Problem 70 - Algebra and Trigonometry

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?

Step-by-Step Solution

Answer

The ladder has 11 rungs, and 401.5 inches of material are needed for the rungs.

Step 1: Identify the width of both the top and bottom rungs

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

Step 2: Determine the decrease in width per rung

Each rung is 2.5 inches shorter than the one below it.

Step 3: Calculate the number of rungs

To find the total number of rungs, use the arithmetic sequence formula:\[ a_n = a_1 + (n - 1)d \]Here, \(a_1 = 49\) inches, \(a_n = 24\) inches, and \(d = -2.5\) inches. Plugging in the values, we get:\[ 24 = 49 + (n - 1)(-2.5) \]Solving for \(n\):\[ 24 = 49 - 2.5(n - 1) \]\[ 24 = 49 - 2.5n + 2.5 \]\[ 24 - 51.5 = -2.5n \]\[ -27.5 = -2.5n \]\[ n = \frac{-27.5}{-2.5} \]\[ n = 11 \]

Step 4: Determine the width of each rung

The rungs decrease by 2.5 inches each time starting from 49 inches. Therefore, the widths of the rungs from bottom to top are: 49, 46.5, 44, ..., 24 inches.

Step 5: Calculate the total material needed

To calculate the total width of all the rungs, sum the arithmetic series:\[ S_n = \frac{n}{2} (a_1 + a_n) \]Plugging in the values, we get:\[ S_{11} = \frac{11}{2} (49 + 24) \]\[ S_{11} = \frac{11}{2} (73) \]\[ S_{11} = \frac{11 \times 73}{2} \]\[ S_{11} = 401.5 \]The total material needed for the rungs is 401.5 inches.

Key Concepts

ladder problem

Ladders used by fruit pickers often have a unique design: they taper from a wide bottom to a narrower top. This makes them more stable at the bottom while making it easier to reach fruits at the top. In this problem, we know the width of the bottom rung is 49 inches and the top rung is 24 inches. By understanding these dimensions and the fact that each rung is consistently 2.5 inches shorter than the one below it, we can solve the problem methodically.

arithmetic sequence formula

An arithmetic sequence is a series of numbers where each term increases or decreases by a constant difference. Here, each rung's width decreases by 2.5 inches from the rung below it, forming an arithmetic sequence. The formula we use to find any term in the sequence is: \[a_n = a_1 + (n - 1)d\] where \(a_1\) is the first term (49 inches), \(d\) is the common difference (-2.5 inches), and \(a_n\) is the nth term, which in this case is the top rung (24 inches). Solving this helps determine the number of rungs:

material calculation

To find the total material required for the ladder's rungs, we sum the widths of all the rungs. Using the arithmetic series sum formula: \[ S_n = \frac{n}{2} (a_1 + a_n)\] Here, \(n\) is the number of rungs (11), \(a_1\) is the width of the bottom rung (49 inches), and \(a_n\) is the width of the top rung (24 inches). Plugging these values into the formula, we get: \[ S_{11} = \frac{11}{2} (49 + 24) = 401.5 \, inches\] So, 401.5 inches of material is required to make the rungs of this ladder.