Question 689·Hard·Math

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A shipping company charges according to a linear formula $S(w)=mw+b$ , where $S(w)$ is the cost, in dollars, to ship a package that weighs $w$ pounds.

A $3$ -pound package costs $8.45 to ship.
A $7$ -pound package costs $14.05 to ship.

What is the cost, in dollars, to ship a $12$ -pound package?

A linear model adds the same cost for every extra pound. When two weights and their prices are given, keep each price paired with its weight, then use a Desmos regression to find the per-pound rate and the fixed charge. Evaluate the model at the requested weight. A quoted price isn't the fixed charge unless its weight is zero.

Hints

Hint 1
Each weight is an input, and its shipping cost is the matching output. Keep the pairs together: what do the two prices tell you when you put their weights into $S(w)=mw+b$ ?
Hint 2
The slope is the cost added per pound; the intercept is the cost at zero pounds. Neither quoted price is automatically the intercept. How can you use both weight-cost pairs to find the slope and intercept?
Hint 3
Once you have the model, evaluate it at the requested weight. In $S(12)$ , the $12$ is the weight going in, and the result is the shipping cost coming out.

Step-by-step

Approach 1: Fit and evaluate the shipping model
Step 1
Turn the two prices into equations
The $3$ -pound package costs $8.45, and the $7$ -pound package costs $14.05. Keep each cost with its weight when you put them into the given formula:
$S(3)=3m+b=8.45,\qquad S(7)=7m+b=14.05.$
Step 2
Find the rate and fixed charge
Type $S(w)=mw+b$ , then $[S(3),S(7)]\sim[8.45,14.05]$ . The $\sim$ tells Desmos to fit both prices at once. Under PARAMETERS, it shows $m=1.4$ and $b=4.25$ . The slope $m$ means $1.40 per pound; the intercept $b$ means $4.25 at zero pounds. A quoted cost is the intercept only when its input is zero. Don't use $8.45 as $b$ : that package already weighs $3$ pounds.
Step 3
Find the 12-pound cost
Type $S(12)$ below those lines. Desmos displays $21.05$ , the cost in dollars when the weight is 12 pounds. So shipping the package costs $21.05. Choice B.

Approach 2: Start from a known price
Step 1
Find the cost per pound
You can skip finding the fixed charge. From $3$ to $7$ pounds, the price rises from $8.45 to $14.05. The rate of change is the price change divided by the weight change. Type $\frac{14.05-8.45}{7-3}$ ; Desmos displays $1.4$ dollars per pound.
Step 2
Add only the cost of the extra weight
Start at the known $14.05 cost for $7$ pounds. A $12$ -pound package is $12-7=5$ pounds heavier, so type $14.05+1.4(12-7)$ . Desmos displays $21.05$ . The shipping cost is $21.05. Choice B.

Question 689·Hard·Math

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