Answer :
Sure, let's analyze and solve the given problem step-by-step.
### Step-by-Step Solution
#### Step 1: Calculate the multiplier for the annual operating cost.
We'll determine how many times more the annual operating cost for an incandescent bulb is compared to an LED bulb.
- Incandescent bulb's annual operating cost: [tex]$54.00 - LED bulb's annual operating cost: $[/tex]7.50
The multiplier for the annual operating cost is calculated by dividing the incandescent bulb's annual operating cost by the LED bulb's annual operating cost:
[tex]\[ \text{Cost multiplier} = \frac{\$54.00}{\$7.50} = 7.2 \][/tex]
Thus, the annual operating cost for an incandescent bulb is 7.2 times more than that of an LED bulb.
#### Step 2: Calculate the total cost for the first year for an incandescent bulb.
The total cost for the first year includes the cost per bulb and the operating cost. We need to account for the fact that an incandescent bulb has a life span of 3 months. Therefore, in one year (12 months), you will need 4 incandescent bulbs.
- Cost per incandescent bulb: [tex]$1.30 - Annual operating cost for an incandescent bulb: $[/tex]54.00
The total cost for using incandescent bulbs for the first year:
[tex]\[ \text{Total cost (incandescent, first year)} = \text{Annual operating cost} + (\text{Cost per bulb} \times \text{Number of bulbs needed}) = \$54.00 + (\$1.30 \times 4) = \$54.00 + \$5.20 = \$59.20 \][/tex]
#### Step 3: Calculate the total cost for the first year for an LED bulb.
The total cost for the first year includes the cost per bulb and the operating cost. An LED bulb has a life span of 10 years, so you only need one LED bulb for the first year.
- Cost per LED bulb: [tex]$25.00 - Annual operating cost for an LED bulb: $[/tex]7.50
The total cost for using an LED bulb for the first year:
[tex]\[ \text{Total cost (LED, first year)} = \text{Annual operating cost} + \left(\frac{\text{Cost per bulb}}{\text{Life span in years}}\right) = \$7.50 + \left(\frac{\$25.00}{10}\right) = \$7.50 + \$2.50 = \$10.00 \][/tex]
#### Step 4: Calculate how much cheaper, in dollars, using LED bulbs is than using incandescent bulbs for the first year.
[tex]\[ \text{Cost difference} = \text{Total cost (incandescent, first year)} - \text{Total cost (LED, first year)} = \$59.20 - \$10.00 = \$49.20 \][/tex]
### Final Answers
1. How many times more is the annual operating cost for an incandescent bulb than an LED bulb?
- The annual operating cost for an incandescent bulb is 7.2 times more than that for an LED bulb.
2. How much cheaper, in dollars, is the total cost (the sum of the operating cost and the cost of bulbs) to pay for the first year if using LED bulbs than using incandescent bulbs?
- Using LED bulbs is \$49.20 cheaper than using incandescent bulbs for the first year.
### Step-by-Step Solution
#### Step 1: Calculate the multiplier for the annual operating cost.
We'll determine how many times more the annual operating cost for an incandescent bulb is compared to an LED bulb.
- Incandescent bulb's annual operating cost: [tex]$54.00 - LED bulb's annual operating cost: $[/tex]7.50
The multiplier for the annual operating cost is calculated by dividing the incandescent bulb's annual operating cost by the LED bulb's annual operating cost:
[tex]\[ \text{Cost multiplier} = \frac{\$54.00}{\$7.50} = 7.2 \][/tex]
Thus, the annual operating cost for an incandescent bulb is 7.2 times more than that of an LED bulb.
#### Step 2: Calculate the total cost for the first year for an incandescent bulb.
The total cost for the first year includes the cost per bulb and the operating cost. We need to account for the fact that an incandescent bulb has a life span of 3 months. Therefore, in one year (12 months), you will need 4 incandescent bulbs.
- Cost per incandescent bulb: [tex]$1.30 - Annual operating cost for an incandescent bulb: $[/tex]54.00
The total cost for using incandescent bulbs for the first year:
[tex]\[ \text{Total cost (incandescent, first year)} = \text{Annual operating cost} + (\text{Cost per bulb} \times \text{Number of bulbs needed}) = \$54.00 + (\$1.30 \times 4) = \$54.00 + \$5.20 = \$59.20 \][/tex]
#### Step 3: Calculate the total cost for the first year for an LED bulb.
The total cost for the first year includes the cost per bulb and the operating cost. An LED bulb has a life span of 10 years, so you only need one LED bulb for the first year.
- Cost per LED bulb: [tex]$25.00 - Annual operating cost for an LED bulb: $[/tex]7.50
The total cost for using an LED bulb for the first year:
[tex]\[ \text{Total cost (LED, first year)} = \text{Annual operating cost} + \left(\frac{\text{Cost per bulb}}{\text{Life span in years}}\right) = \$7.50 + \left(\frac{\$25.00}{10}\right) = \$7.50 + \$2.50 = \$10.00 \][/tex]
#### Step 4: Calculate how much cheaper, in dollars, using LED bulbs is than using incandescent bulbs for the first year.
[tex]\[ \text{Cost difference} = \text{Total cost (incandescent, first year)} - \text{Total cost (LED, first year)} = \$59.20 - \$10.00 = \$49.20 \][/tex]
### Final Answers
1. How many times more is the annual operating cost for an incandescent bulb than an LED bulb?
- The annual operating cost for an incandescent bulb is 7.2 times more than that for an LED bulb.
2. How much cheaper, in dollars, is the total cost (the sum of the operating cost and the cost of bulbs) to pay for the first year if using LED bulbs than using incandescent bulbs?
- Using LED bulbs is \$49.20 cheaper than using incandescent bulbs for the first year.
