Question 657·Hard·Algebra

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A long-distance moving company quotes $1,500 to move a shipment that weighs 10,000 pounds and $2,100 to move a shipment that weighs 16,000 pounds. Each quote consists of a one-time truck rental fee plus a charge that is directly proportional to the weight of the shipment.

Which of the following linear functions gives the quoted cost $C(p)$ , in dollars, as a function of the shipment weight $p$ , in pounds?

A linear cost model fits a quote with a one-time fee plus a charge per pound. The per-pound charge is the slope, or the cost added for each extra pound; the fixed fee is the cost at zero pounds. Turn the quotes into weight-cost pairs, then use a Desmos table regression to find both parts. Don't mistake the total cost of one shipment for the fee.

Hints

Hint 1
The one-time fee is paid once, while the weight-based charge grows with the number of pounds. In $C(p)=ap+b$ , which part should represent each charge?
Hint 2
A point pairs an input with its output. Here the input is shipment weight and the output is quoted cost. What two weight-cost pairs do the quotes give you?
Hint 3
A Desmos regression finds values that fit the pairs in a table. Use $y_1\sim ax_1+b$ , then read the per-pound charge and fixed fee under PARAMETERS.

Step-by-step

Fit the cost model in Desmos
Step 1
Separate the two charges
The charge directly proportional to weight grows by the same amount per pound. Call that rate $a$ ; it's the slope. Call the fee paid once $b$ . The one-time fee belongs in the constant term, not in the amount multiplied by weight. Write the model as:
$C(p)=ap+b$
Step 2
Turn the quotes into weight-cost pairs
The weight is the input, and the quoted cost is its output. So the two quotes give these pairs, written as $(\text{weight},\text{cost})$ :
$(10{,}000,1{,}500)\quad\text{and}\quad(16{,}000,2{,}100)$
Step 3
Find the rate and fee
Enter those pairs in a Desmos table, then type $y_1\sim ax_1+b$ . The symbol $\sim$ tells Desmos to find values that fit both quotes. Under PARAMETERS, it reports $a=0.1$ and $b=500$ . So the rate is $0.10 per pound, and the fixed fee is $500.
Step 4
Write the quoted cost function
Put the rate and fee into $C(p)=ap+b$ :
$C(p)=0.10p+500$
The quoted cost is $0.10 for each pound plus a $500 truck rental fee. Choice D.

Question 657·Hard·Algebra

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