The volume of gas a balloon can hold varies directly as the cube of its radius. Imagine a balloon with a radius of 3 inches can hold 81 cubic inches of gas. How many cubic inches of gas must be released to reduce the radius down to 2 inches?

57 in³

Step-by-step explanation:

Since the volume of gas and the cube of its radius vary directly, we can set up an initial equation to find the constant of variation:

V = kr³, where v=volume, r=radius and k=contant of variation

Given that a balloon with a radius of 3 inches can hold a volume of 81 cubic inches, we can plug these values into the equation and solve for 'k':

81 = k3³ or 81 = 27k or k = 3

Given 'k', now we can find the volume of the balloon with a radius of 2:

V = (3) (2³) = 24 in³

Since the question asked how much gas must be released to reduce the radius, we would find the difference of 81 and 24:

81 - 24 = 57 in³