Question

question 5: shipping costs
a company charges shipping fees based on package weight:

• $10 for packages weighing up to 5 pounds.
• $2 per pound for packages between 5 and 20 pounds.
• $1 per pound for packages over 20 pounds.

this is modeled by:

$$ c(w)=\\left\\{\
LATEXBLOCK0
\ ight. \
LATEXBLOCK1
$$

question 6: cellular data plan
find the shipping cost for packages weighing 3 pounds, 10 pounds,
and 25 pounds. interpret each result in the context of shipping.

Explanation:

Step1: Determine the interval for \(w = 3\)

Since \(0\leq w=3\leq5\), we use the first - part of the piece - wise function \(C(w)=10\).

Step2: Determine the interval for \(w = 10\)

Since \(5 < w = 10\leq20\), we use the second - part of the piece - wise function \(C(w)=10 + 2(w - 5)\).
Substitute \(w = 10\) into \(C(w)=10+2(w - 5)\):

$$
LATEXBLOCK0
$$

Step3: Determine the interval for \(w = 25\)

Since \(w=25>20\), we use the third - part of the piece - wise function \(C(w)=40+1(w - 20)\).
Substitute \(w = 25\) into \(C(w)=40 + 1(w - 20)\):

$$
LATEXBLOCK1
$$

Answer:

• When \(w = 3\) pounds, the shipping cost \(C(3)=10\) dollars. This means that for packages weighing up to \(5\) pounds, the company charges a flat - rate of \(10\) dollars.
• When \(w = 10\) pounds, the shipping cost \(C(10)=20\) dollars. The base cost is \(10\) dollars (for the first \(5\) pounds), and since the package weighs \(10-5 = 5\) pounds more than \(5\) pounds, and the cost per pound for the weight between \(5\) and \(20\) pounds is \(2\) dollars per pound (\(2\times5=10\) dollars added to the base cost).
• When \(w = 25\) pounds, the shipping cost \(C(25)=45\) dollars. The base cost for the first \(20\) pounds: using the formula for \(5 < w\leq20\), \(C(20)=10+2(20 - 5)=10 + 30=40\) dollars. Then, since the package weighs \(25-20 = 5\) pounds more than \(20\) pounds, and the cost per pound for weight over \(20\) pounds is \(1\) dollar per pound (\(1\times5 = 5\) dollars added to the cost of shipping a \(20\) - pound package).