Find the vertical asymptote.
[tex]
\begin{array}{r}
y=\frac{3 x+30}{x-10} \
x=[?]
\end{array}
[/tex]
Answer (2)
Identify the denominator of the rational function.
Set the denominator equal to zero: x − 10 = 0 .
Solve for x : x = 10 .
The vertical asymptote is 10 .
Explanation
Understanding the Problem We are given the function y = x − 10 3 x + 30 and we want to find its vertical asymptote.
Finding the Denominator Vertical asymptotes occur when the denominator of a rational function is equal to zero. In this case, the denominator is x − 10 .
Solving for x To find the vertical asymptote, we set the denominator equal to zero and solve for x :
x − 10 = 0 x = 10
The Vertical Asymptote Therefore, the vertical asymptote is x = 10 .
Examples
Imagine you are designing a bridge. The function y = x − 10 3 x + 30 could represent the stress on a certain part of the bridge as a function of the load x . The vertical asymptote at x = 10 would represent a critical load value where the stress approaches infinity, indicating a point of potential failure. Understanding asymptotes helps engineers design structures that avoid these critical points, ensuring safety and stability. Similarly, in economics, asymptotes can model scenarios where increasing investment yields diminishing returns, helping businesses make informed decisions about resource allocation.
The vertical asymptote of the function y = x − 10 3 x + 30 is found by setting the denominator to zero, resulting in x = 10 . Therefore, the vertical asymptote is at x = 10 .
;
