Question

Consider an ordinary 100W incandescent light bulb and a LED light bulb that produces the same amount of light, but consumes 20W of electricity. Assuming the values for purchase price and lifetime for each bulb, and a cost of electricity as given below, compute the number of hours of usage required for the LED light bulb to become more cost-effective than the incandescent bulb for each case below. If the crossover in cost occurs because a new bulb must be purchased, you may take the number of hours to be equal to the time at which the bulb must be purchased. Note: one year of moderate usage corresponds to ~1000 hours.

Answer 1

To calculate the number of hours of usage required for the LED light bulb to become more cost-effective than the incandescent bulb, we need to compare the total costs of each bulb. The LED bulb uses 80% less energy than the incandescent bulb, resulting in electricity cost savings. By comparing the total costs, we can find the point at which the LED bulb becomes more cost-effective.

Step-by-step explanation:

To calculate the number of hours of usage required for the LED light bulb to become more cost-effective than the incandescent bulb, we need to compare the total costs of each bulb. The LED bulb uses 80% less energy than the incandescent bulb, resulting in electricity cost savings. We also need to consider the purchase price and lifetime of each bulb. By comparing the total costs, we can find the point at which the LED bulb becomes more cost-effective.

Let's use the given values: The LED bulb uses 20W of electricity and the incandescent bulb uses 100W. Assume the cost of electricity is $0.10 per kilowatt-hour. The LED bulb costs $20.00 and the incandescent bulb costs $0.75.

First, we calculate the energy used during the year for each bulb: E = Pt. For the LED bulb, the energy is (20W)(3 hours/day)(365 days/year) = 21.9 kilowatt-hours. For the incandescent bulb, the energy is (100W)(3 hours/day)(365 days/year) = 109.5 kilowatt-hours.

Next, we can multiply the energy by the cost of electricity to find the cost for each bulb. For the LED bulb, the cost is (21.9 kWh)($0.10/kWh) = $2.19. For the incandescent bulb, the cost is (109.5 kWh)($0.10/kWh) = $10.95.

Now we need to consider the initial purchase price and the lifetime of each bulb. The incandescent bulb lasts for 1.08 years (1200 hours) and the LED bulb lasts for 45.66 years (50,000 hours).

Finally, we can calculate the total cost for each bulb, including the purchase price and the energy cost. For the LED bulb, the total cost is $20.00 + $2.19 = $22.19. For the incandescent bulb, the total cost is $0.75 + $10.95 = $11.70.

Therefore, the LED bulb becomes more cost-effective than the incandescent bulb after approximately 545 hours of usage, since the total cost of the LED bulb is lower.

Answer 2

Assuming the the values for purchase price and lifetime for each bulb, and a cost of electricity are:

Incandescent Bulb

Power:100W

Purchase Price: $1.50

Lifetime:
`1000 h \approx 1 year`

Electricity cost: $0.10/kWh

LED Bulb

Power:20W

Purchase Price: $40

Lifetime:
`25000 h \approx 25 years`

Electricity cost: $0.10/kWh

Let's begin with the calculations:

The electric energy
`E` used for a bulb is:

`E=P.t` (1)

Where
`P` is the electric power and
`t` the time the bulb is used.

And the operating cost is:

`Cost=E. Cost (/kWh)` (2)

Where
`Cost (/kWh)` is the Electricity cost in units of
`/kWh`

For the Incandescent Bulb:

`E_(IN)=(100W)(1000h)` (3)

`E_(IN)=100000 Wh=100 kWh` (4) This is the electric energy used

Calculating the cost:

`Cost_(IN)=E_(IN) Cost(/kWh)_(IN)` (5)

`Cost_(IN)=(100 kWh)($0.10/kWh)` (6)

`Cost_(IN)=\$10` (7) This is the electricity cost for incandescent light bulb for 1000 hours

For the LED Bulb:

`E_(LED)=(20W)(25000h)` (8)

`E_(LED)=500000 Wh=500 kWh` (9) This is the electric energy used by the LED bulb

Calculating the cost:

`Cost_(LED)=E_(LED) Cost(/kWh)_(LED)` (10)

`Cost_(LED)=(50 kWh)($0.10/kWh)` (11)

`Cost_(LED)=\$50` (12) This is the electricity cost for LED bulb for 25000 hours

As we can see the use of LED bulb for 25 years (25000 h) is $50, this means in 1 year the cost would be $2.

On the other hand, the use of Incandescent bulb for 1 year (1000 h) is $10.

This means using the LED bulb will result more cost-effective than the Incandescent bulb, although the initial investment is high.