Example 6 from Algebra and Trigonometry, 12.3 The Parabola

Solving Applied Problems Involving Parabolas

A cross-section of a design for a travel-sized solar fire starter is shown in Figure 13. The sun’s rays reflect off the parabolic mirror toward an object attached to the igniter. Because the igniter is located at the focus of the parabola, the reflected rays cause the object to burn in just seconds.

ⓐ Find the equation of the parabola that models the fire starter. Assume that the vertex of the parabolic mirror is the origin of the coordinate plane.
ⓑ Use the equation found in part ⓐ to find the depth of the fire starter.

Figure 13 Cross-section of a travel-sized solar fire starter

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
- ⓐ The vertex of the dish is the origin of the coordinate plane, so the parabola will take the standard form $x^{2} = 4 p y,$ where $p > 0.$ The igniter, which is the focus, is 1.7 inches above the vertex of the dish. Thus we have $p = 1.7.$
  $\begin{array}{l} x^{2} = 4 p y & \begin{matrix} \end{matrix} Standard form of upward-facing parabola with vertex (0,0) \\ x^{2} = 4 (1.7) y & \begin{matrix} \end{matrix} Substitute 1 .7 for p . \\ x^{2} = 6.8 y & \begin{matrix} \end{matrix} Multiply . \end{array}$
- ⓑ The dish extends $\frac{4.5}{2} = 2.25$ inches on either side of the origin. We can substitute 2.25 for $x$ in the equation from part (a) to find the depth of the dish.
  $\begin{array}{l} x^{2} = 6.8 y & Equation found in part (a) . \\ {(2.25)}^{2} = 6.8 y & Substitute 2 .25 for x . \\ y \approx 0.74 & Solve for y . \end{array}$
  The dish is about 0.74 inches deep.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Balcony-sized solar cookers have been designed for families living in India. The top of a dish has a diameter of 1600 mm. The sun’s rays reflect off the parabolic mirror toward the “cooker,” which is placed 320 mm from the base.

ⓐ Find an equation that models a cross-section of the solar cooker. Assume that the vertex of the parabolic mirror is the origin of the coordinate plane, and that the parabola opens to the right (i.e., has the x-axis as its axis of symmetry).

ⓑ Use the equation found in part ⓐ to find the depth of the cooker.

How did it go?

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

ⓑ Use the equation found in part ⓐ to find the depth of the cooker.

Solution

Your turn

Solution

Your turn