Question 3·200 Super-Hard SAT Math Questions·Algebra

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A car-rental company charges a flat fee of $35 for a one-day rental. This fee includes up to 100 miles driven. For miles 101 through 150, the company charges $0.40 per mile. For each mile beyond 150, the company charges $0.60 per mile. An airport surcharge of 10% is applied only to the per-mile charges (not to the $35 flat fee). After the airport surcharge is applied, a local tax of 8% is applied to the entire amount (including the $35 flat fee and any surcharges).

Which equation gives the total cost $y$ , in dollars, for driving $x$ miles, where $x>150$ ?

The cue for piecewise linear pricing is that different miles have different rates. For a trip beyond the last cutoff, count the miles in each tier before applying any percentages. Build the charge in the order given, then use Desmos to find the slope and constant of the line. The main trap is applying a surcharge to the flat fee when it covers only mileage.

Hints

Hint 1
Even when $x>150$ , not every mile costs the highest rate. The middle tier contains 50 miles, and only $x-150$ miles are beyond it. How would you charge those two groups separately?
Hint 2
Adding a 10% surcharge means multiplying by $1.10$ . Apply that multiplier to the per-mile charges only, then add the flat fee outside it.
Hint 3
The 8% tax comes after the airport surcharge, so multiply the whole subtotal by $1.08$ . To match an equation in $y=mx+b$ form, you can use two mileages above 150 to find its slope.

Step-by-step

Build the charge, then find its line
Step 1
Count the charged miles in each tier
Let $p(x)$ be the per-mile charge before surcharges. Miles 101 through 150 account for 50 miles, while $x-150$ miles cost the higher rate because $x>150$ . So $p(x)=0.40(150-100)+0.60(x-150)$ . The first 100 miles are already included in the flat fee.
Step 2
Apply the airport surcharge to mileage
A 10% increase multiplies a charge by $1.10$ . The airport surcharge applies only to per-mile charges, so keep the $35 fee outside that multiplication: $q(x)=35+1.10p(x)$ . Here $q(x)$ is the subtotal before local tax.
Step 3
Tax the entire subtotal
The 8% local tax multiplies the whole subtotal by $1.08$ . Apply each percentage to the amount the problem names, in the order given. The total-cost function is $f(x)=1.08\left[35+1.10\left(0.40(150-100)+0.60(x-150)\right)\right]$ .
Step 4
Find the cost per additional mile
Type the formula for $f(x)$ in Desmos, then type $f(152)-f(151)$ . Both mileages are above 150 and differ by one mile, so this difference is the slope, or added cost per mile. Desmos prints $0.7128$ ; its fraction button shows $\frac{891}{1250}$ .
Step 5
Find the constant and match the equation
For a line written $y=mx+b$ , subtract $151$ times the slope $m$ from the cost at $151$ miles to find $b$ . Type $f(151)-151(f(152)-f(151))$ ; Desmos prints $-45.36$ , or $-\frac{1134}{25}$ with its fraction button. This negative constant isn't a zero-mile rental price: the equation applies only when $x>150$ . Insert the slope and constant: $y=\frac{891}{1250}x-\frac{1134}{25}$ . This gives the total cost in dollars for $x>150$ miles. Choice A.

Question 3·200 Super-Hard SAT Math Questions·Algebra

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