Read the information given below and answer the questions that follow :
The following functions have been defined :
`la(x, y, z) ="min" (x+y, y+z)`
`"le" (x,y,z) ="max" (x-y, y-z)`
`ma (x,y,z) =(1//2) ["le"(x,y,z) +la (x,y,z)]`
What is the value of ma (10, 4, le (la (10, 5, 3), 5, 3)) ?

Options

A. `7.0`
B. 6.5
C. `8.0`
D. 7.5

Answer

To solve the problem step by step, we will follow the definitions of the functions provided and compute the required values sequentially.

### Step 1: Calculate `la(10, 5, 3)`

The function `la(x, y, z)` is defined as:

\[ la(x, y, z) = \text{min}(x+y, y+z) \]

Substituting the values \(x = 10\), \(y = 5\), and \(z = 3\):

\[ la(10, 5, 3) = \text{min}(10 + 5, 5 + 3) = \text{min}(15, 8) = 8 \]

### Step 2: Calculate `le(la(10, 5, 3), 5, 3)`

Now we need to calculate `le(8, 5, 3)` since we found `la(10, 5, 3) = 8`.

The function `le(x, y, z)` is defined as:

\[ le(x, y, z) = \text{max}(x-y, y-z) \]

Substituting the values \(x = 8\), \(y = 5\), and \(z = 3\):

\[ le(8, 5, 3) = \text{max}(8 - 5, 5 - 3) = \text{max}(3, 2) = 3 \]

### Step 3: Calculate `ma(10, 4, le(la(10, 5, 3), 5, 3))`

Now we need to calculate `ma(10, 4, 3)` since we found `le(8, 5, 3) = 3`.

The function `ma(x, y, z)` is defined as:

\[ ma(x, y, z) = \frac{1}{2} [le(x, y, z) + la(x, y, z)] \]

We need to calculate both `le(10, 4, 3)` and `la(10, 4, 3)`.

#### Step 3.1: Calculate `le(10, 4, 3)`

Using the definition of `le`:

\[ le(10, 4, 3) = \text{max}(10 - 4, 4 - 3) = \text{max}(6, 1) = 6 \]

#### Step 3.2: Calculate `la(10, 4, 3)`

Using the definition of `la`:

\[ la(10, 4, 3) = \text{min}(10 + 4, 4 + 3) = \text{min}(14, 7) = 7 \]

### Step 4: Calculate `ma(10, 4, 3)`

Now, substituting the values we calculated:

\[ ma(10, 4, 3) = \frac{1}{2} [le(10, 4, 3) + la(10, 4, 3)] = \frac{1}{2} [6 + 7] = \frac{13}{2} = 6.5 \]

### Final Answer

The value of `ma(10, 4, le(la(10, 5, 3), 5, 3))` is **6.5**.

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