Question

the figure below shows a rectangular prism. one base of the prism is shaded.
(a) write an expression to find the volume of the prism using its length, width, and height. then simplify.
volume of prism: ■■×■■=■■cubic units
(b) find the area of the shaded base. then, using the area of the shaded base, write an expression to find the volume of the prism and simplify.
area of shaded base: ■■×■■=■■square units
volume of prism: ■■×■■=■■cubic units

Explanation:

Step1: Find volume using length, width, height

The formula for the volume of a rectangular prism is \(V = l\times w\times h\). Here, \(l = 6\), \(w = 2\), \(h = 7\). So the expression is \(6\times2\times7\).

$$6\times2\times7=(6\times2)\times7 = 12\times7=84$$

Step2: Find area of shaded base

The shaded base is a rectangle. The formula for the area of a rectangle is \(A = l\times w\). Here, \(l = 2\), \(w = 7\). So the expression is \(2\times7\).

$$2\times7 = 14$$

Step3: Find volume using base - area and height

The formula for the volume of a prism is \(V=B\times h\) (where \(B\) is the base - area and \(h\) is the height). Here, \(B = 14\) (from step 2) and \(h = 6\). So the expression is \(14\times6\).

$$14\times6=(10 + 4)\times6=10\times6+4\times6=60 + 24=84$$

Answer:

(a) Volume of prism: \(6\times2\times7 = 84\) cubic units
(b) Area of shaded base: \(2\times7=14\) square units; Volume of prism: \(14\times6 = 84\) cubic units