A delivery truck carries a total of x bags of organic fertilizer and y bags of compost. Ea… — SAT

A delivery truck carries a total of $x$ bags of organic fertilizer and $y$ bags of compost. Each bag of fertilizer weighs $45$ pounds, and each bag of compost weighs $30$ pounds. When the truck is loaded to exactly $80\%$ of its maximum payload capacity, the relationship between $x$ and $y$ can be represented by the equation $3x + 2y = 240$ . What is the truck's maximum payload capacity, in pounds?

A
$2,880$
B
$3,000$
C
$3,600$
$4,500$ Answer

Answer

The maximum payload capacity of the truck is

4,500

pounds.

The total weight of the bags loaded on the truck is represented by the expression

45x + 30y

pounds. Factoring out

15

from this expression gives

15(3x + 2y)

. We are given that when the truck is loaded to exactly

80\%

of its capacity, the relationship between

x

and

y

3x + 2y = 240

. Substituting

240

for

3x + 2y

in the weight expression shows that the weight of the load at

80\%

capacity is

15 \times 240 = 3,600

pounds. To find the maximum capacity, we set up the equation

0.80 \times C = 3,600

, where

C

is the maximum payload capacity. Solving for

C

yields

C = \frac{3,600}{0.80} = 4,500

pounds.

Step-by-Step Solution

Write the expression for the total weight of the bags in terms of

x

and

y

Total weight

W = 45x + 30y

pounds.

Each of the

x

bags of fertilizer weighs

45

pounds and each of the

y

bags of compost weighs

30

pounds.

Relate the total weight expression to the given equation for

80\%

capacity.

45x + 30y = 15(3x + 2y)

Factoring out the greatest common divisor of

45

and

30

, which is

15

, allows us to write the weight expression in terms of the left-hand side of the given equation

3x + 2y = 240

Calculate the actual weight of the load when the truck is at

80\%

capacity.

W = 15 \times 240 = 3,600

pounds.

Since the relationship at

80\%

capacity is

3x + 2y = 240

, we can substitute

240

for

3x + 2y

in the factored weight expression.

Calculate the maximum payload capacity of the truck.

Maximum capacity

C = \frac{3,600}{0.80} = 4,500

pounds.

The weight of

3,600

pounds represents

80\%

of the maximum payload capacity, so dividing the actual weight by

0.80

gives the total capacity.

Key Concept

Translating a real-world scenario into a linear equation and interpreting its coefficients and constants to solve for unknown quantities.

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