A large tank is designed with ends in the shape of the region between y=x^2 / 2 and y = 15, measured in feet. Find the hydrostatic force on one end of the tank if it is filled to a depth of 5 feet with gasoline ( Assume that the density of gasoline is 42 lb/ft^3 )
y = x^2 / 2
y = 5 (height of gas)
5 = x^2 / 2 (intersection)
x = sqrt(10) intersection
this takes form of the shape of a parabola from -sqrt10 to sqrt10 (about 3.6)
hydrostatic force = integral(42 * depth * width)
depth = (5-x^2/2)
width = x
hf = 42 integral( (5-x2/2)(2x) )
hf = 84 integral( (5x-x3/2) )
hf = 84 [0,sqrt 10] (5x2/2 - x4/8)
hf = 84 ( 5*10/2 - sqrt10 ^4 /8)
= 1050
this is the answer I got and it is incorrect, not sure where i went wrong.
y = x^2 / 2
y = 5 (height of gas)
5 = x^2 / 2 (intersection)
x = sqrt(10) intersection
this takes form of the shape of a parabola from -sqrt10 to sqrt10 (about 3.6)
hydrostatic force = integral(42 * depth * width)
depth = (5-x^2/2)
width = x
hf = 42 integral( (5-x2/2)(2x) )
hf = 84 integral( (5x-x3/2) )
hf = 84 [0,sqrt 10] (5x2/2 - x4/8)
hf = 84 ( 5*10/2 - sqrt10 ^4 /8)
= 1050
this is the answer I got and it is incorrect, not sure where i went wrong.