Chapter 12.2, Problem 64AYU

To determine

To find: 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?

Answer to Problem 64AYU

a. 11

Explanation of Solution

Given:

Bottom rung $=49" \text{wide}$.

Top rung $=24" \text{wide}$.

Each rung is $2.5"$ shorter than the one below it.

Formula used:

The $n$th term of an arithmetic sequence can be found by the formula,

$a_n=a_1+\left(n-1\right)d$   (Formula 1)

Sum of the first $n$ terms of an arithmetic sequence,

$S_n=\frac{n}{2}\left[a_1+a_n\right]$   (Formula 2)

Calculation:

From the given data, we get $a_1=49,a_n=24,d=-2.5$.

a. To find the number of rungs in the ladder:

$24=49+\left(n-1\right)\left(-2.5\right)$   (By formula 1)

$2.5n=49-24+2.5=27.5$

$n=11$

Therefore, the number of rungs in the ladder $=11$.

To determine

To find: 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?

Answer to Problem 64AYU

b. $401.5$

Explanation of Solution

Given:

Bottom rung $=49" \text{wide}$.

Top rung $=24" \text{wide}$.

Each rung is $2.5"$ shorter than the one below it.

Formula used:

The $n$th term of an arithmetic sequence can be found by the formula,

$a_n=a_1+\left(n-1\right)d$   (Formula 1)

Sum of the first $n$ terms of an arithmetic sequence,

$S_n=\frac{n}{2}\left[a_1+a_n\right]$   (Formula 2)

Calculation:

From the given data, we get $a_1=49,a_n=24,d=-2.5$.

b. To find the total number of material required to make the rungs for the ladder:

$S_n=\frac{11}{2}\left[49+24\right]$   (By formula 2)

$=\frac{11}{2}\left[49+24\right]=401.5$