MATH SOLVE

3 months ago

Q:
# 7. Show all work to identify the discontinuity and zero of the function f of x equals 5 x over quantity x squared minus 25.8. The aquarium has 6 fewer yellow fish than green fish. 40 percent of the fish are yellow. How many green fish are in the aquarium? Show your work.

Accepted Solution

A:

Question 1:For this case we have that the function [tex]f (x) = \frac {5x} {x ^ 2-25}[/tex] is undefined or discontinuous where the denominator equals 0.[tex]x ^ 2-25 = 0\\x ^ 2 = 25\\x = \pm \sqrt {25}\\x_ {1} = + 5\\x_ {2} = - 5[/tex]Thus, the function is undefined or discontinuous at +5 and -5.To find the zeros of the function we match the function to zero and clear "x":[tex]\frac {5x} {x ^ 2-25} = 0[/tex]Factoring the denominator, taking into account that the roots are -5 and +5:[tex]\frac {5x} {(x + 5) (x-5)} = 0[/tex]We multiply by[tex](x + 5) (x-5)[/tex]on both sides of the equation:[tex]5x = 0\\x = 0[/tex]ANswer:Discontinuity: + 5, -5Zero: x = 0Question 2:For this case we propose a system of equations:x: Be the variable that represents the yellow fishy: Be the variable that represents the green fish[tex]x = y-6\\x = 0.4 (x + y)[/tex]We manipulate the second equation:[tex]x = 0.4x + 0.4y\\x-0.4x = 0.4y\\0.6x = 0.4y\\y = \frac {0.6} {0.4} x\\y = 1.5x[/tex]We substitute in the first equation:[tex]x = y-6\\x = 1.5x-6\\x-1.5x = -6\\-0.5x = -6\\x = \frac {-6} {- 0.5}\\x = 12[/tex]So, we have 12 yellow fish in the aquarium.[tex]y = 1.5 * 12\\y = 18[/tex]So, we have 18 green fish.Answer:12 yellow fish18 green fish