The base of a rectangular prism is always a rectangle. Therefore, to find the length of BC we will use the formula for the area of a rectangle.
Area=l* wIn our case we can assume that l=AB and w=BC.
Area=AB* BC
Next, let's consider the given information. We know that the area of the base of the prism is equal to 20cm^2. From the diagram we have that AB= x. Using this information and the formula above, we can write BC in terms of x. Let's do it!
Since the shaded region is a side wall of a rectangular prism, it is a rectangle. Therefore, to calculate its area we will use the formula for the area of a rectangle.
Area=l* w
In our case we can assume that l=DC and w=BC.
Area=DC* BC
We are also given that DC= 3BC. Let's substitute this expression into the formula above and simplify.
Area= 3BC* BC ⇔ Area=3BC^2
Finally, we want to express the area of the shaded region in terms of x. From Part A we know that BC= 20xcm.
Volume=Area of the Base*Height
We are told that the area of the base is 20cm^2.
Volume= 20*Height
The height of the given rectangular prism is equal to DC. From Part B we know that DC=3BC and from Part A we know that BC= 20x. Let's bring all of this information together.
Height=DC=3BC=3( 20/x)=60/x
Finally, we can calculate the volume of the prism.
Volume=20*60/x=1200/x
The volume of the prism is 1200x cubic centimeters.
