Question

An architect makes a blueprint for a custom-built house, the customer requests that the room under the roof is constructed at maximum volume. What dimensions for this room should the architect put on his blueprint if the length of the house is 100 feet, the width is 40 feet and the height of the space under the roof is 16 feet (see picture).

Answer 1

The dimensions that the architect should design the room under the roof is to have a length of 100 feet, a width of 40 feet, and a height of 16 feet to achieve the maximum volume of 32,000 cubic feet.

The shape of this attic (the room under the roof) is often a triangular prism, which is the shape you get when a triangle (the cross-section of the roof) is extended along the length of the house.

The volume of a triangular prism (the attic) is given by the formula:

V =
`(1)/(2)`​×base×height×length

Where:

The base is the same as the width of the house = 40 feet

The height of the triangle is the height of the attic space = 16 feet

The length of the house is the length of the prism = 100 feet

The maximum volume of the room under the roof (attic) should be:

V =
`(1)/(2)`​×40×16×100 = 32,000cubic feet

Thus, the architect should design the room under the roof to have a length of 100 feet, a width of 40 feet, and a height of 16 feet to achieve the maximum volume of 32,000 cubic feet.

Answer 2

We will have that the volume will be: