Question 33·SAT Desmos Regression Practice·Advanced Math

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A delivery service estimates the total time $T(v)$ (in hours) to complete a route using the model

T(v)=k+\frac{d}{v},

where $v$ is the average speed in miles per hour, and $k$ and $d$ are constants with $v>0$ .

If $T(24)=9$ and $T(36)=7$ , which choice is the value of $T(60)$ ?

In a travel-time model with a constant term, some time stays fixed while the rest changes with speed. Two known speeds and times give enough information to find both parts. Fit the given model with a Desmos table regression, then evaluate it at the new speed. Scaling the entire time by a speed ratio fails because the fixed part does not shrink.

Hints

Hint 1
The model separates the time into $k$ , which stays fixed, and $d/v$ , which changes with speed. Only one part changes when the speed changes. What equations do the two known times give you?
Hint 2
Each known time is an input-output pair: speed is the input, and total time is the output. Put both pairs in a Desmos table so one set of constants has to fit both.
Hint 3
A regression can fit $k$ and $d$ from both table rows using the formula in the question. After finding them, evaluate the whole model at the requested speed, not just $d/v$ .

Step-by-step

Fit the given model in Desmos
Step 1
Turn the known times into equations
Putting each given speed into $T(v)=k+\frac{d}{v}$ gives two equations. The same $k$ and $d$ must make both true:
$k+\frac{d}{24}=9,\qquad k+\frac{d}{36}=7.$
Only $d/v$ changes with speed; $k$ stays fixed.
Step 2
Find the two constants
Enter the two input-output pairs in a Desmos table: speed in $x_1$ , time in $y_1$ . Then type $y_1\sim k+\frac{d}{x_1}$ . The $\sim$ tells Desmos to fit one shared pair of constants to both rows. Under PARAMETERS, it shows $k=3$ and $d=144$ .
Step 3
Evaluate the time at 60 mph
Call the model $f$ in Desmos: type $f(x)=k+\frac{d}{x}$ , then $f(60)$ . Desmos prints $5.4$ ; its fraction button shows $\frac{27}{5}$ . This includes both the fixed time and the speed-dependent time. The estimated total route time at $60$ mph is $\frac{27}{5}$ hours. Choice B.

Question 33·SAT Desmos Regression Practice·Advanced Math

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