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?
Answers
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