Problem 71 - Introductory and Intermediate Algebra for College Students 4th

Question

Describe the difference between the following problems: How much fencing is needed to enclose a garden? How much fertilizer is needed for the garden?

Step-by-Step Solution

Answer

Fencing a garden is a one-dimensional problem, concerned with the boundary or perimeter of the garden, while fertilizing a garden is a two-dimensional problem, concerned with the total area the garden covers. Therefore, the amount of fencing required will be equivalent to the total length of the garden's boundary, while the amount of fertilizer needed will depend on the total area of the garden.

Step 1: Assessing Fencing Requirements

The problem of how much fencing is needed to enclose a garden is dependent on the boundary length of the garden - how many units around is the garden? If it is a simple rectangular shape, the calculation will be \( 2 * length + 2 * width \), which is the formula for the perimeter of a rectangle. Let's assume the garden is in a square shape of side=length with value of 5 meters. The perimeter is \( 4*length = 4 * 5 = 20 \) meters. The amount of fencing needed is the same as the perimeter, in this case, 20 meters.

Step 2: Assessing Fertilizer Requirements

The problem of how much fertilizer is needed for a garden is determined by the total area of the garden - how many units of space does the garden cover? If it is a simple rectangular shape, the calculation will be \( length * width \), which is the formula for the area of a rectangle. Using the same dimensions of our square garden, the area will be \( length * length = 5 * 5 = 25\) square meters. Fertilizer packages usually indicate how much is needed per square meter; if it's 10 grams per square meter, we'll need \( 25 * 10 = 250 \) grams.