A repeated percent change signals an exponential model: each equal time interval multiplies the current amount rather than subtracting a fixed number. Find the starting value, the percent left after one interval, and the number of intervals elapsed. Use the percent remaining as the multiplier, then divide the total time by the interval length to count how many times it applies.
Hints
- Hint 1
A decrease tells you what is lost, but the model needs what remains. Start with the whole temperature, or , and subtract the percent lost. What fraction remains after each cooling interval?
- Hint 2
The exponent counts how many times the temperature is multiplied by the cooling factor. The factor applies once every minutes, so how many such intervals have passed after minutes?
Step-by-step
Build the exponential model
Step 1Find the factor for one cooling interval
No Desmos needed. The given percent and interval identify the model directly. A multiplier is the number you multiply the current temperature by after each interval. The coffee loses , so subtract that from the whole :
The multiplier is what remains, not what was lost.
- Step 2
Count the cooling intervals
The exponent, the small raised number, counts how many times the multiplier applies. Each interval lasts minutes, so minutes contain intervals. At , the exponent is , meaning one multiplication by . Using instead would count too many intervals.
- Step 3
Write the temperature equation
The coffee starts at degrees Celsius when . An exponential model multiplies that starting temperature by once per -minute interval, so
This models the coffee's temperature minutes after cooling begins. Choice D.