Question 31·SAT Desmos Regression Practice·Algebra

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A courier company models the pre-tax delivery fee $F(m)$ (in dollars) for a delivery of $m$ miles as a linear function.

An 8% sales tax is added to the pre-tax delivery fee.

A promo code reduces only the per-mile charge by 20% (the fixed fee is not discounted) before tax is added.

With the promo code, a 6-mile delivery costs $16.47 in total (after tax).
Without the promo code, a 10-mile delivery costs $23.49 in total (after tax).

Based on the model, which choice is the pre-tax fee, in dollars, for an 8-mile delivery without the promo code?

A linear model separates a fixed fee from a per-mile charge. When one quoted total uses a promo and another doesn't, write a separate after-tax equation for each, changing only the part the promo affects. Fit both equations with a Desmos matched-list regression, then evaluate the regular pre-tax model. Treating the two quotes as points on one unchanged price line mixes different rates.

Hints

Hint 1
In a linear model, the fixed fee is $b$ , and each mile adds $r$ . Write $F(m)=b+rm$ . Which part of that expression does the promo actually change?
Hint 2
A 20% cut leaves 80% of the per-mile rate. Tax comes afterward: its multiplier applies to the whole resulting fee, including the fixed charge.
Hint 3
You have two unknowns, the fixed fee and the regular per-mile rate. Make one equation from each quote, then fit both conditions together rather than treating the quotes as points on the same price line.

Step-by-step

Model both quotes and fit the fees
Step 1
Write the regular pre-tax model
A linear fee has a fixed charge $b$ plus $r$ dollars for each mile, so $F(m)=b+rm$ . The requested pre-tax, no-promo fee will come from this model.
Step 2
Change only the per-mile rate
A 20% reduction leaves $80\%=0.8$ of the per-mile rate, so the promo pre-tax fee is $b+0.8rm$ . The promo changes the rate, not the fixed fee. Multiplying all of $b+rm$ by $0.8$ would discount the fixed fee too.
Step 3
Turn the promo quote into an equation
An 8% tax makes the tax multiplier $1.08$ . The 6-mile promo quote is after tax, so multiply its entire pre-tax fee by $1.08$ : $1.08(b+0.8(6)r)=16.47$ .
Step 4
Turn the regular quote into an equation
The 10-mile quote is without the promo, so it uses the full per-mile rate $r$ . Tax is still added: $1.08(b+10r)=23.49$ .
Step 5
Fit the fixed fee and regular rate
Type $[1.08(b+0.8(6)r),1.08(b+10r)]\sim[16.47,23.49]$ . The matched-list regression pairs each modeled total with its quoted total and fits one shared pair of values. Under PARAMETERS, Desmos shows $b=9.25$ and $r=1.25$ : a $9.25 fixed fee and $1.25 per mile before any discount.
Step 6
Evaluate the requested fee
Type $F(m)=b+rm$ , then $F(8)$ . Desmos shows $19.25$ . Use the regular pre-tax model here: the fee for an 8-mile delivery without the promo code is $19.25. Choice C.

Question 31·SAT Desmos Regression Practice·Algebra

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