A gardener at a nursery is filling pots with soil to prepare to transplant seedlings into these larger pots. Each new pot needs about 27 cubic inches of soil. The amount of soil s (in cubic inches) it takes to fill p pots is given by the function s 5 27p.
a. The gardener is filling the pots from a bag of soil that contains 3456 cubic inches of soil. Graph the function and identify its domain and range. How many pots can be filled from the bag?
b. Suppose the gardener needs to fill 100 pots. Graph the function on the same coordinate plane in part (a) and identify its domain and range. How much soil (in cubic inches) will the gardener need
To answer these questions, let's start by understanding the given function:
The function s = 27p represents the amount of soil in cubic inches (s) needed to fill p pots.
a. To graph the function and determine its domain and range:
1. We can represent the function graphically on a coordinate plane. The x-axis represents the number of pots (p), and the y-axis represents the amount of soil (s) in cubic inches.
2. To find the domain, we need to determine the possible values for the number of pots. Since the number of pots cannot be negative or fractional (you can't have a fraction of a pot), the domain is all non-negative integers: {0, 1, 2, 3, ...}.
3. To find the range, we need to determine the possible values for the amount of soil. Since the amount of soil (s) is given by 27p, the range will be all positive integers that are multiples of 27: {27, 54, 81, 108, ...}.
4. Now, let's plot some points on the graph. For example, if we let p = 0, the amount of soil needed would be s = 27(0) = 0 cubic inches. Another point could be p = 4, where s = 27(4) = 108 cubic inches. Connect these points (0, 0) and (4, 108) to form a line.
5. This line represents the function s = 27p.
Using this graph, we can find how many pots can be filled from a bag of soil containing 3456 cubic inches:
1. On the graph, find the y-coordinate that corresponds to the x-coordinate of 3456. This will determine the number of pots that can be filled from the bag.
2. From the graph, you can see that when x = 128 (approximately), y = 3456. Therefore, the number of pots that can be filled is approximately 128.
b. To determine how much soil will be needed to fill 100 pots:
1. On the same coordinate plane as before, we add another point where x = 100. Find the corresponding y-coordinate.
2. From the graph, you can see that when x = 100, y = 27(100) = 2700 cubic inches. Therefore, the gardener will need 2700 cubic inches of soil to fill 100 pots.
This is how you can approach solving these problems using the function s = 27p and its graphical representation.