Question 838·Hard·Math

Find the fraction left after a $12\%$ decrease:

As a decimal, $88\%=0.88$. That's the model's decay factor: it multiplies the temperature after one full decrease. Type the given function $T(h)=740(0.88)^{\frac{2}{3}h}$ into Desmos.

At $h=0$, the exponent is $0$, so the starting temperature is $740$ degrees Celsius. After one $12\%$ decrease, the temperature is $740(0.88)$. The exponent counts uses of $0.88$, so this target is reached when the exponent is $1$, not necessarily when $h=1$.

Under the function, type $T(h_1)\sim740(0.88)$. The regression symbol $\sim$ tells Desmos to find a number $h_1$ that gives the target temperature; the subscript makes it a number to find rather than a graphing variable. Under PARAMETERS, Desmos shows $h_1=1.5$. So one full decrease takes $1.5$ hours.

The model measures $h$ in hours, but $k$ is in minutes. Type $60h_1$ because each hour has $60$ minutes. Desmos prints $90$, so the temperature decreases by $12\%$ every $90$ minutes. Choice D.