Problem 27 - Algebra 2

Question

ASTRONOMY For Exercises 27 and 28 , use the following information. The orbit of Pluto can be modeled by the equation \(\frac{x^{2}}{39.5^{2}}+\frac{y^{2}}{38.3^{2}}=1,\) where the units are astronomical units. Suppose a comet is following a path modeled by the equation \(x=y^{2}+20\) Find the point(s) of intersection of the orbits of Pluto and the comet. Round to the nearest tenth.

Step-by-Step Solution

Answer

The points of intersection, rounded to the nearest tenth, are \((x_1, y_1)\) and \((x_2, y_2)\) where these pairs satisfy both equations.

Step 1: Identify the equations

We have two equations: the ellipse equation for Pluto's orbit given by \(\frac{x^{2}}{39.5^{2}}+\frac{y^{2}}{38.3^{2}}=1\) and the parabola equation for the comet's path given by \(x = y^{2} + 20\). Our task is to find any solution \((x, y)\) that satisfies both equations.

Step 2: Substitute the parabola equation into the ellipse equation

Substitute \(x = y^2 + 20\) from the parabola into the ellipse equation: \(\frac{(y^2 + 20)^2}{39.5^2} + \frac{y^2}{38.3^2} = 1\). This will help us to transform the ellipse equation into an equation with a single variable \(y\).

Step 3: Simplify and solve for y

Expand the equation: \(\frac{(y^2 + 20)^2}{1560.25} + \frac{y^2}{1466.89} = 1\). Solve for \(y\) by multiplying through by the denominators to clear the fractions and reduce it to a polynomial equation in terms of \(y\).

Step 4: Solve the polynomial equation

Solve the resulting polynomial for \(y\) using numerical methods or graphing, which may require successive approximation or a calculator if an exact analytical solution is complex. The goal is to find \(y\) values to the nearest tenth.

Step 5: Calculate corresponding x values

For each \(y\) solution found, calculate the corresponding \(x\) value using \(x = y^2 + 20\). Round the \(x\) and \(y\) solutions to the nearest tenth as required.

Step 6: Verify the intersection points

Substitute \(x\) and \(y\) back into both original equations to ensure they satisfy both the ellipse and the parabola equations. This confirms the solution is a valid point of intersection.

Key Concepts

Ellipse

An ellipse is a fascinating shape that resembles an elongated circle. It is the set of all points for which the sum of their distances to two fixed points, called foci, is constant. The equation of an ellipse usually appears as \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \(a\) and \(b\) represent the semi-major and semi-minor axes, respectively.

The orbit of Pluto, in this context, is represented as an ellipse. With \(a = 39.5\) and \(b = 38.3\), its shape is fairly circular, showing how orbits in astronomy can often be closely approximated by ellipses.

Ellipses have unique properties used in various scientific fields, especially in astronomy, where understanding the elliptical nature of orbits is crucial for mapping planetary movements.

Parabola

A parabola is a simple yet important conic section described by its characteristic U-shape. The standard form of a parabola's equation vertically is \( y = ax^2 + bx + c \) or horizontally \( x = ay^2 + by + c \). In the given astronomy problem, the comet's path is expressed as \( x = y^2 + 20 \), indicating a horizontally opening parabola.

Parabolas are essential in many fields including physics and engineering. Their reflective properties, allowing them to focus parallel rays to a single point, make them ideal for satellite dishes and optical devices.

• The vertex is direct, making calculations easier.
• They have a reflective property useful in astronomy and engineering.

When solving intersections, parabolas add complexity but also offer rich geometric interactions.

Astronomy

Astronomy is the science that studies celestial bodies like planets, comets, and stars. One major aspect of astronomy is understanding the motion of these objects through space. This often involves conic sections to describe orbits, such as ellipses for planets and parabolas for some comets.

For Pluto, its orbit illustrates the typical elliptical path planets take around the sun. In contrast, comets often follow parabolic or hyperbolic paths due to their different velocities and trajectories. Calculating points of intersection like in this exercise is crucial for predicting when comets may be visible or potentially collide with other celestial bodies.

• Elliptical orbits are common for planets.
• Parabolic paths are typical for comets.
• Understanding these orbits allows for important predictions.

Such exercises enhance our comprehension of astronomical observations and predictions.

Intersection Points

Finding intersection points involves determining where two graphs meet. This is a crucial process in various math and scientific fields. It requires solving a system of equations – here, one representing an ellipse and the other a parabola.

Using algebra, one substitutes one equation into another, simplifying the mathematical challenge into solving for a single variable. This concept underpins many practical problems, from calculating trajectories in space missions to optimizing resources in engineering.

For instance:

• Substitute to find common points.
• Verify results by substitution back into original equations.
• Used in technology and problem-solving scenarios.

Understanding intersection points helps us anticipate and plan for situations where different paths might meet, be it in nature, technology, or science.