Question

A mosaic tile floor is designed in the shape of a trapezoid 30 feet wide at the base and 15 feet

wide at the top. The tiles, 12 inches by 12 inches, are to be placed so that each successive row contains one less tile than the row below. How many tiles will be required?

Short answer

Total number of tiles required is$360tiles.$

Step-by-step solution

Given information:

Let us assume, $1feet=1tile$.

Then, Number of tiles required at the base, $a_1=30tiles$

Number of tiles required at the top, $a_n=15tiles$

Since each successive row requires one less tiles, $d=-1$

$\therefore$ Sequence will be$30,29,28,...15$. It is an Arithmetic Sequence.

Formulas: 

$$\begin{gathered} \left(1\right)n=\frac{a_n-a_1+d}{d}\left[Onsimplifyingtheformula{a}_n=a_1+(n-1)d\right] \\ \left(2\right)S_n=\frac{n}{2}(a_n+a_1) \end{gathered}$$

To find the unknown term " n ":

Substitute the given values in formula (1), we get

$n=\frac{15-30-1}{-1}$

$$\begin{gathered} =\frac{-16}{-1} \\ =16 \end{gathered}$$

To find the required number of tiles to build a floor design:

Substitute the given values in formula (2), we get

$S_n=\frac{16}{2}(15+30)$

$$\begin{gathered} =\frac{16}{2}\times 45 \\ =8\times 45 \\ =360 \end{gathered}$$