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EXAMPLE 3

You can take panoramic photographs using a hyperbolic mirror. Light rays heading toward the focus behind the mirror are reflected to a camera positioned at the other focus as shown. After a photograph is taken, computers can “unwrap” the distorted image into a 360 ° view. EXAMPLE 3.

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EXAMPLE 3

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  1. You can take panoramic photographs using a hyperbolic mirror. Light rays heading toward the focus behind the mirror are reflected to a camera positioned at the other focus as shown. After a photograph is taken, computers can “unwrap” the distorted image into a 360° view. EXAMPLE 3 Solve a multi-step problem Photography

  2. – – y2 7.90 x25.50 y22.812 x2 5.50 b2= c2–a2 = 3.662–2.8125.50 = 1 = 1 EXAMPLE 3 Solve a multi-step problem • Write an equation for the cross section of the mirror. • The mirror is 6 centimeters wide. How tall is it? SOLUTION STEP 1 From the diagram, a= 2.81andc= 3.66. To write an equation, find b2. Because the transverse axis is vertical, the standard form of the equation for the cross section of the mirror is as follows: or

  3. y2 7.90 32 5.50 – = 1 y2 20.83 y – 4.56 ANSWER So, the mirror has a height of– 2.81 – (– 4.56) = 1.75centimeters. EXAMPLE 3 Solve a multi-step problem STEP 2 Find the y-coordinate at the mirror’s bottom edge. Because the mirror is 6 centimeters wide, substitute x = 3 into the equation and solve. Substitute 3 for x. Solve fory2. Solve fory.

  4. for Example 3 GUIDED PRACTICE 6. What If ? In Example 3, suppose that the mirror remains 6 centimeters wide, but that a = 3centimeters and c = 5centimeters. How tall is the mirror ? SOLUTION STEP 1 From the diagram, a= 3 andc= 5 To write an equation, find b2. b2= c2–a2 = 52– 32 = 16

  5. y2 32 x216 y2 9 x2 16 – – = 1 = 1 for Example 3 GUIDED PRACTICE Because the transverse axis is vertical, the standard form of the equation for the cross section of the mirror is as follows: or STEP 2 Find the y-coordinate at the mirror’s bottom edge. Because the mirror is 6 centimeters wide, substitute x = 3 into the equation and solve.

  6. y2 32 32 16 15 4 – = 1 y2 = 22516 – y = ANSWER So, the mirror has a height of– 3 – (– 3.75) = 0.75 cm for Example 3 GUIDED PRACTICE Substitute 3 for x. Solve fory2. Solve fory.

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