Question

the quick ship company uses the function c(w) to calculate the cost in dollars, c, for shipping an item weighing w ounces.
c(w) = \

$$\begin{cases} 0.45w & \\text{if } 0 < w \\leq 16 \\\\ 0.35w + 1.6 & \\text{if } w > 16 \\end{cases}$$

which statement about the shipping rate is true?
shipping at the second rate will always cost $1.60 more than shipping at the first rate.
if the weight is greater than 16 ounces, the shipping rate is $0.35 per ounce, plus $1.60.
to find the charge for shipping a package weighing more than 16 ounces, multiply the weight by $0.35.

Explanation:

Step1: Analyze the first statement

The first rate is \( C(w)=0.45w \) (for \( 0 < w\leq16 \)) and the second rate is \( C(w) = 0.35w + 1.6 \) (for \( w>16 \)). Let's take \( w = 17 \) (which is \( w>16 \)). First rate at \( w = 17 \) (even though it's not in the domain, just for comparison of the difference) would be \( 0.45\times17=7.65 \). Second rate at \( w = 17 \) is \( 0.35\times17 + 1.6=6.95 \). The difference is \( 6.95 - 7.65=- 0.7 \), which is not \( 1.6 \). So the first statement is false.

Step2: Analyze the second statement

For \( w>16 \), the function is \( C(w)=0.35w + 1.6 \). In the slope - intercept form of a linear equation \( y = mx + b \), here \( m = 0.35 \) (which is the rate per ounce) and \( b = 1.6 \) (a constant added). So this means if the weight is greater than 16 ounces, the shipping rate is $0.35 per ounce, plus $1.60. This statement is consistent with the function definition.

Step3: Analyze the third statement

For \( w>16 \), the function is \( C(w)=0.35w + 1.6 \), not just \( 0.35w \). So we can't just multiply the weight by $0.35, we also have to add $1.6. So the third statement is false.

Answer:

If the weight is greater than 16 ounces, the shipping rate is $0.35 per ounce, plus $1.60.