A logistics company distributes T tons of cargo among a fleet of trucks. A standard truck … — ACT

A logistics company distributes $T$ tons of cargo among a fleet of trucks. A standard truck has a maximum capacity of $c$ tons, and a heavy-duty truck has a maximum capacity of twice that of a standard truck. The fleet consists of $x$ standard trucks and $y$ heavy-duty trucks, all loaded to their maximum capacities. If the ratio of standard trucks to heavy-duty trucks in the fleet is exactly $3:1$ , which of the following equations correctly expresses the capacity of a standard truck, $c$ , in terms of $T$ and $y$ ?

A
$c = \frac{T}{4y}$
B
$c = \frac{3T}{7y}$
$c = \frac{T}{5y}$ Answer
D
$c = \frac{T}{2y + 3}$
E
$c = T - 5y$

Answer

The equation

c = \frac{T}{5y}

correctly expresses the capacity of a standard truck.

The total cargo

T

is the sum of the capacities of all trucks. Since there are

x

standard trucks carrying

c

tons each and

y

heavy-duty trucks carrying

2c

tons each, the equation is

T = cx + 2cy

. The

3:1

ratio of standard trucks to heavy-duty trucks means

x = 3y

. Substituting this relationship into the cargo equation gives

T = c(3y) + 2cy = 5cy

. To isolate the capacity of a standard truck,

c

, divide both sides by

5y

to get

c = \frac{T}{5y}

Step-by-Step Solution

Write the equation for the total cargo weight by summing the capacities of all trucks in the fleet.

T = cx + 2cy

The total cargo

T

is distributed among

x

standard trucks, each carrying

c

tons, and

y

heavy-duty trucks, each carrying

2c

tons.

Translate the ratio of standard trucks to heavy-duty trucks into an algebraic relationship.

x = 3y

The ratio of standard trucks (

x

) to heavy-duty trucks (

y

) is

3:1

, meaning there are 3 times as many standard trucks as heavy-duty trucks.

Substitute the ratio relationship

x = 3y

into the total cargo equation.

T = c(3y) + 2cy

Replacing

x

with

3y

eliminates the variable

x

and expresses

T

in terms of

c

and

y

Combine like terms to simplify the expression.

T = 5cy

Adding

3cy

and

2cy

yields

5cy

Isolate the variable

c

by dividing both sides of the equation by

5y

c = \frac{T}{5y}

Division is the inverse operation of multiplication, allowing us to solve the one-step equation

T = (5y)c

for

c

Key Concept

Translating verbal descriptions into algebraic equations and solving one-step literal equations

Estimated Time:2m 0s

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