Lyle is replacing light bulbs in his apartment.
He is using energy-saving bulbs and regular bulbs, and wants to use 21 or fewer bulbs
altogether.
He wants no more than 18 energy-saving bulbs and at least 2 regular bulbs.
The energy-saving bulbs cost $5.95 each, and the regular bulbs cost $7.85 each.
What is the most Lyle can spend on replacement bulbs? How many of each will be use?
s </= 18
r >/= 2
cost = 5.95 s + 7.85 r
s + r </= 21
maximize cost (for some reason) (so use all expensive ones but not to jump to conclusion)
graph it, I did s hor (x axis) and r vertical (y axis)
below line sloping down from (0,21) to ( 21,0) which is is r =21 - s
left of (18,0)
above r = 2
useful intersections are (0,21) and the intersection of s = 18 and r = 21-s
which is (18,3)
calculate
cost = 5.95 s + 7.85 r
at those two points and pick the higher one (obviously regular = 21)
I suspect a typo but gave you the method anyway.
Well, Lyle is quite the light bulb aficionado! Let's do some math to figure out his spending limit and the number of bulbs he needs.
Let's say Lyle uses "e" energy-saving bulbs and "r" regular bulbs. According to the given conditions, we have the following constraints:
1. Lyle wants to use 21 or fewer bulbs: e + r ≤ 21. (Equation 1)
2. Lyle wants no more than 18 energy-saving bulbs: e ≤ 18. (Equation 2)
3. Lyle wants at least 2 regular bulbs: r ≥ 2. (Equation 3)
Now that we have the constraints, let's find the maximum amount Lyle can spend on replacement bulbs.
The cost of energy-saving bulbs is $5.95 each. So, the cost of "e" energy-saving bulbs is 5.95 * e.
The cost of regular bulbs is $7.85 each. So, the cost of "r" regular bulbs is 7.85 * r.
The total spending limit for Lyle is the sum of these costs, which should be maximum:
Total spending limit = 5.95 * e + 7.85 * r. (Equation 4)
Now, let's find the maximum spending limit with the given constraints. We can use trial and error or substitution methods.
Let's start by substituting the value of "r" (using Equation 3) in Equation 4:
Total spending limit = 5.95 * e + 7.85 * 2
= 5.95 * e + 15.70
To find the maximum spending limit, we need to maximize the value of "e" while staying within the given constraints (Equations 1 and 2).
Case 1: e = 18 (maximum value allowed by Equation 2)
Total spending limit = 5.95 * 18 + 15.70
≈ $147.80
Case 2: e = 17
Total spending limit = 5.95 * 17 + 15.70
≈ $141.55
Case 3: e = 16
Total spending limit = 5.95 * 16 + 15.70
≈ $135.30
By observing the pattern, we can see that as the value of "e" decreases, the total spending limit decreases. Therefore, the maximum spending limit occurs when e = 18 (the maximum value allowed by Equation 2).
So, the most Lyle can spend on replacement bulbs is approximately $147.80.
Now, let's determine the number of energy-saving and regular bulbs he will use.
Since the maximum number of energy-saving bulbs is 18, we consider e = 18.
Using Equation 1: e + r ≤ 21
18 + r ≤ 21
r ≤ 21 - 18
r ≤ 3
Based on Equation 3, r must also be at least 2. So, the number of regular bulbs (r) will be 2.
Therefore, Lyle will use 18 energy-saving bulbs and 2 regular bulbs.
I hope Lyle's apartment is now shining brighter than the sun!
Let's solve this step-by-step:
Step 1: We need to find the maximum number of energy-saving bulbs Lyle can use.
Since Lyle wants no more than 18 energy-saving bulbs, the maximum number of energy-saving bulbs he can use is 18.
Step 2: We need to find the minimum number of regular bulbs Lyle can use.
Lyle wants at least 2 regular bulbs, so the minimum number of regular bulbs he can use is 2.
Step 3: We need to find the total number of bulbs Lyle will use.
The maximum total number of bulbs Lyle can use is given as 21.
To find the distribution of energy-saving and regular bulbs, let's assume Lyle uses x energy-saving bulbs and y regular bulbs.
Step 4: We can set up the following equation based on the information we have:
x + y ≤ 21 (Total number of bulbs should be 21 or fewer)
x ≤ 18 (Maximum number of energy-saving bulbs is 18)
y ≥ 2 (Minimum number of regular bulbs is 2)
Step 5: Now, let's calculate the cost of the bulbs.
The cost of each energy-saving bulb is $5.95, so the total cost of the energy-saving bulbs will be 5.95x.
The cost of each regular bulb is $7.85, so the total cost of the regular bulbs will be 7.85y.
Step 6: We need to find the maximum amount Lyle can spend on the replacement bulbs.
By calculating the total cost, we get the equation:
Total Cost ≤ Maximum Spending
Step 7: Bringing the equations together, we have:
5.95x + 7.85y ≤ Maximum Spending
Since the maximum spending is not given, we cannot determine the exact value of the maximum amount Lyle can spend on replacement bulbs or the number of each type of bulb he will use without further information.
To find out the most Lyle can spend on replacement bulbs and the number of each bulb he will use, we can set up a system of inequalities based on the given conditions.
Let's assign variables to represent the number of energy-saving bulbs and regular bulbs Lyle will use. Let's use "e" for energy-saving bulbs and "r" for regular bulbs.
Based on the given conditions, we can set up the following inequalities:
1. The total number of bulbs used should be 21 or fewer: e + r ≤ 21
2. Lyle wants no more than 18 energy-saving bulbs: e ≤ 18
3. Lyle wants at least 2 regular bulbs: r ≥ 2
Now let's solve this system of inequalities to find the maximum amount Lyle can spend and the number of each bulb he will use.
Step 1: Solve the inequalities to find the possible range of values for "e" and "r":
From inequality 2: e ≤ 18
From inequality 3: r ≥ 2
The possible range of values for "e": 0 ≤ e ≤ 18
The possible range of values for "r": 2 ≤ r ≤ 21
Step 2: Calculate the maximum amount Lyle can spend:
To find the maximum amount Lyle can spend, we need to find the scenario in which he spends the most money. Since energy-saving bulbs are cheaper than regular bulbs, we should aim to use as many energy-saving bulbs as possible.
Let's assume Lyle uses all 18 energy-saving bulbs (e = 18) and the remaining bulbs are regular bulbs (r = 21 - 18 = 3).
The cost of the energy-saving bulbs: 18 * $5.95 = $107.10
The cost of the regular bulbs: 3 * $7.85 = $23.55
Therefore, the maximum Lyle can spend on replacement bulbs is $107.10 + $23.55 = $130.65.
In summary, Lyle can spend a maximum of $130.65 on replacement bulbs. He will use 18 energy-saving bulbs and 3 regular bulbs.