Question

the perimeter of a right angled triangle is 56 cm and its inradius is 3 cm.the sides in cm are integers.if the triangle is rotated about its hypotenuse ,what is the volume of solid so generated, in cubic cm?​

Answer 1

Answer:

Triangle is  7 , 24 , 25    

1232 cm³ or 4224 cm³

Step-by-step explanation:

the perimeter of a right angled triangle is 56 cm and its in radius is 3 cm

=> Area of Traingle =  Perimeter * inradius /2  = 56 * 3/2

Area of Triangle = (1/2) Base * Height

(1/2) Base * Height = 56 * 3/2

=> Base * Height = 56 * 3

=> B * H = 168

Hypotenuse = √B² + H²

B + H + √B² + H² = 56

=> B + 168/B  + √B² + (168/B)² = 56

=> B² + 168  + √B⁴  + 168²  = 56B

=> B² - 56B + 168 = - √B⁴  + 168²

Squaring Both sides

   and Solving we get

B³ - 31B² + 168B = 0

dividing by B as B can not be 0

B² - 31B + 168 = 0

=> B² - 24B - 7B + 168 = 0

=> B(B - 24) - 7(B-24) = 0

=> (B-7)(B-24) = 0

=> B = 7 or 24

then H = 24 or 7

hypotenuse = √24² + 7² = √625  = 25

Triangle is  7 , 24 , 25    

Cone will formed after rotating

Volume of Solid = (1/3)πr²h

Now if r = 7  Then h = 24

& Volume = (1/3)(22/7)7² * 24  = 1232 cm³

but if r = 24 & height = 7

Volume = (1/3)(22/7)24² * 7  = 4224 cm³