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Circular Model for Cell Phones

Using a circular model, determine if you are within a cell phone tower's range based on your location. Calculate the additional distance you will be in range if you travel in specific directions.

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Circular Model for Cell Phones

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  1. In the diagram above, the origin represents the tower and the positive y-axis represents north. EXAMPLE 4 Write a circular model Cell Phones A cellular phone tower services a 10 mile radius. You get a flat tire 4 miles east and 9 miles north of the tower. Are you in the tower’s range? SOLUTION STEP 1 Write an inequality for the region covered by the tower. From the diagram, this region is all points that satisfy the following inequality: x2 + y2 < 102

  2. ? 42 + 92 < 102  97 < 100 ANSWER So, you are in the tower’s range. EXAMPLE 4 Write a circular model STEP 2 Substitute the coordinates (4, 9) into the inequality from Step 1. x2 + y2 < 102 Inequality from Step 1 Substitute for xandy. The inequality is true.

  3. EXAMPLE 5 Apply a circular model Cell Phones In Example 4, suppose that you fix your tire and then drive south. For how many more miles will you be in range of the tower ? SOLUTION When you leave the tower’s range, you will be at a point on the circle x2 + y2 = 102 whose x-coordinate is 4 and whose y-coordinate is negative. Find the point (4, y) where y < 0 on the circle x2 + y2 = 102.

  4. 84 + y = y +9.2 ANSWER Becausey < 0, y – 9.2. You will be in the tower’s range from (4, 9) to (4, –9.2), a distance of | 9 – (– 9.2) | = 18.2miles. EXAMPLE 5 Apply a circular model x2+ y2 = 102 Equation of the circle 42+ y2 = 102 Substitute 4 forx. Solve for y. Use a calculator.

  5. for Examples 4 and 5 GUIDED PRACTICE 6. WHAT IF? In Examples 4 and 5, suppose you drive west after fixing your tire. For how many more miles will you be in range of the tower? SOLUTION When you leave the tower’s range, you will be at a point on the circle x2 + y2 = 102 whose y-coordinate is 9 and whose x-coordinate is negative. Find the point (x, 9) where x< 0 on the circle x2 + y2 = 102.

  6. 19 + x = x +4.4 for Examples 4 and 5 GUIDED PRACTICE x2 + y2 = 102 Equation of the circle x2 + 92 = 102 Substitute 9 for y. Solve for y. Use a calculator. Becausex < 0, x ≈– 4.4.You will be in the tower’s range from(4, 9) to(–4.4, 9),a distance of| 4 – (4.4) | = 8.4 miles.

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