Question 14·SAT Desmos Regression Practice·Advanced Math

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In a wildlife documentary, the percentage $S(t)$ of battery remaining in a camera drone $t$ minutes after takeoff is modeled by a quadratic function.

The drone’s battery percentage satisfies $S(0)=100$ , $S(4)=68$ , and $S(8)=20$ .

Which choice is the value of $t$ for which $S(t)=50$ ?

Three input-output facts and the word quadratic point to a model with three unknown coefficients. Fit a quadratic regression in Desmos, then solve for the time when the model gives the requested battery percentage. Match that time to an exact choice. Because time is measured after takeoff, a negative algebraic solution cannot describe the flight.

Hints

Hint 1
A quadratic regression fits a curve of the form $S(t)=at^2+bt+c$ . Each given time belongs in the input column, paired with its battery percentage in the output column. What three pairs does that make?
Hint 2
The battery percentage is the output of $S$ . To find when it reaches $50$ , set the fitted model equal to $50$ . Don't substitute $50$ for the time.
Hint 3
A quadratic equation can have two solutions, but the situation may allow only one. Since $t$ counts minutes after takeoff, what restriction should you put on the time Desmos finds?

Step-by-step

Fit the model, then find the time
Step 1
Pair each time with its battery percentage
Enter $0,4,8$ in Desmos's $x_1$ table column and $100,68,20$ in the $y_1$ column. Each row now pairs a time with its battery percentage: $(0,100)$ , $(4,68)$ , and $(8,20)$ .
Step 2
Fit the quadratic model
Type $y_1\sim ax_1^2+bx_1+c$ . The regression symbol $\sim$ tells Desmos to find the coefficients that fit the three rows. Under PARAMETERS, it reports $a=-0.5$ , $b=-6$ , and $c=100$ , so the fitted model is $S(t)=-\frac12t^2-6t+100$ .
Step 3
Turn the target percentage into an equation
The $50$ is an output, not a time to substitute. To find the input, set the model equal to the requested output. That gives $-\frac12t^2-6t+100=50$ .
Step 4
Find the time after takeoff
Type $-\frac12t_1^2-6t_1+100\sim50\{t_1>0\}$ . Use $t_1$ for the unknown time because $x_1$ already holds the table's times; the braces keep the result after takeoff. Desmos reports $t_1\approx5.6619$ under PARAMETERS, so the drone reaches $50\%$ battery about $5.66$ minutes after takeoff.
Step 5
Match the time to an exact choice
Type $-6+2\sqrt{34}$ . Desmos prints about $5.6619$ , matching the positive time it found. The other expressions give negative times. So the drone reaches $50\%$ battery at $t=-6+2\sqrt{34}$ minutes after takeoff. Choice A.

Question 14·SAT Desmos Regression Practice·Advanced Math

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