Answer:
[tex]\large \boxed{\text{101 pieces}}[/tex]
Step-by-step 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
[tex]A = \sqrt{(s - a) (s - b) (s - c) (s - d)}[/tex]
where s is the semiperimeter
[tex]s = \frac{1}{2}(a + b + c + d)[/tex]
Data:
a = 13 ft
b = 16 ft
c = 13 ft
d = 26 ft
(a) Calculate the semiperimeter
[tex]\begin{array}{rcl}s & = & \sqrt{(s - a) (s - b) (s - c) (s - d)}\\\\ & = & \frac{1}{2}(13 + 16 + 13 + 26)\\\\ & = & \frac{1}{2}(68)\\\\ & = & 34\\\end{array}[/tex]
(b) Calculate the area
[tex]\begin{array}{rcl}A & = &\sqrt{(s - a) (s - b) (s - c) (s - d)}\\\\& = &\sqrt{(34 - 13) (34 - 16) (34 - 13) (34 - 26)}\\\\& = &\sqrt{21 \times 18 \times 21 \times 8}\\\\& = & \sqrt{63504}\\\\ & = & \mathbf{252}\\\end{array}[/tex]
The area of David's floor is 252 ft².
2. Calculate the number of pieces of material
[tex]\text{No. of pieces} = \text{252 ft}^{2} \times \dfrac{\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 $\large \boxed{\textbf{101 pieces}}$}[/tex]
