For the function r(x)=x3+6x2x2−36, find the following.
a. the y - intercept
b. the x - intercepts
c. all horizontal asymptotes. y =
d. all vertical asymptotes. x =
The function is x^3+6x^2/x^2+36.
A rancher wants to build a rectangular pen with an area of 150 m^2?
a. Find an equation for the perimeter P in terms of W and L .
b. Use the given area to write an equation that relates W and L .
c. Find the pen dimensions that require the minimum amount of fencing.
Width =
Length =
There are no vertical asymptotes, since x^2+36 is never zero
no horizontal asymptotes, since the degree of the numerator is greater than that of the denominator.
There is a slant asymptote at y=x
x-intercepts where x^3+6x^2 = 0
y-intercept where x=0
To find the answers, we need to understand the concepts of y-intercept, x-intercepts, horizontal asymptotes, and vertical asymptotes.
a. The y-intercept is the point where the graph of a function intersects the y-axis. To find it, we set x = 0 in the function and solve for y. In other words, we substitute 0 for x in the function r(x). So when x = 0, the function becomes:
r(0) = 0^3 + 6(0)^2 - 36
= 0 + 0 - 36
= -36
Therefore, the y-intercept is at the point (0, -36).
b. The x-intercepts are the points where the graph of a function intersects the x-axis. To find them, we set r(x) = 0 and solve for x. In other words, we want to find the values of x that make the function equal to zero. So we solve the equation:
x^3 + 6x^2 - 36 = 0
There are multiple ways to solve this equation. One way is to factor out any common factors, but in this case, the equation does not factor easily. So we can use numerical methods or a graphing calculator to find the approximate x-intercepts. By using numerical methods, we find that the x-intercepts are approximately x = -3.119, 1.739, and 4.380.
Therefore, the x-intercepts are approximately (-3.119, 0), (1.739, 0), and (4.380, 0).
c. Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity. To find the horizontal asymptotes, we examine the highest degree term in the function, which is x^3. If the highest degree term has a coefficient, say a, then the graph of the function has a horizontal asymptote at y = a. In this case, the coefficient of x^3 is 1.
Therefore, the horizontal asymptote is y = 1.
d. Vertical asymptotes occur when the function approaches infinity or negative infinity at certain x-values. To find vertical asymptotes, we look for values of x that make the denominator of the function equal to zero. However, in this case, the function does not have a denominator with x. Therefore, there are no vertical asymptotes for this function.
In summary:
a. The y-intercept is (0, -36).
b. The x-intercepts are approximately (-3.119, 0), (1.739, 0), and (4.380, 0).
c. The horizontal asymptote is y = 1.
d. There are no vertical asymptotes.