First — talk for 2 minWhat's your family's electric bill like, roughly? Has anyone in your house thought about solar panels? What stopped them, or pushed them toward it? If no one, Mr. Bennett will share.
Two prices, moving in opposite directions
Every year, rooftop solar panels get a little cheaper. Since 1977, the price of a solar panel has dropped from about \$76.67 per watt to about \$0.11 per watt in 2024 — falling roughly 10% per year for almost fifty years. Engineers call this pattern Swanson's law.
Meanwhile, the price of grid electricity creeps up. The U.S. average residential rate was about 9 cents per kilowatt-hour in 2010 and is around 16.5 cents today — growth of about 2.5% per year. One curve goes down. The other goes up.
Somewhere between those two curves, there is a single year where they cross. Before that year, grid electricity is cheaper; after, rooftop solar is cheaper. So in your state, which year does solar win — and what is keeping that year in the future?
Worked Example #1 — The national crossing year (find t)
ProblemSolar cost \(S(t)\) and grid retail rate \(R(t)\) are measured in cents per kWh, with \(t\) = years since 2010. Find the year both costs are equal.
DataSolar starts at 24¢/kWh and decays at factor 0.92 per year: \(S(t) = 24 \cdot (0.92)^{t}\). Grid starts at 9¢/kWh and grows at factor 1.025 per year: \(R(t) = 9 \cdot (1.025)^{t}\). Find \(t\) where \(S(t) = R(t)\).
Word Bank
divide
decay
logarithm
equal
isolate
multiply
growth
ratio
Number Bank
9.08
2.667
0.4260
2019
1.1141
0.0469
8/3
15.7
| # | Reasoning (fill in) | Math |
|---|
| 1 |
Set the two costs . This is the equation that holds at the crossing year. |
\(24 \cdot (0.92)^{t} = 9 \cdot (1.025)^{t}\) |
| 2 |
both sides by \(9 \cdot (0.92)^{t}\) so the bases are grouped on one side. |
\(\dfrac{24}{9} = \dfrac{(1.025)^{t}}{(0.92)^{t}} = \left(\dfrac{1.025}{0.92}\right)^{t}\) |
| 3 |
Simplify the on the left and the base on the right. |
\(\dfrac{8}{3} \approx\) \(= (\) \()^{t}\) |
| 4 |
Take a on both sides to free \(t\) from the exponent. |
\(\log\!\left(\tfrac{8}{3}\right) = t \cdot \log(1.1141)\) |
| 5 |
Divide to \(t\), then evaluate. Add 2010 to convert \(t\) to a year. |
\(t = \dfrac{\log(8/3)}{\log(1.1141)} \approx \dfrac{}{}\) \(\approx\) yr ⇒ year |
Page 1 of 3 · When Does Solar Win? · Algebra 2 · Unit 5 · Lesson 16
Worked Example #2 — Texas crossing year
ProblemIn Texas, solar installs cheaply and grid rates are low. Find the Texas crossing year.
Data\(S_{\text{TX}}(t) = 22 \cdot (0.92)^{t}\) and \(R_{\text{TX}}(t) = 11 \cdot (1.025)^{t}\), in cents per kWh, with \(t\) = years since 2010. Note \(0.92\) is a decay factor: \(b < 1\).
Word Bank
divide
log
simplify
equal
substitute
isolate
add
Number Bank
2
0.301
6.4
2016
1.1141
0.0469
3.2
| # | Reasoning (fill in) | Math |
|---|
| 1 |
Set the two cost functions . |
\(22 \cdot (0.92)^{t} = 11 \cdot (1.025)^{t}\) |
| 2 |
both sides by \(11 \cdot (0.92)^{t}\) to group the bases. |
\(\dfrac{22}{11} = \left(\dfrac{1.025}{0.92}\right)^{t}\) |
| 3 |
the ratio on the left, then the base on the right. |
\(= (1.1141)^{t}\) |
| 4 |
Take a on both sides. Use \(\log_b(\dots)\) or \(\log\) base 10 — same answer. |
\(\log(2) = t \cdot \log(1.1141)\) |
| 5 |
Divide to \(t\). Then add 2010 to get the calendar year. |
\(t = \dfrac{0.301}{0.0469} \approx\) yr ⇒ year |
Activity 2 answers — pick one state per pair, then trade
California: \(30 \cdot (0.92)^{t} = 14 \cdot (1.06)^{t}\). Divide: \(\dfrac{30}{14} =\) \( = (1.152)^{t}\). Take a log: \(t \approx\) yr ⇒ year .
