Members of a high school choir want to attend a national contest. After a sign-up, the choir director
stated that the trip will cost $400 per choir member plus a $1500 fee for the choir to participate.
a) Write an equation to represent the total average cost of the trip per choir member.
b) What is the horizontal asymptote of the function? What does it mean in relationship to this problem?
What is the vertical asymptote? What does it mean in relationship to the problem.
c) Find the total average cost of the trip if 45 members attended.
d) How many members must attend to bring the total average cost to $425 per member.
the total cost for x members is c = 400x+1500
"total average" makes no sense ...
(a) avg cost is a = c/x = 400 + 1500/x = (400x+1500)/x
(c) a(45) = 400 + 1500/45 = 433
(d) 1500/x = 25, so x = 60
a) The equation to represent the total average cost of the trip per choir member can be written as:
Total Cost = (400 * number of members) + 1500
b) In this equation, as the number of members increases, the total cost increases as well. However, there is no limit to the number of members or the total cost. Hence, there is no horizontal asymptote in this problem. A horizontal asymptote represents a limit or a maximum value that the function approaches as the input goes to infinity or negative infinity.
There is also no vertical asymptote in this problem as the number of members is not restricted to any specific range. A vertical asymptote would represent a limit or a point at which the function is undefined.
c) If 45 members attended, we can substitute the value of the number of members into the equation:
Total Cost = (400 * 45) + 1500
Total Cost = 18000 + 1500
Total Cost = $19500
The total average cost of the trip, if 45 members attended, would be $19500.
d) To find out how many members must attend to bring the total average cost to $425 per member, we need to solve the equation:
Total Cost = (400 * number of members) + 1500
Total Cost/number of members = 425
Substituting the value of the total average cost per member, we get:
(400 * number of members) + 1500 = 425 * number of members
Simplifying the equation:
1500 = 25 * number of members
number of members = 1500/25
number of members = 60
So, 60 members must attend to bring the total average cost to $425 per member.
a) The equation to represent the total average cost of the trip per choir member can be written as:
C(x) = (400x + 1500) / x
b) The horizontal asymptote of the function is y = 400. In this context, it means that as the number of choir members (x) gets larger and larger, the total average cost per member will approach $400.
The vertical asymptote does not exist in this context since the cost per member can be calculated for any positive number of choir members.
c) To find the total average cost of the trip if 45 members attended, substitute x = 45 into the equation:
C(45) = (400 * 45 + 1500) / 45
C(45) = (18000 + 1500) / 45
C(45) = 19500 / 45
C(45) ≈ $433.33
So, the total average cost of the trip if 45 members attended is approximately $433.33 per member.
d) To find out how many members must attend to bring the total average cost to $425 per member, we need to solve the equation:
425 = (400x + 1500) / x
First, multiply both sides of the equation by x:
425x = 400x + 1500
Next, subtract 400x from both sides:
25x = 1500
Finally, divide both sides by 25:
x = 1500 / 25
x = 60
So, 60 members must attend to bring the total average cost to $425 per member.
a) Let x represent the number of choir members attending the trip. The total average cost of the trip per choir member can be represented with the equation:
Total cost = (400 * x + 1500) / x
b) The horizontal asymptote represents the long-term behavior of the function as x approaches infinity or negative infinity. In this case, as the number of choir members attending the trip increases infinitely, the additional cost of $400 per member becomes negligible compared to the constant $1500 fee for the choir to participate. Therefore, the horizontal asymptote is y = 1500.
The vertical asymptote represents the restriction or limitation in the domain of the function. In this case, the vertical asymptote occurs when the number of choir members attending the trip is zero, as it is not possible to have a negative or zero number of members. Therefore, the vertical asymptote is x = 0.
c) To find the total average cost of the trip if 45 members attended, substitute x = 45 into the equation:
Total cost = (400 * 45 + 1500) / 45
Total cost = (18000 + 1500) / 45
Total cost = 19500 / 45
Total cost ≈ $433.33
Therefore, the total average cost of the trip for 45 members is approximately $433.33 per member.
d) To find the number of members required to bring the total average cost to $425 per member, we can rearrange the equation:
425 = (400 * x + 1500) / x
Multiply both sides by x:
425x = 400x + 1500
Subtract 400x from both sides:
25x = 1500
Divide both sides by 25:
x = 1500 / 25
x = 60
Therefore, 60 members must attend to bring the total average cost to $425 per member.