The number of bicycles, n, sold at one store in a week is a
function of the price, p, in dollars. So n(p) for p > 80.
The store’s cost, c, in dollars for each bike is a function of the number of
bikes the store sells each week. So c(n) 0.002(n � 2)2 � 80.
(a) Evaluate n(100) and n(180). Why are these values reasonable in this
situation?
(b) Evaluate c(8) and c(48). Why are these values reasonable in this
situation?
(c) Evaluate the cost of each bicycle to the store if the selling price is $120.
(d) Determine the store’s profit per bicycle if the selling price is $120.
(e) Evaluate the total profit if the selling price is $120.
(f) Express the cost of each bike to the store as a function of the selling
price.
(g) Express the total profit in terms of the functions c and n and the
variable p.
(h) Use graphing technology to graph total profit versus price.
Where is the solution??
(a) To evaluate n(100), we need to plug in the value 100 for p in the function n(p). Similarly, to evaluate n(180), we need to plug in the value 180 for p. Since the given condition is p > 80, both 100 and 180 satisfy this condition, and therefore, these values are reasonable in this situation.
(b) To evaluate c(8), we need to substitute the value 8 for n in the function c(n). Similarly, to evaluate c(48), we need to substitute the value 48 for n. These values are reasonable because they represent the number of bikes sold by the store in a week. It is expected that the cost of each bike will increase as more bikes are sold, which is reflected in the formulas.
(c) To find the cost of each bicycle to the store when the selling price is $120, we need to evaluate c(n) at n(p=120). First, we substitute p=120 into the function n(p) to find the value of n at this price. Then, we substitute the value of n into the function c(n) to find the cost of each bike.
(d) To determine the store's profit per bicycle when the selling price is $120, we need to subtract the cost per bicycle from the selling price. We evaluate n(p=120) to find the value of n at this price, and then substitute it into c(n) to find the cost per bike. Finally, we subtract the cost per bike from the selling price.
(e) To evaluate the total profit when the selling price is $120, we first calculate the profit per bike as described in part (d). Then, we multiply the profit per bike by the number of bikes sold, which is given by n(p=120). This will give us the total profit.
(f) The cost of each bike to the store can be expressed as a function of the selling price by substituting n(p) into the function c(n). We replace n with n(p) to obtain c(n(p)).
(g) The total profit can be expressed in terms of the functions c and n, and the variable p by first calculating the profit per bike as described in part (d). Then, we multiply the profit per bike by the number of bikes sold, which is given by n(p). This can be represented as p - c(n(p)).
(h) To graph the total profit versus price, we can first select a range of prices to consider. Then, for each price value, we evaluate n(p) and c(n(p)), and subtract c(n(p)) from p to find the profit per bike. Finally, we multiply the profit per bike by n(p) to get the total profit at each price. Plotting these points on a graph will give the total profit versus price.