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Difficulty: Very hardInterpreting Linear Relationships in Context

A shipping company offers two types of delivery services: standard and express. The total shipping cost, in dollars, for a package sent via standard service is modeled by the function C(w)=1.25w+bsC(w) = 1.25w + b_s, where ww is the weight of the package, in pounds, and bsb_s is a constant representing the flat handling fee. The total shipping cost, in dollars, for a package sent via express service is modeled by the function E(k)=mek+beE(k) = m_e k + b_e, where kk is the weight of the package, in kilograms, and beb_e is a constant representing the flat handling fee.

The rate of change of the express shipping cost with respect to the package's weight, in dollars per kilogram, is 2.42.4 times the rate of change of the standard shipping cost with respect to the package's weight, in dollars per pound. The flat handling fee for the express service is 4.504.50 dollars more than the flat handling fee for the standard service. If it costs 58.5058.50 dollars to ship a package weighing 26.426.4 pounds using the express service, what is the cost, in dollars, to ship a package weighing 2020 pounds using the standard service? (Assume 1 kilogram=2.2 pounds1\text{ kilogram} = 2.2\text{ pounds}.)

Answer: 43 dollars

Answer

The cost to ship a package weighing 20 pounds using the standard service is 43 dollars.
The correct answer is obtained by first calculating the rate of change of the express shipping cost (2.4×1.25=3.002.4 \times 1.25 = 3.00 dollars per kilogram). Next, the weight of the package is converted to kilograms (26.4/2.2=1226.4 / 2.2 = 12 kilograms) to match the express cost function's variable. Using the given cost of 58.5058.50 dollars for the express shipment, the express handling fee is determined to be 58.50−3(12)=22.5058.50 - 3(12) = 22.50 dollars. The standard handling fee is then found by subtracting 4.504.50 dollars from the express fee (22.50−4.50=18.0022.50 - 4.50 = 18.00 dollars). Finally, the cost of a 2020-pound standard package is computed as 1.25(20)+18.00=43.001.25(20) + 18.00 = 43.00 dollars.

Step-by-Step Solution

1
Determine the rate of change for the express service (mem_e).
me=3.00m_e = 3.00 dollars per kilogram
The rate of change of the standard shipping cost is 1.251.25 dollars per pound. Since the rate of change for the express shipping cost is 2.42.4 times this rate, me=2.4×1.25=3.00m_e = 2.4 \times 1.25 = 3.00 dollars per kilogram.
2
Convert the weight of the express package from pounds to kilograms.
k=12k = 12 kilograms
The weight of the package is given as 26.426.4 pounds. Using the conversion 1 kilogram=2.2 pounds1\text{ kilogram} = 2.2\text{ pounds}, the weight in kilograms is 26.42.2=12\frac{26.4}{2.2} = 12 kilograms.
3
Find the express flat handling fee (beb_e) using the given cost of the express shipment.
be=22.50b_e = 22.50 dollars
We are given that the cost of shipping a 1212-kilogram package using the express service is 58.5058.50 dollars. Substituting these values into the express cost function: 3(12)+be=58.50⇒36+be=58.50⇒be=22.503(12) + b_e = 58.50 \Rightarrow 36 + b_e = 58.50 \Rightarrow b_e = 22.50 dollars.
4
Find the standard flat handling fee (bsb_s).
bs=18.00b_s = 18.00 dollars
The express handling fee is 4.504.50 dollars more than the standard handling fee: be=bs+4.50⇒22.50=bs+4.50⇒bs=18.00b_e = b_s + 4.50 \Rightarrow 22.50 = b_s + 4.50 \Rightarrow b_s = 18.00 dollars.
5
Calculate the cost to ship a 2020-pound package using the standard service.
43.0043.00 dollars
Using the standard cost function C(w)=1.25w+bsC(w) = 1.25w + b_s with w=20w = 20 and bs=18.00b_s = 18.00: C(20)=1.25(20)+18.00=25.00+18.00=43.00C(20) = 1.25(20) + 18.00 = 25.00 + 18.00 = 43.00 dollars.

Key Concept

Interpreting slope, y-intercept, and rates of change of linear functions in a real-world context with unit conversions.
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Topics

Interpreting Linear Relationships in Context

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