A shipping company calculates the total cost C, in dollars, to deliver a package of weight… — GRE General Test

A shipping company calculates the total cost $C$ , in dollars, to deliver a package of weight $w$ pounds using the linear relationship $C = kw + b$ , where $k$ and $b$ are positive constants. The delivery cost for a $4$ -pound package is $\$19$ , and the delivery cost for a $9$ -pound package is $\$39$ . If a customer pays a total delivery cost of $\$71$ for a single package, what is the weight of the package, in pounds?

A
$13$
B
$15$
$17$ Answer
D
$18$
E
$21$

Answer

The weight of the package is

17

pounds.

Using the two points

(4, 19)

and

(9, 39)

, the rate of change

k

is calculated as

\frac{39 - 19}{9 - 4} = \frac{20}{5} = 4

dollars per pound. Substituting

k = 4

into

4(4) + b = 19

yields

b = 3

. Setting the linear equation

4w + 3 = 71

and solving for

w

gives

4w = 68

, so

w = 17

pounds.

Step-by-Step Solution

Set up a system of linear equations using the given data points

(4, 19)

and

(9, 39)

4k + b = 19

and

9k + b = 39

The cost model follows

C = kw + b

for weight

w

Subtract the first equation from the second equation to solve for

k

(9k + b) - (4k + b) = 39 - 19 \implies 5k = 20 \implies k = 4

Subtracting eliminates the constant

b

to determine the unit rate per pound.

Substitute

k = 4

into

4k + b = 19

to solve for

b

4(4) + b = 19 \implies 16 + b = 19 \implies b = 3

Finding

b

establishes the full linear equation model:

C = 4w + 3

Substitute

C = 71

into the linear equation

4w + 3 = 71

and solve for

w

4w = 71 - 3 \implies 4w = 68 \implies w = 17

Solving for

w

yields the required weight corresponding to a

\$71

delivery cost.

Key Concept

Linear Modeling and Single-Variable Linear Equations

Estimated Time:1m 30s

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