Subjects algebra

Operation Cost Nets

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Operation Cost Nets


1. Problem 1: Given the operation $\triangle \times \square = \square$, find $\triangle \times 8$.\n Step 1: The operation shows that $\triangle \times \square = \square$, meaning multiplying $\triangle$ by any $\square$ returns $\square$.\nStep 2: This implies $\triangle$ is an identity element of multiplication, i.e., $\triangle = 1$.\nStep 3: Therefore, $\triangle \times 8 = 1 \times 8 = 8$.\nStep 4: We want to represent $8$ in the form $\square \times k$. Since $\triangle \times \square = \square$, the output is $\square$, so $\triangle \times 8 = \square \times 8$.\nStep 5: Match with options: Option (E) is $\square \times 8$, which fits the answer.\n 2. Problem 2: Ice cream cost problem: Ice cream costs 3 dollars, chocolate dipped costs 0.5 dollars extra. Find a formula for 5 chocolate dipped ice creams.\n Step 1: Cost of one chocolate dipped ice cream is $3 + 0.5 = 3.5$ dollars.\nStep 2: Cost of 5 chocolate dipped ice creams is $5 \times (3 + 0.5) = 5 \times 3.5$.\nStep 3: Among the options, this matches option (E): cost = (3 + 0.5) \times 5.\n 3. Problem 3: Nets and boxes. Find which net cannot be folded into a box.\n Step 1: Net (A) T-shape with 5 squares in a row and one above second square can be folded into a box.\nStep 2: Net (B) rectangular with 4 squares in a row and one attached below last can form a box.\nStep 3: Net (C) zigzag with stair-step 5 squares cannot fold into a box because faces won't align properly.\nStep 4: Net (E) cross-shaped with a central square connected on all four sides to one square each can fold into a box.\nStep 5: Net (D) is unclear, cannot analyze.\nStep 6: So answer is (C): zigzag net cannot fold into a box.