Question 36·SAT Desmos Regression Practice·Advanced Math

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A website engineer models the average page-load time $L(n)$ (in seconds) when $n$ identical servers are running by

L(n)=\frac{an+b}{n+h},

where $a$ , $b$ , and $h$ are constants and $n\ne -h$ .

Performance tests show that $L(1)=3$ , $L(2)=\frac{14}{5}$ , and $L(5)=\frac{5}{2}$ . Which choice gives the value of $b$ ?

A rational function has a variable in its denominator. When a model has unknown constants and several tested input-output values, substitute each input and clear the denominator to get one equation per test. Then fit those equations together with a matched-list regression in Desmos. The same constants must satisfy every test. Don’t mistake the whole numerator at one input for a single constant.

Hints

Hint 1
A function value such as $L(1)$ means replace every $n$ in the model with $1$ . After you write the resulting fraction, what equation do you get by multiplying by its denominator?
Hint 2
Each test gives an equation involving the same three constants. Clear the denominator in each equation so the constants appear only in terms that can be added or subtracted.
Hint 3
A matched-list regression pairs entries in two lists by position. Put your three known values in one list and the corresponding expressions in the other, keeping the same order.

Step-by-step

Clear the denominators, then fit the equations
Step 1
Make an equation from the first test
$L(1)=3$ means put $1$ in for $n$ . The model gives a load time at $n=1$ , so $1+h$ isn’t zero:
$\frac{a+b}{1+h}=3.$
Multiply by $1+h$ :
$a+b=3(1+h).$
Distribute the $3$ :
$a+b=3+3h.$
Subtract $3h$ from both sides: $a+b-3h=3$ . This is the first equation for the constants.
Step 2
Make an equation from the second test
For $L(2)=\frac{14}{5}$ , replace $n$ with $2$ :
$\frac{2a+b}{2+h}=\frac{14}{5}.$
Multiply by $5(2+h)$ to clear both denominators:
$5(2a+b)=14(2+h).$
Distribute on both sides:
$10a+5b=28+14h.$
Subtract $14h$ from both sides: $10a+5b-14h=28$ . Keep the $2a$ term: the numerator is not $b$ alone.
Step 3
Make an equation from the third test
For $L(5)=\frac{5}{2}$ , replace $n$ with $5$ :
$\frac{5a+b}{5+h}=\frac{5}{2}.$
Multiply by $2(5+h)$ :
$2(5a+b)=5(5+h).$
Distribute on both sides:
$10a+2b=25+5h.$
Subtract $5h$ from both sides: $10a+2b-5h=25$ . Now all three tests are equations in $a$ , $b$ , and $h$ .
Step 4
Fit one shared set of constants
Type the matched-list regression $[3,28,25]\sim[a+b-3h,10a+5b-14h,10a+2b-5h]$ . Each position pairs an earlier equation’s right side with its left side. Desmos reports $a=2$ , $b=10$ , and $h=3$ under PARAMETERS. These give the original load times $12/4=3$ , $14/5$ , and $20/8=5/2$ , so the model’s constant $b$ is $10$ . Choice B.

Question 36·SAT Desmos Regression Practice·Advanced Math

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