Pennsylvania: \(28 \cdot (0.92)^{t} = 11 \cdot (1.022)^{t}\). Divide: \(\dfrac{28}{11} \approx\) \( = (1.111)^{t}\). Take a log: \(t \approx\) yr ⇒ year .
Ranking (earliest crossing first): → → .
Conclude
The math says PA crossed in 2019. So why is rooftop solar still expensive in PA?
Three states. Three crossing years. Same math — different starting values and growth factors. But the math gives the year a system-level average crosses. The year a real household can afford to install is moved by levers the equation hides.
The model says
California crosses in ~2015.
Texas crosses in ~2016–2017.
Pennsylvania crosses in ~2019.
Real households face
About 65% of a U.S. residential install price is not the panels. It is permitting, inspection, labor, financing, and sales overhead. Germany pays ~\$0.50/W in soft costs. PA pays ~\$1.65/W.
Reflect: Which state would you want to live in if you wanted to install solar today — and what one policy lever (permitting, net metering, community solar) would move your state's crossing year forward the most?
Page 2 of 3 · When Does Solar Win? · Algebra 2 · Unit 5 · Lesson 16
Tier 2 · General Academic Words
- intersect — to cross at one point. Two lines or curves intersect where they share the same input and output.
- equilibrium — a balance point where two forces or values are equal.
- threshold — the moment something crosses from one state into another (like cheap to expensive).
- payback — the time it takes for a savings or earnings to equal the original cost you paid.
- decline — to get smaller over time.
- install — to set up equipment so it works in a building (like putting solar panels on a roof).
- residential — having to do with homes where people live.
- retail rate — the price a regular customer pays, not a wholesale or bulk price.
- average — the typical value when you spread numbers evenly.
- variable — something that can change, like price or time.
- permit — an official paper that says you are allowed to do something.
- policy — a rule or plan that a government or company chooses to follow.
- subsidy — money a government gives to lower the price of something for buyers.
Tier 3 · Solar, Grid, & Math Words
- rooftop solar — solar panels installed on the roof of a home or building to make electricity from sunlight.
- photovoltaic (PV) — the kind of solar panel that turns sunlight directly into electricity.
- kilowatt-hour (kWh) — the unit you pay for on an electric bill. One kWh runs a 100-watt bulb for ten hours.
- levelized cost (LCOE) — the average cost per kWh for an energy source, spread over the whole life of the system. It lets you compare solar and grid fairly.
- grid electricity — power that comes from the wires running into your house from the utility company.
- payback period — how many years until the money you save on electric bills equals what you spent to install solar.
- Swanson's law — the pattern that solar panel costs drop by about 20% every time the world's total panels double — about 10% per year.
- intersection — the point where two graphs cross. The \(x\)-coordinate is the input where the two functions are equal.
- system of equations — two or more equations you solve at the same time. The solution makes every equation true.
- exponential decay — a pattern where a value is multiplied by the same factor less than 1 each step. The amount shrinks fast at first, then slowly.
- exponential growth — a pattern where a value is multiplied by the same factor greater than 1 each step. The amount grows slowly at first, then fast.
- logarithm (log) — the inverse of an exponent. \(\log_b(x)\) answers, "what power of \(b\) gives \(x\)?" Use a log to free a variable from an exponent.
- base (b) — the number being multiplied repeatedly in \(b^t\). \(b > 1\) is growth; \(b < 1\) is decay.
- starting value (a) — the amount at \(t = 0\), before any growth or decay. The \(y\)-intercept of the curve.
- net metering — a billing rule that pays solar owners for extra electricity their panels send back to the grid.
- soft costs — the parts of a solar install that are not panels — permits, labor, financing, sales. About 65% of U.S. residential cost.
Page 3 of 3 · When Does Solar Win? · Algebra 2 · Unit 5 · Lesson 16