An artist creates x small sculptures and y large sculptures. Each small sculpture requires… — SAT

An artist creates $x$ small sculptures and $y$ large sculptures. Each small sculpture requires 3 hours of crafting and 2 hours of painting. Each large sculpture requires 7 hours of crafting and 3 hours of painting. The artist can spend at most 120 hours on crafting and at most 50 hours on painting. Which of the following pairs of small and large sculptures can the artist create under these constraints?

A
5 small sculptures and 15 large sculptures
B
12 small sculptures and 9 large sculptures
10 small sculptures and 8 large sculpturesAnswer
D
15 small sculptures and 12 large sculptures

Answer

10 small sculptures and 8 large sculptures

The option featuring 10 small sculptures and 8 large sculptures is correct because substituting

x = 10

and

y = 8

satisfies both linear inequalities representing the constraints. Specifically, the crafting time of 86 hours is less than or equal to the maximum allowed 120 hours (

3(10) + 7(8) = 86 \leq 120

), and the painting time of 44 hours is less than or equal to the maximum allowed 50 hours (

2(10) + 3(8) = 44 \leq 50

Step-by-Step Solution

Set up the system of inequalities representing the constraints.

The crafting constraint is

3x + 7y \leq 120

. The painting constraint is

2x + 3y \leq 50

. The variables must be non-negative:

x \geq 0

and

y \geq 0

To model the limits on crafting hours and painting hours mathematically.

Substitute the values of each option into the system to verify which one satisfies both inequalities.

For the option with 10 small sculptures and 8 large sculptures (

x = 10, y = 8

3(10) + 7(8) = 30 + 56 = 86 \leq 120

(True)

2(10) + 3(8) = 20 + 24 = 44 \leq 50

(True)

Only a coordinate pair that makes both inequality statements true is a valid solution.

Key Concept

Systems of Linear Inequalities in Two Variables

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Step-by-Step Solution

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