EXAMPLE 3 Solve a multi-step problem Photography You can take panoramic photographs using a hyperbolic mirror. Light rays heading toward the focus behind.

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EXAMPLE 3 Solve a multi-step problem Photography 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 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 = 2.81 and c = To write an equation, find b 2. b 2 = c 2 – a 2 = – Because the transverse axis is vertical, the standard form of the equation for the cross section of the mirror is as follows: y – x = 1 or y – x = 1

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. y = 1 y – Solve for y 2. y – 4.56 Solve for y. ANSWER So, the mirror has a height of – 2.81 – (– 4.56) = 1.75 centimeters.

GUIDED PRACTICE for Example 3 6. What If ? In Example 3, suppose that the mirror remains 6 centimeters wide, but that a = 3 centimeters and c = 5 centimeters. How tall is the mirror ? SOLUTION STEP 1 From the diagram, a = 3 and c = 5 To write an equation, find b 2. b 2 = c 2 – a 2 = 5 2 – 3 2 = 16

GUIDED PRACTICE for Example 3 Because the transverse axis is vertical, the standard form of the equation for the cross section of the mirror is as follows: y – x 2 16 = 1 or y 2 9 – x 2 16 = 1 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.

GUIDED PRACTICE for Example 3 Substitute 3 for x. Solve for y 2. Solve for y. ANSWER So, the mirror has a height of – 3 – (– 3.75) = 0.75 cm y – = 1 y 2 = – y = 15 4