव्यास बराबर भएको एउटा ठोस बस्तु छ, जसको पूरा आयतन 2707 घन से.मि. छ । यदि वेलनाको मात्र उचाइ 80 से.मि. भए उक्त्त ठोस वस्तुको पूरा सतहको क्षेत्रफल पत्तालगाउनुहोस् ।
A solide is made by hemisphare and cylinder having equal radii. The volume of the solide is 2707 $cm^3$. If the height of the cylinder is 80 cm, find the total surface area of the solide.
Solution :
The volume of the solide (V) = 2707 $cm^3$
the height of the cylinder (h) = 80 cm
Total surface area (TSA) = ?
By the formula,
$V=\pi r^2h + \frac{2}{3} \pi r^3 \\ or, 2707 = \frac{22}{7} \times r^2 \times 80 + \frac{2}{3} \times \frac{22}{7} \times r^3 \\ or, 2707 = \frac{22}{7} (80r^2 + \frac{2r^3}{3}) \\ or, 2707 \times \frac{7}{22} = (80r^2 + \frac{2r^3}{3}) \\ or, \frac{18949}{22} = \frac{240r^2 + 2r^3}{3} \\ or, 56847 = 5280r^2 + 44r^3 \\ or, 44r^3 + 5280r^2 - 56847 = 0 \\$
Using advance calculator,

On solving the equation, we get only one posite value of r, i.e. r = 3.23.
Now,
The total surface area of the solide (TSA) = TSA of hemisphare + CSA of cylinder
$ = 3\pi r^2 + 2 \pi rh \\ = 3 \times \frac{22}{7} \times (3.23)^2 + 2 \times \frac{22}{7} \times 3.23 \times 80 \\ = \frac{688.5714}{7} + \frac{11,369.6}{7} \\ = 98.367 + 1,624.228 \\ = 1,722.595 cm^2 \\$