Jiao works at least 36 h/wk. She earns $15/h as a science tutor and $25/h as a teacher. The number of hours Jiao works as a teacher, y, is at least twice the number of hours she works as a tutor, x. This set of inequalities represents the constraints in this situation. x y≥36, y≥2x, x≥0 The objective function is I = 15x 25y, where I represents Jiao's income for a week. What is the minimum amount of money she can earn in a week?



Answer :

To determine the number of hours Jiao works as a teacher, \( y \), we start by setting up the conditions given in the problem:

1. Jiao works at least 36 hours per week in total.

2. Jiao earns $15 per hour as a science tutor and $25 per hour as a teacher.

Let's denote:

- \( x \): Number of hours Jiao works as a science tutor.

- \( y \): Number of hours Jiao works as a teacher.

From the problem statement, we have the following equations based on the total hours and the earnings:

\[ x + y \geq 36 \]

\[ 15x + 25y \] (Total earnings per week)

To solve for \( y \), the number of hours Jiao works as a teacher, we need to consider the minimum scenario where \( x \) (hours worked as a science tutor) is maximized given the constraint \( x + y \geq 36 \).

Assuming \( x \) is maximized while satisfying \( x + y = 36 \) (to meet the minimum total hours requirement), then:

\[ x = 36 - y \]

Now, substituting this into the earnings equation:

\[ 15x + 25y = 15(36 - y) + 25y \]

\[ 15 \cdot 36 - 15y + 25y \]

\[ 540 + 10y \]

To maximize \( y \), we need to satisfy the condition \( x = 36 - y \geq 0 \):

\[ 36 - y \geq 0 \]

\[ y \leq 36 \]

Therefore, the minimum number of hours \( y \) that Jiao works as a teacher is at least:

\[ \boxed{36} \]

This means Jiao must work at least 36 hours as a teacher per week to satisfy the total minimum hours requirement of 36 hours per week and maximize her earnings.Answer:

Step-by-step explanation: