Problem 59 - Contemporary Precalculus

Question

A ladder with nine rungs is to be built, with the bottom rung 24 inches wide and the top rung 18 inches wide. If the lengths of the rungs decrease uniformly from bottom to top, how long should each of the seven intermediate rungs be?

Step-by-Step Solution

Answer

Answer: The lengths of the seven intermediate rungs are 23.25 inches, 22.5 inches, 21.75 inches, 21 inches, 20.25 inches, 19.5 inches, and 18.75 inches.

Step 1: Calculate the difference in length between the first and last rung

To find the difference in length between the first and last rung, subtract the length of the top rung from the length of the bottom rung: Difference = 24 inches - 18 inches = 6 inches

Step 2: Calculate the size of the uniform step

Since we have 9 rungs and only 7 intermediate rungs, there will be 8 steps between these rungs. To find the size of each step, divide the difference in length by the number of steps: Step size = 6 inches / 8 steps Step size = 0.75 inches

Step 3: Calculate the length of the intermediate rungs

Now that we have the step size, we can calculate the length of each intermediate rung by starting from the bottom rung and subtracting the step size, moving up the ladder: Rung 2: 24 inches - 0.75 inches = 23.25 inches Rung 3: 23.25 inches - 0.75 inches = 22.5 inches Rung 4: 22.5 inches - 0.75 inches = 21.75 inches Rung 5: 21.75 inches - 0.75 inches = 21 inches Rung 6: 21 inches - 0.75 inches = 20.25 inches Rung 7: 20.25 inches - 0.75 inches = 19.5 inches Rung 8: 19.5 inches - 0.75 inches = 18.75 inches The lengths of the 7 intermediate rungs are: 23.25 inches, 22.5 inches, 21.75 inches, 21 inches, 20.25 inches, 19.5 inches, and 18.75 inches.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. In other words, when you move from one number to the next in the sequence, you always add or subtract the same fixed amount, known as the common difference. This characteristic makes arithmetic sequences predictable and allows us to calculate any term in the sequence if we know the first term and the common difference.

For instance, if a sequence starts with the number 5 and has a common difference of 3, the successively following terms would be 8, 11, 14, and so on. This pattern is useful in various real-world applications, such as planning budgets over time, arranging equal payments or distances, or determining the size of objects that change incrementally, much like the rungs of a ladder in the provided exercise.

Uniform Decrease

A uniform decrease refers to a situation where a certain quantity is reduced by the same amount each time over a period or across a sequence. This term is commonly used in settings where something diminishes in a steady, even fashion. The concept of uniform decrease is inherently connected to arithmetic sequences. However, instead of adding a constant difference to get the next term as you would with an arithmetic sequence of increasing numbers, you subtract the same value to find subsequent terms in a decreasing pattern.

Let's draw on our ladder example. The ladder’s rung widths decrease uniformly, which means each rung is shorter than the previous one by the exact same length; that similarity in subtraction is the essence of uniform decrease. This predictability is what allows us to compute the width of each intermediate rung efficiently.

Sequence Step Size

The sequence step size, in the context of arithmetic sequences, is synonymous with the common difference. It represents the magnitude of increase or decrease between consecutive terms in the sequence. In the ladder scenario, the sequence step size is the uniform amount by which each rung's width decreases as you move from the bottom to the top.

Calculating the step size is a critical step. Once it is determined, you can easily find any term in the sequence without having to incrementally add or subtract across the entire sequence. In our example, the step size is 0.75 inches, which is the amount subtracted from the width of one rung to get the width of the next rung up on the ladder. Understanding the step size concept not only aids in problems like these but also forms the basis for solving more complex problems that involve arithmetic patterns.