3) A batch of a special detergent is manufactured using two parts of compound Z,

Question

3) A batch of a special detergent is manufactured using two parts of compound Z, one and a quarter parts of compound DE, two and a half parts of compound PS, one part of compound RT, and three parts of compound B. Compound PS costs twice as compound B, compound Z costs one-third as compound PS, Compound DE costs $3 more than B, and RT costs half the cost of Z. If we cannot exceed a $35 cost per batch.
(a) Find the cost for each compound in the batch.
(b) Find the cost, per part, for each compound.

Explanation / Answer

We are given that the composition of a batch of detergent is detergent = 2*Z + 1.25*DE + 2.5*PS + RT + 3*B Furthermore, the relative costs of each compound are given as PS = 2*B Z = (1/3)*PS DE = 3+B RT = (1/2)*Z and that the cost of a batch does not exceed $35. (b) We assume that one part of compound B costs $X, and then rewrite the costs of each of the compounds in terms of compound B PS = 2*B Z = (1/3)*PS = Z = (1/3)*2*B = (2/3)*B DE = 3+B RT = (1/2)*Z = RT = (1/2)*(1/3)*PS = (1/2)*(1/3)*2*B = (1/3)*B Thus, the cost of one batch of the detergent is = 2*(2/3)*X + 1.25*(3+X) + 2.5*(2*X) + (1/3)*X + 3*X = 10.91666*X + 3.75 and since this cannot exceed $35, we can solve 10.91666*X + 3.75 = 35 for X = $2.86 Thus, we can now find the costs, per part, of each of the compounds as B --> $2.86 PS --> $5.72 Z --> $1.91 DE --> $5.86 RT --> $0.95 (a) And thus, for one batch of the detergent, the cost of each compound is B --> 3*$2.86 = $8.58 PS --> 2.5*$5.72 = $14.30 Z --> 2*$1.91 = $3.82 DE --> 1.25*$5.86 = $7.33 RT --> $0.95, which sums to $34.98, thus meets the requirement of the cost not exceeding $35

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