Question

David plans to cover the floor of his room with new material.

The floor is an isosceles trapezoid whose bases are 16 feet and 26 feet
and sides are 13 feet in length. Each piece of new material has an area of 2.5 square feet.
Assuming the pieces of new material can be cut as needed, how many pieces does
David need?
101
126
202
252

Answer 1

`\large \boxed{\text{101 pieces}}`

Explanation:

1. Calculate the area of the floor

The formula for the area of a trapezoid is

A = ½(a + b)h,

where a and b are the lengths of the parallel sides and h is the distance between them.

However, we are not given the value of h, so we must use Brahmagupta's formula.

If the four sides are a, b, c, and d, the area A is

`A = √((s - a) (s - b) (s - c) (s - d))`

where s is the semiperimeter

`s = (1)/(2)(a + b + c + d)`

Data:

a = 13 ft

b = 16 ft

c = 13 ft

d = 26 ft

(a) Calculate the semiperimeter

`\begin{array}{rcl}s & = & √((s - a) (s - b) (s - c) (s - d))\\\\ & = & (1)/(2)(13 + 16 + 13 + 26)\\\\ & = & (1)/(2)(68)\\\\ & = & 34\\\end{array}`

(b) Calculate the area

`\begin{array}{rcl}A & = &√((s - a) (s - b) (s - c) (s - d))\\\\& = &√((34 - 13) (34 - 16) (34 - 13) (34 - 26))\\\\& = &√(21 * 18 * 21 * 8)\\\\& = & √(63504)\\\\ & = & \mathbf{252}\\\end{array}`

The area of David's floor is 252 ft².

2. Calculate the number of pieces of material

`\text{No. of pieces} = \text{252 ft}^(2) * \frac{\text{1 piece}}{\text{2.5 ft}^(2)} = \textbf{100.8 pieces}\\\\\text{David can't buy a fraction of a piece, so he must buy $\large \boxed{\textbf{101 pieces}}$}`