Question

1 for each solid, write down an expression for the volume v:
a
b
c
(diagrams: a sphere with diameter 4x, a triangular prism with height a, base b, length 2c, a cone with radius 3y, height (y - 4))

Explanation:

Part a: Sphere

Step1: Find the radius

The diameter of the sphere is \( 4x \), so the radius \( r=\frac{4x}{2} = 2x\).

Step2: Volume formula for sphere

The volume of a sphere is given by \( V=\frac{4}{3}\pi r^{3}\). Substituting \( r = 2x\), we get \( V=\frac{4}{3}\pi(2x)^{3}=\frac{4}{3}\pi\times8x^{3}=\frac{32}{3}\pi x^{3}\).

Part b: Triangular Prism

Step1: Area of triangular base

The area of the triangular base with base \( b \) and height \( a \) is \( A=\frac{1}{2}\times b\times a\).

Step2: Volume of prism

The volume of a prism is \( V = \text{Base Area}\times\text{Length}\). The length of the prism is \( 2c \), so \( V=\frac{1}{2}\times b\times a\times2c=abc\).

Part c: Cone

Step1: Identify radius and height

The radius of the cone \( r = 3y \) and the height \( h=y - 4\).

Step2: Volume formula for cone

The volume of a cone is \( V=\frac{1}{3}\pi r^{2}h\). Substituting \( r = 3y \) and \( h=y - 4\), we get \( V=\frac{1}{3}\pi(3y)^{2}(y - 4)=\frac{1}{3}\pi\times9y^{2}(y - 4)=3\pi y^{2}(y - 4)\).

Answer:

a. \( V=\boldsymbol{\frac{32}{3}\pi x^{3}} \)
b. \( V=\boldsymbol{abc} \)
c. \( V=\boldsymbol{3\pi y^{2}(y - 4)} \)