Q. 82 - Precalculus Enhanced with Graphing Utilities

Question

Solar Energy The solar electric generating systems at Kramer Junction, California, use parabolic troughs to heat a heat-transfer fluid to a high temperature. This fluid is used to generate steam that drives a power conversion system to produce electricity. For troughs $7.5$ feet wide, an equation for the cross-section is $16y^2=120x-225$.

(a) Find the intercepts of the graph of the equation.

(b) Test for symmetry with respect to the $x$-axis, $y$-axis, and origin.

Step-by-Step Solution

Answer

a) The graph of the equation has an $x$-intercept of $\frac{15}{8}$ and no $y$-intercept.

b) The graph of the given equation is symmetrical about the x-axis but not about the y-axis hence it will not have about the origin. 

Step 1: Determine the x -intercept.

For $x$-intercept, substitute $0$ for $y$ and then solve for $x$.

$\begin{array}{rcl}16{\left(0\right)}^2& =& 120x-225\\ 120x& =& 225\\ x& =& \frac{225}{120}\\ & =& \frac{15}{8}\end{array}$

Step 2: Determine the y -intercept.

For $y$-intercept, substitute 0 for $x$ and then solve for $y$.

$$\begin{gathered} 16y^2=120\left(0\right)-225 \\ 16y^2=-225 \\ y=\pm \sqrt{\frac{-225}{16}} \end{gathered}$$

The value of $y$ is imaginary because inside the square root the value is negative.

Step 3: Determine the symmetry.

Perform the test for symmetry with respect to the $x$-axis. Put $y=-y$

$\begin{array}{rcl}16(-y{)}^2& =& 120x-225\\ 16y^2& =& 120x-225\end{array}$

Since after substituting $y=-y$ into the given equation, the resulting equation is equivalent to the original equation hence the graph of the given equation is symmetrical about the $x$-axis.

Perform the test for symmetry with respect to the $y$-axis. Put $x=-x$

$\begin{array}{rcl}16y^2& =& 120(-x)-225\\ 16y^2& =& -120x-225\\ & \ne & 120x-225\end{array}$

Perform the test for symmetry with respect to the origin. Put $x=-x ; y=-y$

$\begin{array}{rcl}16(-y{)}^2& =& 120(-x)-225\\ 16y^2& =& -120x-225\\ & \ne & 120x-225\end{array}$

Step 4: Write the conclusion

a) There is no $y$-intercept and $x$-intercept is $\left(\frac{15}{8},0\right)$.

b) The graph of the given equation is symmetrical about the $x$-axis but not about the $y$-axis hence it will not have about the origin.