Question

1. find the earnings for giving a haircut and a shampoo to m men and w women. justify your answer.
2. the expression 15x + 11 represents the perimeter of the trapezoid. what is the length of the fourth side? explain your reasoning.

Explanation:

Step1: Calculate earnings for men

For men, haircut is $15 and shampoo is $7. So for \(m\) men, the total earnings from men is \(15m + 7m=(15 + 7)m = 22m\).

Step2: Calculate earnings for women

For women, haircut is $45 and shampoo is $12. So for \(w\) women, the total earnings from women is \(45w+12w=(45 + 12)w=57w\).

Step3: Calculate total earnings

The total earnings is the sum of earnings from men and women. So total earnings \(E = 22m+57w\).

Step1: Use perimeter formula for trapezoid

The perimeter \(P\) of a trapezoid is the sum of all its sides. Let the fourth side be \(s\). Given \(P=15x + 11\), and three sides are \(2x + 3\), \(3x+4\), \(4x - 1\). Then \(P=(2x + 3)+(3x + 4)+(4x - 1)+s\).

Step2: Simplify the left - hand side

\((2x + 3)+(3x + 4)+(4x - 1)=2x+3+3x + 4+4x-1=(2x+3x + 4x)+(3 + 4-1)=9x+6\).

Step3: Solve for \(s\)

Since \(15x + 11=9x+6 + s\), then \(s=(15x + 11)-(9x + 6)\).
\(s=15x+11 - 9x-6=(15x-9x)+(11 - 6)=6x + 5\).

Answer:

The earnings for giving a haircut and a shampoo to \(m\) men and \(w\) women is \(22m + 57w\).