Ladders used by fruit pickers are typically tapered with a...

Name: Problem 66, Chapter 11: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2020-05-09T21:00:49-08:00
Duration: 2 min 49 s
Channel: Julie Silva
Description: 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?

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?

Key Concepts

nth Term Formula

The nth term formula for an arithmetic sequence, given by a_n = a_1 + (n-1)d, is essential for determining the number of terms in a sequence when the first term, the last term, and the common difference are known. This is crucial in calculating the total number of rungs on a ladder with uniformly decreasing lengths.

Sum of an Arithmetic Series

The sum of an arithmetic series computes the total of all terms in an arithmetic sequence and is given by the formula S = n/2 (a_1 + a_n). This concept is used to determine the total material required by adding up the measurements of all the ladder rungs.

Arithmetic Sequence

An arithmetic sequence is a series of numbers or measurements in which each term is obtained by adding a constant value, called the common difference, to the previous term. This concept is used to model situations where a quantity increases or decreases uniformly, such as the lengths of the ladder rungs.

Common Difference

The common difference is the fixed amount that each term in an arithmetic sequence increases or decreases relative to the previous term. In problems involving measurements that taper, the common difference helps determine how each subsequent measurement relates to the earlier one.

Transcript

00:01 So in this problem, we're told that we have a ladder, as i kind of illustrated by our picture here, and we're told that the length of the bottom rung is 49 inches, and the length of the top rung is 24 inches wide.

00:13 We want to know how many rungs does this ladder have if each rung is 2 .5 inches shorter than the one below it.

00:20 Well, let's think of this in terms of a sequence.

00:23 So if our bottom rung is 49 inches wide, and we're going to keep subtracting 2 .5 inches, the next rung, would be 46 .5 inches, and then the one after that would be 44 inches, and so forth, all the way till our last one, which is 24 inches.

00:39 So we want to know how many terms are in this sequence.

00:42 Well, a sub 1 is 49, a sub n is equal to 24, and our common difference is negative 2 .5.

00:50 So we're going to use our formula, a sub n, equals a sub 1 plus the quantity of n -minus 1 times d to find our n value, because n will tell us how many rungs there are.

01:01 Well, let's substitute in our values.

01:03 Well, we're going to have 24 is equal to 49 plus the quantity of m minus 1 times negative 2 .5.

01:12 So to solve for n, i'm going to first subtract 49 from both sides of the equation.

01:18 So 24 minus 49 is negative 25, and it's equal to the quantity of m minus 1 times negative 2 .5.

01:27 Next, i'm going to divide both sides of my equation by negative 2 .5, because then those terms cancel out, and negative 25 divided by negative 2 .5 is equal to 10...

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