Question 154·200 Super-Hard SAT Math Questions·Advanced Math

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A quadratic function models the height, in feet, of an object above the ground in terms of time, in seconds, after the object is launched off an elevated surface. The model indicates that at a time of 4 seconds, the object is 255 feet above the ground. At a time of 11 seconds, the object is 444 feet above the ground. If the object was at a height of 15 feet when it was launched, what is the height, in feet, of the object above the ground 18 seconds after being launched?

A quadratic model can be determined from three time-height pairs. Time is the input and height is the output; the launch height supplies the pair at time zero. Enter the pairs in a Desmos table, fit a quadratic, and evaluate it at the requested time. Don't subtract the launch height: the model already measures height above the ground.

Hints

Hint 1
At launch, no time has passed. If $h(t)$ means the height after $t$ seconds, the starting height gives $h(0)$ . What function values do the two later times give?
Hint 2
A quadratic has three coefficients: the numbers multiplying $t^2$ and $t$ , and the number standing alone. Three time-height pairs let Desmos fit them. Put time in the input column and height in the output column.
Hint 3
After fitting the quadratic, define a function with Desmos's stored coefficients and enter the requested time. You want height above the ground, not how much height the object gained after launch.

Step-by-step

Fit the quadratic in Desmos
Step 1
Turn the heights into function values
Let $h(t)$ be the height, in feet, $t$ seconds after launch. At launch, $t=0$ , so the starting height is $h(0)=15$ . The other times give $h(4)=255$ and $h(11)=444$ . Each time and its height make one input-output pair.
Step 2
Enter the three time-height pairs
Enter $0,4,11$ in Desmos's $x_1$ column and $15,255,444$ in its $y_1$ column, one matching pair per row. The table now holds $(0,15)$ , $(4,255)$ , and $(11,444)$ . Don't swap the columns: height depends on time, not the other way around.
Step 3
Fit the quadratic
Type $y_1\sim ax_1^2+bx_1+c$ below the table. The $\sim$ tells Desmos to find the three coefficients that fit all three pairs. Under PARAMETERS, Desmos reports $a=-3$ , $b=72$ , and $c=15$ .
Step 4
Define the height function
Type $h(t)=at^2+bt+c$ to use Desmos's stored fitted values. This gives $h(t)=-3t^2+72t+15$ , a model whose value at $t=0$ is the given starting height.
Step 5
Find the height at the requested time
Type $h(18)$ , since the question asks for the height 18 seconds after launch. Desmos prints $339$ . The object is $339$ feet above the ground, not $339$ feet above where it started. Choice B.

Question 154·200 Super-Hard SAT Math Questions·Advanced Math

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