Modelling Problems

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James makes wooden rocking horses as part of his business. To calculate the cost (C) of making each rocking horse he uses the model $$C=\frac{k}{n+1}$$ where (n) is the number of rocking horses. The table indicates James’ production costs per horse for varying numbers of horses produced. N 2 5 7 11 C 80 40 30 20 a. Find the value of k in his production. b. Calculate his initial costs. c. James sells the rocking horses for $50 each. How many horses will he need to sell before he begins to make a profit? d. James makes and sells 14 horses calculate his total profit? e. Why would it be unwise for James to rely on this formula when producing large numbers of rocking horses? Justify your answer with appropriate mathematics
a. $$80=\frac{k}{2+1}$$ $$80=\frac{k}{3}$$ k = 240	to find the value of k, substitute in any pair of numbers from the table solve for k
b. $$C=\frac{240}{0+1}$$ = $240	initial costs are when N = 0
c. $$50=\frac{240}{n+1}$$ $$50^{\times(n+1)}=\frac{240}{\cancel{(n+1)}}^{\times\cancel{(n+1)}}$$ 50n + 50 = 240 50n = 190 n = 3.8 ∴ n = 4	Find the break even point, which is where the costs equal the sales solve for n so he will need to sell 4 horses or more as he can’t sell part of a unit
d. Sales: 14 × 50 = $700 Cost = $$\frac{240}{14+1}$$ = $16 per horse Total cost = 16 × 14 = $224	Profit is sales price – cost price Profit = 700 – 224 = $476 Find the value of the sales when n = 14, sold at $50 each Find the cost of making a single horse if he makes 14 horses find the cost of making 14 of them at $16 each
e. when n gets large C becomes unrealistic as it gets closer to zero and it should be obvious that the cost cannot be zero an example of making 1000 horses: $$\frac{240}{1000+1}=0.24$$ the formula says that it will cost him only 24 cents to make a horse, and that is a ridiculously small cost. As you make more, soon the number would become so small, it would be practically zero, which is unrealistic

 

 

 

 

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