Question

select the correct answer.
mr. cutler makes clay bricks and clay tiles to sell at maker fairs. it takes him 10 minutes to make a brick and 20 minutes to make a tile. each brick uses 3 pounds of clay and each tile uses 2 pounds of clay. he has 160 minutes available for making the bricks and tiles and has 20 pounds of clay on hand. if he makes a profit of $2 on each brick and $3 on each tile, how many bricks and tiles should he make to maximize his profit?
a. mr. cutler should make 7 bricks and 2 tiles to maximize his profit.
b. mr. cutler should make 7 bricks and 0 tiles to maximize his profit.
c. mr. cutler should make 2 bricks and 7 tiles to maximize his profit.
d. mr. cutler should make 0 bricks and 8 tiles to maximize his profit.

Explanation:

Step1: Define Variables

Let \( x \) be the number of bricks and \( y \) be the number of tiles.

Step2: Set Up Constraints

Time constraint: \( 10x + 20y \leq 160 \) (simplify to \( x + 2y \leq 16 \))
Clay constraint: \( 3x + 2y \leq 20 \)
Non - negativity: \( x\geq0,y\geq0 \) (and \( x,y \) are integers)

Step3: Set Up Profit Function

Profit function \( P = 2x + 3y \)

Step4: Analyze Each Option

• Option A: \( x = 7,y = 2 \)

Time check: \( 10\times7+20\times2=70 + 40=110\leq160 \)
Clay check: \( 3\times7+2\times2=21 + 4=25>20 \) (invalid, exceeds clay)

• Option B: \( x = 7,y = 0 \)

Time check: \( 10\times7+20\times0 = 70\leq160 \)
Clay check: \( 3\times7+2\times0=21>20 \) (invalid, exceeds clay)

• Option C: \( x = 2,y = 7 \)

Time check: \( 10\times2+20\times7=20 + 140 = 160\leq160 \)
Clay check: \( 3\times2+2\times7=6 + 14=20\leq20 \)
Profit: \( P=2\times2 + 3\times7=4 + 21 = 25 \)

• Option D: \( x = 0,y = 8 \)

Time check: \( 10\times0+20\times8=160\leq160 \)
Clay check: \( 3\times0+2\times8=16\leq20 \)
Profit: \( P=2\times0+3\times8 = 24 \)

Answer:

C. Mr. Cutler should make 2 bricks and 7 tiles to maximize his profit.