An exponential decay model shrinks an amount by the same factor each time its exponent increases by . When you're given an amount remaining and asked for the time, set the model equal to that target, then use a Desmos regression to find the time. Watch the interval in the exponent: a factor applied every few hours doesn't apply every hour.
Hints
- Hint 1
The model gives gigabytes remaining, so the target becomes . Don't subtract from : that would describe gigabytes transferred. What equation do you get when you replace with its formula?
- Hint 2
The exponent counts three-hour intervals: it rises by each time grows by . A Desmos regression can find the time that makes the model equal the target.
Step-by-step
Approach 1: Set the target and solve in Desmos
- Step 1
Turn the remaining amount into an equation
The word remaining means , not . Substitute the given model:
The exponent counts three-hour intervals, not individual hours. Type in Desmos to define the function for the next line.
- Step 2
Find the time that reaches the target
Add . The symbol tells Desmos to fit the unknown input to the target amount; unlike plain , won't become a slider. Under PARAMETERS, Desmos reports . So gigabytes remain after 9 hours. Choice C.
Approach 2: Count the halvings
- Step 1
Count how many times the archive halves
No Desmos needed. Count the halvings in the model. The factor means each step leaves half as much: gigabytes. Reaching the target takes three halvings.
- Step 2
Match the halvings to the exponent
The exponent counts those halving steps, so three halvings mean
- Step 3
Convert the interval count to hours
Multiply both sides by :
It takes 9 hours for gigabytes to remain. Choice C.
Lessons that teach this
- SAT Advanced AlgebraIntermediateCoreSolve exponential equations
- SAT Nonlinear FunctionsIntermediateBuild, identify, and interpret exponential models
- SAT Advanced AlgebraBeginnerUse exponent rules and common bases
- DesmosAdvancedCoreExponential models, transformations, and Log Mode
- DesmosBeginnerSolve one-variable equations