An online retailer determines that the cost to ship a package of weight w pounds is given … — ACT

An online retailer determines that the cost to ship a package of weight $w$ pounds is given by the linear expression $C(w) = kw + b$ , where $k$ and $b$ are constants. Shipping a $3$ -pound package costs $11.50$ dollars, and shipping an $8$ -pound package costs $24.00$ dollars. If the total shipping cost for two packages is $47.00$ dollars, and one of the packages weighs $5$ pounds, what is the weight, in pounds, of the other package?

Answer: 10.6 pounds

Answer

The weight of the other package is 10.6 pounds.

The correct weight of 10.6 pounds is found by setting up a linear cost function

C(w) = 2.5w + 4

using the two data points, calculating the cost of the 5-pound package as

16.50

dollars, subtracting this from the total cost of

47.00

dollars to get

30.50

dollars, and solving

2.5w + 4 = 30.50

for the weight.

Step-by-Step Solution

Set up the linear system from the given costs

3k + b = 11.50

and

8k + b = 24.00

To determine the relationship between weight and shipping cost.

Solve for the slope

k

k = 2.50

Subtracting the first equation from the second eliminates

b

Solve for the intercept

b

b = 4.00

Substitute

k = 2.50

back into the first equation.

Determine the cost of the 5-pound package

C(5) = 16.50

dollars

Evaluate the linear expression

2.50(5) + 4.00

Determine the remaining cost for the second package

C(w_2) = 30.50

dollars

Subtract the cost of the first package from the total cost (

47.00 - 16.50

Solve the linear equation for the second package's weight

w_2 = 10.6

Solve

2.50w_2 + 4.00 = 30.50

for

w_2

Key Concept

Solving Linear Equations

Question

Answer

Step-by-Step Solution

Key Concept

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