Question

question 3 of 5 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 0 bricks and 8 tiles to maximize his profit. d. mr. cutler should make 2 bricks and 7 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 - negative constraints: \( x\geq0,y\geq0 \)

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: \( 3\times7+2\times2=21 + 4 = 25>20 \) (violates clay constraint)
• Option B: \( x = 7,y = 0 \)
• Time check: \( 10\times7+20\times0 = 70\leq160 \)
• Clay check: \( 3\times7+2\times0=21>20 \) (violates clay constraint)
• Option C: \( x = 0,y = 8 \)
• Time check: \( 10\times0+20\times8=160\leq160 \)
• Clay check: \( 3\times0+2\times8 = 16\leq20 \)
• Profit: \( P=2\times0 + 3\times8=24 \)
• Option D: \( x = 2,y = 7 \)
• Time check: \( 10\times2+20\times7=20 + 140=160\leq160 \)
• Clay check: \( 3\times2+2\times7=6 + 14=20\leq20 \)
• Profit: \( P=2\times2+3\times7 = 4 + 21=25 \)

Since \( 25>24 \) and options A and B violate the clay constraint, option D gives the maximum profit.

Answer:

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