A logistics company models its total daily delivery cost C, in dollars, as a linear functi… — GMAT

A logistics company models its total daily delivery cost $C$ , in dollars, as a linear function of the total distance driven $d$ , in miles, given by $C = m \cdot d + b$ , where $m$ and $b$ are constants. The total daily cost increases by $\$180$ when the distance driven increases by $120$ miles. If the total daily delivery cost is $\$450$ when $250$ miles are driven, what is the total daily delivery cost when $400$ miles are driven?

A
$600
B
$630
$675Answer
D
$720
E
$810

Answer

The total daily delivery cost when 400 miles are driven is $675.

The cost function follows the linear form

C = m \cdot d + b

. The marginal rate

m = \frac{\Delta C}{\Delta d} = \frac{180}{120} = 1.5

dollars per mile. Substituting

d = 250

and

C = 450

gives

450 = 1.5(250) + b

, which yields

b = 75

. Substituting

d = 400

into

C = 1.5d + 75

produces

C = 600 + 75 = 675

dollars.

Step-by-Step Solution

Determine the slope (variable cost per mile)

m

from the rate of change.

m = \frac{\Delta C}{\Delta d} = \frac{180}{120} = 1.5

dollars per mile.

The slope of a linear equation represents the constant rate of change between the dependent and independent variables.

Calculate the fixed base cost

b

using the given data point

(d = 250, C = 450)

450 = 1.5(250) + b \implies 450 = 375 + b \implies b = 75

Substituting a known point into the linear model determines the vertical intercept (fixed cost).

Evaluate the linear function for

d = 400

miles.

C = 1.5(400) + 75 = 600 + 75 = 675

dollars.

Applying the fully determined linear equation

C = 1.5d + 75

yields the total cost for the required distance.

Key Concept

Linear Modeling in Two Variables (

y = mx + b

)

Question

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Step-by-Step Solution

Key Concept

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