Answer the following question in detail, the answer is 13. Imagine you have an unlimited supply of rectangular blocks of dimensions 11 cm x 11 cm x 18 cm. These can be made into a tower by stacking them one on top of the other in any orientation.

What is the least total number of blocks required to make two towers so that the first tower is exactly

2 cm taller than the second?

To solve this problem, we need to find the minimum number of blocks required to construct two towers with a specific height difference. The first tower needs to be exactly 2 cm taller than the second tower.

### Step-by-Step Solution:

#### Step 1: Understand the Block Dimensions and Orientation

We have an unlimited supply of blocks with dimensions:
- \(11 \, \text{cm} \times 11 \, \text{cm} \times 18 \, \text{cm}\)

Each block can be stacked in one of three orientations:
1. Height = 11 cm (if the 11 cm x 18 cm face is the base)
2. Height = 11 cm (if the 11 cm x 18 cm face is the base)
3. Height = 18 cm (if the 11 cm x 11 cm face is the base)

For our problem, we can use either 11 cm or 18 cm as the height of each block.

#### Step 2: Represent the Heights of Two Towers

Let:
- \(x\) = Number of blocks used for the first tower.
- \(y\) = Number of blocks used for the second tower.

The height of the first tower (in cm) can be expressed as:

\[
H_1 = 11a + 18b
\]

where \(a\) and \(b\) are the number of blocks oriented with heights 11 cm and 18 cm, respectively.

Similarly, the height of the second tower can be expressed as:

\[
H_2 = 11c + 18d
\]

where \(c\) and \(d\) are the number of blocks oriented with heights 11 cm and 18 cm, respectively.

#### Step 3: Set Up the Equations for Height Difference

The problem states that the first tower must be exactly 2 cm taller than the second tower:

\[
H_1 = H_2 + 2
\]

Substitute the height expressions:

\[
11a + 18b = 11c + 18d + 2
\]

This equation needs integer solutions for \(a, b, c,\) and \(d\).

#### Step 4: Explore Possible Combinations

We want to minimize the total number of blocks used, \(x + y = (a + b) + (c + d)\).

Let's consider a few cases:

1. **Case 1: All Blocks Oriented at 18 cm**  
   If all blocks are oriented to have a height of 18 cm, we have:

   \[
   18b = 18d + 2 \implies b = d \, \text{(no integer solution for height difference)}
   \]

   This case does not work since adding 2 cm is impossible with blocks oriented at 18 cm only.

2. **Case 2: Mixing 11 cm and 18 cm Orientations**  
   Let's try combinations with both 11 cm and 18 cm orientations:

   We set:

   \[
   H_1 = 18 \times 1 + 11 \times 1 = 18 + 11 = 29 \, \text{cm}
   \]
   
   \[
   H_2 = 11 \times 2 = 22 \, \text{cm}
   \]

   Here, \(H_1 = 29 \, \text{cm}\) and \(H_2 = 22 \, \text{cm}\), so the difference is:

   \[
   H_1 - H_2 = 29 - 22 = 7 \, \text{cm} \, (\text{not 2 cm})
   \]

   This doesn't work either.

#### Step 5: Identify the Correct Combination

We aim to find the smallest configuration such that \(H_1 - H_2 = 2\). By checking through combinations:

- First Tower: \(18 \times 3 + 11 \times 1 = 54 + 11 = 65\)
- Second Tower: \(18 \times 3 = 54\)

The difference is:

\[
H_1 - H_2 = 65 - 54 = 2 \, \text{cm}
\]

Thus, the first tower uses 4 blocks (3 blocks of 18 cm and 1 block of 11 cm), and the second tower uses 3 blocks (all 18 cm blocks).

#### Step 6: Conclusion

The least total number of blocks required is:

\[
4 + 3 = 7 \, \text{blocks.}
\]

### Answer

The least total number of blocks required to make two towers such that the first is exactly 2 cm taller than the second is **7 blocks**.

Imagine you have an unlimited supply of rectangular blocks of dimensions 11 cm x 11 cm x 18 cm. These can be made into a tower by stacking them one on top of the other in any orientation. What is the least total number of blocks required to make two towers so that the first tower is exactly 2 cm taller than the second?

Discover how to determine the least number of blocks needed to build two towers with a specific height difference using algebraic methods. This problem involves combinatorics and optimization, ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation