Question

The volume, V cm^3 of a solid is directly proportional to the cube of its radius, r cm. When the radius of the solid is 6 cm, its volume is 905 cm^3.

(a) Find an equation connecting V and r.
(b) Calculate the volume of the solid when its radius is 10 cm.
(c) V = k(r^3), where k is a constant.
(d) V = k/r^3, where k is a constant.

Answer 1

The relationship between volume and radius is given by V ≈ 4.1898r³. When the radius is 10 cm, the volume is 4189.8 cm³. The correct formula representing this relationship is V = k(r³), not V = k / r³.

Step-by-step explanation:

The volume of a solid is directly proportional to the cube of its radius, which can be represented by the equation V = k × r³, where k is a proportionality constant, and r is the radius. To find k, use the given values when r = 6 cm and V = 905 cm³, which means k = V / r³ = 905 / 6³ = 905 / 216 ≈ 4.1898. So the equation connecting V and r is V ≈ 4.1898r³.

For part (b), to calculate the volume when the radius is 10 cm using the established equation, substitute r with 10 to get V ≈ 4.1898 × 10³ which gives V ≈ 4.1898 × 1000 = 4189.8 cm³.

As for part (c), the equation V = k(r³) is indeed correct as it satisfies the direct proportionality of V and r³. Part (d) suggesting V = k / r³ is incorrect, as it implies an inverse relationship between volume and the cube of the radius, which contradicts the direct proportionality mentioned in the question.