A gardner 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=27p.
a. The gardener is filling the pots from a big 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 at most 100 pots. Graph the function on the coordinate plane in part (a). Identify its domain and range. How much soil (in cubic inches) will the gardener need?
a. To graph the function s=27p, we can use a coordinate plane. We'll plot the value of p on the x-axis and the value of s on the y-axis.
The domain represents all the possible values that p can take. In this case, p represents the number of pots, so it must be a positive whole number. Therefore, the domain is the set of positive integers.
The range represents all the possible values that s can take. The function s=27p tells us that the amount of soil s is directly proportional to the number of pots p. Since we can have any positive whole number of pots, the range is also the set of positive integers.
To find out how many pots can be filled from the bag of soil, we need to find the value of p when s=3456. We can solve the equation s=27p for p by dividing both sides by 27:
s/27 = p
Substituting the value of s=3456, we get:
3456/27 = p
p ≈ 128
Therefore, the gardener can fill approximately 128 pots from the bag of soil.
b. To graph the function with a restriction of at most 100 pots, we need to limit the range of p. On the coordinate plane, we'll only plot the values of p from 1 to 100.
The domain remains the same, as p still represents the number of pots, which must be a positive whole number.
The range, however, is now limited to the set of positive integers ≤ 100.
To find out how much soil will be needed, we need to calculate s for p=100 using the equation s=27p:
s = 27 * 100
s = 2700
Therefore, the gardener will need 2700 cubic inches of soil to fill at most 100 pots.
the range of s is [0,3456]
since s=27p, that means the domain is [0/27,3456/27] = [0,128]
(b) now the domain is [0,100]
so, the range of s is [0,2700]