Chapter 4.3, Problem 43E

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7th Edition

Stefan Waner + 1 other

ISBN: 9781337274203

Textbook Problem

**CAT Scans** CAT (computerized axial tomographic) scans are used to map the exact location of interior features of the human body. CAT scan technology is based on the following principles: (I) Different components of the human body (water; gray matter; bone, etc.) absorb X-rays to different extents; and (2) to measure the X-ray absorption by a specific region of, say, the brain, it suffices to pass a number of line-shaped pencil beams of X-rays through the brain at different angles and measure the total absorption for each beam, which is the sum of the absorptions of the regions through which it passes. The accompanying diagram illustrates a simple example. (The number in each region shows its absorption, and the number on each X-ray beam shows the total absorption for that beam.)^{27}

*In Exercises 39-44, use the table and the given X-ray absorption diagrams to identity the composition of each of the regions marked by a letter.*

Type |
Air | Water | Gray Matter | Tumor | Blood | Bone |

Absorption |
0 | 1,000 | 1,020 | 1,030 | 1,050 | 2,000 |

Identify the composition of site *x*.

(The horizontal and slanted beams each pass through five regions.)

To determine

**To calculate:** The composition of marked by a letters *x* with the help of provided diagram and table.

Explanation

**Given Information:**

The provided table is shown below:

Type | Air | Water | Grey matter | Tumor | Blood | Bone |

Absorption | 0 | 1000 | 1020 | 1030 | 1050 | 2000 |

The provided *X*-ray diagram is shown below:

**Formula Used:**

Elementary row operations

Type 1: Replacing the row

Type 2: Replacing the row

Gauss Jordan reduction method:

Step 1: First clear the fractions or decimals if any, using operations of type 1.

Step 2: Select the first nonzero element of the first row as pivot.

Step 3: Use the pivot to clear its column using operations of type 2.

Step 4: Select the first nonzero element in the second row a pivot and clear its column.

Step 5: Turn all the selected pivot elements into a 1 using operations of type 1.

**Calculation:**

Consider the provide table is,

Type | Air | Water | Grey matter | Tumor | Blood | Bone |

Absorption | 0 | 1000 | 1020 | 1030 | 1050 | 2000 |

Consider the provided diagram is,

The three beams pass through the same four square on the left let us label those squares *y*. the vertical beam also passes through other 2 squares which we can label as *z* and *u*.

The *X*-ray can be shown as

To determine the values of each compositions equations can be formed by adding the values along each beams.

Add the values along each beam.

And

Also,

And

The augmented matrix for the given system of equations:

Apply Gauss Jordan reduction method to get the solution of the given system of equation.

Pivot the first nonzero element of the first row and clear its column.

Perform the operations

Pivot the first nonzero element of the second row and clear its column.

Perform the operations

Next pivot the first nonzero element of the third row and clear its column

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