Question

Citrus Ladders: Ladders used by fruit pickers are typically tapered with a wide bottom for stability and a narrow top for ease of picking. If the bottom rung of such a ladder is 49 inches wide and the top rung is 24 inches wide, how many rungs does the ladder have if each rung is 2.5 inches shorter than the one below it? How much material would be needed to make the rungs for the ladder described?

Short answer

The number of rungs in the ladder is 11

Material needed to make the rugs for the ladder is 401.5 inches

Step-by-step solution

Given Information 

The bottom rung of ladder is 49 inches wide and the top rung is 24 inches wide.

Each rung is 2.5 inches shorter than the one below .

Number of rungs

The bottom rung of ladder is 49 inches wide and the top rung is 24 inches wide.

first term is 49 and and last term is 24

Each rung is 2.5 inches shorter than the one below .
difference is -2.5 inches

$$\begin{gathered} a_n=a_1+(n-1)d \\ 24=49+(n-1)(-2.5) \\ 24=29-2.5n+2.5 \\ 24=51.5-2.5n \\ 24-51.5=-2.5n \\ -27.5=-2.5n \\ dividebothsidesby-2.5 \\ n=11 \end{gathered}$$

The number of rungs in the ladder is 11

Materials needed 

To find out materials needed , we need to find out sum

first term =49, last term = 24, d=-2.5 and n=11

$$\begin{gathered} S_n=\frac{n}{2}(a_1+a_n) \\ {S}_{11}=\frac{11}{2}(49+24) \\ {S}_{11}=\frac{11}{2}\left(73\right) \\ {S}_{11}=401.5 \end{gathered}$$

Material needed to make the rugs for the ladder is 401.5 inches